A Framework for Research on Problem-Solving Instruction.pdf
0 likes30 views
The NATO ASI Series presents the results of activities sponsored by the NATO Science Committee, which aims to disseminate advanced scientific and technological knowledge. The series is published internationally by an board of publishers. It provides full bibliographical references and abstracts to over 30,000 contributions from scientists in its various sections, including life sciences, physics, mathematics, behavioral sciences, and applied sciences. Access is provided both online and via a CD-ROM database.
![2
classrooms, (c) a focus on individuals rather than groups or whole classes, and (d) the
atheoretical nature of much of the research.
The Role of the Teacher
Silver [35] has pointed out that the typical research report might describe in a general way the
instructional method employed, but rarely is any mention made of the teacher's specific role. A
step toward structuring descriptions of what teachers do during instruction was taken by the
Mathematical Problem Solving Project (MPSP) at Indiana University with the identification of
several teaching actions for problem solving [38]. Subsequent to the work of the MPSP, we
identified ten teaching actions for problem solving [5, 6]. These teaching actions were selected
by listing the thinking processes and behaviors that the research literature and other sources
suggested as desirable outcomes of problem-solving instruction. We then identified teaching
behaviors that seemed likely to promote these behaviors. Our task was made especially difficult
by the fact that none of the literature on mathematical problem-solving instruction discussed the
specifics of the teacher's role and very little of the research literature on teaching dealt with
problem solving. As reasonable as our effort seemed at the time, we now view it as
incomplete. What is needed is to consider teaching behavior, not simply as an agent to effect
certain student outcomes, but rather as one dimension of a dynamic interaction among several
dimensions.
Observations of Real Classrooms
A few years ago we conducted a large-scale study of the effectiveness of an approach to
problem-solving instruction based on the ten teaching actions mentioned above. The research
involved several hundred fifth and seventh grade students in more than 40 classrooms [6]. The
results were gratifying: students receiving the instruction over the course of a year benefited
tremendously with respect to several key components of the problem-solving process.
However, despite the promise of our instructional approach, the conditions under which the
study was conducted did not allow us to make extensive systematic observations of
classrooms. Ours is not an isolated instance. Good and Biddle [13], Grouws [14], and Silver
[35] have noticed an absence of adequate descriptions of what actually happens in the
classroom. In particular, there has been a lack of descriptions of teachers' behaviors, teacher-
student and student-student interactions, and the type of classroom atmosphere that exists. It is
vital that such descriptions be compiled if there is to be any hope of deriving sound
prescriptions for teaching problem solving.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-17-320.jpg)
![3
Focus on Individuals Rather than Groups or Whole Classes
Much of the research in mathematical problem solving has focused on the thinking processes
used by individuals as they solve problems or as they reflect back on their work solving
problems. When the goal of research is to characterize the thinking involved in a process like
problem solving, a microanalysis of individual performance seems appropriate. However,
when our concerns are with classroom instruction. we should give attention to groups and
whole classes. We agree with the argument of Shavelson and his colleagues [32] that small
groups can serve as an appropriate environment for research on teaching problem solving. But.
the research on problem-solving instruction cannot be limited to the study of small groups.
Lester [25] suggests that. in order for the field to move forward. research on teaching ｰｲｯ｢ｬ･ｾ＠
solving needs to examine teaching and learning processes for individuals, small groups, and
whole classes.
Atheoretical Nature of the Research
The absence of any widely-accepted theories to guide the conduct of research is a serious
problem [1, 17,20]. The adoption of a theory orientation toward research in this area is crucial
if progress is to be made toward establishing a stable body of knowledge about problem-
solving instruction. In particular, we would like to see the development of theories of problem-
solving instruction become a top priority for mathematics education research.
Toward a Solution
An important ingredient in making research on problem-solving instruction more fruitful is the
clear description of the factorsl involved. In this section we provide an analysis of the factors
relevant to the study of problem-solving instruction. This analysis is intended as a structure
within which to design research.
Categories of Factors Related to Problem-solving Instruction
Our first attempt to provide an analysis of the factors that might be related to problem-solving
instruction was made by Lester [24] in his general discussion of methodological issues
lWe are aware that the word "factors" is commonly associated with experimental and quasi-experimental research
methods. Our use of this word should not be interpreted as indicating that we have a preference for this sort of
research to study problem-solving instruction. We have no such preference. Indeed, we believe that progress in
this area will bemade primarily by means ofcarefully conducted ethnographic studies, longitudinal case studies
of classrooms, observations of individuals and small groups, and clinical interviews. The word "factors" is used
here exclusively to indicate what we consider to be key ingredients in the success or failure of problem-solving
instruction.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-18-320.jpg)
![4
associated with research in this area. Lester's analysis was based on the earlier efforts of
Dunkin and Biddle [9] and Kilpatrick [16]. Dunkin and Biddle's work was a broad view of the
important categories of factors associated with research on teaching and was not specifically
concerned with either mathematics or problem solving. Kilpatrick's analysis was restricted to
research on teaching problem-solving heuristics in mathematics. After considering research on
teaching behavior and teacher-student interaction (see e.g., [22, 23, 30, 31, 41, 42]), Charles
[5] developed a refinement of Lester's original analysis. The primary improvement was that
Charles' analysis incorporated two factors which were not present in Lester's work: a teacher
planning category and interactions among categories of classroom processes. The structure
presented below is a further refinement of our previous conceptualizations and other more
recent conceptualizations of teaching. As is true of Charles's categorization, we have chosen to
classify factors on the basis of three broad categories: Extra-classroom Considerations,
Classroom Processes, and Instructional Outcomes. We have also identified a category that cuts
across each of these categories: Teacher Planning. The four categories are discussed in the
following sections.
Category 1: Extra·classroom Considerations. What goes on in a classroom is
influenced by many things. For example, the teacher's and students' knowledge, beliefs,
attitudes, emotions, and dispositions all play a part in determining what happens during
instruction. Furthermore, the nature of the tasks (i.e., the activities in which students and
teachers engage during instruction) included during instruction, as well as the contextual
conditions present also affect instruction. We have identified six types of considerations:
teacher presage characteristics, student presage characteristics, teacher knowledge and affects,
student knowledge and affects, tasks features, contextual (situational) conditions.
Teacher and Student Presa&<; Characteristics. These are characteristics of the teacher and
students that are not amenable to change but which may be examined for their effects on
classroom processes. In addition, presage characteristics serve to describe the individuals
involved. Typically, in experimental research these characteristics have potential for control by
the researcher. But, awareness of these characteristics can be useful in non-experimental
research as well help researchers make sense of what they are observing. Among the more
prominent presage characteristics are age, sex, and previous experience (e.g., teaching
experience, previous experience with the topic of instruction). Factors such as previous
experience may indeed be of great importance as we learn more about the ways knowledge
teachers glean from experience influences practice (see [33]).
Teacher and Student Affects. Cognitions. and Metacognitions. The teacher's and students'
knowledge (both cognitive and metacognitive) and affects (including beliefs) can strongly
influence both the nature and effectiveness of instruction. As a category, these teacher and
student traits are similar to, but quite different from, presage characteristics. The similarity lies
in the potential for providing clear descriptions of the teacher and students. The difference
between the two is that affects, cognitions, and metacognitions may change, in particular as a
result of instruction, whereas presage characteristics cannot.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-19-320.jpg)
![5
Much recent attention has been given to the role of the teacher's knowledge as it relates to
planning and classroom instruction (see e.g.,[33]). Doyle [8] suggests that teaching can be
viewed as a problem-solving activity where the knowledge a teacher brings to the teaching
situation is craft knowledge [34], that is, knowledge gleaned from experience. Doyle suggests
that "teaching is, in other words, fundamentally a cognitive activity based on knowledge of the
probable trajectory of events in classrooms and the way specific actions affect situations" [8,
p. 355]. Thus, there is growing evidence that the knowledge a teacher brings to the classroom
has a significant impact on the events that follow, and that there are a variety ofkinds of teacher
knowledge that need to be considered in research on teaching (e.g., content knowledge,
classroom knowledge).
Task Features. Task features are the characteristics of the problems used in instruction and
for assessment. At least five types of features serve to describe tasks: syntax, content, context,
structure, and process (see [12]). Syntax features refer to the "arrangement of and relationships
among words and symbols in a problem" [18, p. 16]. Content features deal with the
mathematical meanings in the problem. Two important categories of content features are the
mathematical content area (e.g., geometry, probability) and linguistic content features (e.g.,
terms having special mathematical meanings such as "less than," "function," "squared").
Context features are the non-mathematical meanings in the problem statement. Furthermore,
context features describe the problem embodiment (representation), verbal setting, and the
format of the information given in the problem statement. Structurefeatures can be described as
the logical-mathematical properties of a problem representation. It is important to note that
structure features are determined by the particular representation that is chosen for a problem.
For example, one problem solver may choose to represent a problem in terms of a system of
equations, while another problem solver may represent the same problem in terms of some sort
of guessing process (see for example, Harik's discussion of "guessing moves" [15]). Finally,
process features represent something of an interaction between task and problem solver. That
is, although problem-solving processes (e.g., heuristic reasoning) typically are considered
characteristics of the problem solver, it is reasonable to suggest that a problem may lend itself to
solution via particular processes. A consideration of task process features can be very
informative to the researcher in selecting tasks for both instruction and assessment.
Contextual Conditions. These factors concern the conditions external to the teacher and
students that may affect the nature of instruction. For example, class size is a condition that may
directly influence the instructional process and with which both teacher and students must
contend. Other obvious contextual conditions include textbooks used, community ethnicity,
type of administrative support, economic and political forces, and assessment programs. Also,
since instructional method provides a context within which teacher and student behaviors and
interactions take place, it too can at times be considered a factor within this category.
The six types of extra-classroom considerations are displayed in Table 1. It is important to
add that we do not regard these six areas of consideration as comprehensive. It is likely that
there are other influences that may be at least as important as the ones we have discussed.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-20-320.jpg)
![7
content and organization of the textbooks used and, therefore, need not be considered as an
object ofresearch [10].
Category 3: Classroom Processes. Classroom processes include the host of teacher
and student actions and interactions that take place during instruction. We have identified four
dimensions of classroom processes: teacher affects, cognitions, and metacognitions; teacher
behaviors; student affects, cognitions, and metacognitions; and student behaviors. Table 2
provides additional details on each of these dimensions.
Table 2: Oassroom Processes
Teacher Affects. Cognitions & Metacognitions Student Affects. Cognitions & Metacognitions
• with respect to problem-solving phases • with respect to problem-solving
(understanding, planning, carrying out plans, (understanding, planning, carrying out plans,
& looking back) & looking back)
• attitudes about self, students, mathematics, • attitudes about self, teachers, mathematics, &
problem solving, & teaching problem solving
• beliefs about self, students, mathematics, • beliefs about self, teachers, mathematics, &
problem solving, & teaching problem solving
Teacher Behaviors Student Behaviors
Examples: questioning, clarifying, guiding, Examples: identifying information needed to
monitoring, modeling, evaluating, solve a problem, selecting strategies for
diagnosing, aiding generalizing, ... solving problems, assessing extent of
progress made during a solution attempt,
implementing a chosen strategy, determining
the reasonableness ofresults
Both the teacher's and the students' thinking processes and behaviors during instruction are
almost always directed toward achieving a number of different goals, sometimes
simultaneously. For example, during a lesson the teacher may be assessing the appropriateness
of the small-group arrangement that was established prior to the lesson, while at the same time
trying to guide the students' thinking toward the solution to a problem. Similarly, a student may
be thinking about what her classmates will think if she never contributes to discussions and at
the same time be trying to understand what the activity confronting her is all about. For
convenience, we have restricted our discussion of classroom processes to the thinking
processes and behaviors of the teacher and students that are directed toward activities (both
mental and physical) associated with P6lya's [29] four phases of solving problems:
understanding, planning, carrying out the plan, and looking back. That is, we have restricted
our consideration to what the teacher thinks about and does to facilitate the student's thinking](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-22-320.jpg)
![9
Table 3: Instructional Outcomes
Student Outcomes
1. Immediate effects on student leaming with respect to:
a. problem-solving skills b. perfonnance c. general problem-solving ability
d. affects e. beliefs f. mathematical content knowledge
2. Long-tenn effects on student learning with respect to:
a. problem-solving skills b. perfonnance c. general problem-solving ability
d. affects e. beliefs f. mathematical content knowledge
3. Near and far transfer effects
Teacher Outcomes
1. Effects on the nature of teacher planning
2. Effects on teacher behavior during subsequent instruction
3. Effects on teacher affects and beliefs about:
a. effectiveness of instruction b. "worthwhileness" of instructional methods
c. "ease" of use of instructional methods
Incidental Outcomes
Examples: student perfonnance on achievement tests, influence of instruction on student/teacher
behavior in non-mathematics areas, affective changes with respect to mathematics in general,
schooling, ...
effect of problem-solving instruction on mathematical skill and concept leaming. For example,
how is computational skill affected by increased emphasis on the thinking processes involved in
solving problems?
Teacher Outcomes. Teachers, of course, also change as a result of their instructional
efforts. In particular, their attitudes and beliefs, the nature and extent of their planning, as well
as their classroom behavior during subsequent instruction are all subject to change. Each
problem-solving episode a teacher participates in changes the craft knowledge of the teacher
[34]. Thus, it's reasonable to expect that experience affects the teacher's planning, thinking,
affects, and actions in future situations.
Incidental Outcomes. Increased perfonnance in science (or some other subject area) and
heightened parental interest in their children's school work are two examples of possible
incidental outcomes. Although it is not possible to predetermine the relevant incidental effects of
instruction, it is important to be mindful of the potential for unexpected side effects.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-24-320.jpg)
![10
Discussion
There are several factors in the structure described in this paper that deserve particular attention.
In the following paragraphs we discuss these factors.
Extra-clagroom Comidemdom: Teacher's Affects, Cogniliom and Metacogniliom
Research on teaching in general points to the important role a teacher's knowledge and affects
play in instruction. However, they have received relatively little attention in research on the
teaching of problem solving [3, 7, 39, 40]. Future research should consider all of these factors
when real teachers are involved. Questions such as the following need to be investigated: What
knowledge (in particular, content, pedagogical, and curriculum knowledge) do teachers need to
be effective as teachers of problem solving? How is that knowledge best structured to be useful
to teachers? How do teachers' beliefs about themselves, their students, teaching mathematics,
and problem solving influence the decisions they make prior to and during instruction?
Extra-classroom Considerations: Task Features
Although there has been considerable research on task variables (see [12]), it may be time to
consider the very nature of the tasks used for instruction relative to problem solving. In most
research, tasks have been relatively brief problem statements presented to students in a printed
format Thus, research on task variables has considered variables like problem statement length
and grammatical complexity. But, very little attention has been given to the identification of task
variables that should be considered when the problem-solving task is presented, for example,
through a videotape episode from an Indiana Jones movie (see [2]). Or, suppose the real-world
tasks used for instruction were selected from those used in an instructional approach modeled
after the concept of an apprenticeship [21]. Would the task variables of importance be the same
as those examined and discussed in the existing research literature?
Teacher Planning
A teacher's behavior while teaching problem solving is certainly influenced by the teacher's
affects, cognitions and metacognitions during instruction. However, some of this behavior
during instruction is likely to be determined by the kinds of decisions the teacher makes prior to
entering the classroom. For example, a teacher may have planned to follow a specific sequence
of teaching actions for delivering a particular problem-solving lesson knowing that the exact
ways in which these teaching actions are implemented evolve situationally during the lesson.
Or, if the knowledge teachers use to plan instruction is case knowledge, that is knowledge of
previous instructional episodes [8], then we would search for those cases that significantly](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-25-320.jpg)
![11
shape the craft knowledge teachers use as a basis for planning and action. Future research
should consider how teachers go about planning for problem-solving instruction and how the
decisions made during planning influence actions during instruction.
Classroom Processes: Teacher Affects, Cognitions and Metacognitions
Students' cognitive and metacognitive behavior during mathematical problem solving have
received a great deal of attention in recent years (see, for example, [7, 11, 36]). And, there is an
indication that students' attitudes, beliefs, and emotions also are beginning to be the object of
study [27, 28]. Research is needed that describes the thinking a teacher does during the
teaching of problem solving. What does the teacher attend to during instruction? How do
teachers interact with students during instruction? What drives the teacher's decisions at those
times? Are teachers aware of their actions during instruction? Do they consciously assess their
behaviors during instruction? Rich descriptive studies are seriously needed in this much
neglected area. The work ofLampert [19] provides a promising model for consideration.
Classroom Processes: Teacher Behaviors
It is especially important that future research pay closer heed to what teachers actually do during
instruction (i.e., teacher actions). In particular, there is a need to document teacher behaviors as
they relate to the process of solving problems. It would be a significant contribution to our
understanding of how to teach mathematical problem solving if careful, rich descriptions were
prepared of a teacher's actions during each of P6lya's four phases (refer to Table 2). Such
descriptions would help develop a clearer picture of what teachers should do to aid students to
understand problem statements, to plan methods of solution, to carry out their plans, and to
look back (evaluate, reflect, generalize) at their efforts.
Classroom Processes: Student Affects, Cognitions, and Metacognitions
As mentioned earlier, these factors have received substantial attention in the mathematical
problem-solving literature. It would be valuable to know more about the conditions under
which students' cognitions and metacognitions change and when these changes begin to occur.
In a study we conducted [6] continual improvement in students' understanding, planning, and
implementing behaviors (cognitions) occurred during the course of 23 weeks of problem-
solving instruction. However, the design of the study did not allow us to make conjectures
about the aspects of the program that were responsible for the improvement. The effects of a
student's attitudes, beliefs, and emotions during problem solving on problem-solving
performance is another area in need of serious attention. Recent work by Lester, Garofalo, and](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-26-320.jpg)
![12
Kroll [26, 27] illustrates the often strong interaction among affects, cognitions, and
metacognitions. McLeod and Adams [28] provide numerous questions for additional research
in this area.
Classroom Processes: Student Behaviors
As important as it is to begin to study carefully the behavior of teachers during instruction, it is
just as important to observe the behavior of students during problem-solving sessions. In
particular, it would be valuable to know more about the way in which students' interactions
affect their thinking processes and actions. For example, to what extent are students' choices
and use of a particular problem-solving heuristic or strategy influenced by their peers'
decisions? What group processes promote or inhibit successful problem solving? How are
individual differences exhibited during problem solving? What kinds of individual differences
influence success in solving problems and how can these differences be handled by the teacher?
How are students' classroom behaviors influenced not only by their cognitions and
metacognitions during instruction, but how are their behaviors influenced by the attitudes and
beliefs they bring to the lessons? What kinds of student behaviors are attended to by teachers?
How do student-student and teacher-student communications shape affects, beliefs, and
cognitions demonstrated while solving problems and the knowledge students extract from the
experience? Which instructional influences have the greatest effect on a student's actions?
Instructional Outcomes: Students
Both immediate and long-term effects on student learning should be considered with respect to
cognitions and affects. In addition, possible transfer effects (both near and far) should be
investigated. For example, does instruction involving geometry problems have any effect on
students' solutions to non-geometry problems? Does instruction in solving routine story
problems influence students' solutions to non-routine problems? Furthermore, it is not enough
to be content with some vague, general improvement in the ability to get more correct answers
(cf. [36, 37]). Instead, it is essential that researchers look to see if their instructional
interventions actually develop the kinds of student behaviors they were designed to promote.
Finally, how does problem-solving instruction influence the amount and nature of students'
mathematics content knowledge? As an illustration of this last question, consider the situation in
which a third grade teacher has emphasized during the course of an entire school year the
development of various problem-solving strategies. To maintain this emphasis she has found it
necessary to devote less attention than in the past to basic fact drill activities. Has the students'
mastery of these basic facts been affected by this change in emphasis?](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-27-320.jpg)
![13
Instructional Outcomes: Teachers
Lampert [19] illustrates how reflecting on instructional practice influences subsequent planning
and classroom behavior. It would be a useful contribution to document how systematic
problem-solving instruction affects teachers' planning, affects and beliefs. Furthermore it
would be valuable to know how such instruction influences teacher behavior during subsequent
instruction.
A Final Comment
Our analysis of factors to be considered for research on problem-solving instruction is intended
as a general framework for designing investigations of what actually happens in the classroom
during instruction. As mentioned earlier, we recognize that there may be other important factors
to be included in this framework and that certain of the factors may prove to be relatively
unimportant. Notwithstanding these possible shortcomings, the framework we have presented
can serve as a step in the direction of making research in the area more fruitful and relevant.
Finally, at the beginning of this paper we recommended that the development of theories of
problem-solving instruction should become a top priority for mathematics educators. We also
believe that the type of theory development that is likely to have the greatest relevance should be
"...grounded in data gathered from extensive observations of 'real' teachers, teaching 'real'
students, 'real' mathematics, in 'real' classrooms" [24, p. 56]. It is only by adopting such a
perspective that we are likely to make any significant progress in the foreseeable future.
References
1. Balacheff, N.: Towards a problematique for research on mathematics teaching. Journal for Research in
Mathematics Education 21, 259-272 (1990)
2. Bransford, J., Hasselbring, T., Barron, B., Kulewicz, S., Littlefield, J., & Goin, L.: Use of macro-contexts to
facilitate mathematical thinking.. In: The teaching and assessing of mathematical problem solving (R. Charles
& E. Silver, eds.), pp. 125-147. Reston, VA: LEA & NCTM 1988
3. Brown, C., Brown, S., Cooney, T., & Smith, D.: The pursuit of mathematics teachers' beliefs. In:
Proceedings of the Fourth Annual Meeting of the North American Chapter of the International Group for the
Psychology of Mathematics Education (S. Wagner, ed.), pp. 203-215. Athens, GA: University of Georgia,
Department of Mathematics Education 1982
4. Charles, R.: The development of a teaching strategy for problem solving. Paper presented at the Research
Presession of the Annual NCTM Meeting, San Antonio, TX, April 1985
5. Charles, R., & Lester, F.: Teaching problem solving: What, why and how. Palo Alto, CA: Dale Seymour
Publications 1982
6. Charles, R., & Lester, F.: An evaluation of a process-oriented mathematical problem-solving program in
grades five and seven. Journal for Research in Mathematics Education 15, 15-34 (1984)
7. Charles, R., & Silver, E. (eds.): The teaching and assessing of mathematical problem solving. Reston, VA:
LEA & NCTM 1988](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-28-320.jpg)
![18
What would useful research conclusions look like? Who would try to use them?
There is a long history of thoughtful advice about problem-solving, from such eminent names
as Plato, Aristotle, Pappus, Bacon, Descartes, Fermat, Euler, Bolzano, Boole, Hadamard, and
P6lya [29]2. They sought to organize thinking, and some of them even sought to mechanize it,
as well as to assist novices in learning mathematics. The impulse to mechanize thought has
appeared in every generation. The availability of sophisticated computers has stimulated a
burgeoning of activity by numerous workers in the fields of Artificial Intelligence and Cognitive
Psychology, who have likewise contributed to the mechanization of problem solving in
mathematics3 in the idiom of their times.
When research findings are translated into practice, they turn from observation into rules,
from heuristics into content. Attempts to pass on insights become attempts to teach patterns of
thought. Once the "patterns of thought", the heuristics, become content to be learned,
instruction in problem-solving takes over and thinking tends to come to a halt. Dewey put it
nicely, when he said
Perhaps the greatest of all pedagogical fallacies is the notion that a person learns
only the particular thing he is studying at the time. Collateral learning ... may be
and often is much more important than the actual lesson.
Approaching the "content" too directly may not always be wise. In an age noted for its
pragmatism which begins to rival that of Dewey's times, collateral learning may need to be re-
emphasized. To counteract the functionalism of modern educational curriculum specification
and assessment, Bill Brookes, in a seminar, suggested that
We are wise to create systems for spin-offs rather than for pay-offs.
Mathematicians too have been very slow to acknowledge and accept distinctions made by
researchers, fearful perhaps that they will lose touch with their own creativity. People do not,
on the whole, relish being told how they think or how they might think differently.
In order to influence teaching, researchers have traditionally moved to programme
development, almost always text based, with or without training for teachers in the use of the
materials. Even where teachers are supported, it is usually to help them to "deliver the
programme", like some mail-order firm.
My approach is to work from the inside, from and on my own experience. This is not as
solipsistic and idiosyncratic as it sounds, for it draws on results of outer research which speaks
2 See particularly George POlya [29] for a study of the use and teaching of heuristics. For recent surveys, see
also Alan Schoenfeld [32] and H. Burkhardt, S. Groves, A. Schoenfeld, and K. Stacey [2].
3 Particularly notable is the work of Edmund Furse [7]. Furse has managed to build a program that will read
standard mathematical texts (translated into a suitable formal language), and then solve the end-of-chapter
exercises. Its solutions look like those of an expert. Furthermore, there is no mathematics embedded in the
program, only a syntactic pattern recognition engine.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-32-320.jpg)
![19
to my experience. Furthennore, there is a long history of people working effectively in this
way, particularly in the phenomenological and hermeneutic traditions, building as they do on
psychological insight of ancient peoples in India, China and the Middle East. Recently there has
been a notable increase in interest from academics, with attention being drawn to metacognition
and reflection4• Humboldt observed that the essence of thinking lies in abstraction, Dewey
spoke of turning a subject over in the mind, Piaget stressed the role of reflective abstraction,
Vygotsky drew attention to the internalization ofhigherpsychologicalprocesses through being
in the presence of more expert thinkers, Skemp lays great stress on the development of
reflective as well as intuitive intelligence, and Kilpatrick [19] made an explicit call for the
development of self-awareness in mathematics education.
Apart from practical considerations such as that teacher-proof materials have never succeeded,
and that there is no evidence for a single way to teach mathematics, there are three principle
justifications for an approach from the inside:
to appreciate the struggles of another, it is necessary to struggle oneself, and to be
able to re-enter that struggle;
to learn to struggle and to make best use of available resources, it is useful to be in
the presence of someone who themselves is in question, who is struggling to
know;
teaching mathematics is ultimately about being mathematical oneself, in front of and
with pupils, and so in order to develop one's teaching, to work at being
mathematical, it is necessary to develop one's mathematical being.
I do not mean to imply that it is best for teachers to be struggling with the same mathematics as
their pupils, although often it is the case that a teacher who does not know answers can be of
more help than a teacher who does. More precisely, knowing the/an answer can make it harder
to appreciate pupils' thinking, and more likely that you will direct them to your own solution.
Instead of dwelling on mathematical answers, teachers can be more helpful by dwelling in their
own questions about what the pupils are thinking, how their powers can be evoked, what
advice might be helpful, and leave the actual mathematical thinking to the pupils. To do this
requires awareness of their own thinking, which comes from inner research.
To influence pupils in how to come into question, to be in question, and to resolve
questions it is helpful if the teachers are themselves in question, either about some other
mathematics, or about their teaching of mathematics, or both. I concur with Brown, Collins,
and Duguid [1] who advocate a cognitive-apprenticeship vocabulary for describing one form of
effective teaching, and this is consistent with current social-constructivist thinking about the
role and significance of peers and experts in learning. The medieval Guild system, when
functioning effectively, had much to recommend it.
4 Some would attribute this to the dawning of the age of Aquarius. Others. such as Julian Jaynes. [17] would
claim that we are experiencing an evolutionary change in the structure ofconsciousness.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-33-320.jpg)
![20
The difficulty as always is to translate description into action. I do not advocate trying to
describe effective teaching in behavioural tenns so that others can then (supposedly) "do it". All
attempts at this of which I am aware, have failed. What I advocate is establishing a collegial
atmosphere in which teachers work on and develop their awareness of their own mathematical
thinking, and of themselves as teachers.
I stress the collegial, because it is not enough to be a hennit, operating alone behind the
closed door of the classroom. It is essential to test out insights and conjectures with others, to
participate in a wider community of fellow seekers. Such peer-checking is an essential part of
the larger methodology ofthe Discipline of Noticing (see Mason and Davis [28] and Davis [3]).
In this paper I want to illustrate rather than derme what I mean by researching/rom the inside,
and to make some remarks about what I have learned over the past twenty-five years. I shall
offer some mathematical questions5 to illustrate a little bit of what I mean by working from the
inside, and to indicate that it is fundamental to me to be consistent in my approach. To speak
about working fonn the inside is to monger words. To appreciate what I have learned, you
must experience it from the inside. If my words speak to your experience, then all well and
good. Ifnot, then we have both wasted time!
When working with teachers I often suggest at the beginning that I shall be working on
three levels:
on Mathematics,
on Teaching and Learning Mathematics,
and on Ways of Working:
- on Mathematics,
- on Teaching and Learning Mathematics.
Having come to mathematics education from mathematics research, I find much that is similar
between working on mathematics and working on teaching and learning of mathematics,
particularly in the need for particular examples, and for attention to be devoted to how
generalizations are drawn from or seen in the particular.
I draw attention to these three levels because I neither claim to know, nor wish to show or
tell people, how to teach. Rather, I wish to draw out from their experience for conscious
attention the powers that they already possess, and through that awareness, try to help them be
able to do the same for their pupils. Socrates' image of midwife for ideas in Plato's dialogue
the Thaetetus has much to recommend it, though care is needed in isolating the useful features
of the much invoked but frequently misunderstood Socratic Dialogue (see for example [12,
13]).
Instruction, that term so beloved of American educators, is much more subtle than directing
pupils through the stages of some institutional regime. It involves more than being sensitive to
pupils' needs, more than actualizing a few slogans. It is about being yourself, being aware of
5 Mindful that a question is ink on paper, and a problem is a state which arises in a person in the presence of a
question, in a context.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-34-320.jpg)
![21
yourself and your powers, and evoking those same powers in others. Thus teaching and
learning are for me almost synonymous. They are about entering the potential space
(D. Winnicot, quoted in [18]) between you and another, and inviting others to join you in that
space.
Sometimes direct instruction, that is, giving pupils specific instructions as to what to do, is
appropriate. Telling pupils things can indeed be effective (despite the current bad press which
exposition receives at present), especially if pupils are prepared to hear what is being said
because of recent experience which has raised doubts or questions. Sometimes a less direct
prompt to recall prior advice or instruction is enough, and at other times being verbally silent
but physically and mathematically present, or even being physically absent is sufficient. Pupil
dependency on teacher intervention can easily and unintentionally be fostered. Tasks are
sometimes directive, sometimes indicatively prompting investigation, sometimes spontaneously
arising from pupil or group. Samples of attempts to manifest these in print, purely as support
materials, are given at the end of the paper.
Examples of Approaching from the Inside
In this section I offer some mathematical questions, together with a few remarks about what
often emerges when using them. If the questions are too easy, then nothing will be learned
from them. If you get stuck, then there is a chance something can be learned. If you simply
scan or skim through them, then nothing will be gained.
One Sum
Take any two numbers that sum to one. Which will be the larger sum: the square of
the larger added to the smaller, or the square of the smaller added to the larger?
Everyone naturally specializes. Some use simple fractions, some use decimals, and some use
algebraic symbols. I refer to all of these as specializing, since people naturally tum to
something specific which they find confidence inspiring. Pointing this out can help to ease a
transition from number to algebra.
Square Sums
32 + 42 = 52
102 + 112 + 122 = 132 + 142
212 + 222+ 23 2 + 242 = 252 + 262 + 272
Most people find the first statement a well known and unexceptional fact. Juxtaposed with the
second, which is somewhat surprising, interest mounts. When the third is offered, there is](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-35-320.jpg)
![22
conviction that there must be a pattern which continues. Locating that pattern evokes powers of
pattern spotting and expression which lie at the heart of algebraic generalization and thinking.
The assumption of pattern is endemic in mathematics, but not always born out:
32 + 42 = 52
33 + 43 + 53 = 63
... ?
Map Scaling
Imagine that you are sitting at home with a map of your country, and you wish to
scale it down by a factor of a half. To do this, you put a transparent piece of paper
over the map, and using your own home-town as centre, scale every other feature
of the map halfway along the line joining it to your centre. Now imagine that a
friend uses the same map, but using a different centre. Will your two scaled maps
look the same or different, and why?
Many people have incomplete intuitions about scaling. There is something paradoxical about
being able to use any point as centre, and still get the same map. Even experienced
mathematicians can be caught out in a conflict between their educated intuition and their gut
response. A computer programme such as Cabri-geometre [20] enables a simultaneous scaling
of the same line or circle from two different points, and watching the two images develop is
often a surprise even when you know what to expect. The following has a similar effect,
particularly on astronomers:
The Half Moon Inn
In England you often see pub signs showing a vertical half-moon (the straight
diameter is vertical) in a black sky with a few stars shining nearby. When, if ever,
can you see a vertical half-moon?
These questions are useful for opening up discussion about the extent to which intuition can be
educated, and the extent to which early intuitions are always present, if suppressed. Fischbein
[6] takes the view that early intuitions are robust against teaching, and Di Sessa [4] develops the
notion ofpsychologicalprimitives as basic building blocks in constructing stories to account for
things that happen in the world. The development of much of mathematics can be seen as the
gradual refinement of intuitions: for example from discontinuity as a jump in the graph, to a
more complex phenomenon; from a function having a derivative of 0 at a point being flat
nearby, to the possibility that it could have arbitrary slope arbitrarily close to such a point.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-36-320.jpg)
![23
Planets [33]
Consider a collection of identical spherical planets in space. From some places on
the the surface of some of the planets, it is possible to see one or more other planets
in the collection. What is the total·area on all the planets of these places?
This question has proved fruitful because, after successfully specializing to two-dimensions,
most people develop an argument which generalizes only with great difficulty to three
dimensions. After some initial success, it is usually necessary to prompt people to go back to
the two-dimensional case and seek an argument which will generalize.
Remarks about my Approach
Inner research begins with self-observation. This is a constant and on-going process, guided by
the concerns of the people I am trying to assist, and by my own propensities. Teachers often
remark to me that it is very rare that they have the time to work on mathematics themselves,
either alone or with colleagues. I take this as resonance with the notion that it would be better if
they did engage in mathematical thinking, but that conditions are not supportive. Many are the
programmes of in-service support for mathematics teachers; the most significant benefit to
participants of any programme is rarely the content, and more often the opportunity and
stimulus to "indulge" in mathematics, and to compare reflective notes with colleagues. Where
teachers are transfixed by "picking up some idea they can use tomorrow" the benefit is short-
lived, as evidenced by their return next time for another "fix".
Give someone a fish, and you feed them for a day;
Teach them to fish, and you feed them for a lifetime.6
After many years of inviting people to engage in mathematical thinking, it became clear that on
the whole it was much more effective to begin with simply stated tasks. As with any useful
observation about teaching, this is not meant to be an unbroken rule; rather it is meant to inform
choices made in the moment. I am always tempted to start with something "interesting", which
means "interesting to me", and because of the complexity, people find it difficult to separate off
a little bit of their attention to observe themselves. Starting with apparently very simple
questions which invoke the type of thinking which is needed when you get stuck - specializing
and generalizing, conjecturing and convincing, animating the static and freezing the dynamic,
working backwards and working forwards - enables participants to attend to their experience as
well as to the task, and thereby to find discussion and elaboration fruitful. More important than
the completion of tasks is awareness of what you are doing. But attention is usually drawn into
the mathematics. For example, when teachers watch videotaped episodes from classrooms,
6 Purportedly a Chinese proverb, source unknown.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-37-320.jpg)
![24
their attention is drawn initially to the task posed to the pupils, and whether they, the viewers,
can do it. Only when they feel confident about and familiar with the mathematics can they attend
to the acts of teaching.
I struggled for a long time with Gattegno's memorable observation that
Only Awareness is Educable [9].
He has a quite specific meaning which I only partially appreciate, connected with his Science of
Education. This is an approach which has much in common with mine, in that it seeks the roots
of mathematics and of thinking in actions and experience, although Gattegno emphasizes
particularly the actions and experience of the neonate in its cot. I do not pretend to have
mastered Gattegno's ideas, nor to be carrying them further. I simply note similarities. I found
increased meaning in his observation when I augmented it by twoother assertions, informed by
that most ancient of metaphors for the psychological make-up of human beings, the image of
the Chariot? The chariot is drawn by a horse, and has a driver as well as an owner. One way to
read it is as follows:
the chariot represents the body, which needs maintenance;
the horse represents the emotions (the source of motion) which need grooming and
feeding and sometimes require blinkering;
the driver represents the intellect which needs challenging tasks;
the reins represent the power of mental imagery to direct, which need to be kept
subtle and loosely-taut;
the shafts represent the psychosomatic interplay between emotion and the
musculature (see [31]);
the owner represents a different form of awareness that is possible, but only when
the chariot is functioning correctly will the owner deign to use it.
The two extra observations suggested by this image are that
Only behaviour is trainable, and
Only emotion is hamessable.
I capitalize the only in each case, because it is often the only which causes the most strife when
people encounter and contemplate these assertions. The definitiveness often generates interest
and thought, whether in favour or in opposition. The three onlys also act to inform an approach
to the construction of mathematical tasks for pupils, outlined in Griffith and Gates [10].
The positive role of frameworks, and for slogans too [14], is that they can serve to heighten
awareness, to remind you in the midst of preparation or lesson, of some alternatives to your
standard, automatic reaction. The negative side is that instead of in-forming, they come to pro-
form, even de-form awareness. The education of awareness, and the integration of
7 Katha Upanishad 1.3.3-5, see for example S. Rhadakrishnar [30]. A more modem version of the image can
be found in Gurdjieff [111.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-38-320.jpg)
![25
observations through subordination8 are part of the development of being. but the metaphor of
growth suggests a monotonic progression. and this does not conform with my experience.
Often what once seemed sorted out, needs to be re-evaluated and re-questioned. and stimulus to
do this comes through working with colleagues in order to seek resonance with their
experience.
Mathematical questions. and the recall of teaching or learning incidents through anecdotal
re-telling or through video-tape are devices for focusing attention. In recalling an event in
retrospect. there is a temptation to judge. to berate oneself for failing to do something. This is a
waste of energy. To notice something about yourself is like suddenly taking a breath of clean
air. like suddenly coming upon an unexpected vista. The energy which accompanies the insight
has to go somewhere. and it is all too natural to throw it away on judgement and negativity. To
make use of what is observed. it is better simply to note. to "in-spire" the in-sight into oneself.
to accept. By re-calling and re-entering specific salient moments. moments that come readily
and vividly to mind soon after an event. it is possible to take a three-centred approach to
developing one's being. The horse. carriage. reins. shafts and driver can be employed in a
balanced and intentional storing of experience which can serve to inform practice in the future.
This is the theory behind the presentation and recommendations in Mason. Burton and Stacey
[24]. By vividly re-entering a significant moment through the power of mental imagery. you
contribute to a rich store ofexperience to be resonated at a later date. To stimulate the growth of
an inner monitor. it is necessary to direct some of the energy released from noticing. for that
purpose.
What tends to happen is that you become aware of missed opportunity. After the lesson it
suddenly comes to you what you could have done; a few moments after a pupil asks a question.
you find yourself answering in your usual automatic fashion; despite intending to listen to what
pupils have to say. you find yourself telling them what you think; while working on a question
you gradually become aware that you are not getting anywhere. These moments of awareness
come at fll'St in retrospect. The challenge is to move them into the moment. so that you become
aware of opportunities when they are relevant. Strategies and techniques for doing this are part
of the Discipline of Noticing. and have been described elsewhere.
Noticing Action in the Moment
To become aware of your own thinking. and to be able to make use of personal insights when
teaching. it is necessary to be able to catch action in the moment that it happens. But action is
very hard to catch in yourself when you are being successful. It is likely. for example. that you
found at least one of the tasks presented earlier so easy that you simply "did it". The trouble
8 A reference to Gattegno's memorable title. What we owe children: The subordination ofteaching to learning.
Gattegno [8].](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-39-320.jpg)
![26
with success is that although it breeds psychological confidence, it offers no support if and
when things start to get sticky.
It is relatively easy to catch yourself when you are stuck, because attention naturally drifts
from the task at hand to awareness of being stuck. Unfortunately, being stuck does not involve
much action, and again, this state is not terribly helpful in itself. It is an honourable state, full of
potential, and it can be turned to positive advantage, if you recognize and acknowledge it, and
cast around intentionally for advice and assistance. A passing teacher can at this point say many
things to you, some of which you can hear, while others will go right past you. If the teacher
always provides the same sort of help, then you may come to recognize that constant help, or
you may simply come to depend on it
Let me take one class of examples: teacher interventions when pupils get stuck. There is
considerable interest at present in scaffolded instruction in mathematics, in which the teacher,
with Vygotsky's zone ofproximal development in mind, asks questions to locate just where the
pupil's current "zone" lies, and then tries to "operate in that zone". But the sorts of questions
that teachers ask in these circumstances have been described over and over again over the years.
The new metaphors make little difference to what happens, just different ways of reading them.
Edwards and Mercer [5] concluded after extensive study of transcripts that
Despite the fact that the lessons were organized in terms of practical actions and
small-group joint activity between pupils, the sort of learning that took place was
not essentially a matter of experiential learning and communication between
pupils.... While maintaining a tight control over activity and discourse, the
teacher nevertheless espoused and attempted to act upon the educational principle of
pupil-centred experiential learning ... (p.156)
John Holt [15] has a most succinct version in an interaction with Ruth:
We had been doing math, and I was pleased with myself because, instead of telling
her answers and showing her how to do problems, I was "making her think" by
asking her questions. It was slow work. Question after question met only silence.
She said nothing, did nothing, just sat and looked at me through those glasses, and
waited. Each time, I had to think of a question easier and more pointed than the last,
until I found one so easy that she would feel safe in answering it. So we inched our
way along until suddenly, looking at her as I waited for an answer to a question, I
saw with a start that she was not at all puzzled by what I had asked her. In fact, she
was not thinking about it. She was coolly appraising me, weighing my patience,
waiting for that next, sure-to-be-easier question. I thought, "I've been had!" The
girl had learned to make all her previous teachers do the same thing. If I wouldn't
tell her the answers, very well, she would just let me question her right up to them.
Bauersfeld calls this thefunnel effect, as teacher and pupil are together drawn down a funnel of
increasing detail and special cases. The outcome is usually a question so trivial that the pupil is](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-40-320.jpg)
![27
in wonder that the teacher could ask it, but their attention is absorbed by each successive
question and is drawn away from the larger task.
There are alternatives. When you recognize that you have something specific that you are
trying to get pupils to "see" or "say", you can, for example, stop your current questioning,
acknowledge what you are doing, and simply tell them the idea that you have in mind. This
requires more presence and strength than might appear, and that presence and strength are built
up over a period of time through noticing missed opportunities. You can then suggest that they
reconstruct it in their own language, perhaps by repeating it to a colleague. Then they can
explore variations of the idea, and pose themselves more general questions which succumb to
the same approach. That works if they see the relevance of what is said to what they are trying
to do. Unfortunately, when a teacher leaves a group of pupils after an intervention it is often the
case that the pupils act as if the teacher had never been. They carry on as before. I conjecture
that one reason is that teachers remain caught in giving direct advice, much of which does not
relate to the pupils' experience.
More fruitful is a gradual movement from direct advice, though indirect prompts which
evoke recent memories of similar situations, with the overall aim of withdrawing prompts
altogether and detecting pupils using the suggestions spontaneously themselves. Brown,
Collins and Duguid [1] referred to this asfading. I have found it useful to keep in mind a
transition in intervention from directed, to prompted, to spontaneous use of technical terms and
heuristic advice by pupils. When you are stuck on a problem, you tend to be deeply involved in
that problem. When you seek advice, you immediately try to apply what you are told to the
problem, and so tend not to be aware of the shape or form of that advice. In order to assist
pupils to become aware of the existence of useful advice, metacognitive shifts can be effective.
Instead of asking Can you give me an example?, the question What question am I going to ask
you? can draw attention away from the particular to the general, prompting a shift in the
structure of attention [23, 26]. The pupil may not know the first few times, so you can resort to
your original question. But at least you are signalling that the advice itself is of some, indeed
more importance than success on the particular task. In this way, pupils can move from
requiring direct advice, to prompted advice, and eventually, to giving themselves that same
advice. In the process they will have internalized their own mathematical monitor, and
intentionally contributed to the growth of their inner teacher ([16, 27]).
In terms of evaluating the effectiveness of some framework such as directed, prompted,
spontaneous, advice on what to do when you are stuck, or any other proposed to pupils and/or
teachers, I want to know whether people find the ideas informing their practice. But I distrust
asking them directly. Even observing lessons to see if there is some manifestation of the
framework is a form of prompting probe which is likely to distort, and assumes that what is of
value is observable. In seeking to evaluate it is all too tempting to be caught in a cause-and-](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-41-320.jpg)
![28
effect paradigm, in which treatment is administered and pupil reactions noted. But teaching and
learning do not follow cause-and-effect. They are concurrent not consequential. I consider that
teaching takes place in time, through specific acts,
while learning as taking place over time, through maturation.
Evidence for "learning" based on direct instructions and prompts does not approach what pupils
have internalized and use spontaneously. Another way to express this is by the following
assertion
Wounds are to patients, as assessment is to students.
The value of such an assertion is in the challenge that it induces, and the multiplicity of
interpretation, including contradictory views. Just as health is defined by evaluators as "healthy
response to illness", so learning is defined as "appropriate response to assessment". But health
is much more than response to illness, and learning much more than response to examination
questions. Indeed the medical metaphor is deeply embedded in educational rhetoric.
In seeking evidence, I prefer to wait for spontaneous reference and use of related ideas by
teachers when talking or writing to colleagues about what they do. The language they use may
not be the same as that used by me, indeed I would be wary if it were. And when the language
varies, I cannot assert that the expression they come to has been stimulated, influenced or
informed by me. Similarly with pupils, I distrust direct questions and oblique prompts which
try to ascertain whether pupils have integrated advice into their own thinking. or about the value
they have found for themselves [25]. Searching their work for evidence of using a particular
framework ignores unmanifested influence and behaviour that is informed by exposure to that
advice, while use of the terms of some framework may simply be the effect of expecting that
that is what I want to see. I am more impressed by spontaneous use in later courses or in other
work. This approach renders statistical analysis powerless, but when spontaneous
manifestation occurs, it provides evidence ofreal value and significant use. This sort of external
validation is therefore rather unreliable for evaluative purposes.
Validation of the effectiveness of inner research lies with the researcher in seeking
resonance with pupils and colleagues, not just once, but frequently and repeatedly. Re-
questioning becomes part of established practice. Progress is not achieved by building on
certainty, because insights are constantly in need of verification in fresh times and places, and
of restatement in current idioms and language so that people can recognize them in their own
experience.
I conclude with three examples of types of written materials, designed to assist teachers to
sharpen their awareness of their own thinking, and perhaps then to use some modified version
of the tasks with their pupils. The examples are merely ink on paper. What is important is to](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-42-320.jpg)
![30
With an unfolded dragon curve, start at segment 1 and traverse the paper strip
recording an R or an L for right or left turns as you traverse the curve. This
is the turn-sequence.
One way to find a description is by generalizing from several systematically
developed examples. Another is to work from what happens to a turn
sequence when a new fold is made.
A Dragon Curve is the curve obtained as the limit of a sequence of more
"folded" paper dragon curves.
B - General Dragons. The original paper dragon was made by always
folding the right-hand end over the left to make the folds and creases. But at
any stage you could fold the right-hand under the left instead.
Afold-sequence is an infinite sequence of O's and U's, and is to be interpreted
as instructions to fold the right-hand end of the strip Over or Under what is
already there.
There is a 1 to I correspondence between the fold-sequences and the real
numbers on [0, I].
A fold-sequence gives rise to a generalized paper dragon curve, by opening
out the strip of paper and making the creases all form right-angles.
One version is that the dth Dragon Curve is formed from the d-I th by
inserting in front of, between each letter of, and at the end of the tum-
sequence for the d-Ith curve, chosen in tum from L, R , L, ... or the letters
R, L, R, ... depending on whether the dth term of the fold-sequence is 0 or
U.
Work out two different ways
of determining the turn-
sequence of a dragon curve of
degree d in terms of the
previous sequence for degree
d-J.
Which points of a paper
dragon curve are also points
of the limit Dragon Curve?
Show that any sequence
beginning (0, ...J has the
same effect as some other
sequence (U, ...).
Show that two different
sequences both beginning
with 0 give different results.
Think binary.
Specify how to work out the
tum-sequence corresponding
to a given fold-sequence.
Why does this work?](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-44-320.jpg)
![31
Another version is that the next Dragon Curve is fonned from the last by
inserting an L or an R as appropriate in between two copies of the previous
sequence, but transfonned according to whether the letter in the fold-sequence
is 0 or U. For an 0, the sequence stays as it was. For a U, it is reversed and
the letters interchanged.
THEOREM: Dragon curves are (almost) simple. That is, they never cross
themselves, though they do touch at various points.
Example B: Exploration
Why does this work?
Suggestion: to cross itself,
the curve would have to turn
through 360 degrees between
two creases.
This extract is taken from Mason [22] which are materials designed to stimulate teacher's own
mathematical thinking in a context of topics taught in the classroom. There is very little
exposition or even assistance, since it depends on teachers or groups of pupils to work together
puzzling them out with the assistance of a tutor when necessary. "Readers" are expected to
select from the multitude of activities presented, and to explore around them as they see fit.
What is the Problem?
Adding together two numbers is a basic arithmetic operation, as is multiplication.
The idea of what constitutes a number grew, historically, from whole numbers
through fractions to decimals (though the full story is immensely complex as one
might imagine). At each stage, a new notation for numbers was suggested in order
to make thinking or computation easier and, with the notation, the new kinds of
number came (slowly) to be accepted. Each time the old idea had to be integrated
into the new, and the new seen as an extension of the old. Thus, from whole
numbers, the move to fractions is seen as encompassing the whole numbers and so
the arithmetic of fractions must reduce to or extend the arithmetic of whole
numbers. Similarly, since fractions can also be represented using decimal-names,
the arithmetic of decimals must reduce to or extend the arithmetic of fractions. This
process turns out to be more problematic than might be expected!
ｾ＠ WHY NOT? Why is 1.5 + 2.5 not 3.10?
Why is 0.3 x 0.2 not 0.6?
Why is 1.2 x 100 not 1.200?
Why is 3.0/10 not 3?
Why is 3.05/10 not 3.5?](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-45-320.jpg)
![33
complete theory or description ofthe addition of such decimal numbers. To predict the length of
the period of the sum of two periodic decimals, specialize systematically in order to see what is
going on. (Question: Can you have a periodic decimal-name in which the period is infinite?)
You can also convert to fractions and then convert back again of course.
ｾ＠ SAME AND DIFFERENT What is the same, and what is different, about the
products in each of the following rows, and between the rows?
• three 2s, thirty 20s
• thirty-four 12s,
three 20s,
thirty-four 1.2s, three point four 1.2s
ＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭｾ＠
Comments What "rules" are pupils expected to deduce from such patterns? How are they
helped to see connections between the rules?
When a decimal-name is the only or major name of a number available, multiplication of
decimals turns out to be much more complicated even than addition.
ｾ＠ DECIMAL PRODUCTS Begin to carry out each of the following products by long
multiplication, in order to experience the uncertainties inherent in multiplying infinite
decimal-names.
• 0.3 x 1.5, 0.33 x 1.5, 0.333 x 1.5,
• 0.3 x 0.3, 0.33 x 0.33, 0.333 x 0.333,
·1.2 x 0.8, 1.22 x 0.81, 1.222 x 0.818,
0.3: x 1.5
0.3: x 0.3:
1.2: x 0.8:1:
Use your knowledge of fractions to verify what is suggested by the calculator.
ＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭｾ＠
Comments Using long multiplication, at what point in the calculation of 1.2,' x 0.8,""
can you be sure of even the first digit in the product? Just finding the first non-zero digit in a
product of two periodic decimals is not always easy if you confine yourself to some rule for
multiplying the decimal-names witlwUl converting to fractions.
Example C: Worked Examples
One difficulty with neatly recorded solutions by experts is that many pupils never realize the
struggles which even an expert may have been through before reaching a conclusion. That is
why I stress being mathematical with and in front of pupils. Mason, Burton, and Stacey [24]
presented a kind of"resolution with commentary" which has proved popular with students. The
attempt is made to give evidence of dead ends and fresh starts, mistakes and corrections,
conjectures which tum out to need modifying, and so on. In a similar vein, Open University
tutors are encouraged to run technique bashing sessions, in which the tutor works on questions](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-47-320.jpg)
![34
on an overhead-projector, trying to expose as much of their inner chatter and thoughts as
possible while they work. Students are encouraged not to take notes, but to try to enter the
screen, as if the tutor's words were inside their own head.
Providing examples is always tricky, for the recipient may not attend to the features which
the presenter is mentally stressing and which make the example exemplary.
ALL ONES [21]
Which of the numbers 1, 11, 111, 1111, 11111, ... can be a perfect square?
STUCK? Try them on your calculator. Make a conjecture. Which digits do perfect
squares end in?
Resolution
Trying the numbers on a calculator seems a good place to start. The calculator shows that
only 1 seems likely to be a perfect square, but it is far from clear why this may be.
The terms 11, 1111, 111 111, ... are divisible by 11, which suggests looking at the
quotients 1, 101, 10101, ... obtained by removing the factor of 11. These numbers would
have to have a factor of 11 as well, if the original number was to be a perfect square - a lot of
energy can be spent going this route!
Start again. The square roots on the calculator (to two decimal places) are 1, 3.33, 10.54,
33.33, 105.41, 333.33, ....
I could look for a pattern here, trying to make use of the appearance of those threes.
I stop and ask myself which digits can be the last digits of a perfect square. I find that
numbers ending in
1,2,3,4,5,6,7,8,9,0,
when squared, have as their last digits
1, 4, 9, 6, 5, 9, 4, 1, O.
Only numbers ending in 9 or 1 could possibly be square roots of numbers consisting solely
of Is. Dead halt - what now?
Carry the same idea further - what two-digit numbers can be the last two digits of a perfect
square? Careful- that looks like a lot of work. I really want to know whether 11 can be the last
two digits of a perfect square. Trying all the cases as I did for single digits will get an answer,
but so will a little algebra.
Suppose lOa + 9 or lOa + 1 were square roots of an all 1's number. Their squares are
100a2 + 180a + 81 and lOOa2 + 20a + 1.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-48-320.jpg)
![Some Issues in the Assessment
of Mathematical Problem Solving!
Jeremy Kilpatrick
Department of Mathematics Education, University of Georgia, Athens, GA 30602, USA
Abstract: To assess problem solving in mathematics adequately one must address the
narrowing effects of current testing practice and of the continued pressure for efficient
measurement. A new psychometrics is needed. Also, solutions need to be communicated and
to be assessed as communication. A central issue is to make a valid assessment that sees
problem solving as situated and mental processes as multiple and nonlinear.
Keywords: assessment, mathematical problem solving. task validity, process scoring,
measurement theory, communicating solutions
To the extent that problem solving has become a common shibboleth if not necessarily a central
feature of the school mathematics curriculum in recent years, it has raised serious issues of
assessment. Most of these issues concern how problem-solving performance might be
measured, but several relate to assessment in general and others concern how problem solving
is to be understood. This paper addresses some issues of each type as they touch on
possibilities for further thought and research in mathematics education.
Validity of Tasks
In a paper prepared for this conference, John Mason [10] argues for spontaneous evidence that
pupils have learned to solve problems. He claims that asking pupils directly is likely to distort
what they can do and are willing to do. Recent research on situated cognition underlines his
concern. Researchers have raised the question of how reasonable it is to assume that a task is
the "same task" when it is set for pupils to solve as it is when they define and set the task for
themselves.
Newman, Griffin, and Cole [14] tried to make the same task happen in two different
settings. They gave a group of fourth graders (9- to 1O-year-olds) a one-on-one tutorial in
which the child was given four stacks of cards, each stack being of a different color and bearing
1I am grateful to Vema Adams for suggesting the idea of problem solving as a composition task.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-51-320.jpg)
![38
the picture of a different movie star. The child was asked to find all the ways that pairs of stars
could be friends. Children who did not invent a systematic procedure for checking whether
they had all pairs were given hints to lead them to such a procedure. As a check on whether
they had developed some system, a fifth stack was added and the task repeated. Children who
did not arrive at a system were explicitly shown one. Then in a laboratory setting the children
were given four chemicals and asked to find all pairs. Even though the teacher posed the
problem in the laboratory, the children did not take solving that problem as their goal until they
had formed some pairs and wanted to form more. Although none of the children started out by
doing the task, those who finally did were much more successful than on the first task because
they had discovered the task on their own.
The children were not all doing the same task in the laboratory setting, and that led the
researchers to reconsider what had happened in the movie-star tutorial. It became clear that
many of the children may not have been doing the movie-star task at all in the first two trials.
Newman and his colleagues observed that in the usual division of labor, the researcher sets the
task and the subject is expected to do it. They concluded:
When experimenters present a well-defined task to subjects in a standardized way,
they have little chance to observe the subjects' formation of new goals or their
application of a procedure to new situations. [14, p. 173]
What applies to researchers giving laboratory tasks to subjects would seem to apply equally
well to teachers giving assessment tasks to pupils (see also [14]). Spontaneous evidence of
problem formulation and solution activities is valuable and ought to be built into
continuous-assessment programs in mathematics. How planned assessment might be put into a
context that permits an examination of how pupils are formulating goals and how they are
applying procedures to new situations remains as a major challenge. Perhaps the overarching
assessment issue stimulated by this line of research is how to make a valid assessment 0/what
students know and can do in/ormulating and solving mathematicalproblems.
Objective Assessment
The assessment of intellectual performance has a long history. Over 4000 years ago in China,
the emperor was testing his officials every three years [3]. Civil service examinations that
included arithmetic were in place in China by 1115 B.C. and were being widely copied in
Europe and the United States by the 19th century. England underwent an examination fever in
the mid-19th century in which much of the examining changed from oral to written form.
Mathematics always seemed to playa prominent role in innovative examinations (see [7], for
details).](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-52-320.jpg)
![39
The movement to new-type, or objective, testing that occurred in several countries during
the ftrst half of this century tended to stress the measurement of a pupil's knowledge as the
prime index of that pupil's achievement in a school subject. At the concluding conference of the
International Examinations Inquiry [12, pp. 240-252], for example, E. L. Thorndike argued
that for many purposes examinations should be as psychometrically "pure" as possible and
therefore that examinations in mathematics, say, should not be needlessly confounded with
measures of verbal ability or of one's general knowledge of the world. The emphasis on pure
measures and objective scoring, coupled with the behaviorist view that complex thinking can be
decomposed into simpler bits to be learned ftrst, has led to the multiple-choice and
short-answer tests so common today. Such tests emphasize what pupils do not know:
A high score in this kind of test does not infallibly demonstrate the attainment of
what we call a liberal education; but a low score does infallibly demonstrate a lack
of liberal education, because it reveals the absence of the foundation upon which a
liberal education must stand.... One may have a flourishing tree without fruit, but
one cannot have fruit without a tree; knowledge - ample and accurate knowledge
- is the tree on which the fruit we call culture must grow. [M. McConn, quoted in
6, p. 140]
One of the problems we face in assessing problem solving is that most current assessment
techniques are directed toward grading the lumber of the tree of content knowledge rather than
tasting and judging the fruit of problem solving. Pupils have become so accustomed to tests
that look only for knowledge that they are not sure how to respond to requests that they solve
complex problems either as part of class work or an assessment. An issue that has arisen from
the extensive use of objective knowledge-based tests is how to contend with the effects of
current assessmentpractice.
Assessing Products and Processes
Another source of difftculties in assessing problem solving is a failure to distinguish clearly
between different types of problems. Historically, problems appear to have been included in
the school curriculum primarily to provide a means to introduce and practice standard solution
techniques [16]. Consequently, most problems in the curriculum are rather straightforward and
routine. Sections in standardized tests labeled "problem solving" tend to consist of simple
applications of content knowledge to well-structured, well-rehearsed, and stereotypical
problem situations. Such tests may capture the kind of problem solving (or exercise answering)
that is typical of many textbooks and much instruction, but they do not assess problem solving
of a more open and original kind. Efforts to study the "processes" used in answering such
exercises rather than simply assessing their "products" may be somewhat misguided.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-53-320.jpg)
![40
For example, some researchers have developed "analytic scoring schemes" [9, pp. 63--64]
based on "phases" of problem solving similar to the phases used by Polya [15] in How to
Solve It. To the extent that such schemes are applied to the processes pupils appear to use
when they are solving problems by "thinking aloud," the schemes may be helpful to teachers
and researchers in understanding how a pupil is tackling a problem. It is difficult to see the
value of such schemes, however, when the problems are routine and the evaluator is sifting
through the pupils' written work for evidence of "understanding the problem," or "devising a
plan." Any inference about the process a pupil is using is bound to be somewhat shaky and
needs to be checked against other information. Furthermore, the point of solving a routine
problem is to get the official solution that everyone knows is there. It matters not how that
solution has been found; the point is to get it.
In the first U.S. National Assessment of Education Progress that dealt with mathematics,
some of the assessment was conducted by way of interviews. The test constructors thought
that a "real-life" exercise in balancing a checkbook might yield interesting problem-solving
processes. Perhaps it did, but in the context of such an assessment, all that people were really
interested in after the exercise was scored was whether or not the checkbook had been balanced.
The number of examinees who did it one way versus the number who did it another way may
intrigue some researcher but seemed to have essentially no practical value. The same thing
seems to be true of various procedures for solving simple word problems. When such
procedures are studied as part of a research project or to provide a teacher with information
about his pupils, they can be informative. When they yield only descriptive information related
to an assessment, however, they seem relatively barren. The issue for assessment is not so
much what the pupil was thinking while she or he devised a solution as what solution the pupil
offers. Unless they are rephrased, simple word problems typically ask for only a word,
number, or sentence in response. They do not challenge the pupil to compose an explanation or
justification. Perhaps, for example, the National Assessment checkbook exercise would have
been more illuminating if the respondents had been asked not merely to balance the checkbook
but to explain how balancing is done - after they had done it and not while they were
attempting it. The relevant issue is how to assess problem solution in the sense of how and
why the problem was solved the way it was rather than in the sense of what answer was
obtained.
New Approaches to Assessment
Attempts to include more challenging problems in assessment instruments have run into several
complications. Such problems tend to be difficult for pupils to respond to, especially in the](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-54-320.jpg)
![41
context of a timed test. Unless previous instruction has familiarized pupils with similar
problems, they may not know how to begin or what sort of answer is expected. The process of
grading their answers may yield scores whose meaning is uncertain or whose reliability is
shaky.
The California Assessment Program [2] tried out a set of five open-ended questions in its
1987-1988 Survey of Academic Skills at Grade 12. The questions turned out to be rather
difficult: the responses rated as showing "demonstrated competence" were never more than
20% of the total, and from 50% to almost 70% were no response or an "inadequate" response.
The Mathematics Assessment Advisory Committee conjectured that the poor performance had
resulted from the pupils' lack of experience in expressing mathematical ideas in writing.
Several generations of short answer testing in the U.S. appear to have taken their toll on pupils'
facility with open-ended questions.
In the Netherlands, the Hewet Project [8] appears to have had relatively more success than
the California Assessment Project in using alternatives to timed written tests. The Hewet
Project tried a variety of alternatives. One consisted of two-stage tasks in which the pupils took
a timed written test consisting of open-ended and essay questions, the responses were scored,
and the pupils had the opportunity to retake the test at home over several weeks to obtain a
second score. Other alternatives consisted of take-home tasks, essays, and oral tests. Perhaps
because of the different pattern of mathematics instruction and testing in Dutch schools
compared with California schools, but even more likely because of the special instruction
provided in the Hewet Project, the level of performance was relatively high. The 28 pupils
(from 2 teachers) participating in an internal examination earned scores that averaged from 6 to
8 out of 10 on a timed written test, a take-home task, and an oral test. The project did not let
problems of intersubjectivity in grading control the selection of tasks. The criteria, or
principles, for developing alternative assessment tasks were that the tasks should (a) improve
learning, (b) allow candidates to show what they know, (c) operationalize curriculum goals, (d)
not be selected primarily to allow objective scoring, and (e) fit into usual school practice [8, p.
183].
The demand for objective scoring and high reliability are part of the fetish of efficiency that
shapes and directs much assessment today. As Norman Frederiksen [4] observed, however,
efficient tests tend to drive out less efficient tests, leaving many important abilities
untested - and untaught. An important task for educators and psychologists is to
develop instruments that will better reflect the whole domain of educational goals
and to fmd ways to use them in improving the educational process. (p.201)
In a call for "useful assessment," Wolf, Bixby, Glenn, and Gardner [17, p. 35] argue that such
assessment should (a) be multidimensional, (b) capture knowledge and skill as exercised in](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-55-320.jpg)
![42
context, (c) be longitudinal enough to permIt mquiry into the procedures by which
understanding developed, and (d) offer information about the pupil's ability to amplify his or
her thought through connection to tools, resources, and other thinkers. This view of useful
assessment is in harmony with many current efforts by mathematics educators to construct new
techniques and materials for assessment. The difficulty does not appear to be a lack of
creativity but rather our tradition-bound views of how measurement must be conducted. A key
issue is how to deal with pressuresfor efficiency and reliability ofmeasurement.
Measurement Theory
Psychologists appear to be searching not only for new instruments but also new theories to
support the construction of such instruments. "It is only a slight exaggeration to describe the
test theory that dominates educational measurement today as the application of twentieth century
statistics to nineteenth century psychology" [11]. Current test theory treats problem-solving
ability as a single, continuous variable, whereas current cognitive psychology conceives of
problem solving as entailing a variety of processes, including the restructuring of knowledge,
the development of internal representations, and the use of sophisticated strategies to construct
as well as to monitor and critique solutions. Pressures to create efficient test instruments may
have inhibited the development of measurement models to handle the new view of cognition as
nonlinear, dynamic, and context-bound. One recent framework replaces the linear structure of
abilities with a lattice model [5], but much further effort is needed. The issue is how to
develop a new psychometrics.
Communicating Problem Solving
Mathematics educators and researchers concerned with mathematics education need not wait for
new measurement models before they begin investigating new approaches to the assessment of
problem solving. One of the most promising of these approaches is to treat the solving of a
problem as a task in composition. That is, just as in writing a composition, one can distinguish
knowledge telling from knowledge transforming [1], so in solving a problem, one can
observe that some thinking is an almost mechanical execution of a well-practiced procedure,
whereas other thinking operates at several levels to yield an understanding of the problem
through various transformations that eventually produce a solution. When a written account of
a solution to a mathematical problem is required from a pupil, the pupil engages in an activity
much like writing a composition. The pupil needs to plan how the argument will be organized,
what the reader needs to know, and how the various ideas are related. The written solution can](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-56-320.jpg)
![Assessment of Mathematical Modelling
and Applications
Henk van der Kooij
OW&OC. Tiberdreef 4.3561 GG Utrecht, The Netherlands
Abstract: In The Netherlands mathematics education is changing from learning about
structures and mechanic manipulation of formulas into learning how to formulate, how to
structure. The so called realistic approach to mathematics education is strongly process oriented
in stead of product oriented.Students (and teachers) are active in (re)inventing mathematics for
themselves.ln this kind of education important skills to be tested are: choosing suitable
strategies for solving problems, modelling real problems and critical analyzing given models
and given solutions. The HEWET and HAWEX-project have shown that there are appropriate
ways for assessment of mathematical modelling and applications, even in time restricted written
tests (like the final examination).
Keywords: realistic mathematics education, mathematical modelling, applications, HEWET
project, HAWEX project, final examination
The HEWET-project resulted in a new curriculum for the upper grades (age 16-18) of the pre-
university level of secondary education in The Netherlands: mathematics-A. The curriculum is
aiming at those students who are preparing for a study in social sciences at university. Since
1987 nationwide the final examinations are based on this curriculum.
The HAWEX-project led to new curricula (mathematics-A and -B) for the upper grades of
Havo (higher general education, a middle level of Dutch secondary education, see Appendix 1).
Mathematics-B is aiming at the students who are preparing for any technical study in higher
vocational education. Mathematics-A is aiming at those students who are preparing for any
social study, but also at those students who go to work after finishing school.
The first experimental examinations were held in 1989 and 1990.
Although the curricula are different because of the different groups of students they are
aiming at, all three of them are based on the same idea: the realistic approach of mathematics
education. In this approach modelling, applications and problem solving are very important
ingredients. Consequently the assessment of this kind of education must contain aspects of
modelling and applications.
Before turning to the assessment we will first look at some aspects of the education itself.
The general ideas of the realistic approach of mathematics education and the consequences for
the HEWET mathematics-A curriculum are described extensively, respectively by Treffers [3]](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-59-320.jpg)
![46
and De Lange [2]. We will look at the way the ideas of the realistic approach work out in the
classroom.
Mathematization in the Classroom
In the realistic approach mathematization plays a very imponant role. Real world problems are
explored intuitively, resulting in mathematizing that real situation.Imponant activities in this pan
of the process of mathematization are: Organizing, structuring, schematizing and visualizing.
As soon as the problem has been transformed to a mathematical problem it can be attacked
and treated with (more or less advanced) mathematical tools. Keywords of this pan of
mathematization are: representing relations in formulas, proving, refining and adjusting models,
combining and integrating models, formulating a new mathematical concept, generalizing.
A beautiful example of a problem that asks for a number of these activities from students is
the 'Rat-problem', taken from one of the HEWET-booklets. The problem stans with a text from
'Rats', a book written by a well known Dutch novelist Maarten 't Han:
GROWTH OF RAT POPULAnONS. As regards the progeny of one pair of rats
during one year the numbers given vary considerably. In the next chapter I shall
discuss the scanty information supplied by research into the fenility of rats in
nature, but at this point it might be interesting to estimate the number of offspring
produced by one pair under ideal conditions. My estimate will be based on the
following data. The average number of young produced at a binh is six; three out of
those six are females. The period of gestation is twenty one days; lactation also lasts
twenty one days. However, a female may already conceive again during lactation,
she may even conceive again on the very day she dropped her young. To simplify
matters, let the number of days between one litter and the next be fony. If then a
female drops six young on the first of january, she will be able to produce another
six fony days later. The females from the first litter of six will be able to produce
offsprings themselves after a hundred and twenty days. Assuming there will always
be three females in every litter of six, the total number of rats will be 1808 by the
next first ofjanuary, the original pair included.
This number is of course entire fictitious. There will be deaths; mothers may reject
their young; sometimes females are not in heat for a long time. Nevenheless, this
number gives us some idea of the host of rats that may come into being in one
single year.
The simple question on this text:
Is the conclusion that there will be 1808 rats at the end ofthe year correct?
Although the question is simple, the problem turns out to be very difficult.Given as a
homework task no more than 5 or 6 students out of a group of 30 are successful in solving it.
They have to demonstrate and explain their solutions to the other students.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-60-320.jpg)
![47
Four examples ofwell structured schema:
(A)
(B)
(C)
t
-1
0
1
2
3
4
5
6
7
8
9
0
2
3
4
5
6
7
8
9
N(t)
2
2 6
2 6 6
2
2
2
4
4
total number
8
2
8
2
4
8
146
278
536
974
1808
N(t) _ 2 ｻＬｾｾＮ＠
ＭｲｬＭＫｉｾｉｾｉＭＭｾｉ＠ ｾｉＭＫｉｾｉＭＭｉｲＭｲｬＭｲｬ＠
t - -1 0 1 2 3 4 5 6 7 8 9
adults = 4 7 10 22 43 73 139
(D)
adults young] youngO t number offemales
1 0 0 -1
1 0 3 0 1 + 3 =
1 3 3 1 4 + 3
4 3 3 2 7 + 3 =
7 3 12 3 10 + 3x4 =
10 12 21 4 22 + 3x7 =
22 21 30 5 43 + 3x1O =
43 30 66 6 73 + 3x22 =
73 66 129 7 139 + 3x43 =
139 129 219 8 268 + 3x73 =
268 219 417 9 487 + 3x139 =
Figure 1
I
4
7
10
22
43
73
139
268
487
904](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-61-320.jpg)
![(d) Geometrical solution
h 6 SO· h 36 2 4
"6=T5' . =T5= .
so y = 8.4; ｾ＠ and xB follow by
substitution of y = 8.4 in the
formulas
50
6
Figure 6
The HEWET and the HAWEX-project have shown that by the realistic approach of mathematics
education students become good (and sometimes very professional) problem solvers, who are
fully aware of what they are doing and for what purpose. They are critical of the way they are
attacking problems, and also of the way other people do. Very often they surprise their teachers
with more elegant solving strategies than the ones the teachers have in mind.
In order to enable them to arrive at that level , it is very important that every student is
given the opportunity to create and to use his/her own solving strategies, at least until other
strategies (of other students or the teacher's one) are accepted as being more suitable.
A more extensive discussion about these ideas can be found in Gravemeijer [1].
The Assessment
It may be clear that students who have learned mathematics this way, may not be "punished" by
examinations in which only technical and algorithmic skills are tested.
In The Netherlands the final examinations are time restricted written tests. Traditionally the
stated problems are of a very technical kind. The students have to show that they are well
trained in using standard algorithms. This kind of testing only shows what a student does not
know and he is punished for that.
The new curricula are asking for tests in which students can demonstrate they are able
- to choose an appropriate strategy for solving a problem,
- to criticize a given model,
- to integrate different mathematical models.
Part of the HAWEX-project was the description of general goals, knowledge and skills that
can be tested in the final examinations (see Appendix 2).This was not done in the HEWET-
project. It appeared that many teachers did not take these higher goals for granted
Of course very open problems, like the Rat-problem, don't fit in written tests like the final
examinations. Three examples of problems are given to show some possibilities for testing the
stated goals within the final examinations.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-64-320.jpg)
![56
(c)
Figure 13
Very clear and simple:
The volume of one pyramid is (0.5 x 3 x 3) x 4/3 = 6
So the volume is: V = 6 x 6 x 10 - 4 x 6 = 336
Alternative Tests
Final examinations are not the most appropriate tool to test the so called higher goals of the
curriculum.In the HEWET-project several ways of alternative testing are developed [2].
In the HAWEX-project a new phenomena is introduced: a practical examination.
Sixty mathematics-B students did a practical job on Space geometry,working in couples for
three hours. The concrete object was a lamp, built of 8 half cubes. Some of these half cubes are
rotative, so there are many different shapes possible. Each couple had 10 cardboard half cubes
to use, if necessary. Fourteen problems, starting simple and very complex in the end, had to be
solved. Photos and drawings of the different shapes of the lamp were the source for the
problems they had to solve (figure 14). Of course the couples were allowed to deliberate. They
also were allowed to consult the teacher.At the end of the three hours each couple had to hand in
one set of solutions to the 14 problems.
This way of testing offers some advantages over 'normal' tests.
The students have to cooperate. They must convince each other, for they had to hand in
only one solution.Of course this skill can never be tested in a written test.
The problems can be more complex than the ones in the written test, because the students
can deliberate, while the teacher can lend a helping hand, if necessary.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-70-320.jpg)
![A Cognitive Perspective on Mathematics:
Issues of Perception, Instruction, and Assessment
Patricia A. Alexander
College of Education, Texas A&M University, College Station, Texas 77843, USA
Abstract: In this paper, concerns systemic to educational theory and practice are discussed in
relation to the current state of mathematics education research and mathematics instruction.
These concerns are synthesized into three general issues addressing the perception of the
discipline, the nature of schooling, and measurement models and assessment practices.
Potential solutions to these issues are offered.
Keywords: cognitive psychology, knowledge, classroom culture, context, metacognition,
misconceptions, assessment, measurement theory, anchored instruction
Society is presently faced with a staggering number of life-threatening and planet-threatening
problems, such as AIDS, economic stagnation, and environmental abuse. Any solution to these
enormous problems will require logical, rational, creative thought, and systematic investigation;
that is, any solution requires mathematical problem solving. Yet, where will we find those
individuals capable of the formulating the solutions to these and to other problems that have still
to be detected? Finding these individuals is, perhaps, even a greater problem facing our society
today. As Paulos [41] so aptly stated in his bestselling book on innumeracy:
I'm distressed by a society which depends so completely on mathematics and
science and yet seems so indifferent to the innumeracy and scientific illiteracy of so
many of its citizens...(p. 134).
Despite the decades of research activity and aggressive instructional reforms [e.g., 35, 36],
more and more American students seem to be exiting educational programs with inadequate
mathematics knowledge, with a distaste for or a fear of mathematics, or with a cache of
mathematical formulae and terminology that they seem unwilling or incapable of transferring to
unfamiliar, complex, or "real world" situations. These are conditions that underlie the issues to
which the title of this presentation refers.
My presence at this international conference further suggests that these questions of
importance to mathematics education are of broader concern to cognitive researchers](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-75-320.jpg)
![62
investigating human intellectual processing. For the past five years, I have been engaged in
research on the interaction of domain knowledge and strategic processing. This research has
afforded me the opportunity to examine theory and practice in mathematics, as well as in the
domains of human biology/immunology, physics, and social studies. Not surprisingly, this
enterprise has led me to see the current state of mathematics education research and mathematics
instruction in terms of underlying issues; issues that have relevance to a number of academic
disciplines outside of mathematics. Indeed, it would not be inconceivable to say that almost
every academic discipline is currently exhibiting the same general symptoms frequently
attributed to mathematics: declining achievement and rising incompetence among students,
instruction that is decontextualized, routinized, and focused on lower-order cognitive
processing, student disenchantment, and teacher frustration. Rather than diminish the
importance of these issues for mathematics, these shared symptoms hint at the systemic nature
of the affliction, a point I will return to in the conclusion of this presentation.
I concur with Schoenfeld [59] that the issue for research is to understand and abstract the
common symptoms from isolated cases and examples and to use that understanding to create
environments that foster intellectual growth and academic achievement, thereby treating the
affliction. Thus, for the purpose of this presentation, I will merge a number of concerns into
three global issues with relevance to mathematics. Specifically, I will consider how perceptions
of the discipline, the nature of schooling, and measurement models and assessment practices
may negatively affect mathematics learning. Even though I treat these three issues as if they
were discrete, I fully acknowledge that they are highly interrelated topics. That is, one cannot
completely separate questions about the discipline of mathematics from questions about how
mathematics is taught or how it is assessed. The separation, however, does permit me to
identify some global concems that can more readily be discussed here.
Perceptions of the Discipline of Mathematics
Mathematics is a discipline that consists of many related domains. Disciplines can be
distinguished by the fact that they encompass extensive bodies of knowledge that are oriented
toward fundamental principles or generalizations [45, 67]. These fundamental principles form
the cruxes around which domain-specific knowledge is internalized [2]. Domains, by
definition, are fields of study that, like disciplines, consist of declarative knowledge (knowing
that), procedural knowledge (knowing how) and conditional knowledge (knowing when and
where) [39, 55]. Further, from a within-individual perspective, disciplines (and domains) have
both tacit and explicit dimensions, as well as formal and informal components [4]. The formal
component refers to that knowledge which is specifically taught, while the informal pertains to
that knowledge which is acquired from everyday situations or experiences.
Despite these unifying characteristics, disciplines are not all alike [2, 23, 52, 64]. Some
can be characterized as rather principle-driven, procedurally-rich, or well-structured (e.g.,](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-76-320.jpg)
![63
physics). Others can more accurately be described as composite (e.g., social studies) or ill-
structured (e.g., reading) in nature [19, 46]. Such differences are nontrivial and should be
considered important both to practice and research in mathematics. Relatively speaking,
mathematics is a fairly well-structured discipline that is procedurally-rich and which can be
represented by many well-defined tasks or problems [29,71]. These general distinctions,
however, should not obscure the structural variability that exists between the discipline of
mathematics and other disciplines or within mathematical domains. Geometry, for example,
appears to be a domain that is highly visual and spatial in nature, syllogistic in reasoning, and
centered around postulates, theorems, and proofs. Algebra, by comparison, has a much more
sequential and computational character, and is oriented toward determining functional
relationships.
Not only are there valid distinctions to be made between domains, but also within them.
That is, while mathematics domains may fall toward one end of the structuredness continuum or
another, there remains variability within each. Let me again use geometry to illustrate how a
mathematics domain can be at once both well-structured and ill-structured. As a well-
structured domain, geometry may be part of one of two broad families: Euclidean and non-
Euclidean. In both families, we can systematically discuss the relationships between
dimensions, coordinates, angles, and so on, by postulates, theorems, and proofs. In Euclidean
geometry, the sum of the angles of every triangle equals 180 degrees. In a non-Euclidean
geometry, such as that of Lobachevsky, the sum of the angles of every triangle is not equal to
180 degrees. Geometry emerges as a more ill-structured domain when we debate which of the
two families is better applied to investigate variables that describe our world. If we consider
space and time on a scale in which straight rather than curved dimensions are appropriate, then
we will adopt a Euclidean perspective. Yet, if we consider scales in which space and time are
best represented as curved or nonlinear dimensions, then we will adopt a non-Euclidean
perspective. In truth, any mathematics domain can be approached in a more or less well-
structured manner. Introducing uncertainty within any given problem, leaves more room for
diversity, novelty, and creativity in solution, thus making the problem more ill-structured.
The reason for discussing the nature of mathematics here is that, regrettably, there are
numerous indicators that teachers and researchers tend to be short-sighted in their perceptions of
mathematics, tending to see only the well-structured dimensions. Likewise, as I will discuss
later, there is evidence that the structure of the discipline itself has not always been weighed in
generalizing the results of certain research programs.
If we were to deduce the nature of mathematics from the instructional practices that are
evidenced in American classrooms, we would surmise that mathematics knowledge is fixed,
static, and complete. In other words, we would lose sight of the fact that mathematics is, in
truth, developing, dynamic, and incomplete. Based on school practices, we would also infer
that mathematics is not just relatively but entirely a well-structured discipline composed
completely of well-defined problems [52]. As Resnick [52, p. 32] states:](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-77-320.jpg)
![64
Educators typically treat mathematics as a field with no open questions and no
arguments, at least none that young children or those not particularly talented in
mathematics can appreciate.
We would further conclude that mathematical understanding is achieved by the acquisition of
basic mathematics facts and algorithmic procedures and is demonstrated by one's ability to
identify the correct formula and execute it in an automatized and effortless manner. In other
words, mathematics as practiced in the classroom is generally arithmetic; arithmetic that stresses
lower-order cognitive processing [17]. According to Peterson [44],47% of instructional time
is devoted to unguided paper-and-pencil seatwork and 43% to whole class instruction. She
further reports that for the vast majority of this instructional time (85%), students are engaged in
lower-level cognitive processing or not cognitively engaged at all. While this pattern of
unguided practice and low-level cognitive processing occurs in other areas (e.g.,
reading/language arts [18]), there seems even less active involvement or verbal engagement
during mathematics instruction than in these other curricular areas [52]. Unless teachers
possess an understanding and appreciation of the complexity and diversity of mathematics,
unless teachers learn to see the ill-structured aspects of even the most well-structured of
mathematics domains, and unless they identify the opportunities for higher-order mental
processing, there remains little hope that they will engender these same understandings and
perceptions within their students.
How does the nature of mathematics influence research activity? Judging from the research
in expertise and misconceptions, it would seem that with few exceptions [e.g., 37, 60] the
choice of domain or the selection of domain-related tasks is of little or no significance to
reported outcomes. Quite to the contrary, I would argue that the dissimilarity among domains
or domain-specific tasks should be carefully evaluated in discussions of expertise and
misconceptions [2]. Before we accept the "strong domain knowledge - weak general
strategy" arguments espoused in the expert-novice research, for instance, we should remember
that much of this research not only involved more well-defined domains (e.g., physics,
computer science) but also well-structured problems (e.g., solving quadratic equations). As the
research in expertise has reached into less structured fields or tasks [e.g., 40, 73], it has been
more difficult to distinguish novices from experts and it has been harder to overlook the role
played by general cognitive and metacognitive strategies [28].
The research on misconceptions exhibits some of these same problems. Is it coincidence
that the research on misconceptions has occurred primarily in mathematics and science? Are the
patterns in students' mis-understandings in other disciplines as evident, as resilient, or as
detrimental? For one, the selection of well-structured domains may have been influenced by the
availability of tasks for which outcomes are verifiable or refutable. If there is no agreed-upon
"correct" concept, then how can there be a mis-conception to investigate? In educational
psychology, for example, one can argue that cognitivism is a more appropriate theory than
behaviorism to explain human learning. However, while debatable, neither theoretical](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-78-320.jpg)
![65
perspective is truly refutable. The use of well-defined tasks in mathematics and science makes
this verification much easier and pennits one to trace the source of the error to some underlying
misconception.
Nonetheless, I concur with Spiro et al. [64] that the differences in domains probably give
rise to differences in domain-related errors. To achieve a greater understanding of
misconceptions, therefore, we must study conceptual errors within domains and with tasks that
are ill-structured as well as well-structured. Further, we must seek to uncover whether the
cause ofcertain misconceptions relates more to the lack ofcontent-specific knowledge, the lack
ofproblem-solving abilities, or some combination thereof.
In summary, understanding of the discipline of mathematics is central to the health of
mathematics research and education. Research agenda and interpretations of findings, as well
as decisions about what should be taught and how it should be taught in schools, should be
consistent with our understanding of the discipline. In this way, we may avoid the
overgeneralizing or trivializing ofthe effects such inherent differences may have on mathematics
learning and instruction.
The Nature of Schooling and Knowledge Acquisition
There is no doubt that schooling helps to shape students' conceptions of mathematics.
Although I do not support Lave, Smith, and Butler's [33] contention that classroom practices,
more than conceptual limitations or strategic deficits, are the primary source of problems in
learning, there is little question that such practices significantly contribute to the lack of
mathematics advancement. My own observations of classroom behavior and that of others
(e.g., [50]) have shown that teachers are frequently content if their students can complete
stereotypical problems using the routinized arithmetic procedures they were taught. That their
students understand the concepts upon which these procedures are based or recognize their
purpose or value seems of little consequence. That students may perceive the beauty of
mathematics is beyond consideration.
What is it about the nature of schooling and the culture of classrooms that inhibits
mathematics learning? The practice of mathematics in American schools has certain salient
attributes that may provide some clues. First, school mathematics consists of a series of classes
or courses isolated from other academic disciplines. That is, teachers and students understand
"mathematics" to be content-specific knowledge that can be taught and learned without regard to
history, reading, or any other school subjects. Second, not only is mathematics separated from
other academic disciplines but the various fields of study within mathematics are similarly
taught in this disassociated manner. Thus, the high school student comes to believe that
algebra, geometry, and trigonometry are discrete areas of study rather than mathematical
domains related on the basis of shared principles. Finally, students are given little opportunity
to discuss mathematical concepts or externalize their thinking during problem solving [31, 32],](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-79-320.jpg)
![66
nor are students encouraged to construct meaning within the social context of the classroom
[10, 15,25].
The consequence of these instructional rituals are that students equate "doing" mathematics
(i.e., arithmetic computation) with "knowing" mathematics [17, 26]. Further, because the
"doing" of mathematics primarily entails the completion of routine, well-defined, teacher-
prescribed problems, students do not readily see the correspondence between mathematics and
complex problem solving. Since "school" problems do not frequently demand it and since
teachers do not seem to stimulate or reward it, higher-level, strategic processes may fail to
become fully developed. Yet, competence in mathematics, to say nothing of expertise, requires
not only declarative "facts," and domain-specific procedures but also general cognitive and
metacognitive strategies [58]. Without the ability and the motivation to apply general cognitive
or metacognitive strategies, every problem that is the least bit novel or complex would become
an insurmountable obstacle. Despite this realization, the focus on domain-specific facts and
procedures at the expense of strategic processing remains characteristic of classroom practices.
As I [3] and others [43, 49] have argued, it is of little value to persist in the theoretical or
practical separation of domain knowledge and strategic processing. The very fact that a
conference on mathematics and problem solving is needed, however, suggests that the
integration of these two essential knowledges in theory, as well as in practice, has yet to be
achieved.
The concerns about the interaction of domain knowledge and strategic processing are not as
simple as they may first appear, however. More than just realizing that both are essential to
mathematics performance, we must come to understand how the development of one form of
knowledge influences the other. That is, how does the acquisition of domain knowledge
change the amount and type of strategic processing? Are those who know more in less need of
general strategies, for instance? Likewise, how do general cognitive and metacognitive
strategies contribute to the acquisition and transfer of domain knowledge? Are general
strategies, such as analogical reasoning, basic to knowledge transfer and knowledge
restructuring, as has been suggested? [e.g., 3,72]. We also need to investigate how the nature
of the tasks (e.g., well-defined or ill-defined) impacts strategic processing. For instance, are
general strategies more critical with tasks that are ill-defined or novel than they are for tasks that
are well-defined? Such questions about the interaction of domain knowledge and strategic
processing can only be addressed through extended research programs.
Beyond compartmentalizing mathematics by discipline and by domain, and by fostering the
separation of domain knowledge from strategic processing, school mathematics is a weak
representation of the discipline from which it is drawn. Descartes, the 17th century
philosopher, held that mathematics was basic to all knowledge, since it permitted humankind to
view their world in logical and reasoned ways. Schools would also hold mathematics to be
basic, although in school venacular "basic" has unfortunately come to mean "minimal" or
"simplistic." Schools have come to equate mathematics with the procedures of addition,
subtraction, multiplication, and division. What they fail to see is that proficiency in executing](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-80-320.jpg)
![67
these "basic" procedures is only a rudimentary stage toward thinking mathematically, just as the
ability to decode and to recognize words are basic to reading but do not, themselves, constitute
reading. Even the problems that students practice in school are poor approximations of the
complex and ill-structured problems (e.g., predicting earthquakes, forecasting the economy)
that are tackled by contemporary mathematicians [65]. As a result of these and other factors,
school mathematics remains a poor sampling of the discipline that it is intended to represent [30,
50]. Mathematics, ofcourse, is not unique in this regard. Most of us would be hard-pressed to
name any class or course of study that currently represents a "good" sampling of the discipline
from which it arises.
The abstraction of mathematics from realistic contexts is another hallmark of schooling.
"School" problems, for instance, bear little resemblance to the real-world dilemmas that
students encounter; that is, the problems that are the mainstay of students' drill-and-practice
have little in common with students' out-of-school knowledge and experiences [25, 59]. Lave
et al. [33] argue that this deliberate abstraction and decontextualization of mathematics came
about because of the misconception that such abstraction facilitated knowledge acquisition and
transfer.
To this theoretical explanation, I would add a pragmatic one. I would contend that the
abstraction of mathematics also occurs because few teachers are capable of or willing to deal
with the inherent complexities ofmathematics. Data would suggest, for instance, that there are
individuals teaching mathematics who do not have the certification nor the competencies to do
so [35]. In addition, teachers of mathematics are handed instructional materials (Le.,
curriculum guides, texts) that are highly abstracted in that they are only skeletal, simplistic
representations of the discipline. Through this abstraction, mathematics is converted to a
hierarchy of basics that can be symbolically represented, algorithmically manipulated, and easily
verified [30]. Unfortunately, teachers limited understanding of the discipline does not permit
teachers to elaborate or extend those instructional materials adequately; in essence, they are
unable to put any meat on the skeleton they are given. As a result, instruction remains
abstracted and students acquire only a piecemeal knowledge of mathematics. To paraphrase a
line from Shulman [62], teachers teach what they know--and students learn what teachers teach.
I do not wish to suggest that teachers deliberately set out to "dummy-down" the
mathematics curriculum or to inhibit students' learning. To the contrary, these teachers are
performing their perceived role in the culture of the classroom. Certain social and cultural
theorists might well suggest that the culture of the classroom is merely a reproduction, if not
reflection, of the culture of society which is intended to perpetuate a class system [24, 34].
Even from a more conservative perspective, it would seem that teachers are modeling the
mathematical pedagogy, as well as communicating the content-specific knowledge, that they
have internalized as a consequence of their educational experiences. What that culture of the
classroom has taught them is that teacher and textbooks are the authorities and that the task for
the student is to passively absorb what is communicated to them verbally or in print [31]. As
Lampert (p. 32) writes:](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-81-320.jpg)
![68
Even when teachers give an explanation rather than simply stating a rule to be
followed, they do not invite students to examine the mathematical assumptions
behind the explanation, and it is unlikely that they do so themselves...
Even though this practice conflicts with cognitive theory that tells us that one learns best when
actively involved [e.g., 74, 76, 77], this same passivity can be witnessed in other curricular
areas. Instructional approaches such as reciprocal teaching [38], cooperative learning [16], and
cognitive apprenticeships [15] represent attempts to alter the culture of the classroom.
There is yet another explanation for why the simplication or abstraction of mathematics
occurs in school. There seems to exist the misperception that mathematical literacy can only
occur with the onset offonnal instruction, and that prior to formal instruction children have little
conceptual understanding about mathematics. As has occurred with language literacy,
mathematics educators must come to realize that many valuable lessons are learned outside of
the classroom, and that formal instruction can be enhanced by activating students' informal
knowledge base. There also appears to be a serious underestimation of the mathematical
capabilities of young children [22, 53]. The research of my colleagues and I [e.g., 7, 8, 75]
has reinforced that point. Specifically, in our research on analogical reasoning, the majority of
preschoolers we tested evidenced reasoning abilities assumed to be beyond their developmental
capabilities [e.g., 47] when they were presented with a motivating task performed within an
appropriate and motivating setting. Researchers in mathematics have reported similar findings
[11, 12]. Consequently, the more we build upon students' preexisting conceptual and strategic
knowledge in classroom instruction, the more likely we are to foster subsequent advancement.
Measurement Models and Assessment Practices
As I see it, there are two problems facing mathematics and mathematics education with regard to
assessment. The first is the mismatch between current theories of learning and instruction and
models of assessment [14, 50]. The second pertains to the way that assessment is practiced in
schools [30].
The more I investigate human intellectual processing, the more I come to appreciate its
complexity. My recent efforts to test the interaction of domain and strategy knowledge [e.g.,
5], to propose a framework for relating various forms of knowledge [6], and to articulate a
theory of creativity [4] have made me acutely aware of the dynamic, interactive, and
multidimensional nature of processing that occurs within the individual. Just the ability to read
and comprehend a passage in a book or solve a word problem requires an intricate blend of
domain-specific knowledge, and strategic processing. I have also come to recognize that all
individuals, as social beings, must operate within a sociocultural milieu that significantly
influences their actions. Further, as psychological beings, humans dohot frequently operate in
ways that are coldly rational but are greatly affected by their beliefs, goals, interests, and by
their self-perceptions.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-82-320.jpg)
![69
When you map human complexity onto current models of assessment, you begin to realize
that what we are presently able to test is a pale reflection of what we need to know about human
performance. It is likely that we will never be able to match our understanding of human
processing with measures that are equally dynamic or complex. To do so may not even be
desirable. However, I would argue that the simplistic notions of what constitutes mathematics,
as operationally defined by the tests that we currently use, is far from satisfactory. Certainly
there is much more to be known about mathematics than that which appears on standardized
achievement tests [e.g., 54]. What I suggest is that the mismatch between theories of learning
and instruction and models of measurement can be lessened in three ways. Specifically, I
recommend that we: (a) take new looks at traditional mathematical tasks; (b) develop novel
mathematical tasks or tests; and, (c) devise new, cognitive-oriented measurement models.
As far back as 1978, in their research on "debugging," Brown and Burton alerted us to the
fact that even the most commonplace of mathematical procedures, like addition, could be
reexamined from a cognitive perspective. By means of such reexamination, researchers can
become aware of the underlying procedural deficits that result in performance errors. Brown
and Burton, for instance, determined that the errors students made when performing basic
mathematics procedures (e.g., addition/subtraction) were systematic and traceable to particular
procedural misconceptions. VanLehn's [71] research on repair theory and Sleeman's [63]
study of students' misunderstandings in basic algebra are additional examples of researchers
who have undertaken detailed cognitive analyses of traditional mathematical tasks.
Still other researchers have seen fit to devise mathematical problem-solving tasks that vary
greatly from the traditional paper-and-pencil ones we discussed earlier in this paper. Bransford
and the Cognition and Technology Group at Vanderbilt [13], for example, have created
innovative computer-based problem-solving tasks they refer to as "anchored instruction." The
objective of these videodisc activities is to immerse students in a stimulating environment that
relates to their out-of-school experiences, permits sustained exploration, and encourages
problem-solving from multiple perspectives. Our work on the interaction of domain-specific
and strategy knowledge led our research group at Texas A&M to develop a series of domain
knowledge, strategy knowledge, and interactive knowledge measures that permitted evaluation
of subjects' academic performance both unidimensionally and multidimensionally [5]. Research
on realistic or authentic assessment in science and in reading are additional examples of the
efforts to devise innovative tasks that permit richer cognitive analysis of student leaming.
What is clear from these various efforts is that the concern for higher-order problem solving
in the research community and in educational reforms [e.g., 35] is not yet matched by a large
array of assessment tasks that measure problem solving in valid and reliable ways. Perhaps the
lack of a clear or consistent definition of what constitutes "problem-solving," "higher-order
thinking," or "strategic processing" is one barrier to the development of cognitive-based
mathematical assessments [6]. Considering the diversity of interpretations that exist in the
literature [e.g., 3], arriving at such a consistent definition will certainly be no simple matter.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-83-320.jpg)
![70
Still another way to infuse new life into assessment is with the development of
measurement models that deviate from more traditional psychometric approaches. Item-
response theory or IRT, for example, differs from classical true score theory in that IRT is not
sample dependent and considers the entire response pattern of individuals instead of a
composite score. This means that IRT is more robust than classical true score theory with
regard to estimates of item difficulty and discrimination. Although IRT has not been widely
applied to mathematics, it remains a promising measurement model. Building on IRT,
Tatsuoka [68, 69] has devised rule-space analysis, a procedure in which various erroneous rule
patterns are mapped into multidimensional space. Tatsuoka has used the rule-space computer
program to identify "bugs" in students' solution of signed-number addition and subtraction
problems. Multidimensional scaling (MDS) is yet another measurement approach that could
provide a more in-depth analysis of mathematical performance [61]. Pellegrino, Mumaw, and
Shutes [42] have used MDS to compare the spatial abilities of experts and novices. In my own
research, MDS has been useful in determining whether certain types of errors were more
associated with deficits in domain-specific knowledge or strategic processing [1].
Despite these three promising avenues for lessening the mismatch between theories of
learning and instruction and measurement models, the activities being undertaken in the research
arena have had limited effect on school assessment practices. Several factors contribute to this
condition. First, the cognitive techniques I have been describing are still very much in the
developmental stages [23, 77]. Second, the procedures, tasks, and tests described require a
high level ofexpertise that is not typically available among school personnel. Third, even those
problem-solving tasks that seem more appropriate to classroom use, like Bransford's anchored
instruction, are not ready for widespread application. Fourth, many of the measurement models
and error detection tasks require sophisticated technology to implement.
Given the current state of schooling, then, what can be done to improve assessment
practices in mathematics? To this question, I offer several responses. First, educators need to
look beyond the correct score to the thinking and knowledge that is evident within student
responses, whether that response is correct or incorrect. As the research of Brown and Burton,
VanLehn, and my own research [I, 7, 29] has repeatedly shown, students' errors are
nonrandom, often predictable, and frequently informative. In the same regard, the research my
colleagues and I have conducted has made it apparent to us that errors are not all equivalent. In
that research, we have found that error patterns appear to represent either higher or lower levels
of domain or strategy knowledge. Thus, these error patterns become clues to the organization
and accessibility ofstudents' domain-specific knowledge and of their problem solving abilities.
If students' error patterns can be identified, then more effective intervention programs can be
planned.
Yet another way that school assessment practices can be improved is by better preparing
teachers to deal with test construction and test interpretation [27,66]. The tests that teachers
devise will continue to playa critical role in mathematics education. If nothing else, these tests
signal to students what teachers value in the way of knowledge. If teachers talk about the](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-84-320.jpg)
![71
importance of problem solving but continue to measure only lower-level declarative or
procedural knowledge, students will focus on the "to-he-evaluated" content [56].
It would seem, therefore, that teacher training programs should be organized to provide
preservice teachers with the knowledge and skills they need to construct tests that are not only
valid but also informative, related to out-of-school experiences, and oriented toward problem
solving. Earlier I mentioned the value of looking at student errors. Based on the research, it
should also be possible to teach teachers ways to construct tests that permit them to perform
simple error analyses. By constructing tests that would identify procedural "bugs," teachers
should be better equipped to diagnose student misconceptions and to address those in
instruction.
Training programs should also ensure that teachers are knowledgeable about various forms
of assessment (e.g., norm-referenced, criterion-referenced), so that they are able to interpret test
results accurately and communicate those results effectively [66]. More importantly, an
effective training program should work to ensure that teachers are capable of making
appropriate use of test data in planning and implementing instruction. Perhaps if they felt more
secure about devising their own tests and were more knowledgeable about the limitations of
current standardized measures, then teachers would assume greater responsiblity for student
assessment and feel less pressured to rely on standardized test scores [27].
Potential Solutions
Throughout this paper, I have been describing global issues that I believe to be systemic to
education and which seem to be important to the future of mathematics research and practice. I
would be remiss, however, if my discussion did not conclude with some consideration of ways
that these issues can he positively addressed. If the issues are, as I have stated, systemic to
educational research and instructional practice, then the potential solutions to those problems
can also be described as systemic. These potential solutions, then, speak not only to
mathematics but also to other disciplines. Further, they speak to the SOCiological and
psychological dimensions of learning and instruction, as well as the cognitive. For the
purposes of this paper, I will only briefly describe these potential solutions. More in-depth
consideration must be reserved for some other time and place.
Proposal I: Explore Learning and Instruction as Sociocultural Events
As the research on situated cognition [e.g., 25, 59] suggests, learning cannot be separated from
the sociocultural context in which it occurs. As a microcosm of the society in which it operates,
classrooms have their own languages, value systems, and cultures that impact learning. If we
are to foster the acquisition and utilization of intricate mathematic concepts and reasoning](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-85-320.jpg)
![72
abilities, then we must make mathematics thinking and problem solving a natural aspect of the
classroom culture. We must encourage students to engage in discourse about mathematics. As
Lampert [31, 32] suggests, discourse in the classroom as it relates to mathematics should
appear like the discourse of argument and conjecture, with teachers and students working
together for the purpose of "sense-making." In this situation, the engagement in problem
solving is more important than quick resolution. Students come to see mathematics as a process
more than as a set offacts and fonnulae to memorize [57]. Social and cultural theorists might
argue that such a drastic reconceptualization of mathematics education will require a genuine
demand within society for change; a demand that has yet to be made [48].
Proposal II: Conceive of Instruction as a Partnership Requiring the Active
Participation of Students and Teachers Alike
As the writings ofDewey, Bruner, and, more recently, Wittrock, Resnick, and Bransford have
shown us, students who are actively engaged learn better than those who are not. If we persist
in endowing teachers with full responsibility for "telling" students what they need to know,
while students passively sit by absorbing the content, then mathematics learning will remain
limited. Several techniques have been suggested for making students more active and willing
participants in learning. As noted earlier, Collins et aJ.'s [15] cognitive apprenticeship is one
instructional model, as is Palincsar and Brown's [38] reciprocal teaching approach. These
instructional approaches provide for teacher modeling, thoughtful guidance, and a gradual
transfer of instructional responsibility to the student. Along with these approaches, I would
also recommend that there be some explicit teaching of general problem solving strategies with
the choice of instructional approach being made on the basis of the task and the competency of
the students.
Proposal III: Build on the Interests and Experiences of the Students
It is hard to overestimate the impact that interest and motivation play in students' learning [20,
21]. The success of Bransford's videodisc-based "anchored instruction" is largely attributable
to the fact that programs like the Sherlock Holmes series are visually inviting productions. As a
result, students are eager to participate and able to maintain attention. Yet, so much of what we
ask students to do in classrooms is of limited appeal to them and is removed from their out-of-
school experiences. What we need to do is to seek ways to lessen this mismatch [51]. This can
be accomplished by engaging students in problem solving that is contextualized, concrete, and
realistic, and which speaks to issues and topics that interest them.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-86-320.jpg)
![73
Proposal IV: Show Students the Relatedness of Knowledge
Just as we want to build connections between students lives in and out of school, we want to
help them learn to see the relatedness of knowledge. One of the concepts that was evident in the
literatures on the integration of domain and strategy knowledge [3] and on creative problem
solving [4] was that expertise involves the ability to see relationships across tasks, domains,
and disciplines. For the truly expert or the truly creative, there are no artificial boundaries of
knowledge. What these individuals possess is the flexibility of thought that permits them to
draw connections between concepts that are not apparent. In schools, therefore, we must make
efforts to show students how what they learn in mathematics has relevance to what they learn in
history, music, reading, or science. Likewise, we need to help them see how the various
domains of mathematics (e.g., algebra, calculus, trigonometry) are related. If integrated
structures of knowledge are the hallmarks of expertise and creativity, then we should work
toward these ends in our classrooms.
Concluding Remarks
Despite the somewhat somber tone of this paper, I believe that there is great hope for the future
of education. Now, more than at any other time in our history, we have a deep understanding
of the nature of learning and instruction, we are sensitive to the problems that exist, and we are
willing, even anxious, to bring about change. By integrating this burgeoning body of
knowledge with the content of the various disciplines, we can not only continue to learn more,
but assure that future generations will have the ability and the desire to face challenges that are
currently beyond imagination.
References
1. Alexander, P. A.: Categorizing learner responses on domain-specific analogy tests: A case for error analysis.
Paper presented at the annual meeting of the American Educational Research Association, Boston, MA April
1990
2. Alexander, P. A.: Domain knowledge: Evolving themes and emerging concerns. Educational Psychologist
in press
3. Alexander, P. A., & Judy, J. E.: The interaction of domain-specific and strategic knowledge in academic
performance. Review of Educational Research, 58, 375-404 (1988)
4. Alexander, P. A., Parsons, J. L., & Nash, W. R.:. Toward a theory of creativity. Manuscript submitted for
publication 1991.
5. Alexander, P. A., Pate, P. E., Kulikowich, J. M., Farrell, D. M., & Wright, N. L.: Domain-specific and
strategic knowledge: Effects of training on students of differing ages or competence levels. Learning and
Individual Differences, 1,283-325 (1989)
6. Alexander, P. A., Schallert, V. L., & Hare, V. C.: Coming to terms: How researchers in literacy and
learning talk about knowledge. Review of Educational Research in press.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-87-320.jpg)
![The Crucial Role of Semantic Fields in the
Development of Problem Solving Skills in the School
Environment
PaoloBoero
Dipartimento eli Marematica, Universita eli Genova, Via L. B. Alberti,4, 16132 Genova. Italy
Abstract: I will try to frame from a theoretical standpoint, and clarify, the following points:
Which aspects of problem solving are more context sensitive, on the short term (in relation to
the single problem situation), and on the long term (in relation to the development of the
problem solving skills)? Which elements of the context can have more significant effects on
problem solving processes? Which are the most important differences between the
"contextualized" problem solving processes and the "non-contextualized" problem solving
processes? Which teaching actions associated with the chosen contexts can the teacher perform
in order to enhance pupils' results in problem solving?
Keywords: problem-solving skills, fields of experience, context sensitivity, semantic fields,
meaning, representations, theoretical frame
1. Introduction
Extensive research in the past 15 years has emphasized the importance of the "context" for
problem solving processes. In particular, I would like to recall:
- the research in ethnomathematics concerning the development of applied mathematical
problem solving skills outside of the school (see [3, 12]);
- the research on "context sensitivity" in problem solving.
In several cases it concerns basic research of the influences that the "context", evoked in
the problem, may exert on the choice ofproblem solving strategies. In other cases it concerns
comparative research on the effects of curricular teaching choices which are aimed at pressing
for the development, through the contextualization of problems, of problem solving skills
[11]. In both cases the work is based on the assumption that the "context" can have an effect on
the cognitive behavior during problem solving activities, in the first case, as the element which
has a role (along with others) in determining a single performance and, in the second case, as
the element which has an effect upon the development ofthe problem solving skills.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-91-320.jpg)
![78
In this regard I would also like to recall the research work carried out in psycholinguistics
which (see [10)) reveals the sensitivity to the context in the acquisition of connectives, not only
as the element with which to influence a single performance, but also as an important factor for
the growth oflinguistic skills. More generally, referring to Brown's, Donaldson's, Gelman's,
Nelson's researches, French [10] writes:
"Despite the fact that these investigators study quite different domains within
cognitive development, make different assumptions about the origin of particular
cognitive abilities, and describe their results in different terms, they are in basic
agreement about the importance of contextual factors in both the acquisition and
display of cognitive abilities. Their essential premise is that cognitive competence
initially arises within, is embedded within, and is practiced within particular
contexts.....
Last but not least, there is empirical evidence, to which little consideration is given in current
research on problem solving, which concerns the generally good results achieved in problem
solving with teaching projects based on the teaching "by problems" referred to "areas of
interest" that may be identified by, and stimulating for the pupils (refer to various projects
carried out in Great Britain [2], The Netherlands [8, 13,], USA [14], etc.).
I think that the foregoing justifies the interest for research on the role of the "context" in
problem solving, that goes beyond the simple observations available and the results of the
research acquired thus far and provides elements to clarify the following points (the first three,
mainly oftheoretical interest and the fourth ofparticular interest to teaching):
- which aspects of problem solving are more context sensitive, on the short term (in
relation to the single problem situation), and on the long term (in relation to the development of
the problem solving skills)?
- which elements of the context can have more significant effects on problem solving
processes?
- which are the most imponant differences between the "contextualized" problem solving
processes and the "non- contextualized" problem solving processes?
- which teaching actions associated with the chosen contexts can the teacher perform in
order to enhance pupils' results in problem solving?
To study the above problems further I believe that it is necessary:
- to construct a theoretical reference frame that may be used to speak with sufficient
accuracy of the context from the standpoint of its intrinsic characteristics and from pupil's and
teacher's perspective (Refer to paragraph 2).
- to organize extensive long term experimental activities (with many pupils and teachers
involved) which have common characteristics, namely: the presence of"contexts" which require
considerable teaching, gradually proposing problem situations whereby an increasing command](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-92-320.jpg)
![80
pupil evolution, associated to problem situations presented by the teacher and to the objects and
(external) representations immediately available to the pupil, overlap. As regards the internal
context of the pupil, the evolution is continuous and intense and concerns: the ability of the
pupil to perceive (becoming more complete and ordered with time) the external context and
related, intrinsic constraints; the command of systems (increasingly more complex) and of
relations that tie the specific variables ofeach context together; the invariants and schemes [15]
steadily constructed to adapt to problem situations, inside and outside the school, related to a
given context. As far as the teacher is concerned, his internal context evolves too in relation
mainly to the experience which the teacher gradually acquires in the ways in which the pupils
refer to the external context and, at times, even to their knowledge of the external context and
relationships present in it.
This dynamic insight into the context, in particular, takes account of the process of cultural
growth and "maturation" of the pupils, provided under the guidance of the teacher in the school
environment.
We will refer to the "field ofexperience" as the set of three components ofthe context
referred to a sector ofhuman experience (social, material, intellectual...) that may be identified
by the teachers and pupils as one and sufficiently homogeneous from the standpoint of the
"scripts" (actually or potentially) involved; for example, if we speak of "orientation" or of
"industrial production" or of the "measure of time", we delimit "fields of experience" in which
the pupils (at least, from a certain age on, different for each field of experience) can refer to their
experiences, conceptions, etc.
In this paper I will consider only "real world" fields of experience; but the same definition
might be applied also to "purely mathematical" fields of experience (for instance: "numbers", or
"geometric figures", or "algebra"). Obviously, "numbers" or "geometric figures" or "algebra"
become veritable "fields of experience" only after a suitable experience (at the beginning of
compulsory education, pupils' "internal context" is already rich in numerical experiences, but
rather poor in algebraic experiences). For a purely mathematical field of experience, the external
context contains only external representations (formulas, verbal definitions and statements,
diagrams, geometric figures,...).
We may observe that the constraints depending on the teacher and, more generally,
classroom social interactions are not included in my definition of "field of experience"; this
observation suggests possible integrations of the concept of "field of experience", as a concept
referring to the construction of"meanings", in other didactical or psychological theories.
We note that two or more different fields of experience can have parts in common. For
instance, both in the "field of experience" of hand-craft productions and in that of industrial
productions the problem of the formation of production costs is very important. Similarly, the
"sun shadow" phenomenon is referred to when dealing with the aspects of orientation and the
measure of time (with elementary methods that are transparent and easily grasped by 10-11
years old pupils).](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-94-320.jpg)
![81
It is therefore possible to find limited environments (sometimes fields of experience. at
times not easily recognized as such by the pupils) within the "fields ofexperience" that may not
be broken down further and integrate many problem situations, rich of disciplinary
implications. The problem of the "formation of production costs" and the "sun shadow"
constitute two examples in this regard.
We will call these basic, irreducible and meaningful environments "semantic fields" (and
here again. as researchers. we may consider, with their evolution in time. the "external
component", the "internal component" of the pupil and the "internal component" of the teacher).
What are the purposes of these concepts?
As for the purposes of the teaching planning, they may be used to determine sufficiently
homogeneous "environments" in which to develop teaching activities aimed at the construction
of concepts and specific abilities in various disciplines. In particular, the "fields ofexperience"
provide the possibility to organize a flexible curriculum, in accordance with the subject matters
identified by the pupils, with reference to their extra-mural experiences, whereas the "semantic
field" may be used to identify the basic core of concept construction and problem solving
processes, that may be included in different fields of experience according to the particular
circumstance (interests of pupils, particular social-cultural situations, etc.); for each semantic
field precise teaching itineraries may be developed and enhanced through experience. The
concept of "conceptual field" discussed by Vergnaud [15] may be utilized as a further
instrument with which to identify and organize the elements of the disciplinary curriculum into
an ordered structure, the same elements which are to be pursued through "context-related" work
in the fields of experience. In practice, in our work associated with our projects, we realized the
importance of taking into consideration, within the teaching planning, both the standpoint of the
pupils' cognitive involvement (referred to the fields of experience) and the standpoint of the
disciplinary organization of knowledge (represented by the conceptual fields).
Furthermore, the concepts of "field of experience" and of "semantic field" may, for
research purposes, enable a reasonably accurate analysis of various aspects of context-related
concept construction and problem solving processes to be carried out. And it is strictly in this
sense that we will use them in this paper.
3. Method
The elements upon which we will base our analysis are: episodes (that is, moments of work in
the classroom, carefully documented, usually concerning one problem situation. in which it is
possible to follow, for a sufficiently long period of time - at least half an hour - the
behaviors of one or more children); comparative assessments of classes which work differently
within the framework of the same project, or of different projects; cases of pupils followed](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-95-320.jpg)
![84
"Theorems in action" [15]. the example which we have studied to some depth concerns
the distributive property of multiplication in respect to addition and regards the widespread
occurrence of this "theorem in action" in situations in which the 7 years old children must
evaluate the overall cost of (for example) 4 objects costing 320 liras each.
If we give this problem to children who already possess a reasonable amount of experience
in working with money, in real or simulated purchases, many children spontaneously break the
price down into 300 + 20 and then separately consider 4 times 300 liras and 4 times 20 liras. At
the end, they add up the partial sums thus obtained. By proposing similar problems, in parallel,
referred to lengths (for example, the problem of calculating the length of a corridor giving
access to 4 rooms, each 320 cm long,) to classes which, moreover, are involved in the same
project, the distributive property becomes apparent to a lesser extent. The semantic field of the
"goods/money exchange" seems to be able (as a result of the material organization of the
monetary values and of the social practice of price break-down, according to the money
available during payments) to activate the "theorem in action" of distributivity in a relatively
high percentage of children (over 70% of the seven years old).
We have also observed that the "theorem in action" of distributivity does not occur
frequently when we propose the problem situation of "four time 320 liras" in classes with a
poor experience in working with money.
From the developmental standpoint, the relatively early emergence of the distributive
property in the semantic field of the "goods/money exchange" allows pupils to perform complex
numerical strategies, thus contributing to the increasing of the mastery of numbers and the
connected meanings of operations [7].
Procedural aspects. We have analysed various situations in which specific
characteristics of the semantic field (material aspect of the external context, extra-mural
experiences of the pupil, in particular referred to widespread social practices, etc....) seem to
encourage the acquisition of complex procedures in problem situations that have been properly
contextualized.
For example, in class work situations in which the 8 years old children are asked "how
many sheets 0[21 em paper need to be arranged next to each other on the walls o[the room" , it
is very frequent to see them resort to a "covering strategy" (directly, on the wall, or
graphically). This strategy frequently evolves (in further problems) towards a numerical
strategy acting on the measurements. From the developmental standpoint, this allows children
to explore an important meaning of division and to approach an efficient algorithm to perform
divisions (see [7]).
"Planning skills" in the solution to complex problems (which require a proper linkage
of arithmetical or geometrical operations). Among the "planning skills", first and foremost, we
believe that the development and handling of hypotheses are of particular importance. In
research work conducted last year [6], regarding the first signs of these skills, on 65 children](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-98-320.jpg)
![85
from 4 classes under supervision, from Class II to Class IV, in mathematical and non-
mathematical, context-related and non context-related problem solving activities, I was able to
ascertain how the development of hypothetical reasoning skills in problem solving is apparent in
a large proportion of cases, in well integrated problems within the fields of experience of our
primary school project and not in "decontextualized" problems. In particular, this concerns two
types of hypotheses:
The heuristic-exploration hypotheses necessary to iteratively approach the required
solution. They involve, each time, an evaluation of the trials made and an adjustment of the
solution strategy. In a problem concerning the division of a sum of money (for example 38000
liras), necessary for a classroom activity, between all 17 pupils of the class, given to 8 years
old, when the children still do not know or understand the written calculation techniques of
division, many children resort to strategies such as "ifeach childpays 1000 liras, it means that
17000 liras are collected ... too little ..., if every child pays 2000 liras then 34000 liras are
collected ... still not enough ..., if each child pays 3000 liras... 51000 liras... too much, then
/,11 try with 2100 liras..., etc." (see [7, 9]).
The assumptions concerning constraints specific to problem situations (for instance, in a
geometric problem in which an explanation about the general procedure to measure an angle,
drawn on a sheet of paper with an assigned protractor, is required, it is necessary to take into
account the following two possibilities: angle with sides of length greater than that of the
protractor radius; angle with at least one side smaller than ..., and hence formulate a coherent
measurement project for this.)
Furthermore, as far as the complex problems are concerned, we have realized that certain
semantic fields give the children the possibility to participate in the problem construction
operation from within the problem situation (this occurs, for example, in problems in which one
must construct and compare budgets for travel expenses with different means of transportation),
or to make them solve complex problems, on the basis of the overall sense of the problem
situation and extra-mural experiences. The comparisons carried out indicate that these problem
solving experiences modify the capability to deal, productively, with the usual problems of
standard complexity.
Finally, we can consider the solution to complex arithmetical problems without given
numeric data (achieved through the verbalization of the solution strategy, "in general", or even
with the older children, by using algebraic expressions): this is a very important activity for the
effects it has on the development of the design capability, on computer-aided problem solving
and also on the construction of the meanings of the arithmetical operations; it appears practicable
by a high percentage of pupils, only for heavily contextual-related problem situations in the
fields of experience, familiar to the children.
From a general standpoint, "complexity" in applied mathematical problems may concern
the number of choices and steps that the pupil must keep under control (depending on his](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-99-320.jpg)
![88
- experiences and concepts of the teacher as regards the field experience in which one
works: they act in various ways; in particular, we are able to identify their effects on the
formulation of texts of the problems, on the interaction between the teacher and the pupils in
difficulty and on the choice of certain problem situations with respect to others.
6. Differences Between Contextualized and Non-Contextualized
Solving Activities
The elements determined through the previous analysis suggest that there are important
qualitative differences in the ways the pupils deal with problem solving activities, for the case of
contextualized problems and for the case of non contextualized problems. These differences
may be determined by indicators which, in tum, suggest assumptions on the nature of the
differences and their long term effects.
The differences can be explored by giving classes, which essentially follow the same
course, identical and difficult problems (at the limits of the pupils' problem solving capability),
in parallel, in one case as problems that are well integrated in the subject work as it is
developed, and in the other case as fictional problems.
Among the indicators the following can be listed:
- the use of verbal language: for a given structural complexity of the problems, it is more
widespread in strongly contextualized problem solving and, moreover, it is, for the same child,
generally of a different nature in both cases (for many children in decontextualized problem
solving a sequential language structure is often noted, index of the adherence to a scheme [15];
whereas the same children, in contextualized problem solving, produce solution texts rich in
parallel time connections, assumptions, results);
- heuristic strategies: their presence is anticipated and much more extensive in
contextualized problem solving (as we have already shown for the conditions in which the
development and handling of assumptions of the type "I try ..." - see [6] - appear for the first
time);
- behavior of good problem solvers: as already observed by Lesh [11], even in
decontextualized problem solving situations, they refer extensively to the context evoked by the
problem texts;
- behaviour of weak pupils: for the case of a decontextualized problem, if they are able to
solve it, this generally occurs by making use of schemes; otherwise there is a situation of total
standstill.
For the case of a "truly" contextualized problem, we can observe, in addition to the use of
schemes, clever actions being attempted, for example by the drawing of the problem situation,
by recording possible actions on the variable involved ("I take", "I put", "I move", "I measure",
etc.).](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-102-320.jpg)
![89
The mediating action of the teacher can be put to good use on such attempts.
All this suggests the hypothesis that the inclusion of the problem situation within a field of
experience, in relation to the overall work in progress, can favor:
- the development of mental process to internally and externally represent problem
situations related to perceptions of the context and of connections between the variables
implemented practically during the work in the field ofexperience;
- planning activities oriented by the awareness of the aims which the solution to the
problem has in relation to the development of the school work in the field of experience.
The mental work appears less "contracted", less oriented towards the research of acquired
reproducible schemes, and evolving mental environments appear to be created in which images
and hypotheses and reconstructed experiences are handled.
7. The Role of the Teacher in Contextualized Problem Solving in
Semantic Fields
We will take for granted the complexity of the questions concerning the choice of the fields of
experience within which to propose and choose problem situations (refer to [4] and [5] in this
regard).
I would like to pause a while here to discuss the question of the handling of problem
situations in the classroom. On the basis of what has been said in the previous paragraphs, the
constraints and "concrete" aspects of the external context may not be perceived by the pupil; on
the other hand, at times, the conceptions of the pupil or the external representations operate in a
way of forcing the pupil to reach incorrect solutions. It is therefore necessary that the teacher
does not limit herself or himself to propose "stimulating" problem situations for the pupils'
strategy construction processes, but rather to intervene directly to act as go-between the external
context and the internal context of the pupil and as mediator of the culture elaborated by
humanity to date. For this purpose I do not believe it possible that in certain applied
mathematical problem solving situations, referred to certain "scientific revolutions" (Talete's
Theorem, Archimedes Principle for the equilibrium of the lever, Mendelian transmission of
hereditary characters) the simple proposition of opportune problem solving situations has the
effect of conducting towards the elaboration of correct solution strategies and associated
conceptualizations. For example, for the case of the problem of the length of the shadow of the
80 cm long stick, knowing the length of the shadow cast by the 60 cm long stick, it is true that
the pupil may realize, on account of the results of crucial experiences proposed by the teacher,
that the additive model does not work; it is nonetheless difficult that she or he alone is able to
understand that a multiplication model is better suited to the reality!
The role that the teacher must perform for the productivity of her or his work therefore
appears to depend upon the field of experience chosen and upon the particular problem](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-103-320.jpg)
![Cognitive Models in Geometry Learning
Jose Manuel Matosl
ｳｾ＠ de Ciencias de ｅ､ｾＬ＠ Faculdade de Ciancias e Tecnologia, 2825 Monte da Caparlca, Portugal
Abstract: Recent studies on the ways in which we fonn categories of objects indicate that we
admit that there are special elements (prototypes) in most classes. Moreover, these classes of
objects seem to be organized in such a way that apparently it exists a special hierarchical level,
the basic level. that is generally used in communication. Researchers have identified several
types of cognitive models that seem to play an important role in the ways in which we fonn and
organize these categories. The explanatory power of these models is tested in the re-
interpretation of findings from several investigations.
Keywords: cognitive models, scripts. schemata. metaphors. metonymy. mental images.
language and mathematics. geometry learning. van Hiele theory. social cognition
One of the concerns of psychologists and educators has been the ways in which students
abstract ideas and concepts from their experiences. These abstractions help us organize our
experiential world. Gestalt theory conceived abstraction as a reorganization of the field of
perception. Piaget viewed it as the formation of schemas. and cognitive scientists incorporated
in it the mechanisms of generalization. differentiation. and the recognition of patterns.
Abstraction involves. among other things. the formation of categories of objects.
Traditional psychological studies considered that a small set of simple properties is necessary
and sufficient to establish membership of a category. These categories have defining or critical
attributes which determine which elements are members or are not members of a category. Take
for example the set ofred flowers. Boundaries of this category are assumed to be sharp and not
fuzzy. i.e.• given any flower. either it is red or not [14]. In Piaget's studies. for example,
equivalence relationships determine membership to categories. Moreover it assumed that we use
necessary and sufficient properties of each category when perfonning inferences. deductions.
and when we construct a taxonomy of categories. These two assumptions are usually taken to
represent what is called the classical view of categorization.
lThis article is based in part of the author's doctoral dissertation in process of completion at the University of
Georgia and was partially supported by a grant from the Institute of International Education, New York and
another grant from the Ministry of Education of Portugal.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-106-320.jpg)
![94
It is the purpose of this paper to review current psychological perspectives on categorization
and to apply these perspectives to the discussion of research findings on the learning of some
geometric concepts.
Category Theory
The classical view of categories was first challenged by Wittgenstein's work. He pointed out
that the category of games does not have a set of common properties shared by all the members.
Some games share amusement, others luck, competition, skills, or a mix of these. Wittgenstein
conjectured that game was a cluster concept, held together by a variety of attributes, but no
instance contains all the attributes. The category of games has ajamily resemblance structure.
Like family members, games are similar to one another in some, but not in every, ways. Some
categories, like games or numbers, have no fixed boundaries and can be extended depending on
one's purposes. Some categories, like numbers or polyhedra, have central members. Any
defmition of numbers must include the integers, as any definition of polyhedra must include the
cube [14, 19].
The Study of Prototype Effects
Experiments later performed by Rosch and her colleagues empirically confirmed Wittgenstein's
philosophical investigations [32, 33, 34, 35, 36, 37, 38]. Their research proceeded in two
directions: horizontally it looked for asymmetries among members of one category; vertically, it
looked at asymmetries within nested categories [36]. Their work produced evidence of two
phenomena: prototype effects and basic-level effects.
The term prototype effects refers to the experimental finding that in some categories not all
members have an equal condition. Rosch's initial research with color [32] showed that there are
colors (focal colors) that have a special cognitive status. On the one hand, they were preferred
by her subjects as best examples. On the other hand, subjects learned them easier than the other
colors. This finding run contrary to the previous scientific belief that colors are arbitrarily
named, i.e., that the specific colors chosen to be named are determined by language alone. She
called these focal colors cognitive reference points, and prototypes [33]. She later extended her
research to other categories, usually of physical objects, and in each case, she found prototype
effects, i.e., subjects judged that certain members of each category were more representative of
the category than others. She showed, for example, that her subjects believed robins to be very
typical birds, whereas chickens were less typical. These effects, however, were not discovered
in every category. Rosch, for example, was unable to find prototype effects in categories of
actions like walking, eating, etc. [9].](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-107-320.jpg)
![95
Rosch's research also tried to find other consequences of these prototype effects. She
showed that they predict perfonnance on several tasks focusing on the ways in which central
members of a category are related with peripheral members and with the category itself. She
found that: (a) less typical members of a category are less associated with that category; (b)
typical members appear to have an advantage in perceptual recognition; (c) when people think
of a category member, they generally think of typical instances of that category. She also
showed one asymmetry in the ways in which members of some categories are related to others:
(d) subjects considered less typical members to be more similar to more representative examples
than the converse. Finally, she showed that: (e) the categories studied had a structure of family
resemblances [2, 19,43].
Basic Level Effects
Investigations about the ways in which people nest the classification of objects have been the
object of research. Brown has been credited as being the first to present the problem:
We ordinarily speak of the name of a thing as if there were just one, but in fact, of
course, every referent has many names. The dime in my pocket is not only a dime.
It is also money, a metal object, a thing, and, moving to subordinates, it is a 1952
dime, in fact a particular 1952 dime with a unique pattern of scratches,
discolorations, and smooth places. [5, p. 14]
He pointed out that we have the feeling that some of these names are the "real names," the
others being achievements of the imagination. Although we know that Brown's dime is a
"coin," or a "thing," we are compelled to think that its real name is "dime". Moreover these
special names seem to be frequently linked with non-linguistic actions.
Brown's ideas prompted cognitive anthropologists to search for folk taxonomies-the
ways in which cultures use the fonn "A is a kind of B." Berlin and his coworkers (referred in
[19]) examined folk classification of plants and animals of speakers of Tzeltalliving in a region
of Mexico. They found out that, although their infonnants could name animals and plants in a
variety of ways, they tended to use a single level of classification. Berlin called this level of
classification the folk-generic level (or basic level). This level was in the middle of folk
classification hierarchy. Further research on the Tzeltallanguage discovered that most children
initially learn names at this folk-generic level, and later they find out simultaneously how to
differentiate and generalize these tenns [19].
Rosch and her colleagues developed a series of experiments that confinned most of Berlin's
findings [38]. They found that the psychological most basic level was in the middle of
taxonomic hierarchies. Basic-level categories are basic in perception, function, communication,
and knowledge organization. Essentially they found that the basic level is:](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-108-320.jpg)
![96
- The highest level at which category members have similarly perceived overall
shapes.
- The highest level at which a single mental image can reflect the entire category.
- The highest level at which a person uses similar motor actions for interacting with
category members.
- The level at which subjects are faster at identifying category members.
- The level with the most commonly used labels for category members.
- The first level named and understood by children. (...)
- The level at which terms are used in neutral contexts. (...)
- The level at which most of our knowledge is organized. [19, p. 46]
Other researchers [9] also found that, unlike scientific taxonomies, classification in folk
taxonomies is not always exclusive, i.e., instances can share several taxonomic levels.
Moreover, the implicit categorization criteria may vary. Sometimes, for example, we use the
function as the categorization criterion for superordinate category ("clothing"). Other times we
use the unity of place ("furniture"). In other cases, we form categories ("groceries") from a
composition of criteria. There is also evidence that folk taxonomies are not very extensive
[29].
Implications for Cognition
Prototype and basic level effects destroyed the notion that concepts are organized by sets of
necessary and sufficient conditions, and have prompted the development of new cognitive
models that can accommodate the experimental findings. At the core of this theoretical effort is
the notion of mental representations-"a set of constructs that can be invoked for the
explanation of cognitive phenomena, ranging from visual perception to story comprehension"
[14, p. 383]. This section will analyze the ways in which several psychological theories
account for the effects described previously.
The holistic perspective provides the simplest account for prototype effects. This theory
maintains that, for example, a term like "dog" refers to the mental category dog, which is in
itself an unanalyzable gestalt. It assumes that mental categories are composed of templates,
usually imagistic, that are isomorphic to the object they represent, are unanalyzable, and that
implicitly show the relations between the several features or dimensions of the object. An object
belongs to a certain class if it provides a holistic match to the template of the class. Computer
scientists working in pattern recognition have been using this theory. The theory, however,
seems to be limited to the categorization of concrete objects-it is difficult to talk about
templates for categories like "furniture", or for more abstract entities like "justice" [44].
Another theory, the featural approach, supports the idea that human minds use more
elementary categories and only a few of the words that we manipulate code unanalyzable](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-109-320.jpg)
![97
concepts. Rather, most words are labels for mental categories which are themselves sets of
simpler mental categories, usually called features, properties, or attributes. Each concept is
represented by groups of features that have a substantial probability of occurring in instances of
the concept. Some of the proponents of this theory developed the notion of a group of weighted
features. An object is an instance of a category if the sum of its values for each feature is greater
than a given threshold. Each member of a category is further removed from the prototype the
more it differs in highly weighted features [19,44].
Some researchers [23, 24, 31] have designed a similar model using dimensions instead of
features to represent concepts. This view departs from the featural view especially in the
treatment of continuous dimensions like size. Specific instances depart from the prototype in
continuous degrees. If the relevant dimensions of birds were thought to be animacy, size, and
ferocity, a robin, for example, might have a 1 in the feature of animacy, .7 in size, and .4 in
ferocity. A key concept that is a consequence of this approach is the notion of semantic metric
spaces, which are thought to be multidimensional Euclidean real spaces. The vector (1, .7, .4)
in R3 would represent a robin [44], providing a literal meaning to the notion of semantic
distance which is interpreted as the Euclidean distance on a semantic space. Typically,
researchers in this area attempt to determine relevant dimensions, discuss the meaning of
clusters of concepts, or discuss meanings of translations in the semantic metric space. The
semantic distance between an instance and the prototype corresponds to the degree of category
membership.
Critics to these two last perspectives have pointed out that: (a) the knowledge represented in
a concept includes more than a list offeatures, namely the relationships among the features; (b)
the model does not provide for contextual or background effects. The dimensional perspective
also raises additional problems because of its use of semantic metric spaces. The requirement of
orthogonal dimensions, the necessity of the isotropy of the semantic space, the very possibility
of coexistence of concepts and their members in the same space are some examples of the
difficulties of these theories [19,44].
Global theories of cognition have provided ways to accommodate prototype effects.
Rumelhart, for example, proposes that knowledge of each concept is represented by a schema
(he uses the plural as schemata). A schema is "a data structure for representing the generic
concepts stored in memory" [39, p. 34] and contains the relationships among the components
of the concept in question. A very similar construct, a/rame, is proposed by Minsky [25].
Rumelhart describes the features of schemata using four analogies.
First, each schema is a type of informal, private, unarticulated theory about the nature of
events, objects, or situations that we face. The total set of our schemata constitutes our private
theory about the nature of reality and represent knowledge in all levels of abstraction
("Schemata are our knowledge" [39, p. 41]). This means that we are constantly testing this](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-110-320.jpg)
![98
theory. and that we use it to make predictions about unobserved events. According to Minsky
[25]. this is accomplished by an i,gormation retrieval network.
Second. schemata are active processes like procedures or computer programs. As such they
are able to detennine the extent in which they account for the pattern of observations. and are
capable of invoking other subprocedures (or other subframes).
Third. schemata are like parsers that work with conceptual elements. On the one hand. we
are able to find and verify the appropriate schemata. On the other hand. schemata enable us to
find constituents and subconstituents in our observations.
Finally. the internal structure of schemata is like scripts of plays that can be played with
different actors. The scripts correspond to prototypes of the concepts. They have several
variables that can be "associated with (bound to) different aspects of the environment on
different instantiations of the schema" [39. p. 35]. Minsky talks about terminals. which are
"slots that must be filled by specific instances or data" [25. p. 96]. We are aware of the typical
values of these variables and of their interrelationships.
Both schemata and frames provide a similar explanation of prototype effects. Rumelhart
claims that the meaning of a concept is encoded in terms of the prototypical situations or events
that instantiate that concept. and Minsky defines a frame as the representation of a stereotyped
situation. In more specific explanations, both researchers stipulate that each schema's variables
have default values that are responsible for our expectations and other kinds of presumptions.
These default values are "attached loose to their terminals" [25, p. 97], which allow their
replacement by new items that better fit our experiences.
Cultural Models
The previous models we have been discussing still do not account for contextual or background
effects. nor do they provide any explanation for basic level effects. From an educational point
of view, contextual effects are very important to understand children's enculturation into school
mathematics. It is a plausible conjecture that children's conceptualizations depend heavily on the
social and cognitive context in which learning takes place. Students and their teachers live in a
school culture and we can expect that they intersubjectively share mathematical concepts in
some degree. Both students and teachers are also members of larger social groups and we may
expect that they bring the mathematical knowledge of such groups into the school.
A broader approach to mental representations was needed. one that would take into account
the role played by the community of human minds upon the individual. To describe this
common knowledge, cognitive anthropologists and linguists have developed the notion of a
cultural model-"a cognitive schema that is intersubjectively shared by a social group" [9, p.
809] used by American researchers, or of social representations-system(s) of values. ideas](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-111-320.jpg)
![99
and practices with a twofold function: first, to establish an order which will enable individuals
to orient themselves in their material and social world and to master it; and secondly to enable
communication to take place among the members of a community by providing them with a
code for social exchange and a code for naming and classifying unambiguously the various
aspects of their world and their individual and group history [10, 26, 41] used by investigators
working in the sociological European tradition.
A special example of these cultural models is the notion of scripts, developed by Schank
and Abelson in the context of text comprehension, which are cultural models adapted to the
study of events. A script is "a coherent sequence of events expected by the individual,
involving him either as a participant or as an observer" [I, p. 33] and it can be interpreted as an
extension of schemata to dynamic episodes [2, 14]. Scripts may be thought metaphorically as a
cartoon strip. The departure of an instantiation of events from the expected prototypical episode
gives rise to prototype effects. People, for example, have a "restaurant script" that is composed
of a stereotyped set of events that they expect to happen under certain circumstances. It depends
on a sociocultural institution, that is, the existence of a place that serves and sells food [9].
Idealized Cognitive Models
Recently linguists and anthropologists have been converging in their study of cultural models.
An example is Lakoff's work on cognitive models. Lakoff develops an idea of mental
representations that borrows some of the features of Rumelhart's schemata, Minsky's frames,
and Abelson's scripts, and adds linguistic and cultural components. He proposes that we
organize our knowledge by means of idealized cognitive models, and category structures and
prototype effects are by-products of that organization.
To build his construct Lakoff departs from two assumptions shared by the featural and the
dimensional approaches [19]. The first is that goodness of example is a direct reflection of
degree of category membership, that is, subjects' willingness to say that a chicken is not a good
bird implies that chickens do not have a high degree of members of the category of birds.
Although the construct of a graded membership to mental categories can explain some prototype
effects, others are not. A classical example of a category (first presented by Fillmore) that does
not have a graded membership is the category of "bachelor", for which there are clear
conditions for membership. Nevertheless persons like the pope, or Tarzan do not have a clear
status of membership to this category. Lakoff proposes that prototype effects in this category
are produced not because the category is graded, but because we have an idealized cognitive
model of bachelor based in the context of a human society where there are certain expectations
about marriage and marriageable age. The worse the fit between that idealized cognitive model
and our knowledge of the background conditions, the less appropriate we feel that the concept](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-112-320.jpg)
![100
should be used [19, 28]. Later we will discuss the category of even number which is not
graded either.
The assumption, shared by the featural and the dimensional approaches, that prototype
effects mirror mental representations of categories, that is, categories are represented in the
mind in terms of prototypes and degrees of category membership are determined by their degree
of similarity to the prototype, does not capture the complexity of some categories. For example,
the concept of mother is based on a complex aggregate of several models:
- The birth model: The person who gives birth is the mother. C
•••)
- The genetic model: The female who contributes the genetic material is the mother.
- The nurturance model: The female adult who nurtures and raises the child is the
mother of that child
- The marital model: The wife of the father is the mother.
- The genealogical model: The closest female ancestor is the mother. [19, p. 74].
The concept of mother is not defined by necessary and sufficient conditions, and all those
models converge in a prototypical ideal case. Prototype effects can be explained by tensions
between these models in some situations (stepmother, surrogate mother, foster mother, etc.).
Lakoff claims that a major source of prototype effects is associated with our use of
metonymy-"a situation in which some subcategory or member or submodel is used (...) to
comprehend the category as a whole" [19, p. 79]. Social stereotypes, where a subcategory has
a socially recognized status as standing for the category as a whole, are examples of our use of
metonymy. For example, in the United States, the category "working mother" is not a mother
that happens to be working. Rather it is defined in contrast with the social stereotype of a
"housewife-mother" which is defined by the nurturance model. Prototype effects in the case of
a working mother arise from its comparison with only one of the models in the cluster and not
against the whole category. Put in another way, the "housewife mother" usually stands for the
whole category of "mothers".
Consider an unwed mother who gives up her child for adoption and then goes out
and gets a job. She is still a mother, by virtue of the birth model, and she is
working-but she is not a working mother!
The reason is that it is the nurturance model, not the birth model, that is relevant.
Thus, a biological mother who is not responsible for nurturance cannot be a
working mother, though an adoptive mother, of course, can be one [19, p. 80].
Other kinds of metonymic models include: typical examples, ideals, paragons, salient
examples, and others that we will discuss later. Neither the featural and dimensional
approaches, Rumelhart's schemata, Minsky's frames, nor Schank and Abelson's scripts
account for prototype effects that result from metonymy.
Another important source in the construction of all kinds of models is the use of analogy
and metaphor. Each metaphor is based on a similarity between a source and a target domain,](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-113-320.jpg)
![101
together with a source-to-target mapping [19]. Metaphors allow us to extend the similarities
between the domains beyond their initial state and they structure most of our conceptual system.
We use them to map structures usually on the physical and eventually on the mental world into
other domains through imaginative processes [20].
The result of any such mapping, from physical experience in the source domain to
social or psychological experience in the target domain, it that elements, properties,
and relations that could not be conceptualized in image-schematic form without the
metaphor can now be so expressed in the terms provided by the metaphor
[28, p. 28].
Johnson later elaborated this notion, and proposed the construct of kinesthetic image schema-
basic experiential structures that are a consequence of the nature of human biological capacities
and the experience of functioning in a physical and social environment [17]. Reason for
Johnson is no longer detached from human beings as functioning organisms. These image
schemas significantly structure our experience prior to, and independent of, any concepts, and
are responsible for many of the metaphors we use in abstract domains. Examples of these
schemas include: the container schema that consists of a boundary distinguishing an interior
from an exterior; the part-whole schema that involves the whole, the parts and a configuration;
the link schema, where there are two entities and a link connecting them; the center-periphery
schema where a central element is thought to be more important than the periphery; the source-
path-goal schema that includes a source, a destination, a path, and a direction; the up-down
schema; the front-back schema; and the linear order schema [19]. Quinn and Holland [28] argue
that these imagetic schemas, from which metaphors are based, are not only predicated in our
bodily experiences but may also be built upon elements shared by the cultural group.
In summary, Lakoff proposes that the structure of thought in general, and the categorization
in natural languages in particular is characterized by cognitive models that fall in four types
[19]:
(a) Propositional models that specify elements, their properties, and the relations
holding among them.
(b) Image-schematic models that specify schematic images.
(c) Metaphoric models that are mappings from one of the above models in one
domain to a corresponding structure in another domain.
(d) Metonymic models that make use of the previous models and map one element
of the model to another.
By distinguishing among these types of cognitive models, Lakoff is able to propose a process
of creation of complex cognitive models. He argues that there is a "significant level of human
interaction with the external environment (the basic level), characterized by gestalt perception,
mental imagery, and motor movements" [19, p. 269]. This is the level at which people
function most efficiently and successfully using basic-level and image-schematic concepts.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-114-320.jpg)
![102
Categorization of Mathematical Objects
Mathematical categories have been the object of research both by psychologists, linguists, and
mathematics educators. For some of these researchers, mathematical knowledge is a field of
certainty, bound by the laws of logic, and a clear example of analytic truth. Armstrong, L.
Gleitman, and H. Gleitman's investigation on prototype effects [3] provides a typical example.
In an attempt to prove that prototype effects were unrelated to the ways in which we categorize,
they compared categorization of mathematical entities with the categorization of real world
objects. The rationale for this approach was that if prototype effects could be found in
mathematical categories, like "even number", then Rosch's prototype theory should be wrong,
because these prototype effects are unrelated to membership gradience-the category of "even
numbers" has a clear membership rule. Their implicit assumption was that mathematical entities
have a clear, declarative membership rule, and that their subjects were applying it.
This conception of mathematics runs contrary to recent developments in philosophy,
history, and sociology of mathematics [4, 18, 30,47]. As a result of investigating the roots of
mathematical knowledge, researchers in these fields have been proposing that mathematics
knowledge is generated by social interactions and that mathematical truth is intersubjectivily
shared by the community of mathematicians. Lakatos' work, in particular, shows how
mathematicians themselves may not be in agreement over the meanings of mathematical entities,
even when such meanings are provided by definitions. Although Lakatos's field is history and
philosophy of mathematics, he does provide evidence of prototype effects in the category of
polyhedra. In his historical account of the discussions over a precise definition of the concept of
polyhedra, Lakatos shows how some mathematicians have come up with counterexamples of
polyhedra that did not verify Euler's formula, and how other mathematicians would claim that
they were presenting "monsters" and using "wrong" definitions of polyhedra. Moreover there
are central examples of polyhedra-all mathematicians would agree that any definition of
polyhedra should include prototypes like the five platonic solids.
A second point can be made about Armstrong, L. Gleitman, and H. Gleitman's
investigation. Even if we accept that mathematical categories are classical, we would still have
to show that they are thought as such by the subjects themselves. As Gardner pointed out [14],
their research may very well show that even mathematical categories display a structure similar
to other categories. We will further discuss their research later in this paper.
A strong case for the subjectivity inherent to mathematical entities is put forth by Fischbein.
In his review of the role of intuition in thought he gives examples of what he terms analogic and
paradigmatic models in mathematics and physics. [11]. Analogic models are similar to what we
have termed here metaphoric models, and a paradigm, in Fischbein's terminology, is an
instance of a category that is used to represent the whole category and is thought to be a
particularly good example of the category. This last definition shares both the characteristics of](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-115-320.jpg)
![103
a prototypical and a metonymic model. Fischbein also agrees that mathematical categories may
not be classical. He proposes in his book about intuition [11] that when we define a concept we
never do it as a pure logical construct
The meaning subjectively attributed to it [the concept], its potential associations,
implications and various usages are tacitly inspired and manipulated by some
particular exemplar, accepted as a representative for the whole class (p. 143).
Fischbein's point is as much about students as about mathematicians themselves. Fischbein
compares these reasoning processes to Kuhn's paradigms in scientific thought, and calls this
phenomenon "the paradigmatic nature of intuitive judgment" (p. 143).
An Example: the Concept of Number
Rosch [33] studied examples ofcognitive reference points that included vertical and horizontal
lines, and numbers that are powers of 10. Part of her research looked for prototype effects
using linguistic hedges-"terms referring to types of metaphorical distance" (p. 533), like
"almost", "virtually", "essentially", "loosely speaking". She made use, for example of stimuli
as "103 is essentially 100". She found that, within the decimal system, multiples of ten
constitute reference points. Both 97 and 102 were judged essentially 100, but not vice versa,
and both were considered closer to 100 than 100 was close to them. As a by-product these
asymmetries questioned the isotropy of semantic spaces [33].
Analyzing these results from a linguistic perspective, Lakoff [19] adds that the natural
numbers, for most people, are characterized by the words for the integers between zero and
nine, plus addition and multiplication tables and rules of arithmetic. These digits are the central
members of the category of natural numbers, from which the other members are generated. Any
natural number can be written as a sequence of digits, the properties of large numbers are
understood in terms of the properties of the single-digit numbers, and the computations with
large numbers are understood in terms of computation with the single-digit numbers. Each of
the single digits generates subcategories of its own when multiplied by 10, 100, etc. These
results were actually predicted by Wertheimer [52]. He may be credited as being the first to
draw attention to the special place multiples of ten have in our vocabulary: "He is a man in his
thirties," or "X died in the twenties of last century."
Natural numbers are an example of a category composed of some central members and
some rules for generating the other members. Lakoff [19] claims this is a metonymic model,
where the single-digit numbers stand for the whole category. He also claims that the category of
natural numbers itself is a central category in more general categories of numbers. For example
rational numbers are understood as quotients of natural numbers, real numbers as infinite
sequences of single-digits, etc. These other categories of numbers are understood](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-116-320.jpg)
![104
metonyrnically in terms of the natural numbers. Every axiomatic system involving numbers
must include the natural numbers, so their centrality is reflected even in the work of
mathematicians. Data from mathematics educators confirm this "dissolution of hierarchies" [11,
p. 147]. Tall and Vinner [45] report that often students did not regard -{2 as a complex number
although some of them defined real numbers as "complex numbers with imaginary part zero"
(p. 154).
The effects found by Rosch are explained because we use the powers of ten as a submodel
to comprehend the relative size of the numbers, especially in the context of approximations and
estimations. There are also other models that we use to comprehend numbers. For example, in
the context of body temperature, 380 Celsius is a cognitive reference point where fever is
involved, and where American money is concerned, a cognitive model often includes powers of
five (nickels, dimes, quarters). Each of these models produces prototype effects.
It is important to note these prototype effects are not equivalent to graded category
membership. In fact subjects in Armstrong, L. Gleitman, and H. Gleitman's investigation [3]
agreed that the categories of even and odd numbers are well defined. Nevertheless the
researchers found prototype effects using reaction time and ratings. Lakoff [19] claims that
these effects are the result of the superposition of all those models over the even-odd structure
of the natural numbers.
Another Example: Preferred Triangles
Geometry, in particular, relies heavily on metaphors. Let us just look at the terms "altitude",
"height", "base", "length", and "width". We talk about "the altitude of a triangle", "the altitude
of a trapezoid", "the altitude of a parallelogram", but very rarely about "the altitude of a
rectangle" ("length" and "width" are used instead), or "the altitude of a square" (we use "side"),
and never about "the altitude of a rhombus" (altitudes of rhombuses are seldom used). The
same can be said about the term "base". There is "the base of a triangle" but not "the base of a
rhombus". "Base" and "altitude" are also used with solids in the same way. Virtually every
textbook will say that to calculate the area of a rectangle we have to multiply the "length" and
the "width", whereas the area of triangle is computed using the "base" and the "height".
The term "the altitude of' is used in English common language mainly in relation with
mountains. We would prefer the terms "height of a building" or "height of a person" but not
"the altitude of a house". Although we would say that "the plane is at an altitude of 9 Km" we
would not say "the altitude ofthe plane is 9 Km". The underlying message is that we are asking
students to imagine triangles as mountains, whereas rectangles are thought to be like rooms or
football fields (rhombuses apparently are thought to be diamonds or kites). We are in fact using](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-117-320.jpg)
![105
a "mountain metaphor" to work with triangles and a "football field" or a "room metaphor" to
compute the areas ofrectangles.
Although such worldly terminology would be condemned by Hilbertian formalists, its use
guarantees that students attribute meaning to their actions on mathematical objects. But
mathematicians themselves also make an extensive use of metaphors. As Thom puts it, "the
mathematician gives a meaning to every proposition" [46, p. 202].Terms like "manifold",
"fiber bundle", "curvature", "projection", "kernel", "closure", and many others are all evidence
ofmathematicians' concern for meaning.
It is this concern for the attribution of meaning to mathematical entities that the Teachers'
Edition of Merrill Mathematics, Grade 5 expresses when giving recommendations for the
lessons about the computation of the areas of the rectangle and the triangle:
Make sure each student understands the relationship between length times width for
a rectangle, and base times height while computing the area of a right triangle.
Otherwise, students will not be as apt to use what the already know to solve these
new problems (p. 388).
The attempt to use what students already know is exactly the purpose of the educational use of
the mountain or the football field metaphors.
The mountain metaphor may only be part of the picture. Several researchers have been
reporting students' preference for the upright/horizontal position of geometric figures [8, 12,
13, 51, 54], and in one case [8] one of the informants (Bud) distinguished among several
triangles by the directions they were pointing. Some researchers [13] have interpreted these
phenomena as "perceptual difficulties" (p. 137) but this description does not provide specific
information. We would argue instead that it is a cognitive, not perceptual, problem produced by
the interaction of several cognitive models. The up-down schema in Johnson's terminology
[19] may account for the preferred orientation, and a metaphor mapping the human act of
pointing to some of Bud's triangles may help to explain his answers. We will discuss these
points later.
The use of such processes is a necessary and unavoidable characteristic of thought. It
facilitates students' identification of the relevant elements and their relationships, and permits
their integration with previous knowledge [27]. There is however an unwanted side effect to it.
It is hard to imagine into which direction is an obtuse triangle pointing. Moreover, in the culture
of school mathematics it is irrelevant where triangles are pointing. Students also tend to have
problems when they attempt to apply the mountain metaphor to triangles that are not in the
"mountain position" (no side horizontal), or that do not look like mountains (obtuse triangles
with a horizontal side other than the larger side). A non-obtuse triangle in a "mountain position"
seems to be the cognitive reference point [19] employed by students. Prototype effects are](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-118-320.jpg)
![106
likely to occur with different triangles, as shown by Vinner and Hershkowitz [51] and Wilson's
[54] investigations that included the concept of the altitude of a triangle.
Concept Images and Concept Definitions
The construct of prototypes has also been used by researchers in geometry learning. Some
researchers have reported that students' choice of examples of geometric concepts and their
defmitions of the same concepts do not match [7, 13,21,55]. The construct of concept images
has been proposed by Vinner and some of his colleagues [45,48, 51] as an explanation for
those findings.
A concept image is "the total cognitive structure that is associated with a given concept"
[45, p. 152] and is composed of the images associated with that concept together with a set of
properties and processes. For example, the concept image of a function may include a picture of
the graph, a picture of the algebraic expression that defines the function, together with the
students' defmition of function. Concept image is contrasted with concept definition, which is a
verbal defmition that accurately explains the concept [48] and may differ from the mathematical
definition [51]. Vinner distinguishes between formal and informal learning, and claims that in
the later we need a concept image and not a concept definition. Concept definitions introduced
by means of a definition will remain inactive or eventually be forgotten. In a specific intellectual
task only portions of the concept image are actually evoked (temporary or evoked concept
image). These portions might be contradictory and produce conflict in one person's mind when
these opposing portions of the concept image are used simultaneously [45,48].
These researchers have been using this construct to interpret the finding that often visual
identifications and drawings made by students do not match their definitions [15, 49, 50, 51].
Although Vinner and his colleagues did not perform these investigations within the framework
of categorization theory, Hershkowitz [15] recently attempted a reinterpretation of their findings
consonant with categorization theory and van Hiele theory. We will discuss her proposals
elsewhere (see.[22]).
Applying the construct of concept images to the problem of the mismatch between the
choice of examples and the definitions we could say that the students' concept images and
concept definitions were not matching, and the concept image was taking precedence in
identification or production tasks. However, we would still fail to explain the incompleteness of
the definitions, the absence of the distinction between necessary and sufficient conditions, the
ambiguity of the terms, and how a contradiction between an imagetic and a propositional
representation of concepts could occur in students' minds.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-119-320.jpg)
![107
Learning of Some Geometric Concepts
In this section we will analyze four areas that have been the focus of recent research in
geometric learning. In each of them researchers have detected evidence of students' non-
standard mathematical knowledge. For each of these areas we will attempt to look at them
through the eyes of category theory and provide an explanation of the sources of that
knowledge. This analysis will permit us to uncover prototype effects with roots in distinct
cognitive models, namely image schematic, metaphoric, metonymic models, and scripts.
The Influence of Visual Prototypes and Metaphoric Models
Prototype effects caused by image schematic models, characterized by a gestalt of a geometrical
figure, are well known by researchers [13, 15,21,40,42,51,53]. They are the most simple
cases ofprototype effects. The characteristics of these prototypes can be summarized by:
(1) a preferred position; namely triangles, squares, rectangles, and parallelograms
must have a horizontal base [13, 21, 51, 53];
(2) symmetry; for example, obtuse triangles with their bases in a smaller side are
not recognized, or a right triangle is thought as a half-triangle [6,13,51];
(3) an overall balanced shape; namely students do not recognize "skinny" triangles,
"pointy" triangles, or extremely small squares [6, 13, 15].
These characteristics provide a good description of the prototypical geometrical figures.
Moreover, these gestalts do not require some characteristics that are significant from a standard
mathematical point of view. For example, sides may be curved or "crooked" [6, 13,40]. There
is also some evidence showing that students form such image schematic models even when
only a verbal definition is given [51]. Students also show a substantial agreement about these
models.
In fact it may happen that in some cases these image schematic models are intermix with
some metonymic models. In the case of triangles, Vinner and Hershkowitz [51] present
evidence suggesting that the "overall balanced" isosceles triangles are taken metonymically as
best examples of the whole category of triangles. This idea can be extended to other categories.
It is reasonable to conjecture that a whole set of"overall balanced" rectangles may stand for the
whole category of rectangles.
There is an overall agreement about the sources of these models. As we have seen
previously, upon entering school, children are able to identify balls, cans, boxes, and other
shapes. Usually in school they learn the names of two-dimensional geometric figures. The first
type of objects is composed of real world objects. Children learn them by manipulation or
observation, and they are capable of identifying them regardless of their position. Objects of the
second kind tend to be learned and used mainly in school. Research has shown [13] that](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-120-320.jpg)
![108
geometric figures are usually presented in pictures that match the three characteristics of
students' mental images described above: preferred position, symmetry, and balanced shape. In
summary, children's prototypes are heavily influenced by the best exemplars shown to them in
the school environment.
Image schematic models, however, do not present a complete picture. There is evidence
that there are also perceptual problems involved in the identification of geometrical figures,
namely in the perception of right angles. Vinner and Hershkowitz [51] have shown some
evidence suggesting that isolated right angles or right angles included in right triangles are more
difficult to identify when none of the sides is horizontal. Some of their subjects used the
strategy to turn the figure so that they could accomplish a better identification of the right
angles.
Other models may also be involved in the identification of geometrical figures. Previously
we mentioned a metaphoric model used (or implicitly used) by the mathematical community
when dealing with triangles, the "mountain metaphor". Here we will analyze two examples of
students' metaphoric models. Burger [6], for example, reports that Bud (one of his subjects)
explained that some of his triangles were different from others because they were "pointing that
way [to the right, or down]" (p. 52). The idea that triangles point to a direction is what Johnson
[17] would call a metaphoric model based on our kinesthetic image schema of pointing. For
Bud, triangles (at least some triangles) are embodied, i.e., some of their properties "are a
consequence of the nature of human biological capacities and of the experience of functioning in
a physical and social environment" [19, p. 12]. This way to think about triangles is not Bud's
particular model. Rather it is a social model that we intersubjectively share, because we are all
able to understand Bud's point. In some contexts we ourselves would be willing to say that a
triangle is pointing to a direction.
Fuys, Geddes, and Tischler [13] report an example of another metaphoric model. One of
their subjects (Gene), when asked if a square was a rectangle, answered "Na, that's a box"
(p. 83). Of course Gene knew that, literally, a square is not a box. He was using a box as a
metaphoric model of a square. Gene also thought that "the sides of a rectangle" referred to the
vertical sides, whereas the horizontal sides were not "sides" but "top" and "bottom". Again he
was using a metaphoric model. When we use English words to denote objects in the world that
look like rectangles we may make this linguistic distinction. Both these metaphoric models were
based on his experiences with objects on his environment.
Effects Due to Prototypical Actions
Investigations focusing on students' difficulties in drawing elements on a figure have also
found prototype effects. Previous explanations have interpreted these phenomena as prototype](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-121-320.jpg)
![109
effects produced by comparisons with a prototypical image of the expected drawing. In this
section we will argue for a more dynamic interpretation that takes into account the actions that
students are expecting to perform when asked to draw elements of geometric figures. We will
focus on two important cases: drawing the altitude of a triangle and drawing the diagonals of a
polygon.
Researches on students' drawing of the altitude of triangles have focused on the
determination of the characteristics of the triangle with which students experience more
problems. Vinner and Hershkowitz [51] asked students to draw an altitude of 14 triangles. The
triangles varied on their orientation, their type (isosceles, right, and obtuse), and whether the
altitude to be drawn was going to be inside or outside the triangle. Their results show that the
orientation has almost no effect on students' ability to draw the altitude. However, altitudes that
fell on the side or outside the triangle, and triangles that deviate from isosceles triangle had a
negative impact on students' performance. Vinner and Hershkowitz were able to produce a
statistically significant sequence of increasingly difficult triangles on which to draw an altitude,
as a consequence of their research. From the easiest to the more difficult, the sequence is: (a)
isosceles triangle (non-equilateral) with altitude falling on the side that has different length, (b)
scalene triangle with altitude falling inside the triangle, (c) obtuse triangle with altitude falling
outside the triangle, and (d) right triangle.
These researchers conducted a similar investigation using the diagonals of a polygon [15,
16]. It showed that in the case of concave polygons, only the diagonals inside the polygon, and
that did not contain any side, were drawn.
The models involved in these investigations are distinct from the ones previously described,
because they involve action. The participants were not using image schematic models
exclusively. They were expecting to perform a sequence of actions familiar in a certain context.
This sequence of familiar actions fits exactly the definition of script mentioned previously [1].
An interpretation of Vinner and Hershkowitz's research using scripts may say that the
typical script for drawing the altitude of a triangle occurs in the context of isosceles (non-
equilateral) triangles. Students then seem to attempt to adapt this script to the other cases. When
given an isosceles triangle the student draws the altitude of the triangle so that it falls
perpendicularly on the middle of the side that has different length (74% of the students were
able to do it). When the triangle is quasi-isosceles this script is still maintained, but it breaks
down for many students producing prototype effects when the triangle is considerably non-
symmetric (only 40% of the students answered correctly). In the case of the isosceles triangle
the altitude coincides with the median and with the perpendicular bisector. This is no longer the
case when the triangle does not resemble an isosceles triangle. The original script is changed by
the students into two incompatible scripts. A considerable number of students choose to draw
the median (20%) whereas a smaller number (7%) draws a perpendicular bisector to the side.
The original script breaks down for an even greater number of students in the last two cases](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-122-320.jpg)
![Examinations of Situation-Based Reasoning
and Sense-Making in Students' Interpretations
of Solutions to a Mathematics Story Probleml
Edward A. Silver, Lora J. Shapiro
Learning Research and Development Center, University of Pittsburgh, Pittsburgh, PA 15260, USA
Abstract: This paper discusses a series of five studies examining over 800 students' solutions
to division-with-remainders story problems. The findings from the studies suggest that
students' difficulty in solving these problems is due, in large part, to a failure to engage in
situation-based reasoning and sense-making in interpreting computational results obtained when
solving the problem. Data from recent studies not only has provided direct evidence to support
this hypothesis but also suggested that students' dissociation of sense-making from school
mathematics is a major barrier to their engaging in or displaying the reasoning that leads to
correct solutions.
Keywords: context sensivity, division story problems, remainders, semantic processing,
sense-making, situation-based reasoning
Over the last twenty or thirty years, considerable attention has been paid to understanding the
nature of skilled problem solving, particularly in mathematics. Among the many significant
findings from this research is the importance of semantic processing in the initial stages of
mathematical problem solving [5, 7]. Relatively little attention, however, has been paid to
semantic processing at later stages of problem solving. In this paper, we discuss a series of
empirical research studies that have specifically examined such semantic processing, as it
occurs in children's interpretations of problem solutions.
The research described here has concentrated on children's situation-based reasoning and
sense-making involved in their interpretations and solutions to division-with-remainders story
problems. These problems are particularly rich contexts in which to study situation-based
reasoning and solution interpretations because different problems can be represented with the
same symbolic division expression despite that fact that they have different answers, the
1Some of the research described here was supported by a grant to the Learning Research and Development Center
from the U. S. Department of Education for the Center for the Study of Learning. The opinions expressed here
are those of the authors and do not necessarily represent the views of the center or the department](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-126-320.jpg)
![114
determination of which depends on aspects of the situational context and the quantities involved
in the problem. For example, each of the following problems can be represented by the same
expression, 100+40, but each has a different answer.
Mary has 100 brownies which she will put into containers that each hold
40 brownies.
1) How many containers can she fill?
2) How many containers will she use for all the brownies?
3) After she fill as many containers as she can, how many brownies
will be left over?
Unlike the case for most other story problems encountered by students in elementary school,
sense-making is not an optional activity in solving these problems, since merely performing an
accurate computation cannot ensure arriving at a correct solution.
Divison-with-remainder problems are not only cognitively complex; they are also quite
difficult for middle school students to solve. Children's difficulty in solving them has been
well-documented in a number of national and state assessments. For example, findings from
the Mathematics portion of the National Assessment of Educational Progress (NAEP) [11]
showed that only 24% of a national sample of 13-year-olds was able to correctly solve the
following problem: "An army bus holds 36 soldiers. If 1,128 soldiers are being bused to their
training site, how many buses are needed?"
To better understand the basis for the observed difficulty students have in solving division-
with-remainder problems, a series of studies has been conducted with students in sixth, seventh
and eighth grades. The overall findings suggest that a major factor in students' failure to solve
division story problems with remainders is their failure to engage in semantic processing in the
later stages of problem solving. In particular, students apparently fail to relate their
computational results to the situation described in the problem; i.e., they fail to engage in
situation-based reasoning about the problem and the solution. Each of the studies described
here has contributed to a deeper understanding of the role of sense-making in students' problem
solving, and each is discussed briefly in the remainder of this paper. More detailed reporting of
the procedures and findings cilll be found in the cited papers by Silver and his colleagues.
Early Studies
In the earliest study of students' difficulty with division-with-remainder problems [14], it was
hypothesized that students were failing to attend to relevant information implicitly represented in
the problem situation but not explicitly stated in the story text (e. g., no one is to be left behind;
on one bus there may be some ｾｰｴｹ＠ seats). Several multiple-choice problem variants were
created that made this relevant information more salient, and the performance of about 160](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-127-320.jpg)
![115
middle school students was examined on these variants. Unlike other research on the role of
semantic processing in mathematical problem solving, the focus in this study was on enhancing
students' ability to relate the problem solution and the story context rather than on enhancing
initial mappings between the story text and a mathematical model. The results of this study
demonstrated that students' performance could be significantly improved by making explicit
certain implicit information in the problem or in the required solution.
In subsequent research, Silver and his colleagues [15, 16] continued to use multiple-choice
tasks to examine students' difficulties with division-with-remainders problems. In these
studies, over 500 students' performances were examined on three division problems, similar to
the "brownie problems" presented earlier. The results indicated that students' performance on
each type of problem was improved by the solving of the related division problems. In general,
the results were consistent with the explanation that enhanced performance was due to students'
increased sensitivity and attention to the relevant semantic processing involved in the target
problem solution. In particular, experience with the related problems may have drawn attention
to the need for relating the computational result either to the story text or to the story situation in
order to obtain a final solution to the problem.
Taken together, these results and the NAEP assessment findings suggested that students'
failure to solve the division story problems was due, perhaps in large part, to an incomplete
mapping among referential systems that were relevant to the problem. In particular, it appeared
that students might map successfully from the problem text to a mathematical model (in this
case, probably a division computation), to obtain a computational answer within the domain of
the mathematics model, but fail to return to the problem's story text or to the situation to which
the story text referred in order to determine the correct solution. The lack of semantic
processing after obtaining a computational result was hypothesized as a major factor in the
incorrect solutions generated by many students. Figure I presents a schematic representation of
this hypothesized version of a student's unsuccessful solution attempt for this type of problem.
In the case of a successful solution, it was hypothesized that the student would move from the
story text to the mathematical model, and then after obtaining a computational result, map back
from the mathematical model to the story text or to the story situation in order to decide what
solution would make sense, given the contextual constraints imposed by the problem situation.
In these early studies, the validity of the hypothesized models of correct and incorrect
solution processes was suggested on the basis of indirect evidence available from examinations
of students' performance on multiple-choice items. In order to validate these models more
directly and to extend the examination of children's solutions and interpretations of division-
with-remainder problems, further investigations have been conducted using interviews with
individual students and open-response written task formats with groups of students. Two of
these more recent investigations are discussed in the next section.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-128-320.jpg)
![116
... ---------- ..
Figure 1. Schematic representation of hypothesized unsuccessful solution
Recent Studies
Using a structured individual interview, Smith and Silver [19] examined the problem-solving
performance of 8 middle school students on the following problem:
Students at Greenway Middle School will go by bus to their end-of-year picnic at
Kennywood Park. There will be a total of 1,128 students and adults. Each bus
holds 36 people. How many buses are needed?
Subjects participated in a 15-30 minute individual interview during which time they were given
the problem, asked to read it aloud and requested to think aloud while solving it. Upon
completing their solution, students were asked to critique an alternative numerical solution to
the problem. Students who gave the correct response of 32 buses, were told that some people
think the answer is 31 1/3 and they were asked to evaluate that answer. Alternatively, if
students' who gave an incorrect response (e.g., 31 1/3), were told that some people think the
answer is 32, and they were asked to evaluate that answer.
The interview protocols from this study brought to light some interesting facets of students'
sense-making with respect to the division problem. Furthermore, the interviews revealed that
some students who gave answers that would have been incorrect, if they had chosen them on a
multiple-choice task were able to offer interesting and valid interpretations of the apparently
incorrect numerical answers - interpretations which would have remained invisible in
multiple-choice responses.
The second study sought to expand the findings of the interview study by using a paper-
and-pencil, open-response question with a larger sample of students. Although not providing
the same level of direct access to students' thinking and reasoning about the problem that](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-129-320.jpg)
![117
individual interviews could provide, the open-response question format was viewed as a
desirable alternative to multiple-choice questions for use with a large sample of students. Silver,
Shapiro and Deutsch [18] presented a similar task to a mixed ability sample of 195 students at a
large urban middle school:
The Oearview Little League is going to a Pirate game. There are 5402 people,
including players, coaches and parents. They will travel by bus and each bus holds
40 people. How many buses will they need to get to the game?
Examination of the interview protocols and written responses of the students in both studies
revealed several similar findings and observations with respect to the nature of students'
solution processes and interpretations, to possible factors contributing to students' lack of
success in engaging in situation-based reasoning and semantic processing with respect to
problem solutions and interpretations, and to the validity of the proposed hypothesized models
ofsuccessful and unsuccessful problem solutions.
Solution Processes and Interpretations
Solution Processes. Prior research had not attended to the solution processes used by
students as they solved division-with-remainder problems. Therefore, these studies provided an
opportunity to examine directly which of the many possible solution processes (e.g., drawing
pictures, forming sets, etc.) were used by students to solve the problems. Although many
solution processes could have been used, it was found that all students used an algorithm or
combination of algorithms to solve the problems. Moreover, the vast majority of students used
the long division algorithm, despite the fact that it was more difficult. Approximately one-third
of those using long division made a calculation error. There was, however, a significant
minority who used other algorithmic procedures, such as repeated addition or repeated
subtraction. These students tended to be better able to execute the procedures flawlessly, and
were somewhat more likely to obtain a correct solution
Interpretations. In the interview study, three of the eight students provided unprompted,
spontaneous interpretations of their whole number solutions in terms of the number of buses
needed. This result was similar to that found in the written study, where about one-third of the
students presented appropriate interpretations of their numerical calculations. The
interpretations in this category included expected explanations, such as "a whole number of
buses was needed because you cannot have a fraction of a bus" or "you need an extra bus so
everybody can go". Situation-based interpretations of alternative solutions did appear, however
they were generated by a relatively small number of students. For example, one student in the
2Two other versions of this problem were used with dividends of 532 and 554.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-130-320.jpg)
![120
emphasis on particular calculation procedures or notational form is likely to impede students
from correctly solving problems in which an interpretation of the numerical response is needed.
If issues of mathematical formalism are paramount in the students' attention during problem
solving, then a strong motivation for interpreting the numerical result is less likely to exist.
Moreover, if a student is narrowly focused on issues ofrequired procedural or notational form,
then the student may not even recognize the need to interpret the obtained solution. To engage
in such processing, a student must perceive the need to interpret the numerical solution. The
fact that many students were concerned more with fonn than function may be a reflection of the
imbalance in the middle school curriculum and the predominance of instruction focusing on rote
computational procedures.
Students' tendencies to use memorized procedures rather than sense-making to solve the
problem and their reluctance to share their creative interpretations and alternate solution
processes, probably reflects their views of what is considered appropriate mathematics -
prescribed algorithmic procedures that have little or no connection to real world, non-classroom
considerations. Students' tendencies to dissociate formal, school mathematics from sense-
making have also been noted by Cobb [8] and Schoenfeld [13]. These tendencies may also be
indicative of an unwritten didactical contract [4] between students and their mathematics
teachers - a contract which obliges teachers to provide for students procedures and knowledge
that can be memorized and obliges students to apply these procedures without reference to any
of their other non-classroom experience or knowledge. The existence of such a contract would
explain the apparent dissociation of school mathematics from sense-making for many of the
students sampled and the apparent dissociation of personal invention, creative thinking and
situation-based reasoning from acceptable mathematical activity for many others. Until students
see a richer connection between their situation-based reasoning and the kind of thinking that
occurs in mathematics classrooms, they are likely to be inhibited from being successful in
solving problems that require sense-making.
Hypothesized Solution Models
Based on the above findings, there is considerable direct evidence that students' failure to
interpret their computational results is a major barrier to their obtaining a correct solution. The
data thus provide general support for Silver's hypothesis that students' inabilities to
successfully obtain a solution to the division-with-remainders problem was due to their failure
to engage in semantic processing the later stages of the solution process. Similarly, the
proposed model of problem-solving success was also generally supported by the responses of
students who solved the problem correctly, since their solutions tended to reveal an appropriate
interpretation of the numerical solution.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-133-320.jpg)
![122
January 26, 1991) presented students with division-with-remainder problems using two
different formats for their administration: in one format students critiqued the written work and
solution of a hypothetical student, and in the second setting they acted out the actual ordering
of buses for a school trip. Although limitations in the design of the study hinder interpretation
and generalization of their results, the finding that students exhibited different kinds of
reasoning on the two tasks are nevertheless intriguing. We are currently analyzing students'
responses to alternative paper-and-pencil formats from another study specifically designed to
elicit interpretations by having students consider and evaluate a proposed solution to the "bus"
problem.
Another potentially promising avenue of investigation would involve the extension of the
hypothesized models ofcorrect and incorrect solutions to multi-step problems. These problems
are difficult for children and, like the division-with-remainders problems, a fundamental source
of the difficulty may be the semantic processing requirements imposed by the problems. In
order to complete successfully a chain of computations necessary to produce a successful
solution to a multi-step problem, the solver may need to interpret the result of each calculation
with respect to the ultimate goal of the problem and/or even with respect to the situation
described in the problem. Thus, it is likely that models proposed here can be generalized to
multi-step problem solving.
In addition to continued research related to the general issue of understanding how and
when students connect mathematics to situations, the findings of these studies suggest the need
for research on curricular and instructional changes in mathematics classrooms. Consistent with
calls (e.g., National Council of Teachers of Mathematics, [12]) for increased instructional
attention to mathematical reasoning and problem solving, as well as greater emphasis on
communication and explanations, a number of efforts are underway to create mathematics
programs and classroom environments in which these features are emphasized. For example,
teachers and researchers involved in schools in the QUASAR (Quantitative Understanding:
Amplifying Student Achievement and Reasoning) Project [17] and teachers involved in other
efforts with similar goals (e.g., [3, 6, 9, 10]) are developing and implementing challenging
instructional activities in which sense-making and communication are featured as an integral
part of the curriculum. Findings from these research programs will serve to further our
understanding of the complex relationship between mathematical problem solving, situation-
based reasoning and sense-making - this understanding is critical to our meeting the goals of the
broad-based reforms currently called for in mathematics programs and assessments.
References
1. Baranes, R., Perry, M., & Stigler, J.: Activation of real-world knowledge in the solution of word problems.
Cognition and Instruction 6(4), 287-318 (1989)
2. Boero, P., Ferrari, P., & Ferrero, E.: Division problems: Meanings and procedures in the transition to a
written algorithm. For the Learning of Mathematics, 9(4),17-25 (1989)](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-135-320.jpg)
![126
verbal language very much in simple problems (when a good representation, possibly iconic, of
the problem situation, is often a good starting point to work out an answer) or when
'deterministic' algorithms are available which allow them to perform some calculation with no
need of analyzing the data carefully before planning a strategy. In other words, verbal language
is a powerful tool in order to design strategies in complex problem situations, as its application
seems more general than methods based on iconic representations, which generally do not
work, by themselves, in all the problem situations; some cases of pupils who can easily solve
simple problems but come to a standstill when dealing with more complex ones may be a clue in
this direction. Moreover, the systematic use of verbal language in the analysis of problem
situations and the design of a strategy may prevent pupils from dangerous attitudes and
behaviors such as performing any algorithm whatever without attending to the data and so on.
On this account in our project pupils are forced to use verbal language in problem solving also
by means of suitable problem situations (such as problems without numbers and so on).
As for (iv), it is crucial the transition from a strategy built and performed step by step to a
strategy described in verbal language, with the connections between the successive steps made
explicit (in other words, from a strategy as a tool to a strategy as an object). The discussion and
comparison of strategies has proved an important factor of improvement of problem solving
performance, in particular as regards those pupils who generally meet with difficulties when
dealing with new problem situations.
The skills involved in this transition are also relevant as prerequisites for computer aided
problem-solving. Actually, it has been observed [10] that pupils aged 13 to 14, in their first
approach to informatics, meet with difficulties mainly related with the verbalization processes of
their thinking, the mastery of connectives and specific linguistic structures and the management
of different languages. In particular, the use of connectives to describe processes within
situations logically organized according to constraints not depending upon the pupils and the
teacher (such as the working of a machine) may be an intermediate step to a more formal use of
them in programming.
1.2. Making Hypotheses and Problem-Solving
Many clues suggest that, to plan a strategy as a whole, it is crucial the mastery of some
connectives, in order to explicitly relate distinct statements, and, in particular, the possibility of
making and managing hypotheses. This happens, for example, in the transition from children's
trial and error strategies designed step by step to the mastery of a more effective, algorithm for
division (such as the so-called 'Greenwood' - or 'canadian' algorithm) [5]. In this situation
pupils when trying to plan their strategies are forced to take into account, for each trial, the
possibility of a positive or negative remainder. This happens also when pupils need to change](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-138-320.jpg)
![127
their strategy (because the first one they have tried to plan does not work) or to perform the
same strategy on different data. In brief, there is a great deal of situations in which the mastery
of advanced syntactical forms (in particular, conditional forms) is useful in order to design,
manage, change, perform or compare a resolution procedure. Examples of this kind will be
discussed in 3.2.. This is interesting enough for research, as in the last years we have observed
some correlation between problem-solving skills and the ability at producing conditional forms
in suitable settings. These aspects are discussed in § 4.
1.3. Goals of the Research
The main purposes of the research I am reporting are:
- to state the conditions under which the production of conditional forms is more
frequent;
- to analyse the functions of hypothetical reasoning in problem-solving;
- to analyse the interplay between the mastery of complex syntactical forms (including
conditional forms) and problem-solving skills;
- to discuss the interplay between the production of hypothetical reasoning and the
comprehension of logical connectives, with particular regard to implication (as studied, for
example, by O'Brien at al. [12].
2. The Framework of the Research
2.1. The Educational Context: the Stress on Verbal Language Acquisition
In all the paper, until further notice, I will refer to pupils who have experienced the Genova
Group's Project for primary school. Related to the object of this study, we point out the
following characteristics of the Project (a further information on the educational context in
which the research is carried out can be found in [1, 2, 5, 8]):
-long-term planning of the educational work in all the subject areas (usually with the
same teacher from grade 1 to grade 5);
- familiarity with verbalization processes, due to activities such as reporting in written
form the strategies used to solve a problem as well as a discussion or an experiment performed
in class;
- stress on the construction of linguistic competence to describe procedures or
relationships among facts; in particular, the stress on the mastery of connectives (such as
'before', 'after', 'when', 'while', 'because', 'if...then', ...), focusing on the semantic point of](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-139-320.jpg)
![128
view; for example, this is done, from the age of 6, by means of activities like verbal reporting in
situations that are logically organized according to constraints not depending upon pupils or
teachers (such as the working of a machine);
- acquisition of the algorithms for addition, multiplication, subtraction and division at the
end of processes of progressive schematization and generalization of the heuristic strategies
built by the pupils.
2.2. Methodological Questions: Making Hypotheses Versus Using Conditional
Forms
From the beginning of the research we have dealt with the problem of the interplay between
making hypotheses (as a mental process) and the linguistic forms through which hypotheses are
expressed. It seems that they should not be mechanically connected. The educational context
described in 2.1. (and in particular, the stress on verbalization processes and descriptions of
processes and situations organized according to external constraints) may induce pupils to
report their thought more accurately than in other classes and, from the other hand, may prevent
them from a superficial or semantically unrelevant use of connectives.
As regards the first issue, pupils' written reports are produced so that Ericsson and
Simon's conditions on the reliability of verbal reports [7] are satisfied, since pupils are only
requested to describe their procedure in general and not to perform selective, generative or
inferential processes. Verbalizations are generally performed while pupils are dealing with a
problem or immediately after they have completed the task.
Anyway, we have found pupils who (very likely) make hypotheses but do not report them
when verbalizing their procedures (what does not implies the unreliability of the reports - for
details see § 5.) and others who write down conditional forms with another function than
making hypotheses. For example, there are pupils who write down conditional forms even if,
strictly speaking, they are not related to any hypothesis (see § 3.2.); there are also children (at
the age of 11-13) who can use complex syntactical forms (and, in particular, conditional forms)
when talking freely with some particular adult person, such as their grandmother and so on, but
they cannot use them to describe any relationship or process. It is well known the influence of
'conversational styles' on the production and comprehension of linguistic forms, and in
particular connectives (for example see [13]). These behaviors are much more frequent out of
our project.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-140-320.jpg)
![129
2.3. Experimental Design
Through all the paper I refer to research carried out within the Genova Group's Project, based
on a lot of materials collected in the last years. These materials, produced by pupils from the age
of 8 to 10, are:
- children's verbal traces written down during the resolution process of applied
mathematical problems, with or without numerical data;
- children's written reports of their own procedures for applied mathematical problems,
with or without numerical data;
- children's written descriptions of everyday-life processes (such as the preparation of a
cup of coffee and so on);
- other texts produced in non mathematical situations;
- recordings of interviews (related to how children think they solve problems).
3. Hypothetical Reasoning: Conditions and Functions
3.1. Conditions for the Production of Hypothetical Reasoning
Through all the paper, by 'spontaneous' I mean freely expressed in an educational context, as
sketched in 2.1., which deliberately guide pupils' behaviors to the production of complex
linguistic forms. The spontaneous production of conditional forms in problem-solving seems to
depend upon the following factors:
- the context of the problem: if pupils regard the problem situations as artificial and do not
master the meanings involved, they often produce stereotyped answers (if the problems are
trivial) or come to a standstill (if they are difficult); in situations like these pupils hardly engage
themselves in the endeavor of stating working hypotheses or analyzing data and conditions
which are inherent to the situation. Very often pupils, even if they can find a solution, do not
compare it in a critical way with the situation (see also [3]);
- the nature of the task: for example, hypothetical reasoning hardly can be found when the
task is to solve a problem which needs only recalling and applying a resolution procedure the
pupil has already learn and used in analogous situations, or when the resolution procedure is
constructed by a sequence of steps closely related each other without the arising of alternatives
or difficulties, or when the numerical data distract the pupils from reflecting on the procedure;
- the time of the performance: when the verbalization is performed while a problem is
being solved, a wider presence of hypothetical reasoning and other syntactically complex forms
has been observed; if it happens after a resolution procedure has been found, I have observed a](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-141-320.jpg)
![130
great amount of sequentially-structured texts, with a wide usage of temporal connectives (such
as 'then', 'afterwards' and so on), which are most likely referred to both temporal ordering and
logical consequence.
In other words, conditional forms have been found mainly in arithmetical problems (when
children do not yet know algorithms to solve them) or in complex geometrical problems (with a
crucial presence of figures). No conditional form has been observed in simple problems when
children already know the algorithms to compute the arithmetical opemtions. The availability of
a "deterministic" algorithm often prevents children from producing well organized
verbalizations, with use of conditional or other complex syntactical forms. The need for explicit
reasoning may emerge if the pupil realizes that no algorithm he knows works to get a solution
(and he does not come to a standstill) or if he is comparing and discussing the procedures
produced in class. More generally, the sort of cognitive detachment and the stress on
communication purposes realized when comparing strategies seem to foster the production of
complex syntactical forms.
3.2. Functions of Hypothetical Reasoning in Problem-Solving
The following functions, which are not clear-cut and do not exclude each other, have been
recognized:
(A) individuation of conditions upon which depend, at least partially, the resolution
strategies of a problem; this happens often when pupils deal with problems involving actual
measurements or other 'strong' interactions with physical world; for example, in a problem
involving the measurement of an angle (dmwn on the paper) with a goniometer, the pupils must
deal with the length of the sides of the angle, which may be too short to perform the
measurement immediately: "ifthe sides are too short, I must extend them..."; or when planning
an experiment (related to the measurement of the shadows of a nail in different places) pupils
must deal with a lot of problems (the thickness of the board, the length of the nail, the
horizontality of the board,...) they take into account by means of conditional forms: " If the
board is not as thick as the other, do not drive in it the nail by more than 2 cm ..." ; "If it
happen that the board is lying on the gravel, then level the ground until it becomes ...."; "If
you cannot go on levelling it..., then lay the board .. .". In situations like these pupils seem to
use conditional forms to express hypotheses in order to become progressively aware of the data
and the conditions involved in the problem. For details, see [8].
(B) explorative hypotheses, which are found often in the transition from trial-and-error
strategies designed step by step to more effective algorithms or procedures. For instance, in the
problem of sharing out the total cost of a trip out of the pupils of a class, some of them make
hypotheses as follows:"The total cost of the trip is 74000 lire and there are 19 ofus. If any](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-142-320.jpg)
![131
child pays 1'000 lire it makes 19'000, too little; then let us try with 2'000 lire: it makes
38.000. It is not yet enough. If any child pay 10'000 lire, it makes 190'000, which is too
much, ..." ; so they go on, by trial and error, and come to "little less than 4'000 lire" and
then, specifying "how much one has to take off" they come to "about 3'900 lire").
When building a strategy to compute how many sheets of 21 cm need to be arranged next
to each other on the walls of the room in order to cover the length of the wall completely, good
problem-solvers use hypothetical reasoning to manage the possibility of a positive or negative
remainder: "/ try with 34 sheets 21 cm long; ifit is larger (than the wall), / try with a lower
number, or, if/have spared some space on the wall, / try with a greater number ...tt [8].
Explorative hypotheses are crucial in the planning of a resolution strategy: the pupils can
put themselves in a particular case (stating some particular hypothesis) and try to draw some
conclusion; this may be useful as a step of a process of progressive approximation to the
solution or just to make explicit some relationship involved in the problem situation. In the
example at the beginning of this paragraph the first trial ("Ifany childpays 1'000 lire it makes
19'000") is not only the first step in an approximation process, but also helps the pupil to
grasp the mathematical relationship involved in the problem.
(C) "justification" hypotheses, which are made by some children after they have chosen
and performed a strategy. For example, some children, related to the task of scale
representations of the height of all pupils in the classroom on a sheet on a sheet 42 squares
large, write reports like the following: "The tallest pupil is Roberto, who is 140 cm tall; if/
take 7 cm = 2 squares, Roberto wants 40 squares, all right; this worksfor all the other children
as well,for if/consider another pupil. he is shorter than Roberto, and his height becomes less
than 40 squares". The first 'if' is related to an 'explorative' hypothesis, whereas the second
one has a function at least partially different, since the pupil has already found his answer and
needs no further 'exploration' but only 'logically' justifying what he/she has done. The second
'if' selects a situation which is 'potentially' dangerous for the whole procedure and allows the
pupil to focus on it. This use of conditional forms may be found also in 'working' mathematical
proofs (not in formal deductions!).
(D) Change of data; after the construction of a strategy to solve a given problem,
conditional forms are used to deal with the same problem situation with different initial data:
"we have made 20 biscuits and we have spent...; ifwe make 30 biscuits, we spend...". In this
situations, pupils progressively become aware that they need not to build once again the whole
procedure, but that it is enough to perform the same procedure on different data. They
sometimes remark that it is not necessary "to clear all the blackboard", but it is enough to clear
the old numbers, write down the new data and perform the calculations without modifying the
procedure. This function of conditional forms is closely related to the emergence of the
algorithm as an autonomous object, separated from the data on which it is performed. For
details see [8].](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-143-320.jpg)
![132
(E) 'pleonastic' hypotheses; in some situations, mainly in complex problems or in
problems involving proportion or comparisons, there are children who regard the data as
hypothetical even though they are given under no condition. For example, some pupils write
down reports such as: "iftoday is April 28, from April 28 to may there are 8 days" or even "if
a child must pay half the full fare, I multiply the childfare by the number ofchildren" and so
on. There is some evidence supporting the relevance of complexity: for example, if the task is to
state how much meat a dog eats in a month, knowing that he eats 3 hg every day, no pupil use
pleonastic 'if' to manage the multiplication: "The dog eats 3 hg every day, so in a week he eats
3 x 7 = 21 hg, ..."; if the question is made more complex by introducing further questions
(related to the cost of meat and so on) some of the same pupils write down reports such as "If
the dogs eats..., in a week he...and ifmeat costs..." . This behavior has been found in a lot of
different problem situations. More research is needed to explain all this. The 'selectivity' in the
use of this fonn seems to exclude that this 'if' does not correspond to any mental process of the
pupil or to any hypothesis. One may conjecture that pupils uses pleonastic 'if to insert the data
in a sort of complex elaboration structure which allows them to focus on data and, at the same
time, on some inferential step they want to point out. There is possibly a connection between
this 'pleonastic' use and the use discussed in (D); nevertheless, pleonastic uses of conditional
fonns have been found also before pupils have dealt with problems with modified data.
(F) Comparison of strategies; sometimes, in our classes, children, after they have solved a
problem individually, are requested to compare their strategies each other; in this situation,
fonns of hypothetical reasoning as the following have been observed: "I have done this way...
if I had done as Claudia I should have found:... Then the reasoning ofClaudia needs longer
and more difficult calculations than mine.. .". This the only context in which even poor
problem-solvers can produce conditional fonns spontaneously and frequently.
4. Verbal Reports, Hypothetical Reasoning and Problem Solving
Skills
I have observed a good correlation between problem-solving skills and the ability at producing
conditional fonns when describing their problem-solving strategies [8]. In particular, all pupils
classified as good problem-solvers can use conditional fonns (in problem solving or in other
contexts), even they do not always use them when reporting a resolution procedure.
I remark that by 'good problem-solvers' I mean pupils who are able to give acceptable
solutions to most of the problems (either contextualized or not) they are administered during the
year, not regarding too much the quality of the resolution processes or the reports. By 'poor
problem solvers' I mean pupils who almost never are able to design some strategy to solve
complex problems and often meet with difficulties even when solving simple problems.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-144-320.jpg)
![134
At this regard, a first analysis of pupils' written reports of their strategies, in a lot of
situations including arithmetical and geometrical word problems (at the age of 9-10), has shown
two main patterns of resolution procedures, with regard to the structure of the text. There are
pupils who write down their procedure (while or after solving the problem) as a sequence of
their own operations: "Before I compute, then I do ... and afterwards I find .. .". They hardly
state some relationship concerning the problem situation explicitly and do not use complex
syntactical forms often but organize their reports by means of only temporal connectives (such
as 'before', 'afterwards', and so on). This temporal organization of the text seems to embody
even the 'logical' organization of their procedure. Some of these pupils are very clever at
solving complex problems and often succeed even in changing quickly their strategy when the
first they have tried does not work, and it is likely that they actually make hypotheses when
solving a complex problem even they do not report them.
On the other hand, there are pupils who write down some relation among the 'objects'
involved in the problem situation, and pay less attention to the temporal organization of their
performance. Among 4th graders the first group is larger than the second and in either group
there are good and poor problem solvers. The second style seems more useful in order to
improve performance. Actually, all the poor problem solvers following the first style who
improve their performance during the year change their style, for example introducing some
statement (sometimes as an equation or a diagram, more often in verbal language) to describe
the problem situation or to organize their procedure from the 'logical' - not only temporal - point
of view. On account of this, the mastery of some syntactical construction, which allow children
to state different kinds of relationship among facts, such as, for example, causal relationship,
has proved to be a factor of improvement in problem-solving performance for either group.
In geometrical problems we have found a wide production of conditional forms.
Geometrical figures have emerged as crucial in order to mediate between the problem situation
and the need to plan a strategy. One figure may embody a complex procedure (such as the
iterative construction, square by square, of a polygon satisfying some given properties) or an
'action-proof' (such as the equivalence of 2 polygons). This function of spatial representations
may allow even some poor problem-solvers to make some hypothesis even though they do not
yet master the corresponding linguistic forms (for details, see [6]).
6. Hypothetical Reasoning and the Learning of Logical Connectives
A thorough discussion about the learning oflogical connectives is perhaps out of the scope of
this volume. A lot of studies has shown the difficulty of pupils of almost any age in
understanding logical connectives, in particular implication (for example, [12]). These](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-146-320.jpg)
![135
difficulties might be justified by the tension between the everyday-life meaning of connectives
(in natura1language environment) and the 'logical' one, which are quite different. I want only to
sketch one example in which the research I have reported and the research on the learning of
logical connectives have different outcomes. Some results of Markovits [11] show that the use
of drawings does not improve at all children's understanding of implication (in a 'logical'
sense). On the other hand, in our project a significant number of pupils, at the age of 10, can
manage conditional forms in order to organize a resolution procedure in a large set of problem
situations, and the presence of drawings seems to improve their ability in making hypotheses,
as happens in geometrical problems. This may be justified by the fact that from a (classical)
logical point of view the semantics of an implication 'p->q' is a purely combinatorial function of
the truth-values of the propositions p and q, whereas pupils seem to use spontaneously
conditional forms with a much richer meaning, including temporal and spatial meaning. This
meaning is partially conveyed by everyday-life verbal language, whose semantics are quite
different from the semantics of the formal languages of mathematical logic. This may explain
the 'curious' findings of Markovits, as far as most likely the use of 'statical' drawings may
focus on the combinatorial, truth-functional meaning of implication, whereas verbal language
alone may anyhow preserve some other kind of meaning more understandable by pupils.
7. Further Remarks
From the data I have reported in this paper seem to emerge the following ideas, which are
relevant from the didactical point of view:
- processes of 'progressive schematization' from informal children's strategies to more
effective algorithms, as described, for example, in [5] and [14], are very useful not only in
order to achieve the conscious mastery of algorithms but also as a ground for the construction
of crucial skills concerning the design and management ofprocedures and reasonings as well as
programming;
- activities inducing pupils to reflect upon their own actions and thoughts, by means of
comparison in the classroom or other, are very useful as far as they allow pupils not only to
perform their procedures and reasonings but to regard them as autonomous products; anyway,
the possibility ofreflecting should be deliberately built, through a long term educational work;
- the ability at 'spontaneously' producing procedures and reasonings seems to depend
more upon the mastery of the specific meanings of the problem situation than upon its
complexity itself; the awareness of the specific meanings is very important as regards the
mastery oflinguistic connectives; for more details on the relevance of 'semantic fields' see [4];
- children's procedures and reasonings are strongly related to time; most pupils represent
them as processes with a relevant temporal dimension; only few pupils can represent problem](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-147-320.jpg)
![136
situations by means of 'static' relationships without a temporal dimension; on the other hand,
the mastery of time as a basis for the representation of processes seems to be related to the
mastery of time as an explicit variable of the problem; for more details on the function of time
see [9].
References
1. Boero, P.: Acquisition of meanings and evolution of strategies in problem solving from the age of 7 to the
age of 11 in a curricular environment. In: Proceedings of the 12th Conference of the International Group
for the Psychology of Mathematics Education, vo1.l, 177-184, 1988.
2. Boero, P.: Mathematical literacy for all: experiences and problems. In: Proceedings of the 13th Conference
of the International Group for the Psychology of Mathematics Education, vol.1, 62-76, 1989.
3. Boero, P.: On long term development of some general skills in problem solving: a longitudinal
comparative study. In: Proceedings of the 14th Conference of the International Group for the
Psychology of Mathematics Education, vo1.2, 169-176, 1990.
4. Boero, P.: The crucial role of semantic fields in the development of problem-solving skills in the school
environment. (in this volume), 1991.
5. Boero, P., Ferrari, P. L., & Ferrero, E.: Division Problems: Meanings and procedures in the transition to a
written algorithm. For the Learning of Mathematics 9(3), 17-25 (1989).
6. Bondesan, M. G, & Ferrari, P. L.: The active comparison of strategies in problem-solving: an exploratory
study. In: Proceedings of the 15th Conference of the International Group for the Psychology of
Mathematics Education 1991.
7. Ericsson, K. A., & Simon, H. A.: Verbal reports as data. Psychological Review,.87, 215-251 (1980).
8. Ferrari, P. L.: Hypothetical reasoning in the resolution of applied mathematical problems at the ages of 8-10.
In: Proceedings of the 13th Conference of the International Group for the Psychology of Mathematics
Education, VoU, 260-267, 1989.
9. Ferrari, P.L.: Time and hypothetical reasoning in problem solving. In: Proceedings of the 14th Conference
of the International Group for the Psychology of Mathematics Education, vo1.2, 185-192, 1990.
10. Lemut, E. & Ferrero, E.: On the linguistic prerequisites of computing literacy. In: Proceedings of the
IFIP Technical Committee, 3.139-144, North Holland 1988.
11. MarkovilS, H.: The curious effect of using drawings in conditional reasoning problems. Educational
studies in mathematics,17, 81-87 (1986)
12. O'Brien, T.C., Shapiro, BJ. & Reali, N.C.: Logical thinking - language and context. Educational studies
in mathematics, 4.201-219 (1971)
13. Rumain, B., Connell,I. & Braine, D.S.: Conversational comprehension processes are responsible for ...•
Developmental Psychology, 19 (4),471-481 (1983).
14. Treffers, A.: Integrated Column Arithmetic According to Progressive Schematization. Educational Studies
in Mathematics, 18, 125-145 (1987)](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-148-320.jpg)
![138
competencies, so that they actually become more effective perfonners on future occasions.
Learning has taken place when the student possesses, after the activity, some ability,
competence, knowledge or skill which he did not have before. Thus, this paper is addressing,
first, the question what is to be learnt in mathematics lessons, and secondly, how it may be
most effectively learnt.
Applied and Pure Mathematical Processes
Mathematics has two aspects, roughly fitting the traditional labels applied and pure. First, it is a
means of gaining insight into some aspect of the environment. For example, the exponential or
compound growth function gives us insight into the way in which a population with a given
growth rate grows over time - first slowly and then with an increasingly rapid rate of increase.
Some ofthese propenies are encapsulated in well known puzzles - such as that of the water lily,
doubling its area each day, where will it be the day before it covers the whole pond? - or in the
frequent exhortations of our financial salesman to consider how a modest investment might
grow. Another related example is that of the decrease of the rate of inflation - which many
people believe means that prices are coming down. In the home environment, a little
knowledge of the symmetry group of the rectangular block will tell us when we have turned the
mattress on the bed as many ways round as we can; and a modest knowledge of probability and
statistics will help us to interpret advertising claims about what toothpaste seven out of ten
movies stars use, and not to be excessively hopeful that our next child will be a boy if we have
already produced three girls. These are all 'useful' aspects of mathematics - and note, by the
way, that they all depend on the application of conceptual awareness, not on any technical skill;
they are useful in the same way as is the knowledge gained in most of the subjects of the
curriculum - history, geography, literature, science - that is, deriving from knowledge of
some key facts and explanatory concepts.
The second aspect of mathematics is somewhat less loudly commended in public
nowadays. It is the pure mathematical aspect which it shares with art and music, the solution
and construction of puzzles and problems, and the enjoyment of recognizing and making
patterns. Mathematical problems in newspapers and magazines still attracts a following, and we
might speculate that the capacity to appreciate mathematics as an art to enjoy is initially present
in most people, though it often gets suppressed by distasteful school experiences.
These two modes of interaction of people with mathematics, representing the applied and
the pure mathematical approaches, have been identifiable throughout history as the mainsprings
of mathematical activity.
Freudenthal [5] also distinguishes applied and pure mathematical processes:](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-150-320.jpg)
![139
"Arithmetic and geometry have sprung from mathematising part of reality. But soon, at
least from the Greek antiquity onwards, mathematics itself has become the object of
mathematising... What humans have to learn is not mathematics as a closed system, but
rather as an activity, the process of mathematising reality and if possible even that of
mathematising mathematics."
More briefly,
"Mathematics concerns the properties of the operations by which the individual orders,
organizes and controls his environment." [6, p. 157]
Components of Mathematical Competence
The capacity to do mathematics involves several different kinds of acquisitions. We need to
distinguish (a) between skills, conceptual structures and general strategies; (b) between symbols
and meanings, and (c) between mathematical concepts or structures and the contexts which
embody them; with the notion of transfer of a concept, learned in one context, to its application
in another.
Skills include all the common computational algorithms - for the 4 rules, for moving the
decimal point, geometrical constructions, equation-solving rules. They are routines consisting
of a number of steps and knowledge of them implies the ability to move from one step to the
next fluently.
Conceptual structures are interconnected sets ofrelationships. The concept of place value
is such a structure; it includes the fact that the figure after the decimal point represents a tenth,
that moving the point to the right multiplies the number by powers of ten; it governs the rules
for carrying, and the rules for putting numbers in order of size.
Strategies (including both strategic skills and strategic concepts) are plans which guide a
person's choice of what skills to use or what knowledge to draw upon at each stage in an
activity of problem solving or investigation. Identifying data and conclusion, trying easier
numbers, proving, are examples of general strategies. Fuller treatment of the learning of
strategies appears in Bell [I], and of all these topics in Bell, Costello and Kuchemann [2].
These acquisitions can be classified on three dimensions: first, as schemes governing
actions or knowledge (skills, concepts); secondly, according to the type of connectivity they
possess, skills being mainly linear, though with branches, concepts being more like
interconnected networks; thirdly, according to the degree of generality, the main body of
mathematical concepts and skills being at a lower level than strategies, which govern the way in
which the mathematical concepts and skills are deployed.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-151-320.jpg)
![140
Symbols and Meanings
The relation between symbol-manipulations and their meaning in terms of the concepts denoted
is at the heart of mathematics. It is probably also responsible for much of the difficulty of
teaching mathematics, since the symbol-manipulations are the most visible part of the activity,
and it seems natural to teach them directly without realising that comprehension of the
underlying conceptual transformations is important for effective permanent learning. A
demonstration of the distinctiveness of these two aspects and also of the need to ask carefully
"how does the pupil view the question", is given incidentally in an experiment on the learning
of fractions by 9-10 year olds [7]. In one of their tests the following items appeared.
and:
Look at the square in the top picture.
Four of the ten equal parts are
shaded. Now look at the bottom
picture. This square must have the
same amount shaded.
How many of the five equal parts
should be shaded so that the sam&
amount will be shaded in both squares?
The squares are unit squares.
/1/1/1/1/1/1
00000
00000
00000
If there are six triangles for'every
fifteen circles, how many triangles
would there be for five circle.?
00000
The later, corresponding items are:
4 • .Q and
10 5 15 5
Find the number that goes in
the box.
It seems likely that at least by some pupils the symbolic questions were treated as requests to
manipulate the fraction symbol according to the rule "divide the top and bottom by the same
number". It is also possible that in the diagrammatic case of the fraction item the answer was
obtained by filling up the total area in the second square equal to that shaded in the first. If this
is so, in neither case was the concept of a fraction necessarily involved in the thinking. The low
correlation (about 0.37) between pupils' results on the two forms of test confirm that to the
pupils the two types of problem are substantially different.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-152-320.jpg)
![141
The ability to be able to use symbolisms effectively, working within them when
appropriate, but also to be able to re-establish the links with the underlying concepts when that
is necessary, needs extensive experience in making the translations both ways.
Structure and Context
The most characteristic typical mathematical activity consists of the recognition of a relational
structure in a context, and the use of known properties of that structure to effect a
transformation which reveals some previously hidden aspect of the situation. For example,
consider the problem of finding the number of litres of petrol equivalent to the 5.5 gallons held
by the tank of a Mini, given that a litre is 0.22 gallons. One has to recognise the operation of
division implied by the situation, and then use the relevant algorithm to reveal the answer, 25
litres. This is a quite typical, if elementary, fragment of mathematical activity. Another
problem might require the original size of a poster, reduced by a scale factor of 0.22 to give a
magazine picture 5.5 cm high. Again the operation required is division, but the type of
structure is no longer that of quotition, nor of repeated subtraction but rather as the inverse of
multiplying by a scale factor. The frrst problem might in fact be solved by a pupil by adding
sufficient 0.22s together to give 5.5; if the conversion factor had been 2.2 instead of 0.22 this
would be even more likely. The second is most unlikely to be solved by any such process.
Thus the aspects of an operational problem, as perceived by the pupil, depend on structural
properties of the context and on the particular numbers.
For further discussion of these factors, in particular the effect of familiarity of context and
the transferability of structural knowledge across contexts see Bell, Costello and Kiichemann
[4, Ch. 2].
To secure the leaming of a articular skill, strategy or concept it is first necessary to present
the leamer with tasks which bring them into play. There is thus a need in the curriculum both
for broader tasks, such as extended investigations or practical problems or open situations,
which bring general strategies into playas well as a possibly indetenninate range of particular
mathematical concepts; and also more sharply focused tasks which require the use of a
particular concept, such as place value, or rotational symmetry. In fact, most tasks call into
play both general strategies and particular knowledge. What is learned from them depends on
what is focused upon, what is attended to. It is at this point that the teacher's intervention is of
crucial importance, in directing the students' attention to those aspects of the situation which
they are intended to learn, making them aware of what they've done and how they have done it.
What we are asserting here is that the two basic mechanisms of learning are (a) insight, and (b)
repetition.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-153-320.jpg)
![142
Each new insight adds a further link to the conceptual structure; repetition (with appropriate
variation) forges the link more firmly. Repetition develops fluency of response, but as
Brownell [3, 4] showed in his research on drill in number facts, without attention to increasing
insights, its effects fade quickly.
New insights may be consonant with existing cognitive structure, or may conflict with it.
In the former case, they are assimilated; in the latter, they may be rejected, or ignored, or
deflected into something different which does conform - until the disequilibrium or cognitive
coriflict is intolerable, and reorganization and renewal of the cognitive structure takes place, in
which the new ideas and the old combine in a new synthesis. Piaget described this process as
that by which an organism learns through interacting with its environment. We have used this
as the basis of a teaching methodology which uses coriflict-discussion - the provocation of
cognitive conflict focusing on known or likely points of misconception, followed by resolution
through group or class discussion. A number of particular types of task have proved valuable
for this purpose. We shall describe some of them, and report the results of experimental
teaching using this methodology, which has become known as Diagnostic Teaching.
Diagnostic Teaching
This project was (and is) concerned in developing methods of providing cognitive conflict and
discussion focussed on key conceptual obstacles revealed by research into pupils'
understanding of several important topics.
As the title implies, this method begins by conducting conversations, interviews and tests
with students, looking carefully at their mistakes and using these to help to understand the ways
of thinking that lie behind them. Next comes devising and trying out teaching ideas and
methods aimed at overcoming particular misconceptions, and testing sometime afterwards to see
how successful they are.
Work has covered several known problem areas of the curriculum - directed numbers,
decimal place-value, choice of operation with decimal numbers, algebra - but subsequently
colleagues have been applying the same principles and method to a variety of other curriculum
topics including graphical interpretation, shape and place-value with primary children,
probability with sixth-formers. Other outcomes of the work have included some new insights
into students' understanding in each of these fields. Here we want to discuss the kinds of
classroom-task which have emerged as particularly helpful in generating the conflict and
discussion needed to promote learning. Making up questions, 'marking homework' (using a
real piece of homework or a specially constructed one), filling in tables, and games have all
proved useful, and ways of combining group and whole-class discussion have been explored.
But before describing these tasks, we shall indicate the conceptual field in which we were
working and the particular misconceptions we found.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-154-320.jpg)
![Pre-Algebraic Problem Solving
Ferdinando Arzarello
Dipartimento di Matematica dell'Universita di Torino, Via Carlo Alberto 10, 10123 TORINO, Italia
Abstract: Typical solution processes of (pre-)algebraic problems live dialectically between two
opposite polarities: procedural and relational. The fonner is a-symmetric; is ruled by "the logic
of when"; is close to the meaning of symbols; its main epistemological style is arithmetic. The
latter is symmetric; is ruled by "the logic of iff'; is syntactic, insofar concrete meaning have
evaporated; its epistemological style is anti-arithmetic. But procedural thinking allows pupils to
do concrete experiments and get feedbacks from the problematic-situation; and this, in the long
run, is useful in order to jump to relational polarity, where more formal algebraic manipulations
can be done.
Keywords: pre-algebra problem solving, procedural polarity, relational polarity, arithmetic,
algebra, epistemological obstacles, validation, a-validation, algebra teaching
Introduction
This paper is devoted to discuss last two years' researches of the author in pre-algebraic
problem solving; a precise definition of what pre-algebraic means will be given later. For the
moment. good examples of pre-algebraic problems are those in Appendix 2 (except problem 0),
where pupils are requested to discover some rule from numerical data, which they have
elaborated themselves (see also [7,14]).
Solution's processes of pre-algebraic problems in pupils from 10 to 16 years old (and
over), presuppose and include all the aspects of the solution's processes of arithmetic
problems, such as have been described in Arzarello [I, 2]. Particularly, the notion of
conceptual model, is basic to understand and describe pupils' perfonnances. (A brief sketch of
this notion is given in Appendix 1).
But even if the genns of pre-algebraic thinking are already present in the processing of
many arithmetic problems (see [14]), things are obviously more complicated in pre-algebra than
in simpler standard arithmetic problems. Two main novelties enter, namely:
a) Arithmetic is an epistemological obstacle for constructing a pre-algebraic knowledge (for
the notion of obstacle, see [3]): conceptual models, which pupils have elaborated in previous,
purely arithmetic activities, become too rigid in the new context and block those typical ways of
thinking, which have an algebraic flavour, and allow them to attack the problem properly. To](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-167-320.jpg)
![156
be more precise, in pre-algebraic problem solving, pupils are requested to use more than one
model at once, and also to use them with an epistemological style, which is the exact opposite
of the arithmetic one. So they may lose the control and can no longer integrate their models in a
global strategy. For example, in problem 2, after the "local" law has been found relatively
easily (namely, something like Pn+1 = Pn + n+1), even those pupils who already know the
relation for the sum of first n integers, do not see it in the foregoing formula: numerical data
which they have elaborated (and usually arranged in a table) and the formula are not put
together. The very arithmetic epistemology inhibits students from looking at numbers and
variables in the right manner, that is Arithmetics inhibits algebraic thinking.
b) Integration of this obstacle in a new knowledge, when and if it happens, modifies the
main features of validation (for a discussion of this notion, see [4, 6, 11]): a more "abstract",
less "tangible" and new form of validation appears, which will be called a-validation or weak
validation. This change is a very delicate point: it represents an obstacle, which involves mainly
metacognition, and may inhibit pupils from solving pre-algebraic problems.
For example, in problem 3b, whose algebraic content is mixed with geometry (i.e., number
geometric configurations), 15-16 years old pupils find very difficult to manipulate the formula
they have got for Tn [namely n(n+l)/2] to show that 8Tn +1 is a square (a formal job which
they find easy, if explicitly requested to do it as a manipulation of formulas): some do not do
anything at all; others use a great ingenuity to prove the claim by means of non-algebraic
methods (Le., using geometrical models: see Figure 1).
• II II II • • •
• • II II • • II
• • • II • II II
• • • + • • •
II II • II • • •
II • • II II • •
• • • II II II •
Figure 1](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-168-320.jpg)
![157
The aim of the paper is to give empirical evidence to points a) and b), illustrating empirical
research made with 10 years old pupils to university students, while solving (pre-)algebraic
problems.
Obstacles
It is very well known that many pupils, from middle school up to the university, solve algebraic
problems more in a syncopated style than in a symbolic one (see 7, 8]). 16 years old pupils do
not yet use the algebraic code spontaneously while solving simple algebraic questions (e.g., see
the very well known problem 0 in Appendix 2), even if this may cause them troubles and long
detours (I have verified awful not-algebraic detours also with my undergraduate mathematics
students, while solving problems 3 and 4). Their preferred conceptual models are still
arithmetic ones and this may cause conflicts with the algebraic way of thinking. Moreover, the
only way, which seems to allow them to master semantically the situation is mainly when they
can represent their model using a syncopated language, which allows them to manage things
avoiding conflicts between the situation and the model they are using. The point is that algebra
is understood only in abstract, as a formal and general method, but it is not concretely used as a
method of justification and generalization in specific situations; namely, generalization as an
effective and operative method is used with great difficulty by pupils while solving algebraic
problems. Even if they seem to manage formalism, they do not use it spontaneously in concrete
problematic situations and seem to live more comfortably with its substitutes, namely those
arithmetic conceptual models, where the meaning of involved things is expressed more directly
by the syncopated language of arithmetic: see the sketchy discussion on problem 2. For another
example, see problem 4. Even if it may seem very difficult at a first glance, the problem can be
managed relatively easily by 15-16 years old pupils up to a certain level. In fact pupils can
grasp the rule for So describing it in a syncopated arithmetic way; so they can postpone
algebraic reasoning till the end (when most of them get lost), which allows them to control
semantically the situation for a long time. On the opposite, their teachers have attacked the
problem immediately with algebraic means and some have got serious troubles; for most of
them only a detour to arithmetic syncopated language and to a procedural style (see later, for
this notion) has settled the question.
Students generally learn at school how to solve specific problems with algebra (e.g., using
unknowns), but meet major difficulties when required to use it as a symbolism to express
general solutions, to discover, generalize and prove laws behind numerical relations (as in
Appendix 2 problems). The main difficulty consists in integrating formal (algebraic) algorithms
with arithmetic models. Also here one is faced with the phenomenon of opposition between
model and situation, so that to overcome difficulties, one must adapt the model to the different](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-169-320.jpg)
![158
situation ｾ､＠ get a new model, possibly unstable (see the discussion in Appendix 1). But this
phenomenon has a different conceptual character. The opposition is not between the problem
and the model, which is not adequate; the point now is that conceptual arithmetic models are
adequate to attack the situation, but only from an arithmetic point of view: the opposition now is
between the very way in which one has the habit of using the model, and the necessity of
looking at it in a different, more abstract fashion. Conceptual models must be transformed, but
a partial modification is not enough: they must be inserted into a more general framework. We
have seen that in problem 2 many students do not see things, even if they are under their eyes,
and this happens in all problems we have tested. The difficulty is to look at arithmetic things
with a new (algebraic) eye; so to say, pupils must learn to speak after a new code.
The main novelties of this operation have been pointed out by C. Laborde in her thesis [9J.
More precisely, they represent the main functions of the algebraic code, namely
individualization and linkage. The transformation of conceptual models required to face
(pre)algebraic problems are mainly these. But this kind of operation costs very much to pupils,
who like better to use more stable models. even if inadequate (in this sense, the solutions of
problem 3 are typical), namely to substitute such algebraic functions with extralinguistic
properties. The language of pupils faced with problems like those in Appendix 2 is plenty of
linkages to different aspects of the real space-time situation in which they act, while producing
their own solution of the problem. In their protocols it is easy to find a melting pot, whose
ingredients are both mathematical and extrarnathematical, extralinguistic, and so on. Typically,
mathematical objects are referred to by means of subject's actions (e.g., processes of calculus
made by the subject him/herself, or by somebody else); algebraic laws are put into the flowing
stream of time. All this makes it easier for pupils to remember the meaning of formal things
they are speaking of, and to control semantically the situation. This make it possible for them to
use mainly stable, arithmetic conceptual models, even if their use has only a local character
(i.e., they allow the construction of recurrence laws, not explicit global formulas) and costs a
lot in term of memory; this style decreases very slowly in the course of years: I have found a
relatively low difference between 10 years old children, who have solved problem 1 and my
undergraduate mathematics students, when engaged in solving problems 2,3, and 4.
Pupils (but also mathematicians' speeches) express the meaning of mathematical objects,
relating them in some way to: subject's actions, the very processes of their constructions and
generations, every other extrarnathematical information about them. This is a major root of the
obstacles to symbolization and to purely syntactic manipulation of symbols (particularly when
condensed in literal blocks: see [12]); in fact, this causes an evaporation of extra-mathematical
data and, consequently, a possible dramatic loss of meaning. Evaporation is one of the main
features needed in order to develop (pre-)algebraic models, from arithmetic ones. Without
evaporation, no one can get the general (global) formulas for solving problems 2, 3, 4; but this
may cause big conflicts with arithmetic models, which have allowed pupils to construct and](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-170-320.jpg)
![159
grasp local formulas for the same problem. For example, in problem 2 (discussed above), to
see the formula which expresses the sum of fIrst n integers in the local law of generation, pupils
must forget the meaning of things and to look in another way to the very formula, they have
written.
The Double Polarity
Every solution of a pre-algebraic problem, like those in Appendix 2 lives dialectically in a
double polarity (see [13], for a theoretical framework of this kind of ideas). From one side,
there is the subject, who solves the problem, with his actions in the flowing of time: the
algebraic code is interpreted essentially in a procedural manner.. So arithmetic models can be
easily adapted and integrated to this way of thinking. From the other side, the algebraic code
must be interpreted in an absolute way, independently from the actions of anybody: it is a
contemplation of relations and laws sub specie aeternitatis; neither procedures nor products of
actions are involved: in fact only the abstract-relational aspect remains and its privileged code is
symbolic. For example, 10 years old pupils who attack problem I, fIrst mimic concretely
messenger's and prince's trips, and score in some way the exact time when they meet each
other; after the third meeting, concrete manipulations become too diffIcult, so they try to foresee
"the rule" and check it (procedural polarity, using both additive and multiplicative arithmetic
models). When requested to justify the general rule, they have found (essentially, some formula
which involves powers of 3, but expressed in multiplicative language), things are viewed from
a relational point of view, and their reasoning can be summed up as follows (I use letters, like
d, S, etc. for conciseness' sake; children like better other detours, e.g. the use of concrete
numbers as variables): "suppose the messenger and the prince separate after d days, say in
point S; it takes d days more to the messenger for reaching the castle and coming back to S; in
the meanwhile the prince arrives at a point I; in the end, exactly in d days the messenger reaches
him at a point E, symmetric of S with respect to I". The transition from procedural to relational
polarity is even more marked for strategies of solutions in problems 2,3,4 (protocols from 14
to 16 years old students show a big jump: only the best reach relational polarity without any
help).
The main features of the double polarity with respect to the birth and development of
algebraic thinking can be sketched as follows.
Procedural polarity is a-symmetric, has a privileged direction, is controlled by means of
tense adverbs and prepositions; its logic is the "logic of when". Relational polarity is ruled by
logical and equivalence laws, which are typically symmetric: its logic is the "logic of if and only
if'. The former allows a strict semantic control: extra-linguistic facts link steadily the speech to
subject's actions in the flowing stream of time. The latter, on the contrary, is typically syntactic:](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-171-320.jpg)
![161
extra-mathematical tracks, that is eliminating this very polarity (evaporation), which is the
semantic base of syncopated algebra. Sometimes, a main product of this process is what I call
condensation, a typical phenomenon of algebraic thinking. It means that using the symbolic
code, one can write concisely and expressively the amount of information of a term, whose
complexity cannot be easily ruled with the syncopated language of arithmetic: typical products
of condensation are global general formulas. Condensation is not the result of a process as
much, but rather a general attitude with respect to the very process (what I call a procedure).
Condensation marks deeply the passing from a procedural moment to a more abstract and
relational one; it appears at once in the strategies of solutions of pupils and, as all processes
which happen on the spot, is very difficult to analyze. For example, in all examined cases, most
pupils who worked successfully at our problems, could solve them passing from procedural to
relational polarity, with a sudden change of their style. However, even when explicitly asked,
they were not able to explain what had happened. Only in problem 3 (and in similar ones),
some pupils who had got the "formula" arguing in not algebraic terms (for example, using
geometric patterns like those ofFigure 1) were able to explain it with a lot of details.
Some Final Remarks
To conclude, some general remarks and some sentences which yet need to be supported by
empirical evidence.
The discovery-construction of an algebraic rule is not a trivial process of generalization
from particular to general, but it is stirred by the strained connections between the two
polarities. Typically, the dialectic between the two polarities marks the birth of algebraic work.
Relations between the aims of algebra to answer a question and the very algebraic work
have been identified (see [11]) as the roots of validation (of the product) and control (of the
procedure). In fact, aims live in a procedural dimension, while algebraic work develops in a
relational one: control and validation sometimes involve both, sometimes there is a form of
weak validation, where only the second polarity seems to be involved. In this case a real
validation is very problematic; one can get it only at a very abstract level: I suggest for this sort
of weak validation the name of a-validation. For example, this type of validation has been
observed, even if in an unstable form, in all pupils who have solved algebraically problems 2,
3: in all these cases a-validation has been accompanied by condensation.
At this point, it is possible to give the promised definition ofpre-algebra: it is every activity
which involves both polarities: the first in an effective way, the second in a more problematic
way (and it is so not only because arguments are so, but also because of the "didactic
contract"). In other words, in pre-algebra the procedural aspect is still prevailing, but it is not a
quiet way of life: the tension with relational aspects is smouldering under the ashes.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-173-320.jpg)
![162
Last, but not least, some comments from the point of view of mathematics teaching.
Previous observations on algebraic thinking make sense as far as algebra is not taught as a
set ofpurely mechanical and formal manipulative rules, but is used as a tool to solve problems,
to justify pieces of reasoning, etc. The given problem must be an intellectual challenge for the
pupils, which motivates and sustains all their work; consequently, it may contain the roots for
transforming knowledge, provided one keeps an eye on such long-term aspects of teaching
mathematics as the habit of contrasting and discussing in the classroom (formal mathematics
can grow only on the base of a mathematics of spoken and written words). A typical example
of this teaching style is the devolution of problematic situations to pupils (see [11]). In
devolution (of a stimulating problem) are the roots of the continuous transformations of the
problem - necessary to solve it - as well as of the dynamic process from problem posing to
problem solving and vice-versa; devolution marks also the building of fictitious worlds, where
pupils can do experiments and have feedbacks, which allow them a fruitful interplay between
the problem and their own conceptual models.
As well as the mastery of an algorithm may contract reasoning, so the use of a syncopated
language makes it possible to elaborate and integrate one's knowledge into new conceptual
models, where new algorithms possibly have their own roots. The natural consequences of this
point for mathematics teaching are that, in order to base a good construction of algebraic
knowledge, it is better to postpone formalization, to anticipate pre-algebraic problems, and to
stimulate in pupils the developing of numerical experiments.
Appendix 1 - Arithmetic Conceptual Models
A summary of the notion of conceptual model is sketched, in order to make clear some
references given in the paper.
Following Lesh [10], a conceptual model is conceived as a system which integrates the
organized mathematical knowledge (both concepts and processes) with the activity of problem
solving in concrete problematic situations.
Conceptual models are used by children from the very beginning in their activities of
problem solving. Here is an example. To solve the following additive problem:
(A) "Yesterday at noon there were 28 degrees (Celsius); today there are 4 more.
How many degrees?",
many children use a counting on model (see [5]). That is, they count 29,30,31,32 (perhaps
using 1,2, 3,4 fingers while counting).
Sometimes the problem can be solved directly by very simple, immediate and interiorized
conceptual models (e.g., problem A with counting on). In such cases the models are very close](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-174-320.jpg)
![163
to the intuitive and direct meaning of the words in the problem (in the example, 4 more means
+4). But sometimes this is not the case and the model at disposal of the pupil may not be
adequate to represent and solve the new problem; so it must be transformed and translated into a
fresh one, possibly at a more fonnallevel.
Examples:
(B) "Today at noon there are 32 degrees; yesterday there were 28 degrees. How
much more today than yesterday7";
(C) "Today at noon there are 32 degrees, 4 more than yesterday. How many
degrees yesterday at noon7";
(D) "Six months ago there were 32 degrees (Fahrenheit). Today there are 54
degrees more. How many degrees7".
In such cases the model counting on does not work directly. It must be transformed and
adapted to the new situation (cases B, C) or it must be translated into a more formal and
sophisticated model (case D). The major point now is that the text and the context of the
problem may help or oppose such transformations, which, consequently may be performed in
conformity, neutrality, or opposition with respect to them.
For example, using the classification discussed in Carpenter [5], typical change or compare
problems like a + b =7 can be solved using basic models, which are in conformity with the
(standard) text of the problem. Confonnity means that the problematic situation, the text and the
conceptual model are isomorphic copies ofeach other (so these are easy problems).
In other cases (for example, in change or compare problems like a ±7 = c), basic models
must be deeply reorganized in order to solve the problem: the produced model is not any longer
a copy either of the problematic situation or of the text. It is already a reflected representation of
the problematic situation, drawn out from the text, which from its own appears opaque. I call
this situation of neutrality. Of course, models got in such a situation are less stable than those
got in a situation of confonnity.
Things become more difficult with change or compare problems like 7 ±b =c. In such
cases pupils do not have at their disposal other concrete models, which can fit to the new
problems. So they are forced to adjust their models to them, but unfortunately the text and the
context of such problems make things difficult. In fact, they must make their changes in a
situation of opposition with respect to the text of the problem. This makes such models very
unstable and explains the difficulty of these problems: first, they are got as the result of a long
chain of conceptual transformations; second, they are not good models, because of the
opposition with the text. To manipulate properly such models, pupils must work at a more
formal level than in previous cases, where conformity or neutrality makes it possible to mimic
reality in a more concrete fashion.
Conceptual models integrate concepts and processes in the organized network of
mathematical knowledge. For each field of problems (in the sense of Vergnaud), the relative](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-175-320.jpg)
![164
network is generated starting from some basic models, adapting them to the context and text of
specific problematic situations, by means of translations and transformations.
A conceptual model can be mastered by a pupil with more or less rigidity or instability: the
former, if (s)he has low flexibility in reorganizing her/his models to new situations; the latter,
when the various models generated by basic ones are poorly connected each other.
It is clear that conceptual models may be studied from two points of view, at least:
a) in connection with pupils' organized mathematical knowledge;
b) with respect to the plethora of examples in a given field of problems (varying
also the text and the context).
The former supplies the so called semantic structure of problems and is a typical conceptual
analysis, concerned with the meaning of formalized language in mathematics. The latter, on the
opposite, focuses on processes and natural language and tries to pick out:
(i) basic conceptual models used by pupils;
(ii) typical transformations and translations which reorganize basic models in order
to adapt them to concrete problematic situations.
Conceptual models relative to a field of problems are properly described only by crossing both
kinds of analysis.
Appendix 2 - Problems
Summary of problems quoted in the paper: in brackets information on the age of pupils, to
whom the problem has been posed. For space's reasons, the form in which problems are
quoted is not the same as that given to pupils.
o. Take three consecutive numbers, calculate the square of the middle one, subtract
from it the product of the other two; what is the result? now change numbers,
and try again.... Explain if and why the result is always the same. [13 -16 years
old].
1. A prince decided to make a trip along his land and started with his followers. In
one day they travelled 50 km. Next morning, the prince sent back a messenger to
his castle, while he continued his trip. The prince went on travelling 50 km every
day, while the messenger rode 100 km each day. How long a time before the
messenger reached his prince? And if the history goes on, with the messenger
who rides back and forth from the prince to the castle 100 km a day, while the
prince goes on 50 km every day, how long does it take to meet the second, the
third, the nth time? [10-12 years old].
2. With a single cut, a big pizza can be divided into 2 parts; with 2 cuts (suppose to
do straight cuts), a pizza can be divided at most into 4 parts; with 3 cuts one gets](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-176-320.jpg)
![165
at most 7 parts, etc. Which is the maximum number Pn of parts one can get with
n cuts? [14 years old and over].
3a. Square numbers ｾ＠ are:
n 1
On:
•••••
•••• •••••
••• •••• •••••
•• ••• •••• • ••••
• •• ••• •••• •••••
Figure 2
2 3 4 5
4 9 16
.....
6 7
25 36
Qn + 1 = Qn + 2n + 1 (recurrence formula)
Qn =n2 (explicit formula)
3b. Triangular numbers Tn are:
•
• ••
• •• •••
• •• ••• • •••
• •• ••• •••• • ••••
Figure 3
n 1 2 3 4 5 6 7
T .
n . 1 3 6 10 15 21
Find recurrence and explicit formulas for Tn .
49 ...
28 ...
Show that 8 times a triangular number plus one equals a square number.
3c. Isosceles numbers In are:
••••••
•• •• ••••
• • • • • •
• • • •
Figure 4](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-177-320.jpg)
![166
n 1 2 3 4 5 6 7
In: 3 7 13 21 31 43 ...
Now solve: "The set Q of square numbers and I of isosceles numbers are not
disjunct: the number 1 belongs to both. Find other numbers, which are both
square and isosceles. How many are they? Justify your answers". [16 years old
and over]
4. There are n people at a round table, and we eliminate every second remaining
person until only one remains. For example, for n=lO, the elimination order
(starting to count from n.1) is: 2, 4, 6, 8, 10, 3, 7, 1, 9 and 5 survives.
Determine the survivor's number, Sn . [14 years and over].
References
1. Arzarello, F.: Strategies and hierarchies in verbal problems. In: International Congress on Mathematical
Education, Short communications, p. 15. Budapest 1988
2. Arzarello, F.: The role of conceptual models in the activity of problem solving. In: Actes de PME XIII, pp.
93-100. Paris 1989
3. Brousseau, G.: Les obstacles 6pistemologiques et les problemes en math6matiques. Recherches en Didactique
des MatMmatiques 4(2),165-198 (1983)
4. Brousseau, G.: Fondements et m6thodes de la didactique des matMmatiques. Recherches en Didactique des
Math6matiques 7(2), 33-115 (1986)
5. Carpenter, T. P.: Learning to add and subtract. In: Teaching and learning mathematical problem solving (E.
A. Silver, ed.). Hillsdale, NJ: Lawrence Erlbaum Associates, 1985
6. Chevallard, Y.: La transposition didactique, La Pensee Sauvage: Grenoble, 1985
7. Harper, E.: Ghosts of Diophantus. Educational Studies in Mathematics 18,75-90 (1987)
8. Kieran, C.: A perspective on algebraic thinking. In: Actes de la l3erne Conference Internationale PME, Vol.
2, pp. 163-171. Paris 1989
9. Laborde, C.: Langue naturelle et &riture symbolique, These d'Etat, Grenoble, 1982
10. Lesh, R: Applied mathematical problem solving. Educational Studies in Mathematics 12,235-264 (1981)
11. Margolinas, C.: Le point de vue de la validation: Essai de synthese et d'analyse en didactique des
math6matiques, These, Universite J. Fourier, Grenoble 1989
12. Norman, F. A.: Students' unitizing of variable complexes in algebraic and graphical contexts. In:
Proceedings of the VIII Annual Meeting of PME-NA (G.Lappan & R Even, eds.), pp. 102-107, 1986
13. Sfard, A.: On the dual nature of mathematical conceptions: reflections on processes and objects as different
sides of the same coin. Educational Studies in Mathematics 22,1-36 (1991)
14. Vergnaud, G.: L'obstacle des nombres n6gatifs et I'introduction 11 I'algebre, Construction des Savoirs:
obstacles et conflits, ARC, Ottawa, 1989](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-178-320.jpg)
![170
Somehow, we must act in such a way that attitudes of students and teachers and the public
all begin to change. Further, we must do this in a way that does not manifestly threaten what
students and teachers and the public think ought to happen in schools. Thus, simple minded
cries for new curricular content will not do. Curricular content is probably the most salient
aspect of what goes on in schools, and any call to change the manifest content is bound to
arouse suspicion and distrust. This is not to say that the curriculum should not change, but
rather it is a word of caution about pinning extensive hopes on the effectiveness of a strategy
that is based on changing curricular content.
By the same token, it will not do to simply call for a longer school day, or a longer school
year or more homework. While these changes in themselves are no doubt desirable, it is not
clear that they will have the kind of qualitative effect we seek to have on the teaching and
leaming of science and mathematics. These subjects are not taught adequately anywhere in the
world. The most likely outcome of American youngsters spending as much time in school and
doing as much homework as Japanese youngsters is that American students will do as well as
Japanese youngsters in solving the problems that are used on the cross national tests of
scientific and mathematical competence. As desirable as that may be, that is not the kind of
change that is needed in the US. The kind of change that is needed in the US is precisely the
kind ofchange that is needed in Japan and indeed, everywhere.
We need to begin to educate youngsters to conjecture and to pose problems and to question
evidence and authority. We need to have succeeding generations ask naturally and
spontaneously about everything they see in the world around them, "What is this a case ofl".
Where can we turn, if not to completely new curricular content and not to dramatic
expansions of time in school? I would like to suggest that there is a kind of answer provided by
technology that may help in just the ways we have outlined. I hasten to add that I am not
referring to traditional CAl uses of microcomputers but rather to a use which is conceived of
within and consistent with an entirely different pedagogical framework, i.e. guided inquiry. It is
to this form of computer use in education that I now turn my attention to.
A New Role for Information Technology
Information technologies offer the possibility of constructing "intellectual mirror" software
environments [1] that can scaffold the posing of powerful problems. In this section I will
describe some of properties of such environments, as well as discuss some of the implications
and consequences of the use of such environments in classrooms.
What is "intellectual mirror" software and why do I believe that software of this sort offers
a reasonable promise of helping to move us in the direction discussed earlier in this paper?](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-182-320.jpg)
![175
triplet of measures such as SIDE SIDE SIDE or ANGLE SIDE ANGLE. In either
case the program "knows" what kind of triangle the user intends to work with.
Suppose, on the other hand, that the user had to input a triangle by clicking on
three points on the screen. Suppose further, that in an attempt to enter an equilateral
triangle, the three points that the user clicked determined a triangle with angles of
59, 60 and 61 degrees. What should the program conclude?
The power of an "intellectual mirror" lies in its neutrality with respect to the actions of the
user. It must simply be a mirror that reflects, with as little distortion as possible, the
consequences of his or her actions without attempting in any way to make inferences about the
users' intentions. Imprecision in specifying the "object" of mutual concern to the user and the
program is inimical to the achievement of this goal.
I would like to argue that this need for an abstract description language that allows the user
to communicate with the software environment is an asset rather than a disadvantage in terms of
our long-term educational objectives. Despite all of the rhetoric surrounding the need for more
"hands-on" experience in the education of our youngsters, we should not lose sight of the fact
that "hands-on" experience is merely a stepping stone to a more important goal, namely "minds-
on" experience. In a deep sense, getting youngsters to abstract generality from the particularity
of their own experiences is among our most important ultimate aims as educators.
Some Concluding Remarks
In writing this paper, I have made a variety of assertions and analyses about the desired role of
problem-posing and conjecture making in mathematical and scientific education. These
assertions and analyses have been based on the experiences that my colleagues and I have had,
both directly and indirectly, with large numbers of geometry students and teachers using the
Geometric Supposer. There have been many reports [2] of the effects of this mix of new tools
and habits of mind with traditional curriculum content on teachers and students. The settings
from which these data are drawn vary widely as do the students and the teachers within those
settings.
The outcomes can be categorized in many ways. There are performance measures using
traditional assessment instruments. Using these measures there are small differences in
performance almost always in favor of the students who have become accustomed to making
and exploring conjectures. More important, however, and probably more illuminating are
different outcome indicators. These students come to school early in order to work on "their"
mathematics on their own time. They discuss and argue mathematics with one another, and
they end up taking further mathematics courses that they would otherwise not have taken.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-187-320.jpg)
![Problem Solving in Geometry: From Microworlds to
Intelligent Computer Environments
Colette Labordel , Jean-Marie Laborde2
1Department ofMathematics and Statistics, Concordia University, Montreal H4B lR6, Canada
2Laboratoire LSD2, IMAG, Universite Joseph Fourier-CNRS, BP 53 X, Grenoble, France
Abstract: The role of the computer environment on the problem solving processes is
investigated in two kinds of situations in geometry: (a) situations in which the computer giving
"objective" feedback is used as a tool and (b) situations in which the computer provides an aid
based on an evaluation of the performance of the student (guided activity). In the first kind of
situations, we analyze to what extent the constraints and feedback of a computer environment
may affect the solving processes and the kind of solution elaborated by the student. After
presenting the general principles underlying what an intelligent help provided by a computer
environment can be, an example is proposed in the case of a specific geometrical task for which
a prototype "Hypercarre" has been designed.
Keywords: computer environment, learning environment, guided discovery, feedback,
problem situations, geometry, geometrical figure, inductive and deductive reasoning, model of
student's knowledge
I - Problem Situations: Assumptions and Starting Points
The role of problem solving in mathematics learning has been very often emphasized. Starting
from this claim, the purpose of this paper is to identify the features of problem situations in
computer based environments which may affect the solving processes and consequently favour
the construction by the learner of new solving tools since this construction can be viewed as the
potential source ofknowledge acquisition.
It is generally assumed that solving processes occurring in problem situation depend on
interactions between three main elements: the student as a cognitive subject, the mathematical
problem, and the context. The fact that the computer is part of the context may lead to strong
changes into these interactions and therefore affect two kinds of processes:
- The "devolution" of the problem [4], i.e. to what extent he/she is really in charge
of finding a solution with his/her own intellectual means and all his/her knowledge
- The solving processes. As Pea [13] has stated, computers play not only the role
of conceptual amplifiers but also of conceptual reorganizers as providing new](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-189-320.jpg)
![178
facilities and therefore new solving tools. The same task which can be performed in
a paper and pencil environment with certain solving strategies may require in
computer environment completely different strategies.
In the design of teaching-learning sequences, teachers can play on these changes to improve the
learning processes. For this purpose, a better knowledge of critical aspects of the role of
computer environments is required.
We address this question in the specific case of the teaching of geometry. Geometry
provides an appropriate field of investigation because of the important role played by external
representations (usually called figures) and the new ways of using these representations which
are made possible by software specifically designed for geometry.
We will consider two types of problem situations in computer based environments:
- Situations in which the teacher does not intervene, i.e. neither suggests ideas,
nor corrects mistakes or erroneous procedures: these situations are called in French
"adidactical situations" [4], the student is involved alone in the solving process and
is alone in charge of finding a solution;
- Situations in which the teacher or the computer as playing the role of the teacher
provides the student with feedback taking into account an hypothetical state of the
knowledge of the student which is inferred from the actions of the student at the
computer and from his/her solution.
Some criteria will be considered in this paper to analyze both kinds of situations. They were
chosen because of their potential impact on the solving processes developed by the students:
- To what extent is the problem open-ended? We focus our attention on problems
which students can tackle with their prior knowledge but cannot completely solve
without elaborating new solving tools. Routine problems, or problems too hard for
the students, preventing them from any possible search for a solution are not
considered here. The type of the question may affect the open nature of the
problem. Questions providing the answer may lead students to use any means to
come to the answer and in a sense may prevent them from really entering the
problem: students are partially freed of the responsibility of the answer. A kind of
question in the Varignon problem (see § III.I) like "show that MNPQ is a
parallelogram whatever the quadrilateral ABeD is" does not induce the same
solving strategies as a more open ended question calling for conjectures like "how
to choose D such that MNPQ is a parallelogram ?" .
- The tools which are available or the actions or operations which the student may
perform directly (primitive actions); the solving strategies heavily depend on these
features of the context of the problem.
- Feedback provided by the situation enabling the student to become aware of the
possible incorrect nature of his/her answer; this kind of feedback is a factor of
evolution of the solving processes and consequently of the possible impact of the
problem situation on the learning of mathematics by the student.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-190-320.jpg)
![179
II - The Specific Case of Geometry
11.1 - The Nature of the Production of the Student
Geometrical tasks call for different kinds of productions. Students may have a better control of
certain productions than of other ones. In particular they may be more easily aware of the
incorrectness of their production in certain cases. Producing a proof and producing a diagram
strongly differ from this point of view. Perception allows the student to infer information about
the diagram he/she has constructed and to control it, whereas it is very difficult for a student to
make a judgement about a proof he/she has elaborated1. Construction problems or optimization
problems (for example, fmding the shortest distance between two varying interrelated points or
the largest area of a varying shape) are such problems where a perceptual control is made
possible.
11.2 - The Role of Perception in Solving a Geometrical Problem
Perception is one of the possible tools which can be used to solve geometrical problems. But
the power of geometrical knowledge lies in that it allows to solve problems which cannot be
solved only by a perceptual activity. Therefore tasks in which perception provides the solution
differ from a learning point of view from tasks which cannot be entirely solved by perception.
The latter ones can be used to foster the learning of geometrical knowledge and to be designed
so that perception may constitute a feedback giving information to the student about the path to
a solution, or the correctness of his/her solution In these situations the learning environment is
so organized that perception is not the only instrument of solution but is also an instrument of
validation.
11.3 - The Ambiguous Status of Diagrams: Drawing and Figure
Figures in geometry playa complex role which cannot be reduced to illustrating geometrical
knowledge. They have both a function of a "signifier" and a function of a "signified". Parzysz
[12] has introduced the distinction between "dessin" (drawing) and "figure" (figure). As
material representations (drawing) figures give rise to visual impressions while pointing out on
theoretical concepts. The theoretical and perceptual aspects may interfere, they may fit in each
other or they may be conflicting, as shown by Duval [5]: a drawing can give rise to a visual
1This does not mean that students cannot control the proofs they elaborate in using diagrams and changing
conditions of the problem. However it is not a spontaneous behaviour and has to be learnt.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-191-320.jpg)
![180
impression leading to an analysis of the figure which is not adequate for the mathematical
problem, for example leading to break: up the figure into constituents which are not the right
ones to be considered for solving the problem. In the Varignon problem (see § flI.1), the
students are generally attracted by the sides of the initial quadrilateral and not by its diagonals
because these latter are not drawn on the figure, but the solution is based on the consideration
of these diagonals.
Another element of the complexity of the notion of figure derives from its intended and
implicit generality: a figure does not refer to one drawing but to an infinity of drawings. What is
invariant in this class of objects are the relations between the objects. Furthermore only some
geometrical relations are relevant for the problem, and sometimes they are less visible on the
drawing than other ones. For example, the size of angles and sides of a triangle does not playa
role on the property ofintersection of the right bisectors of the sides of the triangle.
This is a real problem for students, when they move from the geometry of observation
(geometry of drawing) to the geometry of proof (geometry of figure) as it has been observed by
Schoenfeld [14]. The obstacles caused by perceptual aspects of drawing have been for a long
time brought out (cf. for example Zykova [14] or Fisher [18]). Recently they have been
summarized by Yerushalmy and Chazan [17] in three categories:
- The particularity of a drawing may lead students to include irrelevant
characteristics in the drawing;
- Standard drawings cause difficulties in interpreting non-standard drawings:
students were much better at recognizing right angles in an upright position, than
when the right angle was "at the top";
- The inability for students, to see a drawing in different ways to attend selectively
and sequentially to parts and whole, as Mesquita [11] could observe it in her
experiments with students of middle school.
Visual possibilities of computers have been used to design softwares materializing the
multiplicity of drawings attached to the same figure, and/or the notion of variability of a
drawing, briefly in materializing the theoretical notion of figure (which is sometimes called
"class of figures" as in the French curricula) as for example Geometric Supposer (presented in
Yerushalmy & Chazan [17] and in Schwarz [15]), Geocon [2], Cabri-geometre (presented in
Laborde & StriiBer, [9]): drawings are no longer stereotypes, critical cases can be visualized, a
geometrical situation can be considered from several points of view. In this kind of software, a
necessary condition for a construction to be correct is that it produces drawings preserving the
expected properties (for every position of the drawing in Geometric Supposer, when one
element ofthe drawing is dragged in Cabri-geometre).](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-192-320.jpg)
![181
The geometry of this kind of software seems to better materialize incidence geometry than
paper and pencil geometry in so far as correctness of a drawing depends only of incidence
properties and not of elements such as the size of the drawing.
III - Problem Situations in a Computer Environment Without
Teacher Intervention
111.1 - Influence of the Features of the Software on Elaborating the Solution
We consider here mainly the heuristic phase of the solving process. Two criteria have been
defined: (a) available primitives in the program, and (b) feedback provided by the program.
They will be discussed below.
The actions made possible by the software guide the student in his approach to the problem
since even if the student does not carry out a trial and error approach trying each possible
action, he/she is constrained by the ways of action offered by the software. It will be illustrated
here by the example of the Varignon problem already used by Artigue [1] and Gras [7] in
teaching experiments based on computer environments.
The Varignon problem. The problem is the following: ABCD is a quadrilateral.
M, N, P, and Qare midpoints of its sides (see Figure 1).
1 - How to choose D such that MNPQ is a parallelogram ?
2 - How to choose D such that MNPQ is a rectangle ?
3 - How to choose D such that MNPQ is a rhombus?
4 - How to choose D such that MNPQ is a square?
It is clear that the use of software enables the student to produce several drawings and that he
will be easily convinced of the fact that MNPQ is a parallelogram, more easily than in a paper
and pencil environment with only one drawing. In the teaching experiment in a paper and pencil
environment reported by Gras [7], the teacher was led to ask the students to draw several
drawings in various positions replacing in some sense the facility offered by the computer.
Using this problem with both programs Cabri-geometre and Geometric Supposer in a
teacher training at Concordia University (January 91-ApriI91), we could also observe that the
evidence given by the figure was clear to the students with both programs. But after using
Geometric Supposer some of the students concluded that MNPQ could only be a parallelogram
if ABCD was a parallelogram, or a kite or a trapezium. It came from the fact that a quadrilateral
is drawn randomly by Geometric Supposer after the user selects one of the possible categories](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-193-320.jpg)
![182
A M
D
ｾ＠ __________ｾｾ＠ __________--4C
p
Figure 1
among the following: parallelogram, trapezoid, kite, quads/circles (i.e. quadrilateral containing
an inscribed circle or circumscribed by a circle), your own. The "your own" possibility actually
is seldom used by students, firstly because it is the last one in the list (!) and secondly because
it requires the user to give measures of sides, diagonals, or angles to let draw the quadrilateral.
The problem being proposed only in a pure geometrical setting without indications of measures,
an implicit contract prevents students2 from using measures: a change of setting is often
perceived by students as a break ofcontract (We refer here to the notion of"didactical contract"
as introduced by Brousseau, [4], i.e. to the implicit rules underlying the mutual behaviours of
students and teacher). The only students who used the "repeat facility by deforming the present
shape" offered by Geometric Supposer could be aware of the invariance of the property even if
the initial particular shape was not preserved. In this teacher training the students had little time
to become familiar with the program and few of them managed to use it extensively. It is a sign
of the importance of the introduction phase to the use of a program; the mere fact to give to a
student a program does not imply a spontaneous use of all possibilities by him/her (this can
vary from one package to another).
It is also interesting to contrast the use of different programs for the second question with
this one for the third question. Artigue [1] reports that after the first question students using
BucHde (a programming language made of LOGO commands and LOGO procedures)
performed such a big effort in writing a correct procedure producing the drawing that they
considered the problem as achieved after the first question and had to be strongly pushed by the
teacher for looking for particular configurations of MNPQ. They did not succeed in solving the
questions with BucHde: they often obtained particular parallelograms MNPQ but not
2 Students are able to decide alone a change of setting if they are taught to practice this but it is not a
spontaneous behaviour; a teaching about moving from one setting to another one leads to a modification of the
didactical contract](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-194-320.jpg)
![183
deliberately, and they were unable to find the corresponding conditions on ABCD. The indirect
way of choosing D (either by coordinates in a procedure or in a discrete way by placing D on
the screen with an optic pen for each drawing) made very tedious the search for conditions only
by means ofthe program. That is probably the reason why the teacher finally decided to achieve
the work on these questions in a paper and pencil environment.
With Geometric Supposer the teacher students were lead to the same kind of partially
correct answers as they were for the first question. They concluded that if ABCD is a rhombus,
MNPQ is a rectangle and if ABCD is a rectangle, MNPQ is a rhombus. This answer does not
cover all possible solutions which are given by the following properties: if and only if the
diagonals of ABCD are perpendicular, MNPQ is a rectangle, if and only if the diagonals of
ABCD are equal, MNPQ is a rhombus. In this case the "repeat" facility by deforming the
present shape and by moving step by step the vertex D was without help for those who tried to
use it because it destroyed the configuration ABCD giving the expected MNPQ and because it
was too difficult to move in a discrete way the vertex D in preserving the relation between the
diagonals. As in the case of Euclide the facilities of the program did not lead to an economical
search.
With Cabri it is easy to obtain the particular configurations for MNPQ by dragging the
vertex D, the continuous dragging enables the student to deliberately and rapidly produce the
intended drawing. It does not imply that students automatically find out the necessary and
sufficient geometrical conditions but Cabri offers the possibility to make observations on the
realized configuration. The role of the teacher should also be important in this case: it consists
of teaching the students how to extract information when using the dragging mode in order to
be able to solve the geometrical problem. This point is discussed in § III.2.
Concerning the feedback given by the programs, it is clear that the perceptual feedback
worked very well for the first question since every student was convinced to obtain a
parallelogram. Artigue [1] notes that perceptual feedback was often misleading for questions 2
and 3: students must devise some tests to be sure that they obtained a particular parallelogram
MNPQ and very often the tests disqualified the expected property. In the same way the
Supposer and Cabri offer the possibility to check the validity of visual impression by the
measuring facility: measuring of lengths and angles. It is interesting to note the interplay
between the different kinds of feedback given by these programs and to observe that students
were not satisfied by the only visual aspect of the drawing. We think that it is possible to extend
this interplay to the interaction between inductive and deductive reasoning in the case of
software offering a continuous dragging of the drawing.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-195-320.jpg)
![184
111.2 Interaction between Inductive and Deductive Reasoning
One of the dangers of software which has often been mentioned is of inductive kind: the. student
would naturally conceive theoretical objects or relations from the only perception. Mathematics
teaching could very easily avoid the deductivist danger but meet the inductivist danger peculiar
to experimental sciences as Lakatos wrote [10, p. 74]: "On the other hand those who, because
of the usual deductive presentation of mathematics, come to believe that the path of discovery is
from axioms and/or definitions to proofs and theorems, may completely forget about the
possibility and importance of naive guessing. In fact in mathematical heuristic it is deductivism
which is the greater danger, while in scientific heuristic it is inductivism". Lakatos explains that
in deductivism as well as in inductivism, the notion of counter example is not taken into
account. It seems that the danger of a pure inductivism enhanced by visual possibilities of
computers may be avoided if students can be taught how to look for visual counter examples,
and make inferences about the reasons giving rise to counter examples that is about the initial
conditions of the figure which are not satisfied by the counter example. That was the idea
underlying the facility "by deforming present shape" offered by Geometric Supposer but it
cannot be used in practice because of the lack of direct manipUlation: it is impossible to handle
in a precise way the use of arrows That is also one of the ideas underlying the dragging mode
of Cabri which can be really achieved through the direct manipulation. Feedback offered by the
dragging mode of Cabri-geometre is of double nature: (a) giving evidence of the incorrectness
of the solution, and (b) providing information on relations between various elements of the
figure.
- If a drawing is not constructed by means of geometrical relations but is only
visually correct, its properties are not preserved by the dragging mode which keeps
only geometrical relations used for constructing the figure or which can be deduced
from them. The dragging mode is a potential means of producing counter-examples
to a construction and thus provides the students with a kind of control on their
productions.
- It is also possible to extract information from the changes of a drawing through
the dragging mode. Let us come back to the Varignon problem. The question is to
identify the quadrilaterals ABCD leading to a rectangle MNPQ. It is very easy to
visually produce a rectangle MNPQ while dragging the vertex D and observing the
variations of the measure of one angle of MNPQ. This measure is continuously
indicated on the drawing when using the facilities "mark an angle" and "measure".
Observing the produced rectangle very often does not give any idea of the reasons why it is a
rectangle. But when dragging D in controlling its trajectory the user can infer the dependence
relations between D and MNPQ. If D is dragged so that line BD is invariant, MNPQ remains a
rectangle. If D is moved aside the initial line BD, MNPQ is no longer a rectangle. The solver
must then look for the fixed or invariant elements of the drawing: the direction BD and the](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-196-320.jpg)
![185
B
c
o p
Figure 2
points A, Band C. This first step is of perceptual nature. A second step starting from this
visual information involves the search of invariant relations between MNPQ and ABCD, and
the inference of a link between MNPQ and the diagonals of ABCD since AC is fixed and BD
remains globally invariant when D is dragged in keeping MNPQ as a rectangle. The solver is
then lead to draw the diagonals of ABCD. The third step is of deductive kind and consists in
analyzing the figure in a traditional way: what are the relations between AC, BD and the sides
of MNPQ? Such a process combines the two aspects of a proof distinguished by Hanna [8],
since the proof "that proves" is the third step coming at the end of "an approach looking for
insight into the connections between ideas", which according to Hanna is a characteristic aspect
of "a proof that explains".
In a paper and pencil environment, as mentioned by Artigue and Gras, as an effect of the
didactical contract, the students generally do not draw the diagonals of ABCD if they are not
asked to do it. But the visual impression does not lead to the deductive solution as long as the
diagonals are not drawn.
In the described use of Cabri we would like to stress that the changes in the solving process
brought by the dynamic possibilities of Cabri come from an active and reasoning visualization,
from what we call an interactive process between inductive and deductive reasoning. A passive
visualization even of dynamic process is of little help. It is when analyzing the changes of the
drawing under the dragging mode, and not only in seeing them, that the solver may find out
some geometrical relations between elements of the figure. This way of solving the problem is
strongly related to the fact that the solver does not work on the drawing but really on the figure;](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-197-320.jpg)
![186
in this approach the figure is considered as a set of relations between variables and the dragging
mode is a powerful means of externalizing these relations.
It is interesting to note that a specific program for this problem with the same kind of
dragging mode has been designed by Giorgiutti and experimented with students of grade 9 and
younger students [7]. The relation of orthogonality between the diagonals of ABCD through the
use of the dragging mode was perceived by almost all students but Gras notes that in the third
question the isometry of the diagonals is not so easily found out (probably because the
corresponding trajectory ofD is more complex: it is a circle). Gras also notes that students must
be prompted to develop a complete deductive proof in a final step.
The same interactive process between deductive and inductive approach has be organized
on the following problem by in a teaching of geometry including the use of Cabri in a 8 grade
class (the teacher was Capponi and the school is located in the surroundings of Grenoble): the
right triangle.
Let ABC be a right triangle with A vertex of the right angle. D is a point on BC. Let be I
and J the feet of the perpendicular lines drawn from D to AB and AC. How to choose D to
minimize the length of IT?
Such an active and reasoning way of elaborating a solution is of course not spontaneous
and must be learnt by the students. It must be part of a teaching of geometry involving the use
of computers. It would be illusory to believe that such a teaching is easy but the resulting
learning is certainly very powerful in tenns of conceptual knowledge in geometry. This leads us
to focus on time aspects implied by the use of computers.
111.3 - Long Term Learning and Teaching
The computer by its capabilities is a powerful tool of solving problems but because making
possible complex actions introduces a new complexity. An elaborate interpretation of the
dragging mode and of its effects is certainly constructed by the students through a long process
made of interactions with various problem situations on the computer. According to the notion
of "experience field" proposed by Boero [3] we think that before being an experience field for
the student, a software is a field of experimentation as part of the environment in which
problem situations given to the students take place. In interaction with the use of the software
for solving problems, the student constructs a subjective experience ("internal component of the
experience field" according to the tenninology of Boero) of the software which becomes richer.
This dialectic evolution of the software as both experimentation field and experience field is a
long tenn evolution in which the teacher plays a decisive role through the choice of problem
situations.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-198-320.jpg)
![187
IV - From HyperCarre to a General ITS Architecture
IV.1 • Short Description of HyperCarre
Starting in March 89 a group of researchers including both authors as well as Bernard Capponi,
Rudolf Strii6er and Vassilios Dagdilelis started an experiment on the following ideas: to link
two existing tools, Cabri-geometre and HyperCard to achieve a system able to guide a limited
construction task in geometry. The group consists of researchers in mathematic education,
teachers, and computer scientists. Its aims were:
• to structure the knowledge in terms of tasks (problems to be solved) - in a
reference to a constructivist hypothesis;
• to emphasize exploring type activity - in reference to a microworld environment;
• to provide the student with specific help.
The task proposed to the student consisted of constructing a square from a given line-segment
as intended side in the microworld of Cabri-geometre, i. e. to realize on the screen using the
tools of Cabri the shape of a square implicitly retaining its characteristic properties when
dragging one of the endpoints of the initial line-segment. This task was selected as satisfying
some important characteristics
• relatively "simple" task, despite the existence of many different possible solving
strategies (for the grade 5-6);
• unambiguous visual feedback;
• existence of so-called "related task" (see below);
• evidence of some misconceptions.
For a presentation of the initial research and of a report on the classrooms experimentation of
the realized prototype refer to [9].
IV.2 • A Proposal
Considering a problem to solve like in the square task it appears that solving the problem
involves knowledege at different levels [16]
• the perceptive familiarity about the geometrical objects involved in the task (in our
example the squared shape),
• the ability to make explicit characteristics of objects involved in the task (here the
definition of a square in terms of right angles and equal length sides) (theoretical
knowledge),
• the ability of effective performing of actions in a sequence which results in an
object with the expected properties (procedural knowledge).](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-199-320.jpg)
![Task Variables in Statistical Problem Solving
Using Computers!
J. Dfaz Godino, M.C. Batanero Bernabeu, A. Estepa Castro
Departamento Didactica de 1a Matematica, Universidad de Granada, Campus de Cartuja, 18071 Granada, Spain
Abstract: In this work an analysis of some task variables of statistical problems which can be
proposed to the students to be solved on the computer, are presented. The objective of this
didactical - mathematical analysis is to provide criteria of selection of the said problems, directed
to guiding the student's leaming towards the adequate meanings of the statistical notions and to
the development of their ability to solve problems.
Keywords: teaching statistics, data analysis, statistical laboratory, computers in mathematics
teaching, task variables
Computers and Learning of Statistics
The use of computers in mathematics teaching is a phenomena which is more and more
stimulated by the teachers and researchers in Mathematics Education. The impact which it will
have in the design of the curriculae of this decade is shown in documents like the NCTM
Standards [5]. Nevertheless, this use creates problems in the field of research of the Didactics
of Mathematics which can be summarized in the following questions:
(1) How does the availability of computers affect the contents which should be taught?
(2) What new problems, based on the use of these tools should be proposed to the
students?
(3) What didactic consequences are produced when this change is carried out?
In particular, these questions are especially important in the field of Applied Statistics, due
to the growing demand for formation on the part of pupils of different specialities, whose
interest in the subject is essentially instrumental. The fact that a package of programs has been
used in the Statistics teaching at those levels, can produce changes which are necessary to
foresee and assess about the mathematical contents to be taught: certain rules and properties (as
for example, the algorithms of abbreviated calculus or the use of the distribution tables) loose
their validity. Other topics arise as a consequence of the availability of software, such as the
1This report fonns pan of the Project PS88-0104, granted by Direcci6n General de Investigaci6n Cientffica
(MEC), Madrid](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-205-320.jpg)
![194
exploratory data analysis. The way in which certain topics can be approached can also change:
for example, the possibility of carrying out a multiple regression iteratively including or
excluding cases during the process of analysis.
On the other hand, the same statistical packages are converted into objects to be studied.
Likewise, it is necessary to study the sequencing of the traditional type of teaching and
laboratory activities with a computer, in such a way that the main aim of the statistical learning
is not substituted by that of the learning of the calculus packages, thus producing a phenomena
of "glissement metadidactique" [I].
The computers and the whole range of resources which this instrument provides, offers the
pupil new uses of the statistical concepts and procedures, and so, new meanings for the same.
However, within the potentially unlimited set of uses, we should ask ourselves: which ones
should we select? With what criteria should we choose some problems or others for the
different groups ofpupils? The didactical-mathematical analysis of the specific tasks variables to
be developed in the computer environment are shown as a prior and essential step to be able to
construct significant didactical situations.
Role of Problem Solving in Teaching Statistics with Computers
Stanic and Kilpatrick [6] indicate that the role of problem solving in the mathematical curriculum
has been characterized within three modalities: as a context to reach other didactic goals, as a
skill to be learnt and as an art of discovery and invention. Within the first modality the goal
aimed at could be: the justification of the contents to be taught, to stimulate the motivation of the
pupil, to provide a vehicle for the transmission of knowledge and the reinforcement of the
learning by practice and this can even be used as a recreative activity. Each one of these roles
can be affected, in the case of the problems referring to statistical concepts and procedures, by
the use of computers.
If the pupil has a computer available then he is released from the task of calculating the
different statistics and from carrying out the graphic representations of the distributions.
Moreover, it is possible, in the didactic time which is usually available, to solve a greater
number of problems given the great processing speed of the computers. As a result of this, it is
possible to wonder what the type of practical activity which can be proposed to the pupils is,
and what roles could this method of problem solving play in the learning.
We can classify these activities to be carried out with the computer in an overall way, in the
three modalities which are described as follows:
(1) Discovery, using experimentation and simulation. of some of the
mathematical properties of the distributions or their statistics. as would be for example. the
convergence of the frequency polygon to the density function by increasing the number of cases
and subdividing the intervals.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-206-320.jpg)
![195
When a certain mathematical property is taught, the role which in the deduction of the same
has had the intuition and the observation of this property in particular cases, is very often
forgotten. In the field of probability and statistics, the computer is a very powerful instrument to
be able to provide an intuitive approximation, prior to the deductive proof of many properties.
This type of activity would reinforce the role of problem solving as a vehicle for the learning.
(2) Invention of questions from a set of data by the student himself. One of
the main difficulties which is put to the statistician in the analysis of the data provided by a user,
is that often the person who has collected the observations, does not know what can be obtained
from the same. This implies that in many of the cases the fact that a greater or lesser number of
variables and observations than were necessary for the analysis have been collected. So, it is in
our interest to propose to the pupils that they ask questions relative to a given file to increase
their capacity of significant and solvable interrogation on a data set. This capacity may be
developed using a special type of problems and it is made easier with the availability of
computers.
(3) Solving data analysis problems. The capacity to analyze data can not only be
considered as a skill which is necessary in many professions, and which to a great extent
justifies the teaching of the statistical concepts. It may also be seen as an art, since there are
many levels in this capacity. Moreover, the work with files of real data is a source of motivation
for the pupil. Finally, an adequate sequencing of this type of problem may be an effective
means of learning or reinforcement of concepts.
Task Variables in the Data Analysis Problems
From the classification of variables of the problems given by Kilpatrick [4], in Table 1 we show
some task variables, corresponding to the data analysis problems. In the following sections we
describe these variables in detail and we propose specific examples of different types of
problems which can be considered from this classification.
Data Files: Way in which they are Obtained and their Structure
As Jullien and Nin [3] indicate, to choose the set of data on which teaching should be based, the
domain of the reality chosen should be relatively familiar to the pupil and sufficiently rich to
make questions of didactical interest arise. From these applications, a first group of task
variables which are going to condition the types of problems which can be proposed to the
students, arise. These variables are the following:](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-207-320.jpg)
![202
G4: Comparison of related samples, that is to say, the comparison of the
distributions or statistics of different variables in the whole file:
PROBLEM 5: Which of the variables "number of pulsations at rest" and "number
of pulsations after 30 press ups" has more dispersion? What is the average
difference of pulsation between before and after doing the press ups?
This type of problem implies the repetition of one same process of calculation with different
variables in the complete file, and so there is not a selection process. This corresponds to the
study of the differences between related samples, which is also equivalent to that of the
association between one variable included in the set of data and another implicit qualitative
variable.
H: Opening of the problem: In the previous sections we have identified several
different categories of variables; likewise, from the examples shown, we can also deduce a
great range of variability in the opening of the problems: in general, there can be several
methods of solution and even several possible correct solutions for the problems considered. In
particular, one interesting type of problem is to propose to the pupils to choose the most
adequate among several possible solutions to a given problem. For example, to decide what is
the most representative measure ofcentral value of a certain characteristic, or as in the following
case, in what the solution is in some extent subjective:
PROBLEM 6: Build several frequency histograms for the PTS in pocket variable
(SURVEY file). In your opinion, what is the most adequate amplitude of interval?
Reason your answer.
Final Notes
In this paper some of the specific task variables of the data analysis problems have been
described, showing the diversity of problems which give rise to the classification presented.
This variety will be even greater when the study which we have initiated here is completed with
that of other task variables, non specific in the field of statistics, described in the research about
problem solving.
Moreover, it is necessary to take into account the relevant role for the pupil's learning of
the variables linked to the role of the teacher and to the situations in which the problems are
solved. As Dfaz Godino et al. [2] show, it is not enough for the student to face the solution of
the realistic problems of data analysis, provided with powerful resources of calculation, to be
able to acquire a better understanding of the conceptual mathematical objects. In this research
we point out the limited comprehension of the notion of statistical association achieved by the
pupils after a teaching process based on the intensive use of package of data analysis programs.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-214-320.jpg)
![206
However, when we speak of problem-solving, we should keep in mind that there is often a
mismatch between our aims, as mathematics teachers, in giving students problems, and their
own view of the nature of problem-solving activities. For students, the goal of problem-
solving work is often focused on the ends and not the means. Their agenda is governed by
practical rather than theoretical considerations and, as N. Balacheffreminds us, their concern is
"to be efficient not rigorous; it is to produce a solution, not to produce knowledge" [2]. It is
this issue, in particular, that will animate my evaluation of the role of the computer in problem-
solving. I will look at the other side of the 'feedback-at-no-cost' coin and raise the question
whether it, at times, leads to unintended and undesirable effects on the solvers. In particular, I
will give examples from our research that point an accusing finger at solvers' reliance on
computer feedback as the cause of an undue emphasis on 'producing solution' rather than
'producing knowledge'.
ACT I: The Art of Avoiding Concepts and Techniques by Developing
Brute Force Calculations Which Enable One to Travel More Lightly
In preparing for a recent course for teachers on problem solving, I have had a second look at
some papers and reports on research conducted in the late 70's. One of those was a final
technical report of M.G. Kantowski [5] on two exploratory studies on the use of heuristics in
problem solving. One of the reports is among the earliest studies to examine the effects of the
use of computers on heuristic processes, in this case in the solutions of number theory
problems. The report contains summary information as to the kind of use made of the computer
(writing a complete program, a partial program, or using it strictly as a calculator) across the 10
students who participated in the study and across the 15 problems. Kantowski reports that
when students were asked after solving each problem whether they would have solved it
differently without the computer, "in high percentage of cases the students said they would not
even have attempted to solve the problems if the microcomputer had not been available" (My
underline). For example, 80% of the students said they would not have tried the following
problem:
"Find the five digit decimal integer ABCCA whose Cth power is the fifteen digit
integer CCCCCDEBFEGFGFA".
Indeed the percentage of students who used the computer as a calculator in solving this problem
was also given as 80%.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-217-320.jpg)
![208
the exponent in the problem) and that it did not take them many computational trials to settle on
C = 3. The values of the other digits were probably obtained by either via computations or by
extracting some implicit data from the problem ( such as "the end digit of A3 is A"). We can
say that, from their perspective, the students used solution strategies which were optimal in
terms of using the computer for finding a solution, i.e. they met the condition ofefficiency over
that of rigour as stated in the above citation of Balacheff. But the availability of the computer
has changed the character of the problem. It certainly made it into an easier-to-solve problem
and we need to ask ourselves whether too much of the essence of the problem has been given
away. Since the availability of a computer naturally led to strategies involving computational
trial-and-error, the issue to consider here is about the status of empirically derived results such
as C = 3 for the solvers. If it was simply accepted by the solvers as a fact then we may ponder
if it is any different than giving C=3 as part of the problem-statement. On the other hand, such
a result may have triggered a post factum awareness that C=3 is derivable directly from the
givens by thinking of ABCCA as a number between 104 and 105. We could then say that the
computationally-based solution has led to strategic knowledge about how to handle such
problems, so the use of the computer has eventually resulted in 'producing knowledge' rather
than simply producing results. We would then have the expectation that students would be
weaned of their reliance on calculators if several problems of the same type were given over
time.
The situation here is quite different than, say, using software for geometry problem-
solving where the computer provides empirical evidence of relationships which then have to be
proved deductively. In the case of the ABCCA problem, the computer can provide most of the
information about the digits and a tool for verifying that the answer is correct. Its use
encourages a certain computational strategy which is effective for coming up with a correct
answer but may fall short of providing insights about the nature of the problem. It is this aspect
that seems to be addressed by the mathematician Michael Attiyah who commented on the
limitation of the use of computers [1] by speaking of mathematics as "the art of avoiding brute-
force calculation by developing concepts and techniques which enable one to travel more
lightly". For Attiyah, no doubt, 'travel' refers to a life-long journey through mathematics. But
for our students, 'travel' may be a short journey through a single problem. As the title of ACf
I suggests, the availability of the computer gives legitimacy to their feeling that the exact
contrary of Attiyah's statement is true.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-219-320.jpg)
![209
Intermission
Since 'problem-solving' is a general banner which subsumes many different types of activities,
I would like to delineate several general categories of problem-solving activities which are part
of the scenarios discussed in this paper. First, let me point to the obvious. In some problem-
solving activities, the ends rather than the means are important. We may be in need of some
results and we would be willing to beg and steal in order to get them. In such cases, it matters
little if the results are obtained via elegant methods or by a brute-force exhaustive search. This
is the practical aspect of problem solving and one which is not necessarily intrinsic to the
problem but rather depends on the context in which the problem comes up. It is not only
evident in the domain of so called 'real life' or applied problems. Finding some normal
subgroups of a particularly nasty permutation group may be as practical a problem to a 'pure'
mathematician as computing some trajectories of ballistics for an 'applied' one.
But the purpose behind most problem-solving activities within the school setting tends to
have a slightly different focus. As teachers of mathematics we give our students problems that
fall into the following, rather broad, categories:
a. Problems to reinforce some newly learnt techniques and concepts;
b. Problems to challenge students - cultivate interest, curiosity, excitement as well as
develop problem-solving skills;
c. Problems to develop some new mathematical knowledge (about objects, relationships,
properties);
One difficulty with problem-solving in instructional settings is to convince students to
share the same goals as we have (when we make our goals explicit) and the use of computers in
problem-solving may (unintentionally) make the job of convincing even harder. For example,
the ABCCA problem falls squarely into category b., but as suggested in the first act above, the
computer may have deflected the instructional aims by rendering the problem less challenging
and by fostering strategies which are appropriate only in computational environments.
The next two acts involve the use of computers in the solution of problems which fall into
category c. I will start by extracting several episodes from the Logo studies which I conducted
with Carolyn Kieran and Jean-Luc Gurtner [3].
ACT II: Successful Computationally-Derived Solutions as Obstacles
to Generalization
Here I consider some very common Logo tasks, namely those of reproducing figures with n-
fold rotational symmetry. The implicit aim of such tasks is for students to arrive at the](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-220-320.jpg)
![210
relationship governing n, 360, and the angle of rotation, hence the tasks fall into category c.
described above. In the case I will be discussing, the solvers are 12-year olds who, through
their previous 12 hours of work with Logo, have some notion of the relation of 360 to a
complete rotation, though this knowledge manifested itself as a 'theorem-in-action' [6] rather
than as an explicitly-stated knowledge.
Before examining the nature of the solutions of these specific tasks, there are comments to
be made about the effects of the particular Logo context on problem solving. It has been our
experience that work with Logo fosters in pupils a particular set of beliefs that guide their
problem-solving behaviour. In particular, the metaphor of 'drawing with the turtle', which is
successful in initiating children into this computer environment, often leads to a too strong
association of the activity with that of hand drawing. Hence, children tend to conceive of goals
in 'more-or-Iess' terms so they will consider as proper solutions their productions of figures,
even if these only approximate the task figures. The situation is also confounded by their
awareness that figures appearing on the screen or on printouts are often distorted and so a high-
fidelity replication is not really 'part of the game'.
There is another level of complication with children's work, not specific to Logo, but
which has to do with the very presence of geometric figures. It is well-documented that
students have difficulties discerning the essential characteristics of a given figure. For example,
in the task illustrated in Figure 1 below, our intention is to give a prototype of a figure with 3-
fold symmetry (the children already possess a Logo-procedure to construct an 'arm').
Implicitly, some of the visible attributes of the figure such as its size, orientation and its
particular location are not relevant for the task of writing Logo instructions to reproduce the
figure on the screen. The important feature, from our perspective, is the invariance of the angle
of rotation between the adjacent arms of the figure. But to the solvers, the salient features of the
figure might be quite different. From our experience, the most striking is the fact that one of the
arms is in a vertical position. But children might equally well focus on the three arms emanating
from the same point, or possibly, on the vertical symmetry of the figure. Thus, in this
geometric-computer context, not only is the criterion for having reproduced a figure guided by
the belief that the screen output need only be approximate, but the figure to be reproduced is
often misinterpreted.
In time, we became aware of the need to be explicit, both about the given and the goal
governing a task (e.g. "all the turns are the same in this figure"), so our 'experimental contract'
was more transparent. Our insistence that the solvers' solutions had to meet all the stated
conditions of a problem meant that we were increasing the level of demands on the solvers. We
hoped that the that this would result in an increasing awareness of embedded relationships in the
task that could be exploited in the solution.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-221-320.jpg)
![211
Figure 1
The children's solutions of such 'well-structured' tasks such as those of rotational
symmetry were marked by rather consistent solution strategies. For example, among the
rotation tasks, the fIrst one involved six-fold symmetry. A typical solving behaviour can be
nicely exemplifIed by that of Ben, who began by constructing the vertical arm and continued
with a repeated 45-tum. Once he noticed that the number of atms on the screen fIgure was
getting too large, he changed his solution to one involving the sequence of 45, 90, 45, 45, 90
turns (the sixth tum being unnecessary for completing the fIgure) which, because of the vertical
symmetry of the resulting fIgure, he considered as a solution. When it was pointed out to him
that the turns were not all equal, he tried repeated turns of 50 then 65 then 55 and finally settled
on 60. At this point, we might say that the status of his understanding of the nature of the
problem is not very different than the empirical observation that C =3 in the ABCCA problem.
The number simply works.
So far, the above problem-solving behaviour could be termed as a reasonable and efficient
use of the computer, as feedback has led to adjustments till the correct rotation was found.
However, in contrast to the situation described in ACT I, in this study we intervened in a way
that we had hoped would direct attention towards some explicit relationships (so Ben, for
example, was prompted till he ended with the REPEAT 6[Arm Tum 60] construct). But more
importantly, we also looked for evidence of 'producing knowledge', at least in some form of
inductive reasoning, as we observed Ben's and the other children's behaviour over five other
tasks involving rotational symmetry (and over 4 weeks) . Despite the similarities in the task,
Ben never attempted to wean himself of the reliance on the computer in order to fInd the
appropriate angle of rotation. In fact, he approached each of the subsequent tasks 'from scratch'
using the same strategy involving the cycle guess ｾ＠ feedback ｾ＠ patch-up. This behaviour was
rather typical of many other children whom we have observed over the years. What was
perhaps more striking in Ben's case was that even though he was a boy-scout and had the
contexualised knowledge that "360 makes a full circle" from using a compass, he never brought
this knowledge to bear in his work on the rotation tasks. The availability and the 'no cost' of](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-222-320.jpg)
![213
Figure 2
The full analysis of these tasks is found in [3]. Let me point out that despite the apparent
simplicity of the figure, this is not a trivial task. First of all, the computer environment was
constrained so the allowable moves consisted only of drawing rectangles and of displacing the
turtle laterally (in particular, no line segments could be drawn). A solution which met both
conditions (congruency and centering) required the coordination of the two magnitudes of the
rectangle. In particular, regardless whether the stem or the bar rectangle was constructed first,
the turtle had to be displaced by a distance (see boldfaced segment in Figure 3) which equals
half the difference of the base and height of the rectangle, in order to 'interface' correctly with
the next rectangle. Clearly, coming up with the relationship Ibase-heightl!2 required anticipating
where the second-constructed rectangle will appear in relation to the first.
Figure 3
After being given a printout of the figure, the children were asked to write a complete solution
(as a procedure) prior to working on the computer. This meant that their initial use of the
computer was strictly for verifying their first attempt at a solution. However, once started on the
computer, they generally resorted to direct-mode work, relying exclusively on visual feedback
and generating a cascade of patch-up strategies. Because these centering tasks involved
multiple conditions the patch-up strategies, in turn, often moved the attempted solution further
and further away from the goal. Unlike the rotation tasks, the inefficiency of these strategies
for the centering tasks might have triggered a more analytical approach and an effort to draw out
the relevant relationships in the figure, though we saw very little evidence of such switch of
strategy.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-224-320.jpg)
![214
Once again, it is useful to compare the above solving behaviour with those of students in a
somewhat analagous non-computer problem situation. If we look at Figure 2 above, it satisfies
two explicitly stated conditions - the rectangles are congruent and they are centered with respect
to each other. With this task as well as the other centering tasks, our solvers often focused on
one condition which failed to hold and in the process of trying to fix it, they would 'de-
structuralize' other conditions that were already met. For example, in the Tee task, if a solution
had two congruent rectangles but a stem which was not centered relative to the bar, the stem
was reconstructed with a narrower base which made it look more centered but violated the
congruency condition (see Figure 4).
Figure 4
This behaviour seemed to us quite reminiscent of students of the same age working in a non-
computer setting on problems such as:
"In a parking lot there are 40 vehicles. Some are cars and some are motorcycles.
Altogether there are 100 wheels. How many vehicles of each kind are there in the
parking lot"
Our analysis of the solution process for the above problem [4] has also pointed to a constant
shift of attention from one condition (100 wheels) to the other (40 vehicles) and to a difficulty in
coordinating both conditions. However, when we compared the solving behaviour in these two
very different settings, we were mainly struck by one thing. In the computer environment there
was almost a total absence of 'blockages'. Yet, blockages are extremely common phenomenon
of paper-and-pencil solutions of non-routine problems. Our computer solvers were never
stuck; every visual feedback immediately suggested some patch-up action. The quickness of
the feedback seems to have prompted an equally quick respond. Our solvers rarely gave up;
they just kept on going until they felt either that they had solved the problem or that an
approximate solution was 'good enough'.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-225-320.jpg)
![220
microcomputers as tools for the students themselves, where the students engage in meaningful
problem solving activity [7, 13]. As we now know, one of the most well researched computer
environments in education, Logo, can become the means to generate learning situations where
children use ideas first and then discriminate and generalize them [3, 5]. In this way, the
learning of content originates from, and is closely related to, experiences which are personally
meaningful to the children [8]. The computer environment is then used as "scaffolding" [4] to
achieve goals even when ideas and concepts are understood only partially and locally. More
recently, Logo based microworlds have been developed, where children may use specific
primitives as a support in focusing on specific content areas. These primitives are either very
fundamental, so that children may only build with them [11, 17], or more complex, so that
there is also ground for discriminating the ideas embedded within the primitives themselves
[2, 6, 12].
Moreover, although not fully acknowledged in early Logo research, the teacher is an
indispensable factor in this process; we now know that children might employ specific concepts
when working with Logo, but they are not often aware of the power of these, or even that they
are using them at all [1, 9, 10]. A teacher who understands which concepts children are
learning, may encourage them, for instance, to reflect on, generalize and use these concepts in
different contexts [3, 5].
But how can this come about in a classroom where the teacher has been transmitting
knowledge since the beginning of his/her career and the children have been consequently
conditioned to passively sit and listen and then try to memorize at home? What is most likely to
happen when this teacher is given the technology to use Logo in the classroom and is
"encouraged" to do so, given a user friendly manual type book?
Computer environments like Logo, often fall victims to their power and flexibility as
learning tools since they may easily be used even in a classroom where knowledge is
transmitted by the teacher and the computer application becomes just another chunk of
information the children have to memorize. It is often the case that the issue of the way Logo
might be used does not even arise; the computer is just locked up in a cupboard, or the teacher
will pass the "responsibility" to an enthusiast colleague, who in tum will teach enthusiast
children - those who most likely would use computers anyway. In many cases where the
schools can afford it, a computer "expert" is hired to teach Logo (!), most of the time having
little or no experience with educating children or teachers. In any case, there is a tendency for
the use of computers to become a privilege of the few.
Large scale implementations of the above research ideas about how to exploit Logo as a
problem-solving tool for all children in normal classrooms, within the day-to-day function of a
prescriptive educational system, consequently present serious problems that ideal research
situations do not have to face. If environments like the above are to be created, educational
principles have to be questioned, teachers have to be trained, often in contradiction to what they
have learned so far. Furthermore results cannot be measured in the same ways as before since](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-231-320.jpg)
![221
conventional measuring instruments and expectations of the children's performance would be
irrelevant [15].
Providing children with the opportunity to take control of their learning and to solve
personally meaningful problems in classrooms thus has to be a process of negotiation between
innovation and relevance. For instance, there is no recipe for the "right" way to use Logo in the
classroom or the "right" way to train teachers and changes in educational practice will certainly
not come about as a result of short-term training [14]. Allowing children to learn by doing and
teachers to reflect on their children's learning and perhaps to re-formulate their practice, may
currently be the most important educational potential for which such open-ended computer
environments may be used. Within this framework, although what is actually taught may vary
widely, it is always of educational value.
This paper discusses an attempt at a long-term injection of a pupil-centred problem-solving
pedagogy into the culture of a primary school, involving all the school's teachers and children,
within an educational system predominantly based on the prescriptive transmission of
knowledge from teacher to children. While claims to any outstanding success are not intended,
the continual negotiation of the above ideas and arguments within the entirety of the school's
society (all non-enthusiasts included) may be worth sharing.
The project is taking place in a private primary school in Greece and is now at the end of the
final year of its planned six-year duration. Inevitably, the school itself being one of the
privileged private schools in the country does not facilitate straightforward generalizability of
the project's experience to primary schools within the state system. However, the project is
meant to generate potential for a pedagogically-aware use of technology through Logo within
the Greek educational system. Thus, the early availability of technology and specialized teacher
education in one school may be exploited in order to provide a bank of ideas for wider
application in the near future.
An Outline of the Program's Main Features
There are two main factors which enabled the program to take place: the acquisition of the
technology (details are given below) and the availability of one hour per week for each class to
work with the computers for all classes from the 3rd to the 6th year inclusive.
However, there was no possibility to find extra hours for teacher training apart from 10/20
hours before the beginning of each school year and on occasions within the school program
where the teachers of one grade had a free period (e.g. when the children did their P.E. or their
English). Furthermore, the teachers were not expected by the school's direction to receive
training for and to undertake an additional topic to teach as an add-on to their already heavy
schedule. It was therefore essential to begin using the computers as a tool for educational
objectives which were already present. This was already a lot to take on, since none of the](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-232-320.jpg)
![222
teachers had had any experience with computers. There were issues of technical know-how to
be resolved, but more importantly, the teachers had to learn to use Logo in order to generate
learning situations amongst their pupils according to Logo's philosophy. Learning more about
the content of Logo programming and Logo mathematics would come gradually for the
teachers, possibly shaping their intervention strategies vis a vis the children.
All 24 teachers of the school are taking part (16 of them each year) together with all the 500
or so children from the third to the sixth year of primary inclusive (age range 8 to 12). The main
aspects of the project maintained throughout its duration are: a) teacher training; b) curriculum
development; c) informal evaluative research on the development of the teachers' strategies and
the children's learning.
After a pilot first year, implementation, evaluation and development were concurrently
maintained throughout the project. The main educational features of the project are analyzed in
more detail below.
Educational Objectives
The experience of a classroom of 20 children learning Logo during the pilot year highlighted
their dependency on the teacher for learning, their reluctance to explore an idea voluntarily or
solve a problem and their overwhelming inexperience at working within a process of
constructive communication amongst themselves [12]. The results from the pilot year, the
available conditions for implementing the program (one teaching period a week for each class,
very little time available for teacher training) and the aim to exploit Logo's educational strengths
were consequently the main factors in formulating the educational objectives with which the
program began. These were to generate classroom environments encouraging:
a) active thinking (e.g. to solve own problems);
b) initiative (in thinking, creativity and decisions);
c) cooperation (cognitive, affective, social).
These objectives were not mediated to the teachers in a prescriptive manner. The way in which
they are phrased above is the result of them having been reformulated through discussion with
the teachers who subsequently perceived them as an educationally meaningful reason for using
Logo. It is evident from the objectives themselves that the initial focus of the program was on
setting up and establishing a child-centred, investigational classroom atmosphere, in
comparison to what was common practice in the day to day school curriculum. A gradual
injection of content within this process, in the form of Logo programming ideas such as the use
of iteration, procedures, subprocedures and variables, began later; implicitly at first, and then
developing into an organized and explicit "curriculum" (see below). However, content is not
only perceived as a curriculum of specific Logo ideas, slowly integrated within the program's
working structure. The teachers are also using their developing knowledge on Logo](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-233-320.jpg)
![224
Importance is given to the children's presentations of their investigations. Each presentation
essentially has three parts; a) a written essay on problems met during the project, points of
interest and issues referring to the group's cooperation; b) a written record (or a printout) of the
commands and procedures written during the project; c) a graphics dump of the constructed
shape(s), design(s) or figure(s). In effect, therefore, each investigation is completed in six
weeks.
The children's presentations are useful in four different ways. Firstly, it is an integral part
of the children's Logo work, since they use three different symbolic methods (writing, Logo
commands, graphics) to express a meaningful reality, also an important aspect of working in a
turtle graphics environment [13, 16, 18]. Secondly, as a tangible result of the children's work,
which facilitates reflection, social mediation in the classroom and the opportunity for the teacher
to provide further feedback to the children. Thirdly, the presentations are an important part of
the collected data for evaluative research, since they provide a picture of both the children's
work and the teachers' strategies through their written comments. Finally, the presentations are
a means by which the program's objectives and the children's work is mediated to interested
parties outside the school's everyday life, i.e. the parents, the board of directors, other schools
and educators, educational conferences.
Content
As the project progressed, it became common experience amongst the teachers that what the
children learned in the Logo classroom was to a large extent not strictly intended or expected by
the teachers themselves. However, an intended content was developed gradually by the writer
in negotiation with the teachers, as a function of what was actually going on in the classroom
and of the teachers' and children's growing expertise with Logo programming and technical
know-how. The content focused on the features of the Logo programming language in the role
of thinking tools, in the sense that the children would be provided with these tools and
encouraged to use mathematical and programming ideas during their investigations and
structure their written essays in their presentations. The content for each year (age group) is
outlined below. The actual Logo commands and programming methods are being introduced to
the children in between investigations. During investigational work, the teachers only
encourage a group ofchildren to use a particular idea (e.g. procedure) if and when they feel that
the group will see the need for it.
a) year 3 (8-9 years old)
- The children play turtle in the playground and have preliminary experience with
the computers (but not within the structure of "investigations").
- They complete two investigations in total. The focus is on the working method,
the classroom dynamics, the writing of presentations and the detailed explanation of
what is meant by an "investigation". The Logo content is mainly direct-driving.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-235-320.jpg)
![226
a low profile, and was restricted to school working hours. Under this framework, all the
teachers agreed to give it a try--even those who were far from enthusiastic about "computers".
From the outset, the main concern regarding the training of the teachers was to encourage
them to take control of the way in which they·would implement the above educational
objectives. Thus, the reason for focusing on process, was for the teachers to use Logo as a tool
with which to teach something that they considered familiar, relevant and useful in relation to
their experience so far: they felt that the children had very little chance to engage in active
thinking, take initiatives and cooperate in their normal classrooms and that these experiences
would be of great value to them. Once a culture of children building control over their learning
could be established, with the teachers feeling that they were using this new tool meaningfully,
then it would be time to progressively throw more emphasis on exploring the mathematical and
programming ideas embedded in the Logo language. In effect, focus on content was made
explicit during the final two years of the program.
Research and Evaluation Objectives
One of the objectives of the program is to study the longitudinal social and learning
characteristics built within the dynamics of the Logo classroom. A year-to-year record of the
program's progress is kept and analyzed, on the one hand with the aim to evaluate and
reformulate the program itself and on the other with the aim to collect a corpus of data from
which more formal research questions will be developed. The research problem has therefore
been given two dimensions:
1. To study the extent and the way in which the program's educational objectives
are implemented and reformulated year by year.
2. To formulate research questions concerning the development of the teachers'
strategies, the children's learning processes, what meanings they bring to their
working environment and the social dynamics at the level of the classroom as a
whole and at the level of one group of children.
Data Collection and Evaluating Method
As mentioned above, using the computers was a normal everyday function of the school and
teachers and children have not been asked to spend time on top of their working schedule.
Within this framework (apart from the pilot year), research has been informal, data collection
has not required any extra activities on the part of teachers or children and the hitherto general
research objective has been to evaluate the program's progress in the view ofreformulating it.
A detailed account of the collection of data and some summative results from the first three
years of the project can be found in [II]. In brief, the following data has been collected.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-237-320.jpg)
![230
same name to all of these ("Invisible") could serve as a root experience for generalizing the
notion of interfacing procedures, something that does not come about automatically with
children [13]. Furthennore, next to the commands of each section, there would be a drawing or
a printout referring to the commands-a pre-amble to the injection of modularity to their
programming (in example 3, involving 6th year children, structure and modularity are much
more apparent in the children's work). Finally, the screen dump would show their final
product.
The teacher's strategy during the investigation was to allow the children to decide what they
would do and not to intervene in a prescriptive manner, or to readily provide requested or
unrequested factual information, as long as the team seemed to be engaged in the project.
However, as the children write, they finished the letters of the Greek word for "Easter" earlier
than the end of the investigation time. The teacher intervened at that point to encourage the
children to use their creativity and amongst them negotiate expanding their investigation by
adding something relevant to their construction.
The teacher's comments on the essay aim to encourage the children to impose some
structure on the topics they refer to, a problem that is very common in the children's writing in
the school. Not surprisingly, the children have great difficulty in structuring the topics they
write about - most often they either go into extreme detail and stop when they get tired, or they
write vague generalities like the first phrase in the example. In the commands section, it is
evident that there is a lack of discrimination between giving a written title to each section and
showing the graphics it produces, perhaps due to the coincidental matching of the two in this
case. The teacher has consequently asked for the position of the turtle to encourage the children
to think about the structure of their work not only on their commands but also on the graphical
outputs of these commands. Furthermore, she has given them food for thought on using the
REPEAT command, i.e. noticing whether there are similar patterns in their shape.
In general, the teachers' strategies concerning the program's objectives have been
developing mainly as a result of their own common sense, teaching experience and familiarity
with their pupils' specific needs. The seminars and training they receive function as an
opportunity for reflection and for a gradual advancement of their own experience with the
technical know-how and the programming and mathematical content. Finally, the developing
"curriculum" is discussed, and reformed at the beginning of each year.
The Process of Children's Learning
Example 2 is used to discuss the dialectic between the children's working process and their
learning of content, in this case using procedures to program the computer. The example shows
the presentation of a first investigation in the 5th year, the children aged 10 - 11. The
prerequisites here include the use of procedures, but the extent and the way in which they are
used and ofcourse the topic for investigation, is up to the children.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-241-320.jpg)
![231
This was the children's third investigation with procedures (the previous ones were carried
out when the children were still in the fourth year). In the earlier investigations, the children
used procedures only as a means to store information and economize in time and effort; for
instance, the contents of their procedures would be long columns of direct-drive style
commands, only one procedure written within a teaching period and used the following week
as a means to start off where they had left. So, during the investigation in the present example,
the children had progressed to using procedures to sectionalize and begin to structure their
work, even though they still considered procedures as product. Research has shown us
elsewhere [7] that children spend time using procedures as product before gradually shifting
their focus to perceiving procedures as process.
The presentation in the example, reveals how the children began with a concrete aim,
incorporated the writing of procedures in the process of constructing a small part of their project
at a time and discovered after the third session that the figure was too large to fit the screen.
This is the point where their coherent use of programming broke down. Far from realizing that
all that was needed was, for instance, a parameter change in one of the written procedures, the
children started to construct their figure all over again in a new procedure (so that the
prerequisite of using procedures would be fulfilled). What is more, due to time pressure, they
apparently thought it would be quicker to incorporate all the commands in one procedure, in
effect falling back to direct driving within a procedure. A study of the geometrical ideas they
used in their programs indicates that this "regression" was not due to the difficulty they had
with the figures' mathematics; their final procedure (BILLY) indicates that they had worked out
a limited but functional method to change the size of circles by changing the turning quantity of
the turtle (children's localized and limited generalizations of mathematical ideas is discussed in
the following section). So the children's falling back to a more naive use of procedures seems
to have been a step in the process of learning the power of a programming tool by means of
concrete consequences arising from the way in which the tool is used.
Example 2
a) responsible/or the disk: D. M.
b) responsible/or the note book:F. K.
c) responsible/or the presentation book:M. X.
d) responsible/or the writing book o/important notes: M. K.
Investigation 1
Our team is made up by F.K., M. K., D. M. and M.T. During the first day of our
investigation each one of us would say some idea and in the end we decided to
make a bomb. In the first two sessions we had problems about how to make the
circle since we had forgotten from last year. When we made the circle, we made the
fuse. We saw that the screen was not big enough for us and half of the fuse came
out underneath. So we made the circle smaller. After many difficulties and problems](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-242-320.jpg)
![232
which we mention above we managed to make our drawing, the original one. In the
end of the fourth session we printed it
The first day we went to the computers we had many ideas in our minds about what
drawing we would make but in the end we decided to make a bomb. The remaining
time we tried to make the circle but we didn't manage it. The second day, after we
went to the computers we started to draw the circle. That day we had problems with
the procedures but with the help of our cooperation we solved that problem of ours.
The third day we decided to make the fuse. After we made it we saw that half the
fuse came out underneath. But the bell rang and we didn't have time to fix it. The
fourth day we corrected the circle and we made it smaller. In this way, our fuse
came out nice.
lOCH:
lO'MQ
PU
BK50
LT90
FD30
RT90
LT90
FD15
REPEAT 360 [ FD 1 RT 1)
REPEAT 90 [ FD 1 RT 1)
LT90
RT90
FD55
PD
EN>
TOBILl.V
PU
BK50
PD
EN>
REPEAT 360 rFD 1 LT ?I
REPEAT 90 [FD 1 LT 2)
RT90
REPEAT 25 (FD 1 RT 2)
PU
FD5
PD
FD5
PU
BK5
RT90
PU
TOSIX
RT40
REPEAT 40 [FD 1 RT 1)
RT90
PU
FD5
PD
FD5
END
REPEAT 6 [FD 1 RT 1]
LT45
PD
FD5
BK5
RT90
LT 180
LT45
PU
REPEAT 9 [FD 1 LT 1)
RT 135
LT45
PO
FD5
PE
BK5
LT55
FD5
PD
BK5
PU
Figure 2
In this case, it is not clear whether the teacher saw the opportunity to encourage the children to
reflect on their work and to gain some insight into the notion of a procedure as an entity, i.e.
that once written, it always works just as a Logo primitive and can easily be changed or
debugged. Instances such as the above arise very frequently, and the teachers are developing a
means of interpreting the children's work and intervening at moments when factual information
would be meaningful to the children, as in the above case. However, it is not a coincidence that
the "curriculum" for the fifth year focuses on procedure "processing", i.e. editing and
debugging, as part of the children's working method. After having had a first experience with
the syntax and the know-how of writing procedures and using disks to store them and
subsequently using procedures in "naive" ways, most children are ready to bring meaning to](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-243-320.jpg)
![234
concerning tum have a similar pattern, as indicated by the relations they attached between the
number of repeats and the turtle's tum; procedures K2 and K3 involve a total tum of 360
degrees (K being heading-transparent), but procedures K5 and K6 do not seem to relate the two
quantities.
TOK4
TOK:A
FO:A
BK :A
RT90
FO :A
BK:A/2
LT90
FO:A 14
BK:A/2
EN)
TOK2
REPEAT 4 [K 40 RT 90]
EN)
TO
PU
LT
PO
K1
90
REPEAT 4 [ K 60]
END
TOK3
REPEAT 4 [K2 RT 90]
END
I
J.
ｾｐｅａｔＵ｛ｋＶＰ＠ FO 15 LTOO FDSO AT 00]
TOK6
10KS
AEPEAT 8 [K60 FD 15 LT90 FD 25 AT 135)
END
TOK7
K60
PU
F075
RT90
F030
RT90
PO
K60
END
REPEAT 60 [K 60 FO 15 LT 90 FO 50 RT 135]
EN)
Figure 3](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-245-320.jpg)
![235
Finally, the children's limited use of programming and mathematical ideas in connection with
the way they write about their project, indicates their lack of reflection and understanding of the
internal structure of the procedures they themselves build, a finding which supports other
related studies [2, 9]. In their essay, the children write about the need to "work" in order to
write their initial procedure, but refer to the other procedures (which incorporate much more
complicated ideas) as "easy", possibly a result of their exploratory ad-hoc interfaces between
iterations of the figure 4, and their resistance to reflect on the precise consequences of those
interfaces on the graphical output. A likely reason for this may have been that two members of
the group seemed to struggle to keep up with the third, as indicated in their essay. The
children's reflection on their cooperation, however, is an important means by which they learn
to negotiate in the process of their investigational work. Phases of readjustments in their
cooperation, as well as phases of effective communal work as a result of discussing
cooperation problems (the children's writings in example 2 indicate that they were in such a
phase), are an integral part of the program's process-oriented objectives and of course
encouraged by the teachers.
Discussion
"Teaching" the Change of Control
Establishing the use of the computers to bring about and encourage pupil-controlled problem-
solving activity, by no means came automatically. This, of course, was not due to some
specific conceptual difficulty teachers or children had with understanding how they could use
the Logo environment to practice an alternative way of working. What took time was the
process of making the important issues socially explicit within the classroom. For instance, the
teachers gradually developed a meaningful way to communicate to the children why they were
not readily giving them factual answers and why they would often throw the responsibility of a
situation back to the children themselves; encouraging the children in example 1 to continue
with their investigation, but to take responsibility for what they would do next is an example of
an explicit negotiation of this issue both on the part of the teacher and on the part of the
children, since they clearly wrote about the episode in their presentation.
The Working Structure's Receding from Object to Tool
The working structure of carrying out "investigations" and then presenting them on paper was
not immediately employed to encourage investigational work. Indeed, the main reformulation of
the program after the first year, was for the teachers to exemplify the rules of the game to the](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-246-320.jpg)
![238
References
1. Hillel, J. & Kieran, C.: Schemas used by 12 years old in solving selected turtle geometry tasks. Recherches
en Didactique des Mathanatiques, 8(12), 61-103 (1987)
2. Hillel, J.: Procedural thinking by children aged 8-12 using tunIe geometry. In: Proceedings of the Tenth
]nternational Conference for the Psychology of Mathematics Education, pp. 433-438. London 1986
3. Hoyles, C.: Scaling a mountain - A study of the use, discrimination and generalization of some mathematical
concepts in a Logo environment. European Journal of Psychology of Education, 1(2), 111-126 (1986)
4. Hoyles, C.& Noss, R.: How does the computer enlarge the scope of do-able. In: Mathematics. Proceedings
of the Second Logo and Mathematics Education Conference. University of London Institute of Education
1986
5. Hoyles, C. & Noss, R.: Children working in a structured Logo environment: From doing to understanding.
Recherches en Didactiques de MathOOlatiques, 8(12),131-174 (1987)
6. Hoyles, C. & Noss, R.: Seeing what matters: Developing an understanding of the concept of parallelograms
through a Logo microworld. In: Proceedings of the Eleventh ]nternational Conference for the Psychology of
Mathematics Education, pp. 354-359, Montreal 1987
7. Hoyles, C. & Sutherland, R.: Logo mathematics in the classroom. London: Routledge 1990
8. Lawler, R. W.: Computer experience and cognitive development. A child's learning in a computer culture.
New York: Ellis Horwood 1985
9. Leron, U.: Some problems in children's Logo learning'. In: Proceedings of the Seventh ]nternational
Conference for the Psychology of Mathematics Education 1983
10. Kynigos, C.: The tunIe metaphor as a tool for children doing geometry. In Learning Logo and Mathematics
(C. Hoyles & R. Noss, eds.), Cambridge: MIT Press in press
11. Kynigos, C.: From intrinsic to non-intrinsic geometry: A study of children's understandings in Logo-based
microworlds, Unpublished Doctoral Thesis, University of London Institute of Education 1989
12. Kynigos, C.: Can children use the turtle to understand Euclidean ideas in an inductive way? In Proceedings of
the Fourth International Conference on Logo and Mathematics Education. Israel 1989
13. Noss, R.: Creating a mathematical environment through programming: A study of young children learning
Logo, Doctoral Thesis, published by University of London Institute of Education 1985
14. Noss, R. & Hoyles, C.: Structuring the mathematical environment: The dialectic of process and content. In:
Proceedings of the Third Logo and Mathematics Education Conference. London: University of London
Institute of Education 1987
15. Papert, S.: Computer criticism versus technocentric thinking. In: Proceedings of the Logo 85 Conference
1985
16. Papert S. et al.: Final Report of the Brookline Logo Project, (part 2), M]T Artificial Intelligence Laboratory
1979
17. Sutherland, R.: A longitudinal study of the development of pupils' algebraic thinking in a Logo
environment. Doctoral Thesis published by the University of London Institute of Education 1988
18. Weir, S.: Cultivating minds: A Logo casebook. London: Harper & Row 1987](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-249-320.jpg)
![Cognitive Processes and Social Interactions in
Mathematical Investigations
Joao Pedro Ponte, Joao Filipe Matos
Departamento de ｅ､ｵｾＬ＠ Faculdade de Ci!ncias, Universidade de Lisboa, Campo Grande, Lisboa, Portugal
Abstract: Mathematical investigations may be important educational activities, proving to be
useful in the development of mathematical ideas. The principal difficulties may concern
students' content knowledge, reasoning processes, and general attitudes and appreciation. This
paper refers to a computer based investigation undertaken by three eight-grade students,
discussing in special their cognitive processes and social interactions.
Keywords: investigations, computers in mathematics education, cognitive processes,
strategies, conjectures, social interactions
In mathematical investigations students are placed in the role of the mathematicians. Given a
rich enough and complex situation, object, phenomenon or mechanism, they try to understand
it, to find patterns, relationships, similarities, and differences leading to generalizations.
Mathematical investigations range from quite elaborated and complex tasks, that may require a
considerable amount of time to carry out, to smaller activities that may arise by consideration of
a simple variation on a well-established fact or procedure.
Mathematical investigations share common aspects with other kinds of problem solving
activities. They involve complex thinking processes and require an high involvement and a
creative stand from the student. However, they also involve some distinctive features. While
mathematical problems tend to be characterized by well defined givens and goals, investigations
are much looser in that respect. The first task of the student is to make them more precise, a
common feature that they share with the activity of problem posing.
In the process of carrying out a mathematical investigation it is possible to distinguish
activities such as define the objective (what are we trying to know?), set up and conduct
experiences (what happens in such or such specific instance?), formulate conjectures (what
general rules may we propose?), and test conjectures (what may be critical experiences to
ascertain the value of this conjecture? Is it possible to make a proof?).
The realization of mathematical investigations has become a fairly popular curricular
orientation [5, 10]. However, little is known about what happens in the course of mathematical
investigations, specially if carried out in school settings (a point also made by [5, p. 94]). What](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-250-320.jpg)
![240
kind of profit do students take from them? What difficulties do they find? What constraints do
they put on the teachers' role?
The computer, used as a tool, has been proposed as a very useful instrument for carrying
out mathematical investigations. It encourages the realization of a large number of experiences,
allowing the exploration of quite non-trivial situations and issues. It is also of great interest to
know what specific features it may bring to this mathematical activity. These are some of the
questions that we set ourselves to respond in this study.
Pedagogical Context and Research Methodology
As in any other educational activity, in carrying out mathematical investigations, it makes a big
difference the way things are designed and organised. We need therefore to clarify the general
context of the episode, the way the activity was proposed, the role of the teacher (in this case
one of the present authors), and the idea that the students made of their own role in this process.
The activity analysed in this study was carried out in an extra classroom setting at a school
involved in the MINERVA Project (Pole DEFCUL), during the school year of 1988/89.
The students were 8th graders who voluntarily enrolled to work with computers, in the
school computer centre, in a specified weakly time slot of 2 hours. Their relation with the centre
lasted for the whole school year, working in Logo activities and projects. These students had
previous contact with Logo the year before, in the mathematics classroom and before this
episode they developed programming activities for six months, most of them based in projects
of that they designed themselves.
One the students, Maria, was a very good achiever in mathematics and in the other
subjects. The other two, Nuno and Victor, were medium towards low achievers in most school
subjects.
One of the present authors was in the school computer centre during this time slot for the
whole school year and was well known of the students. The activity discussed in this study was
proposed in one session that took place in 18 April of 1989.
The activity was based in a recursive Logo procedure with three variables which draws
peculiar kind of shapes. It shows in an interesting way the effect of recursion in a geometrical
procedure [1, 3, 9]. The students were given a sheet of paper with the procedure, a sample
screen output, and a general formulation of the task (see figure 1). Besides, the researcher
explained the purpose of the investigation and worked out one or two examples with the
students.
During the activity the researcher had a strong interaction with the group, specially in the
beginning and in two or three moments. In the rest of the time the students worked just by
themselves. The activity ended with a general discussion of the results between the students and
the researcher.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-251-320.jpg)
![242
produced, including a scheme describing the main stages of the work of the students within the
activity. From there new observations of the tapes were made in order to clarify new aspects
and produce the final written account.
A Framework to Discuss Mathematical Investigations
Several activities can be identified during the course of an investigation. These can be organised
within three main phases of work which will be now discussed in detail: (a) formulation of
objectives, (b) definition of strategies, (c) reflection on the experiments carried and formulation
and testing of conjectures.
Formulation of Objectives. An investigative task may suggest the setting of a great
multiplicity of objectives. Some may be more general, and others refer to aspects of detail.
Some may be more precise and others more vague. The ability to formulate precise research
objectives is one of the most essential aspects of the ability to undertake investigations.
Significant questions about the setting of the objective of an investigation by the students
are:
-How is the research objective initially formulated?
-Are there turning points in the process of conducting an investigation that can be referred
to change in the overall objective? What can be said about them?
-Are there more general aspects concerning the way they look at the situation that may
change in the process of the investigation?
Professional experience, supported in the literature, asserts that students tend to be not very
good on formulating research questions to investigate in a spontaneous way. Even when
provided with starting points, they may have difficulty in seeing what more general questions
may be asked to extend simple cases already explored [2]. This should be hardly surprising in
view of the overwhelming tradition of teaching well organised and formalised knowledge, that
students are supposed to acquire, and not introducing them to the process of constructing
mathematical knowledge themselves. That is, to teach students "answers" paying no attention to
the "questions" they are supposed to correspond nor to the way they were constructed.
In mathematics teaching the tasks are usually given to the students completely formulated.
What are sensible or senseless questions to ask, what are interesting or trivial questions, etc, is
something to which no attention is usually given. Setting research objectives is therefore one of
the aspects in which students show great difficulty.
Definition of strategies. Strategies used in the course of an investigation refer to three
aspects. The first concerns the representation of the situation (including the identification of key
features and the choice of a suitable notation). The second concerns the key decisions about the
sequence of experiences to carry out, indicating a general line of reasoning. The third has to do](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-253-320.jpg)
![243
with specific tools that are used to construct and interpret the experiences. Significant questions
can be asked about these three aspects:
-Is the representation appropriate (in the sense that it describes important aspects of the
situation)?
-How is the organization of experiences? Are they relevant for the sought objectives? Are
they systematic?
-Is the "technical knowledge" of the students preventing them of devising and organizing
a sensible strategy?
Devising appropriate representations and mathematical notations has been widely
recognized as an essential element for carrying out mathematical investigations [4, 8, 13]. Not
all the representations of a given situation can offer the same insight. Some offer more than
others. It is common that students develop more than one kind of representation and fluctuate
between them [2].
Investigations are often regarded as good starters for mathematical work. However, it
should not be overlooked the fact that "investigational work often rewards mastery of
mathematical technique with success, and punishes mathematical inaccuracies heavily" [13, p.
114-115].
Reflecting on the experiences and formulating and testing conjectures. The
realization of experiences should lead to a reflection on the situation, gaining insight on it,
perhaps revising some aspects of the earlier conceptualization and hopefully to doing some
conjecturing.
The results of the experiences performed can be used to better understand the situation and
draw up conjectures. The conjectures, once formulated, need to be tested.
The processes of conjecturing and testing form a cycle that may run for several times.
Sometimes the students come out the cycle to modify some aspect of the set up of the
experiences. Sometimes the students may feel the need to come even earlier and modify the
overall research goal.
Testing can also take different forms. It can be test of specific chosen cases, testing of
random cases, or attempts to a proof.
Besides our interest in these aspects of the process of conducting investigations, we were
also specifically concerned with two further issues: (a) The role of the computer in mathematical
investigations and (b) Social interactions.
The role of the computer. These investigations where proposed to the students
assuming that the computer would be used to help performing them. If fact, in this activity, it
would be difficult to see the work being carried without the computer.
One should therefore ask what are the consequences of using the computer in the working
processes of the students. Some of the possible consequences of the computer concerning this
kind ofmathematical work are well-known:
-It allows a great number of experiences, encouraging strategies where making a good
number of experiences is an integral part.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-254-320.jpg)
![244
-It allows feedback in different kinds of representations.
-It facilitates the dialogue, since it becomes a new pole of attention. What is done in the
computer is not individual property but public.
Ifthe students are programming themselves, as often happens in Logo activities, the act of
translating an intuition to a program makes it become more obtrusive and therefore more
accessible to reflection [II]. However, in this case the program was already made and the
students just interfered with it for changing some of its minor features.
One should note that the computer offers means of representation that are powerful but
limited. With the computer it is possible to do many things, some of them quite extraordinary.
But computers are limited in what they allow to represent, and they may prove to be unsuitable
for some purposes.
Social interactions. One of most common features of the use of computers in
mathematics education is a change towards group work. Investigations tend also to be
suggested to be performed in groups preferably to individually. However, there are satisfying
and less than satisfying situations of group work. Another important partner in the learning
process is, of course, the teacher.
Therefore significant questions are for example:
-How does the relationship with the teacher and the colleagues interfere (positively,
negatively) with the development of the task?
-How far is carried the process of arguing? Do the students articulate arguments or just
statements? Is there listening to others' arguments?
-Is there a search for group consensus or one takes the lead and determines the course of
the group?
-What seem to be the implications of the situation of social interaction among students
(what seem to be positive effects? negative effects?)
-Why do some students seem to have more initiative than others? Why do some students
seem paralysed? Why are some students apparently not able to take profit from the fact that they
are in a social interaction situation?
The Investigation on Inspirals
In this activity we may distinguish 8 different segments, in which there was a significant
turnover in the course of the events. All of the transitions between segments are characterized
by a change in the composition of the group.
Segment 1. The task begun with two students, Maria and Nuno, and the researcher, who
handed the sheet with the situation, presented it in general terms, formulated the objective, and
gave a suggestion to get them started. This segment lasted for about 2:30 minutes.
The objectives stated in the sheet concerned the nature of the figures that it is possible to get
and asked for a theory about the number and the kind of enrollments. These objectives were](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-255-320.jpg)
![252
a completely unrelated thing, giving rise to no discussion among them, and then they just
moving on. Finally, even these interactions became less frequent.
In general, Maria tended to assume that Nuno was making an interpretation similar to
her's, or she did not even try to understand his view on the successive results obtained at the
computer. Victor had no relevant intervention in the development of the activity.
Maria led the investigative process, making most of the suggestions which were then acted
upon as collaborative decisions, or by making the decision on her own.
From segment 4 on, until the end of the activity, Maria took an even more important role,
taking decisions and reflecting on their results. Nuno was accompanying her activity, but not
really intervening in the decision making. And from time to time he was speaking with Victor of
subjects unrelated to the activity.
The absence of collaborative work was identified since the beginning of segment 4. With
the progress of the investigation Maria seemed to become more confident about what she was
doing and assume that the other students would not be of much help. In this way, the attempt to
an explanation of the behaviour of the turtle by Nuno in segment 4 was lost as a discussion
opportunity.
Personality factors may constitute the main reason for the nature of the students'
interactions. Additional reasons may have to do with their increasing awareness of the
complexity of task and also with the fact that the researcher, in his moments with the group, had
more interchanges with Maria.
Although absent for most of the time, the researcher (who was supervising the work of
other two groups) had an important role on this activity, mostly in the definition of the
objectives, in the adoption of general strategies, and in the suggestion of specific strategies (in
this case "computer tricks").
Conclusion
One should not underestimate the difficulties of the students in investigating complex situations.
We know that making significant discoveries in mathematics is difficult enough for
mathematicians [6, 7] and we should not forget that they are strongly motivated for their
subject. Rich environments, like this one, entail many complexities and students are likely to
find many embarrassments with them and are not necessarily highly motivated for mathematics
[12].
But, by the other side, such difficulties have their reverse. They provide good
opportunities for discussion and reflection, reveal misconceptions, and promote an awareness
of global issues that may become significant for the progress of the students. What happened in
this episode with rotations over 180 degrees is quite illustrative in that respect.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-263-320.jpg)
![Aspects of Computerized Learning Environments
Which Support Problem Solving
Tommy Dreyfus
Institute of Mathematics, University of Fribourg, Switzerland1
Abstract: Examples of students' problem solving processes using computerized learning
environments are described in some detail. These descriptions serve as basis for abstracting
features of learning environments that support specific aspects of problem solving, mainly
control processes such as planning, switching one's point of view, or deciding to work
backward; flexible approaches during conjecturing or search for solution paths are also focussed
on. The following software features were identified as particularly relevant: tool character, non-
evaluative feedback, and operational structure; this structure should include a limited number of
operations, predominantly transformations modelling the underlying mathematical domain.
Keywords: computerized learning environments, problem solving, problem posing, planning,
control, conjectures, flexibility, software tools, operational structure, Stereometrix, Triple
Representation Model
Introduction2
According to Schoenfeld [4], a problem-solving situation is one where the solver does not have
easy access to a solution for solving a problem but does have an adequate background with
which to make progress on it. Schoenfeld investigated students' behavior while they thought
about problems with substantial mathematical content. He asked questions such as how the
problem solver decides which mathematical knowledge to access, and how to use it. These are
questions of heuristics and control in problem solving, but also questions of resources - in
particular the mathematical resources available. Students often have a hard time with anyone of
these aspects: heuristics, control, and resources; but difficulties become compounded, when all
three need to be handled simultaneously. Computerized learning environments can help students
handle such situations by providing mathematical resources in a directly accessible form and by
making available to the students tools that support their heuristic steps and their control of the
problem solving process. In this paper, three examples of problem solving processes with
IOn leave from the Center for Technological Education, Holon, Israel
21 would like to acknowledge the contributions of Nurit Hadas and Baruch Schwan to this paper. Our interaction
during the time we collabprated on the development of and research with Stereometrix and TRM has greatly
influenced the formation ofthe ideas presented here.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-266-320.jpg)
![256
computerized learning environments will be described; these descriptions will then serve as a
basis for discussing general features of the environments that are supportive of heuristics and
control during problem solving.
Spatial Geometry
The Stereometrix Software
Stereometrix is a computerized learning environment for spatial geometry, in particular for the
geometry of standard solids such as cubes, pyramids, and prisms. Work within Stereometrix is
operational, i.e., it is organized in operations to be performed by the student. Two main types of
operations are available in the environment: Constructions and transformations. Constructions
change a given solid, e.g. by adding points or lines to it. The most important construction which
can be carried out by means of the software is that of the perpendicular from a point to a line or a
plane. Others are copying segments, finding midpoints, connecting or disconnecting points,
bisecting angles and intersecting lines. A rather particular "construction" enables one to change
the shape of a solid by moving a point in such a way that incidences are preserved.
Transformations do not change the solid but its position in space and thus its representation on
the screen; in particular, it is possible to rotate the solid around either of three fixed axes; but it is
also possible to rotate it by specifying the desired final position, namely specifying a plane that
shall be parallel to the screen plane. Stereometrix also has a replay feature, which allows one to
save the sequence of operations one carries out and then reuse this sequence on the same or on a
different solid.
Stereometrix has been used with some success to help students visualize three-dimensional
solids, to support them in forming basic concepts of spatial geometry such as the angle between
a line and a plane, to encourage them to formulate and check conjectures about solids, and to
assist them in solving problems of varying complexity. Some of these problems are computation
problems as they might occurin a standard curriculum; others go beyond that level, and should
be considered as rather extensive projects. A more detailed description of Stereometrix and its
possible uses has been published elsewhere [2], Here, we will concentrate on the description of
two episodes that highlight the support that Stereometrix gives during problem solving
processes.
Example 1: Problem of the elusive volume: Given a right prism ABCDEF with an
isosceles triangular base ABC, compute its volume from the following data: The
base AC of the base triangle ABC, the base angle a=CAB of the same triangle and
the angle g between one of the two equal "walls", ADEB, and the diagonal AF of
the third wall, ACFD (see Figure I),](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-267-320.jpg)
![258
pencil drawings made by students at such a stage are often inexact and thus confusing;
moreover, such drawings are fixed, whereas the computer screen image can easily be
dynamically turned to be viewed from an angle from which the most important details can be
seen best. This is the region within which the relationship between the given data and the
missing altitude has to be built; and by implication, the importance of this region goes far
beyond: this is the region, within which possible plans for solving the problem have to be
established and their feasibility checked. The students are thus given a tool which enables them
to take control of their actions and to think at a level of planning: For instance, if I am given g,
and g is an angle in triangle AFG, I could possibly compute other quantities in that triangle; but
the missing altitude DA=FC belongs to triangle AFC, of which I know only AC. But wait: AF is
common to the two triangles...
Above, specific features of the Stereometrix environment and their importance for planning
and control have been pointed out; Stereometrix also incorporates some general features of
learning environments which have a bearing on problem solving: Tool character of the software,
students' control over their actions, advantages offered and limitations imposed by the
operational character of the environment, and type of feedback. These will be discussed below.
Example 2: Problem of the intersecting altitudes: Do the four altitudes of a
tetrahedron meet in a single point?
This problem is non-trivial by most standards. In fact, Davis [1] cited Coolidge to the effect that
most highly educated mathematicians do not know the answer. While it is not difficult to
convince oneself that the altitudes of a regular tetrahedron intersect in a single point, anything
beyond that is anybody's guess. In other words, whereas the "elusive volume" problem is a
textbook problem with a guaranteed answer, the tetrahedron problem puts the solver in a
situation where heuristics will necessarily playa central role: Not only is the answer unknown,
the problem might not even be properly fonnulated. This is a problem solving situation, in
which one first has to come up with a conjecture. Some experimenting might help to find such a
conjecture, but for experimenting one needs tools which are adapted to the problem without
being too constraining. Stereometrix provides construction and transfonnation tools in spatial
geometry without either evaluating or severely constraining the user's actions. It is not difficult
to decide what to do frrst: One wants to check many tetrahedra, construct the altitudes in each
one of them and check whether they intersect. Two features of Stereometrix help in this
endeavor: The replay feature which enables one to do the lengthy construction of the four
altitudes one single time and have it repeated automatically on all the following tetrahedra; and
the special rotation which enables one to tum a specific plane of the tetrahedron into the screen
plane and check whether the suggested intersection point exists there. But more than that: It
would be nice if this experimental heuristic stage were organized in a systematic way. And
again, like any well organized learning environment, Stereometrix provides the structure, the](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-269-320.jpg)
![260
Functions
The Triple Representation Model Microworld
The Triple Representation Model (TRM) is a computerized learning environment about
functions. The environment as well as an associated curriculum and several possible activities
have been described in some detail elsewhere [7]. Its description here will therefore be limited to
what is necessary in order to describe its use as a problem solving tool.
Work within TRM is possible in three modes: Table, Graph and Algebra; each mode
corresponds to a functional representation. The work within any mode is operational, like in
Stereometrix. The most important operations are Find-Image in the tabular representation, Draw
in the graphical one, and Search in the algebraic one. Once a function f is defined, Find-Image
yields the value off(x) for any given x in the domain that the user chooses to specify, and lists
(x, f(x» in a table. Draw generates a graph of y=f(x) on the screen; before using Draw, the
student has to specify a window of values for x and y within which the graph is to be drawn.
The Search operation enables the student to check algebraic conditions for a large number of
equidistant values of x, for example: "From a to b step d: if f(x»C print x", where the student
has to fix the lower bound a and the upper bound b of the search, the step d, the type of
comparison (>, <, = or ;t), and the goal value C. The Search operation prints on the screen the
values of x for which the condition is true.
Passage between representations in TRM is semi-automatic: The student has to specify the
mode (representation) in which (s)he wants to work but is free to switch to another
representation at any time; the link to other representations is established by operations named
Read, which allow one to consult results previously obtained in one representation while
working in another one. Thus information may be consciously transferred from anyone
representation into any other one, but such transfer is not made automatically by the software.
The features that the student has to explicitly specify the representation and explicitly transfer
information between representations are based on pedagogical design considerations: Beginning
students should be consciously in control of their choices and actions. Below, it will be shown
that these same features are directly related to corresponding control over problem solving with
TRM.
Example 3: Solving a third order equation: Find a solution of 3x-x3=1 with an
accuracy of 10-4.
Students were presented with this problem when they were about three months into a TRM
based introductory functions curriculum; at that stage, they were fairly familiar with the relevant
features of TRM and the multi-representational notion of function as underlying TRM, but they
had not yet solved any equations beyond linear ones. Many of them quickly wrote a Search](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-271-320.jpg)
![261
operation of the fonn "From a to b step d: if 3x-x3=1 print x", with different choices of a, b, and
d. Search conditions of this type did not yield any answers. Although they did not understand
the real reasons for this (namely that the solution of the equation is irrational whereas all values
of a+nd are rational), they were thus forced to face the problem whether there is a solution to the
equation at all. The possibilities of TRM gave them concrete ideas how to proceed at this stage.
The structure of TRM is important here: The three separate representations of TRM imply the
suggestion to use another representation, or, in tenns of the problem to be solved, to rather
radically change the point of view. From the point of view of the problem to be solved, this is an
action of control. Such actions are explicit in TRM: The student has to consciously choose the
representation to be used. Such actions at the control level are supported through the design of
the software; they were also strongly used in the accompanying curriculum. Many students thus
switched representations at this stage, and decided to draw a graph of y=3x-x3• From the graph,
they could learn that there are three solutions to the equation, one of them between x=O and x=1.
At the same time, they had the opportunity to experience the power of using several
representations in conjunction when solving a problem. On a more remote level, and from the
mathematics educator's point of view, they had the opportunity to experience the heuristic
efficiency of actions at the control level. Obviously, the problem was not solved for them yet.
One way to progress toward a solution was to zoom in graphically until a solution was located
with some accuracy, say between 0.3 and 0.4, and then to return to the algebraic representation;
then there was an opportunity to discuss why the original Search operation did not produce the
solution, and to suggest a more sophisticated one such as "From 0.3 to 0.4 step 0.0001: if 3x-
x3>1 print x". The first x which satisfies this condition, x=0.3473, is the solution with the
desired accuracy. But it is more important here, to stress the combined effect of the curriculum
and the software on students' problem solving behavior. Students were, over a period of several
months, regularly presented with carefully chosen and sequenced problems which exposed them
to situations in which actions of control were necessary. They were supported, by the
tremendous power of TRM to use these opportunities to take action at the control level; although
we have not directly examined to what extent they became conscious of such action, we did
observe 42 students, one by one, at the end of the instructional period in a problem solving
situation; of these, 32 successfully solved a maximum problem of rather high difficulty [6].
Relevant Features of Learning Environments
One of the most frequent phenomena observed when students are asked to solve complex
mathematical problems is that they are inactive, they seem to be stuck; two possible reasons for
being stuck are that they do not have any idea what they could do next or that they do not dare to
try an idea they might have. The examples described above point to the potential of computerized](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-272-320.jpg)
![264
are more radical than those offered by Stereometrix: TRM not only allows one to choose a
completely different representation for the same function but, in fact, has been designed
specifically with this purpose in mind; such change ofrepresentation is the most important single
operation offered by TRM; and systematic interviews have shown that students familiar with
TRM have used this feature extensively in solving complex problems [5]. It is through such
radical change of point of view that students were able to progress toward solving the third order
equation; and it is because of the easy availability of such change of representation through TRM
that even beginning students had access to powerful problem solving techniques. Again, the
operational organization of the environment, the availability of transformation operations are
obviously essential here. It is an open question whether for questions of flexibility, it is essential
that control be with the student. One could imagine an environment similar to TRM, in which
several representations would always be presented on the screen and all information would be
transferred automatically from one representation to the others. It is an interesting question how
such a change would affect students' problem solving behavior, in particular their concurrent
use of several representations.
Conclusion
In summary, a number of features of learning environments support students in the problem
solving skills of planning (control), conjecturing (heuristics), and flexibility (control and
heuristics). These are the following features: the software is a tool, organized into operations;
often, such operations have the character of transformations which help changing one's point of
view. The number of operations is limited (in accordance with students' age and ability). The
students thus can control their actions, and take decisions within a manageable framework. The
feedback of the software is only in response to student action and is neutral: It does not evaluate
the students' actions. Thus, it does not directly influence their planning; it gives "objective"
mathematical information, and this information supports the student in decision making, without
prejudicing the decision.
Several aspects critical for problem solving in computerized learning environments have not
been discussed in this paper; some of them will now be mentioned in brief remarks that do not
do justice to their complexity.
(i) It has been stressed above that the feedback is objective in the sense of not
evaluating the students' actions. Some care in interpreting the word "objective"
needs, however, be taken: the feedback is necessarily influenced by choices made in
the design of the software, in particular by how mathematical concepts are
modelled, which representations are implemented, how a solid is represented. The
model in the learning environment will always be a model of the mathematical](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-275-320.jpg)
![265
concepts, and not more than that. This problem is often neglected; one notable
exception is Cabri-gwmetre (see [3]).
(ii) The argument for having a limited number of operations in a learning
environment can be inverted: Because some transformations and constructions have
not been included in Stereometrix, they will most probably not be used by the
students. Similarly, because some operations on functions are not available in
TRM, students working with TRM are unlikely to even think of them. Thus, each
learning environment, in addition to offering a choice of possibilities, also imposes
limitations and constraints. Obviously, this is unavoidable; therefore the designer of
any learning environment has to take certain decisions which are likely to be
influenced by considerations of learning goals, student population and curriculum.
The solution of non-routine problems may but need not be among the goals.
(iii) It has been repeatedly stressed above that students are supposed to be familiar
with a learning environment before attempting to use it for solving complex, non-
standard problems. While this seems a fairly trivial requirement, it has some far-
reaching implications: Because of their character of not imposing any action on the
student, learning environments by themselves do not teach. They only make sense
within a curriculum. This may be a problem based curriculum [8]. Such a problem
based introductory functions curriculum has been developed for TRM [7].
Heuristics and control during problem solving activity demand the availability to the problem
solver of a set of appropriate mental tools. Often, students are weak problem solvers, because
they lack such tools. Several cases have been discussed, in which the availability of concretely
modelled mathematical operations within a familiar computerized learning environment helped
students solve complex non-standard problems. The mathematical operations of the learning
environment served as tools for the student which they were able to shape into the necessary
mental tools. The operations are concrete and accessible to the students, the mental tools are
abstract and removed, but apart from this they are often closely related: For example, the
operation of switching a representation is a concrete action in the environment, and a parallel but
rather removed mental tool in the abstract formulation. In many cases, the students succeeded to
map the ones to the others and thus to construct meaning (see also [5]). Operations in a learning
environment become cognitive tools for the student problem solver. Thus the fact that the
software puts at their disposal a limited number of well defined, specific operations from which
to choose (like from a set of tools in a toolbox) leads to the students being stuck less often and
less deeply: they have means to extricate themselves from being stuck and to progress toward
solving problems in a meaningful way.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-276-320.jpg)
![268
General Model
We consider an autonomous agent that possesses knowledge and is capable of applying a part
of this knowledge to try to solve problems. An agent can be a person (a student, a teacher), an
idealized person (like the Anderson's ideal student) or computer software (or a part of
software).
Algebraic Problem Domains, Behaviors
Definitions. An algebraic problem domain2 is an n-uple ( 'll' , lB , lP , 18 ) in which:
'll' is a problem type (like equation solving),
lB is a set of well formed expressions,
lP is a set of problems which is a subset of lB,
18 is a set of possible behaviors.
A behavior is a search tree developed by the reasoning process of an agent in which nodes
are expressions. The root is an element of lP (the problem to be solved), the other nodes are
generated by the application of transformations to nodes of the tree according to strategies. Each
transition of the tree is labelled by the applied transformation.
A successful behavior is a behavior in which at least one node is labelled by the indication
solved (which means that the problem is solved and the nodes labelled by solved contain
solutions). A failed behavior does not include any solved node and is labelled by the sentence
the problem is unsolvable (which means that the agent has the knowledge to recognize some
states as unsolvable) or the sentence I abandon (which means that the agent does not want to
continue searching for a solution).
A partial behavior is a part of a behavior. Given a behavior, there are many classic ways to
generate partial behaviors: transitions may be not labelled, the tree may be reduced to one of its
branches leading to a solution, it may have not termination indication, etc.
Remarks. This definition of behavior includes the productions usually erased or struck
off in a traditional paper-pencil resolution by a person, but it does not involve all the elements
that can be seen on a sheet of paper, in particular remarks about properties of expressions like
this is a polynomial ofthird order or remarks about strategies like now I will try to reduce are
not included in this definition.
The nature of the links in the tree is succession: step A generates step B. Different meanings
can be associated to these successions: production of equivalent problems by the application of
transformations, generation of a subproblems, etc.
2 In comparison with Dillenbourg and Selfs definition [9] of domain problem, we introduce the problem type
and we make explicit the concept of behaviour for algebra.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-279-320.jpg)
![270
Knowledge State
Definition. We consider Balacheffs definition [3] of knowledge state of an agent in an
algebraic problem domain in which: (1) we exhibit the behavioral knowledge component as
Dillenbourg and Self [9] do; (2) we abandon the temporal indexation.
A KS (knowledge state) is described by a n-uple ( C , B , L , P ) in which C is the
conceptual knowledge, B is the behavioral knowledge, L a set of signifiers, P a set of
problems. C , B , L are effective tools for solving the problems of P.
B is decomposed in ( T , M , Q, S ) where:
T is a set of transformations,
M is a set of matching procedures,
Qis a set of calculus procedures,
S is the strategic knowledge,
C contains the concepts underlying B and a solution predicate s.
The existence of pieces of knowledge described here is necessary for any agent capable of
solving problems: as he produces a behavior, he has to apply transformations to expressions,
which involves knowledge in transformations (T), knowledge to determine when
transformations are applicable (M), knowledge to apply transformations (Q), and knowledge to
choose transformations (S). The existence of the solution predicate s is necessary to evaluate
new nodes and to stop the reasoning process. The form of pieces of knowledge and the
connections between them can be modelled in many ways. Some will be proposed later.
Any KS can include correct and incorrect knowledge at any level: correct behavioral
knowledge, bugs in procedural knowledge (bugs in Buggy [6], mal-rules in LMS [16, 17],
bugs in ACT [1], misconceptions [18], etc.).
A KS can be deterministic or not. In the second case, the strategic knowledge includes a
random component, two resolutions of the same problem can generate different behaviors.
Links between a KS and its associated algebraic problem domain. W e
consider an algebraic problem domain (T ,18, P ,18) and a KS, KSI=( C, B ,L, P) for an
agent, in this domain.
KSI has the basic function of being applicable to the problems of P. The application of
KS I to a problem of P starts a reasoning process which generates a behavior with success or
failure. P is composed of problems on which KS I is effective. L contains signifiers usable for
the agent, 18 contains signifiers usable for the problem domain.
Example. For factorization of polynomials defined above, here is a partial description of
the KS of an imaginary agent IA.
BEHAVIOURAL KNOWLEDGE
Transformations:
factor A out of B](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-281-320.jpg)
![272
EXAMPLE OF BEHAVIOUR
Given the problem <factor (X-2)(X-I)+(X-2)(X-3)-18> an example of partial
behavior of this KS is:
(X-2)(X-l)+(X-2)(X-3)-18
factoring out X-2 in (X-2)(X-l)+(X-2)(X-3)
(X-2)(2X-4)-18
factoring out 2 in 2X-4
2(X-2)(X-2)-18
reduction of (X-2)(X-2)
2(X-2)2_18
factoring out 2 in 2(X-2)2_18
2[(X-2)2-9]
factoring (X-2)2-9 as a difference between two squares
2(X-2-3)(X-2+3)
reduction ofX-2-3 and X-2+3
2(X-5)(X+l)
solved
This tree is a one branch tree; links between nodes are implicit.
Organization of KSs
Behavioral knowledge has to be organized in order to be efficient. It can be structured in
different ways by general concepts like heuristics, goals, tasks, or plans. We present below
different organizations (in a nonexhaustive form).
All the strategic knowledge in heuristics. We define heuristics as small pieces of
knowledge used in making choices.
In an all the strategic know/edge in heuristics organization, we consider that T and M
contain no strategic component and that all the strategic knowledge has the form of heuristics.
Heuristics are small pieces of knowledge. This means that no heuristic envisages the entire
situation and that a choice is the result of the application of several heuristics.
The KS of agent IA has this characteristic if we suppose that there is no strategic aspect in
the matching knowledge M.
This organization is well structured according to separation into strategic/non-strategic
knowledge. Strategic knowledge can be described easily because of the small size of its
elements, strategic knowledge is not structured. This organization is not very efficient:
transformations are envisaged on many subexpressions and heuristics have to deal with more
information.
A part of strategic knowledge in transformations. Conditions can be attached to
transformations. For example, the transformation A2_B2 -> (A-B)(A+B) is applicable to 9-4
according to some matching knowledge. This application is not interesting for many problem](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-283-320.jpg)
![273
domains, the transfonnation may be associated with the condition the expression is not
constant.
Goals. Strategic knowledge can use explicit goals and subgoals. This is familiar in AI
conceptions in many domains with reasoning processes that realize decompositions of a goal
into subgoals and construct trees (for example and/or trees). In algebra, goals are generally
associated with other strategic feature, in particular with plans.
Static plans. We define static plans as predefined combinations of actions to realize one
goal. A static plan is a recorded piece of knowledge and not the result of a planning process
(which is a dynamic plan). Static plans can have different structures: combinations of actions
can be limited to sequences or can be more complex, including alternatives. For example plans
can be generated with or without the constraint ofinvoking only actions that succeed.
CAMELIA [19] is a system using plans for solving problems. In CAMELIA plans can be
very complex and can invoke actions that fail. Actions consist of immediate actions or subgoal
choice.
An example of a CAMELIA plan in the domain of primitive calculation is:
FOR the calculation of the primitive of F in a variable R
IF the main operator ofF is + DO
I) generate a variable R1
2) instantiate R with 0
3) FOR EACH term T ofF DO
a) calculate the primitive ofT in RI
b) add RI to R
4) free RI
This plan is an executable piece of knowledge for the calculation of the primitive of a sum. It
can be applied to any sum (which requires the use of a FOR EACH operator in the plan). It
invokes subgoals: calculation of primitives of tenns, addition of expressions. It can fail (if a
primitive cannot be obtained).
The ALGEBRA TUTOR [2]3 is another system using plans and goals for tutoring a student
solving an equation. Plans are sequences of actions which consist of immediate actions or
subgoal choices.
An example of an ALGEBRA TUTOR plan is:
IF the goal is to rewrite an equation with a subexpression distributed
THEN set as subgoals
(1) find the coefficient associated with the subexpression
(2) multiply the parenthesized part by the coefficient
(3) replace the subexpression by the production.
3 See also the TEACHER'S APPRENTICE [12] which is the previous name of the same system.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-284-320.jpg)
![274
Dynamic plans. Dynamic plans are plans which are elaborated during the resolution
process, taking into account the search space and the goals. They are generated by strategic
knowledge. They can generally be seen as skeletons of the search space (or of a part of the
search space) that will be developed afterwards. As far as I know, this kind of plans is not
implemented in the current ll.Es in algebra.
Tasks. We define a task" as an executable set of knowledge to realize one goal. The
difference between static plans and task is the following: a task involves all the knowledge
concerning one goal, it can be seen as a sub-KS; a static plan involves only a part of the
knowledge concerning one goal, this knowledge is organized in a procedural way.
In the current version, APLUSIX has an organization with tasks, static plans and heuristics
for the knowledge used when examples of resolution are generated. Each task has plans and
heuristics; plans invoke tasks or immediate actions. For the factorization of polynomials, the
following constraints have been chosen: plans involve only sequences of actions; actions
always succeed.
An example of an APLUSIX plan in the domain of factorization ofpolynomials is:
IF
AND
AND
AND
TIlEN
a subexpression E of the problem is a sum
an expression U can be factored out in a part ofE
the partial factorization of U in E produces a new expression V
V is a possible factor of another part of E
factoroutU
arrange the result of the factorization
factor out V
arrange the result of the factorization
This plan invokes four successive tasks. When the plan is applied, factor out performs the
selected factorization, arrange realizes some usual developments and reductions after a
factorization. Each task may be realized in several steps.
Control in static plans and tasks. Static plans and tasks correspond to a
proceduralization (or compilation) and structuration of knowledge. The natural way to install
control in static plans is to oblige their execution, i.e., when a static plan is started, no control is
applied, the next step is always executed when no failure occurs. This is the case of the
examples presented below. This kind of control is not psychologically plausible without certain
constraints: if a subtask involving complex reasoning is started, nothing can stop it (it may
succeed or fail within too much time and may even enter an infinite loop). ALGEBRA TUTOR
and APLUSIX solve this problem by using only small subtasks.
4 We use the AI meaning of task [8] which is the description of a process or a process which accomplishes a
rask in the general meaning of wk.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-285-320.jpg)
![276
Production of examples. Many ILEs envisage a short learning stage. Then a reference
KS is a good model for the production of examples. When a wider learning stage is to be
treated, the reference knowledge generally evolves so that a unique reference KS is maladapted.
This evolution can be realized by considering a sequence of KSs. It is this way we used in
APLUSIX. This method is not very suitable because on one hand using few KSs implies
important differences between them and on the other hand using a lot of KSs is difficult to
manage. The genetic graph [11] uses an evolutionary reference KS: a unique representation
implements several KSs with links allowing to find what knowledge can be learnt at any time.
This model has been used in the WUSOR software in a domain involving a small set of
knowledge in a monotonous context. It is an interesting idea, but the elaboration of
evolutionary KSs for large sets of knowledge in non monotonous contexts is a complex
problem6.
Production of explanations Explanations can be generated using reference KSs.
These are model-based explanations. First, a behavior of a KS is an explanation. More
explanations can be given during a problem resolution, at factual and strategic level, using
explanation knowledge. Second, explanations can be produced in describing reference KSs.
We can separate explanation models according to their means and their goals:
- With one reference KS and explanation knowledge, one gets an epistemic explanation
that can exhibit features of a reasoning process according to general characteristics (like
importance of a piece of knowledge in a KS) or describe a reference KS.
- With an evolutionary reference KS, it is possible to take into account the differences
between the current KS and the previous ones, giving priority to recently learned knowledge
for the production of explanations. This is a genetic explanation. It needs to know the previous
KSs used or to form hypothesis on them.
- With a student's model, one can get a personalized explanation in which information on
the student, differences between the reference KS and the student's model are used to select
pertinent components of explanationsfor that particular student.
What is the correct model for an ILE? It depends on the context in which explanations have
to be elaborated. For example, if a teacher uses an ILE in a classroom, epistemic explanation is
pertinent when the reference KS fits the classroom level; genetic explanation ought to be better
if the evolutionary reference KS can be adapted to the evolution of the classroom. Of course,
when a student learns alone with an ILE, a personalized explanation is the best model; however
its quality depends on the quality of the process building the student's model.
6 The learning ofalgebra involves a nonmonotonous evolution of the references knowledge, examples: (1) when
only rational numbers are known, X2-2=O is an equation without a solution; when irrational numbers are
known, it has solutions; (2) in factorization of polynomials, when the discriminant is not known, developing
second order polynomial is a mediocre strategy; when the discriminant is known, it is a good strategy.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-287-320.jpg)
![277
Control of the student. When a tutor has to evaluate the behavior of a student, he has
to show what is correct/incorrect, legaVillegal, authorized/forbidden according to the didactic
contract [4]. In a traditional context, the didactic contract is mainly implicit; in an ILE, it must
be entirely specified. Its task is to accept or refuse the action (or the request) of the student, and
to give appropriate feedback.
One way to control the student's actions is the model tracing methodology [2]. Model
tracing uses a deterministic reference KS and a set of bugs (which is a KS*) to evaluate the
student's action. With this methodology, the student is not allowed to use a path different from
the reference KS. In fact model tracing is suitable for learning skills in problem domains in
which skills are sufficient. When no deterministic reference KS is available or when a
discovery approach is used, the control of the student's actions requires another methodology.
We propose to model the control with two KS*. The first is the epistemic control which
verifies the syntax of expressions, the correctness of the transformations, the applicability of
transformations on expressions, etc. The second is the didactic control which executes controls
not directly related to the scientific knowledge. For example:
Factor out X-2 in (X-2)(X-5)+(X-3)(X-I) is judged as an incorrect request at
epistemic level.
Factor out X-2 in (4X-8)(X-5)+(3X-6)(X-I) is judged as a correct request at
epistemic level; it can be judged as an incorrect one by the didactic control if the
didactic contract requires all factors to be identical for the factor out operator
(which is a plausible didactic contract at some learning stage).
Guiding the student. Guiding the student can be envisaged according to several
principles. Using a deterministic reference KS, the model tracing methodology achieves strong
guiding. At the opposite, controlling the student's actions according to a didactic contract
without adding guiding, allows the student to evolve in a sort of algebraic microworld.
Given a problem, a reference KS is basically capable of generating its own behavior with,
at any node, the best action to proceed. If the reference KS is an extended KS i.e. it has in
addition the capability of evaluating any action in any behavior, a parametrized model based
guiding can be designed as follows: at any time, the student's action is compared to the
reference KS action in terms of distance, the student's action is accepted if this distance does
not overstep a chosen level (the guiding parameter), otherwise it is refused.
Help. We consider here the help asked for by a student when solving problems in an ILE.
A model based help consists in using a reference KS to provide this help. When no strong
guiding is realized, an extended KS is required for providing this help because the KS has to
evaluate a behavior different to his own.
Help also can be based on performance by using a reference KS to try to solve the problem
from the current node without taking care of the others. This kind of help can be effective,
however its effect is in finishing the current resolution; a model based help is supposed to
produce knowledge.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-288-320.jpg)
![278
The APLUSIX Project
APLUSIX is an IACI project in algebra, currently developed in the domain of factorization of
polynomials. This domain is treated at three levels and involves complex strategies. Currently,
the software appears as a learning environment including two learning modes: the learning-by-
example mode in which the system shows how it solves exercises and can be asked for
explanations at a factual or strategic level, and the learning-by-doing mode in which the student
solves exercises. Experiments have been conducted in France in March 1990 and in April 1991.
APLUSIX is implemented on a Macintosh. It uses a LISP environment (Le_Lisp) and the
inference engine SIM [13].
Other learning environments have been developed in algebra during the last few years.
Most of them are concerned by equation solving at a rather low level of knowledge in which
few context-free heuristics [7] are sufficient for solving problems. This is the case of
ALGEBRALAND [5] and ALGEBRA TUTOR [2,12], two systems involving two different
orientations. In ALGEBRALAND the student uses operators in a menu to transform and solve
equations (this interaction mode is close to the learning-by-doing mode of APLUSIX). The
student can perform backtrack when he wants. The search space is clearly represented as a tree
at the interface, reifying the resolution process of the student. Strategies have to be discovered
by the student. In ALGEBRA TUTOR the student activity is controlled by a tutor which uses
an ideal student model and buggy rules. The main purpose of the work with this system is to
learn skills according to the ACT theory [1]. The model tracing methodology is used for
controlling and guiding of the student.
The Learning-by-Example Mode
The model in factorization of polynomials. Factoring polynomials is modelled as a
general task in which a reference KS organization based on static plans and tasks has been
chosen. Plans are sequences of actions and have, in that particular context, the following
properties: when a plan is chosen, it cannot fail; a plan generally does not lead to the solution;
no strategic reasoning is processed between two actions of a plan. Actions are calls to general
tasks or well known subtasks like development ofA,factoring A out ofB, and are executed
when the possibility of being executable has been remarked. Subtasks develop steps of calculus
according to the granularity required in the KS. Plans are classified and chosen by heuristics.
Three reference KSs have been defined by high school teachers corresponding to beginners,
intermediate and high-level reference knowledge.
A first example of a plan at a high-level has been presented above.
Second example of a plan at a high-level:
IF a subexpression E of the problem is a second order polynomial
THEN develop E
factor the result](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-289-320.jpg)
![280
This heuristic proposes to perform a backtrack to a less factorized step when the current step is
weak. Of course, in that case, if there are other steps with an N factors expression and some
interesting plans, the backtrack will be done to one of these other steps in order to maintain the
factorization already obtained. An example of the use of this heuristic is shown in figure 1
(backtrack after step 2).
The problem solving process. At each cycle, the problem solving process first
studies the current step. Heuristics associate a quality to each envisaged plan (according to a
symbolic scale (very-weak, weak, medium ,fair-well, well, very-well)) in order to find the
best one.
According to a cognitive economy principle, the problem solVIng process tries to find the
best plan for the current step without studying all possible plans. This has been implemented by
giving each plan a maximum quality (which is the highest quality the plan can reach) and by
considering them as follows:
the problem solving process first envisages the plans having maximum-quality =
very-well and applies heuristics on them. Then, if one envisaged plans has quality =
very-well (real quality), no other plan is envisaged for the moment, at this step. IT
not, plans having maximum-quality = well are envisaged, heuristics are applied to
them and the system looks for a plan with quality =well among all the plans
envisaged at this step. This process continues until the best plan has been found.
When the best plan of the current step is found, heuristics are used to choose between the
current step and other steps (between going on and backtracking). Stability rules are first
applied: if the best plan of the current step is good according to some concepts, the current step
is chosen and no backtrack is envisaged. In the other case, a backtrack is envisaged, other steps
are updated (as some plans have been applied on most of them, their best remaining plans have
to be found again) and heuristics choose between all these steps. This reasoning process has
been achieved by M. Sardi [15].
The interaction mode. In the learning-by-examples mode, APLUSIX uses the
behavioral knowledge of the chosen reference KS to generate, step by step, a search tree and to
display at each step a transformation.
APLUSIX allows the leamer to search for explanations about the matching and the strategy
used. This helps learners to verify their conjectures, if they have any, or to formulate explicitly
the reasons for choosing an action. As the advance in the solving process is under the learner's
control, he can ask questions about the matching and about the strategies at every step of the
resolution.
Explanations concerning the matching. Questions about the matching are asked
when the learner does not understand how an extended matching is performed. APLUSIX
gives explanations by presenting a description of the rule and the critical intermediate
transformations. Here is an example of explanation (with an intermediate level):](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-291-320.jpg)
![281
3X2_12=3(X2_4)
I applied the rule A2_B2 --+ (A+B)(A-B)
to X2_4 with A=X , B=2
Explanations concerning the strategy. Questions about strategy may be either of the
type why is it a good action? (applying a rule or backtracking), or of the type I would propose
another rule I will indicate, explain to me why it is not the best way to proceed? Figure 2 shows
an example in which the learner wants to know why APLUSIX has decided to generate the
state 2 using the rule difference between 2 squares.
The status of each plan is determined by heuristics implemented as meta-rules. A plan may
be: (1) instantiated, chosen and applied, (2) instantiated and kept in memory, (3) instantiated
and discarded, (4) not yet instantiated. As several heuristics may be applied to determine the
status of a plan, the trace of the heuristics used to solve a given problem is not necessarily the
best way to explain how the solution was reached. This problem arises for most expert systems
at the factual level [10] and at the strategic level [8]. In APLUSIX it concerns basically the
strategic level: explanations are epistemic explanations, they are produced by extracting the
most significant information from the reasoning process.
FACTORIZE
Uhy this backtrack?
Uhy HOT a backtrack?
difference bet.een 2 squares X2 -4
2 (5-2X)(X-2)(X+2)+8(2-X'+4X2_12X+9
fach'r,zatlon of Xl-4 is very ir,teresting •
fact(,rizat,on of X.!-4 produces the factor X-2
X-2 is also a factor in 8(2-X)
Figure 2: Answer to "why this action?" The explanation consists here in indicating the chosen plan.
In the first version of the system (which had no plan), explanations were generated by a
reasoning process which operated on a knowledge base of 90 rules, and on the trace of the
strategic reasoning [14]. In the current version, the explanation module has not yet been
implemented. It will be a reasoning process and will give information depending on the](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-292-320.jpg)
![Problem Solving: Its Assimilation
to the Teachers's Perspective
Paul Emest
School of Education, University ofExeter, SL Luke, Heavitree Rd, Exeter EXt 2LU, England
Abstract: It is argued that "problem solving" has multiple meanings. according to the teacher's
mathematics-related belief-system or perspective. A model of teacher belief-systems is
proposed. It is argued that the central feature is a personal philosophy or conception of
mathematics. and that this determines the teacher's understanding of the nature of problem
solving. Also. mediated by other factors. it influences classroom behavior. Some supporting
evidence is cited.
Keywords: problem solving. multiple meanings. teacher beliefs, philosophy of mathematics.
absolutism, fallibilism
Introduction
It could be said that there is a problem-solving bandwagon roIling. Many influential reports
published in the USA, UK and elsewhere, such as the Agenda/or Action [25], the Cockcroft
Report [6], the Standards [26] and the National Curriculum in Mathematics [10], all strongly
endorse a problem solving approach to school mathematics. These recommendations have been
made for over a decade - three decades, if we include the 'discovery learning' of the 196Os.
There are powerful arguments behind these endorsements, which I will not rehearse here. But
the fact is that most mathematics teaching remains routine and 'instrumental'. More often than
not, children are given a method for carrying out a type of task, and then many graded exercises
to practice and reinforce the method. Each task has a unique correct answer. Of course such
procedures do generate some 'relational' and well as 'instrumental' understanding on the part of
the learner, and perhaps some strategic skills. But the primary focus of such exercises is the
successful acquisition and deployment of procedures, and the acquisition ofrelational
understanding or strategic skills is incidental.
So what is going on? Why the disparity between the problem solving recommendations
and the routine and convergent nature of much of school mathematics? There are many](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-297-320.jpg)
![288
reasons, including institutional resistance to change, vested interests behind the status quo,
individuals' resistance to change, be they teachers, learners, parents or others, and so on. In
this paper I want to pick out one strand to explore, which in my view has received insufficient
attention. That is the assimilation of problem solving to the teacher's perspective, by which I
mean the teacher's mathematics-related belief-system. Teachers have different beliefs about the
nature of mathematics and its teaching and learning, which powerfully affect their classroom
practices. In some form or other, this relationship is now widely accepted, and I elaborate on it
below. But one unremarked consequence of this, I wish to claim, is that problem solving is
understood differently by different teachers, in accordance with their beliefs. A demonstration
of the lack of unique meaning and the diversity of interpretations given to the term 'problem-
solving' by teachers and others might go some way towards explaining how widespread
espousal of problem-solving can co-exist with its widespread rejection in practice.
The Meaning of Problem Solving
Problem solving has been a widespread part of the rhetoric of British mathematics education
since Cockcroft [6]. Worldwide, problem solving can be traced further back, at least to
Brownell [3] and Polya [28], and probably earlier. (In fact Descartes' Rules for the direction of
the mind of 1628 constitute a remarkably complete set of problem solving heuristics.) One of
the difficulties in discussing problem solving is that the concept is ill defined and understood
differently by different authors. However, there is agreement that it relates to mathematical
inquiry. Thus, there are a number of preliminary distinctions which can be usefully applied, for
it is possible to distinguish the object or focus of inquiry, the process of inquiry, and an inquiry
based pedagogy. In this section, I shall explore these different aspects of problem solving.
The Object of Inquiry
The object or focus of inquiry is the problem itself. One definition of a problem is Ita situation in
which an individual or a group is called upon to perform a task for which there is no readily
accessible algorithm which determines completely the method of solution... It should be added
that this definition assumes a desire on the part of the individual or the group to perform the
task." [22, p. 287]. This definition indicates the non-routine nature of problems as tasks which
require creativity for their completion. This must be relativized to the solver, for what is routine
for one person may require a novel approach from another. It is also relative to a mathematics
curriculum, which specifies a set of routines and algorithms. For a problem is only non-routine
if it belongs to the complement of the scope of a given set of routines. The definition also](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-298-320.jpg)
![289
involves the imposition of a task on an individual or a group, and willingness or compliance in
perfonning the task. The relationship between an individual (or group), the social context, their
goals, and a task, is complex, and the subject of the Soviet Activity Theory which has
developed from the work of Vygotsky.
The Process of Inquiry
In contrast with the object of inquiry is the process of inquiry itself. If a problem is identified
with a question, the process of mathematical problem solving is the activity of seeking a path to
the answer. However this process cannot presuppose a unique answer, for a question may have
multiple solutions, or none at all, and demonstrating this fact represents a higher order solution
to the problem.
The formulation of the process of problem solving in tenns of finding a path to a solution,
utilizes a geographical metaphor of trail-blazing to a desired location. Polya elaborates this
metaphor. "To solve a problem is to fmd a way where no way is known off-hand, to find a way
out of a difficulty, to find a way around an obstacle, to attain a desired end that is not
immediately attainable, by appropriate means." [17, p. I]. This metaphor has been represented
spatially [12, figure 8]. Since Nilsson [27] it has provided a basis for some of the research on
mathematical problem solving, which utilizes the notion of a "solution space" or "state-space"
representation of a problem [which] is a diagrammatic illustration of the set of all states
reachable from the initial state. A ?state? is the set of all expressions that have been obtained
from the initial statement of the problem up to a given moment." [22, p. 293]. The strength of
the metaphor is that stages in the process can be represented, and that alternative 'routes' are
integral to the representation. However a weakness of the metaphor is the implicit mathematical
realism. For the set of all moves toward a solution, including those as yet uncreated, and those
that never will be created, are regarded as pre-existing, awaiting discovery. Thus the metaphor
implies an absolutist. even platonist view ofmathematical knowledge.
Inquiry Based Pedagogy
A third sense of problem solving is as a pedagogical approach to mathematics. Cockcroft [6]
endorsed such approaches under the heading of 'teaching styles', although the tenninology
employed does not make the distinction between modes of teaching and learning. In considering
problem solving as an inquiry based pedagogy,number of further distinctions can be made. A
range of different approaches can be generated by distinguishing the roles of the teacher and the
learner, as in Table 1. In the literature a number of terms are used across the whole range of](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-299-320.jpg)
![291
Theorizing the Teacher's Mathematics-Related Belief-System
In this paper I want to stress the importance of the teachers mathematics-related belief-system,
their perspective on mathematics and its teaching and learning, which the title of the paper refers
to, with regard to the implementation of problem solving in the classroom.
In the past few years there has been a growing recognition of the importance of teachers'
beliefs and conceptions [4], and in particular on their mathematics-related belief systems [8, 13,
30]. What underlies this new emphasis is the hypothesis, by now reasonably well confirmed,
that teachers' belief systems are crucial determinants of their classroom practice. Teachers'
conceptions of the nature of mathematics and their personal theories about its teaching and
learning are very important factors in determining how they teach mathematics in the classroom,
and in particular, the place given to problem solving in their teaching. This is not to deny the
vital importance of many other factors, such as teachers' attitudes and confidence, the extent of
their knowledge of mathematics, and of course the social context in which they work.
Figure 1 provides a simplified model of some of the relationships involved. It represents
one central belief, the teacher's conception of the nature of mathematics, as underpinning two
pedagogical components of the teacher's belief system, the theories of teaching and learning
mathematics. These in tum have an impact on practice in the form of the models of learning and
teaching mathematics which are enacted. One use of resources, namely that of mathematics
texts, is singled out as important, for texts embody an epistemological approach to mathematics,
and the extent to which their presentation and sequencing of school mathematics is followed is a
crucial determinant of nature of the implemented curriculum [8,15]. In particular, texts which
embody an algorithmic approach to mathematics teaching can represent a major obstacle to the
introduction of a problem solving pedagogy in the classroom. The downward arrows in the
figure show the direction of influence on behaviour. The content of the upper positioned belief
components is reflected in the nature of the lower positioned components. Because the enacted
models are interrelated, as are the espoused theories of teaching and learning, this is represented
in the figure as horizontal links drawn between them.
As the figure illustrates, the impact of the espoused theories on practice is mediated by the
constraints and opportunities provided by the social context of teaching (Clark and Peterson,
1986). The social context has a powerful influence, as a result of a number of factors including
the expectations of others, such as students, their parents, fellow teachers and superiors. It also
results from the institutionalised curriculum: the adopted text orcurricular scheme, the system of
assessment, and the overall national system of schooling. The social context leads the teacher to
internalise a powerful set ofconstraints affecting the enactment of the models of teaching and
learning mathematics. According to Lacey [18] the influence of the social context and significant
others in it leads to the 'socialisation' of teachers. If the espoused beliefs of the teacher are at](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-301-320.jpg)
![293
mathematics as a discipline. Three groupings or types of philosophies of mathematics can be
distinguished: absolutist, progressive absolutist and conceptual change or social constructivist
philosophies of mathematics [7, 14].
Absolutism
The most widespread philosophies of mathematics are absolutist, which view it as a body of
fixed and certain, objective knowledge. Logicism, fonnalism and to a large extent platonism are
absolutist [1]. Most of these views are foundationist, regarding the truths of mathematics as
being based on certain logical foundations. One consequence is that the structure of
mathematical knowledge is seen to be hierarchical, building upwards from its logical
foundations by chains of logic and definition. Another consequence is that mathematics is seen
as objective, neutral, culture and value free.
Progressive Absolutism
The progressive absolutist position also views mathematics as made up of certain, objective
knowledge. But in addition, it accepts that new knowledge is continually being created and
added by human creative activity. Confrey [7] distinguishes this view, which underpins
Popper's [29] epistemology. It also describes the intuitionist philosophy of mathematics, which
places human activity as central in the creation of mathematics, and which argues that its logical
foundations are never complete [16]. Thus a key feature of this view of this position is that it
emphasizes the human processes of knowledge getting in mathematics, as much as their
product. Progressive absolutism can be identified as the philosophy of mathematics implicit in
the ideology of progressive education [14]. From this perspective, the certainty of mathematical
knowledge is not questioned, but the creative role of human activity in extending it is
acknowledged. This is partly why this position emphasises the process and creative human
aspects of school mathematics. (It also stems from the ideological model of childhood adopted
by this position.)
Fallibilism
The fallibilist philosophy of mathematics is largely due to Lakatos [19, 20], but is also to be
found in Davis and Hersh [9] and Tymoczko [32]. This is a social view of mathematics which](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-303-320.jpg)
![297
Empirical Evidence Supporting Thesis 2
A number of studies offer partial confirmation of Thesis 2 and its sub-theses. A growing
number of these are referred to in the literature, including recent surveys of research on teacher
beliefs in mathematics, such as those of Thompson [31] and Brown, Cooney and Jones [2].
The empirical studies quoted below are representative selections, chosen on the basis of their
explicit treatment ofconceptions of mathematics and problem solving in the classroom.
Lerman [21] describes how open-ended problem posing and solving work has been
introduced into school mathematics in Britain as a consequence of the new GCSE examination
for 16 year olds with coursework assessment of 'investigations'. He reports how the intentions
of this innovation have been subverted by some teachers' view that there is a unique correct
outcome to these tasks, betraying an underlying absolutist philosophy of mathematics. This
supports Thesis 2.1.
Marks [23] provides a detailed case study of a single high school mathematics teacher
Sandy. Sandy views problem solving, by which he means non-routine inquiry into received
problems, as central to mathematics. "It is not...Problem Solving, a topic to be squeezed in
somewhere between Fractions and Quadratic Equations. It is problem solving with a small
"p"...woven into the fabric of mathematics rather than stamped on top" (p. 2). He also states
that it is central to the teaching of mathematics; high school algebra in his case. Sandy's
classroom practice bears this out, with his frequent use of high-level questioning and his
emphasis on heuristic methods. The case study gives every indication of that Sandy has a
progressive absolutist conception of mathematics. Consequently, in view of his pedagogical
beliefs and practices, it supports Thesis 2.2.
A number of further studies offer evidence supporting both sub-theses 2.1 and 2.2.
Thompson [31] provides a case study of three teachers which support these theses. In only two
of the cases (Lynn and Kay) is problem-solving explicitly mentioned. Lynn has an absolutist
conception of mathematics described in terms such as: "cut and dried", and "exact, predictable
absolute and fixed" (p. 116). She sees classroom problem solving as "recall[ing] the
appropriate method or procedure...correctly apply...and obtain the right answer." (p. 117)
This fits perfectly with Thesis 2.1. In contrast Kay sees mathematics as certain but
"continuously expanding...and undergoing changes to accommodate new developments" (page
113), a progressive absolutist view. Her conception of problem solving in the classroom andher
observed practice was to "encourage the students to guess and conjecture...allow them to
reason on their own rather than show them how to reach a solution or an answer" (p.
114).During teaching Kay made explicit reference to the heuristic processes of mathematics. All
inall, she exemplifies Thesis 2.2 very well.
Dougherty [11] studied II teachers of grades 4-6. In brief, ten of these were found to have
some sort of absolutist conception of mathematics (emphasising mathematics as rules and](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-307-320.jpg)
![298
procedures, as a tool, or as step by step methods}. Each of these teachers viewed problem
solving as the more or less routine application of learned methods and procedures. Thus these
ten teachers confirm Thesis 2.1. The remaining teacher in the sample viewed mathematics as
"experiential and not a static body of knowledge" (p. l23), indicative of progressive
absolutism. This teacher also viewed classroom problem solving as 'solving thinking
problems', that is as a creative activity. Classroom observation also showed an appreciation of
divergent student thinking, and students taking responsibility for contributing meaningful
explanations and presentations. This teacher is confirmatory of Thesis 2.2.
It is harder to find empirical evidence in favour of Thesis 2.3, because of the infrequent
realisation of a problem-posing pedagogy in practice. Mellin-Olsen [24] implicitly endorses a
fallibilist, socially based view of mathematics and gives convincing examples of a problem-
posing mathematics curriculum as realised in Norway. This fits well with Thesis 2.3, but is a
carefully argued academic case with practical illustrations, rather than an empirical case study.
Elsewhere I have referred to the combination of a fallibilist philosophy of mathematics with a
problem-posing pedagogy as indicative of a 'public educator' ideology, and illustrated its
adherents, support and implementations, from the literature [14].
One example of classroom practice based on a fallibilist view of mathematics is the
constructivist teaching experiments carried out by Paul Cobb, Terry Wood and Ema Yackel (see
for example [5]). These experiments strategically complied with the expectations of the school
context, such as teaching the normal Grade 2 curriculum. They adopted a small-group
problem-solving pedagogical approach in which children discussed and contrasted their own
solutions and methods, rather than relying on the authority of the teacher for arbitration. Thus
this example is similar to the strategic compliance discussed under Thesis 2.3.
I should close this section by admitting that having pieced together some confirmatory
evidence, as I have done, does not constitute a scientific test of my theses. For I have not
sought to falsify them. However, what I offer is a plausible but conjectured explanation, with
some theoretical and empirical support.
Conclusion
The key point I have tried to make in this paper is that the terms 'problem' and 'problem-
solving' vary in meaning according to the perspective of the speaker. In particular, I have
offered a theory based on the assumption that the teacher's personal philosophy of mathematics
is the major determinant of what the teacher means by problem solving in school mathematics. I
would like to suggest this as an area of research that could be fruitful in the future. What do
teachers (and student teachers) mean by problem solving? How do their meanings relate to the
mathematics-related belief-systems, especially their philosophies of mathematics? If my](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-308-320.jpg)
![Computer Spreadsheet and Investigative Activities:
A Case Study of an Innovative Experience
loao Pedro Ponte, Susana Carreira
Departamento de ｅ､ｾＬ＠ Faculdade de Ciencias, Universidade de Lisboa, Campo Grande, Lisboa
Abstract: This paper analyses an experience undertaken by a group of teachers, who
introduced the computer in a 10th grade mathematics classroom for carrying out problem
solving and investigational activities. The main question of interest is the discussion of
significant issues and unexpected situations that emerged within this innovative process,
specially regarding the reactions of the students. Using a case study methodology [1], the
sources of data were interviews with the teachers, visits to the school, and the learning and
dissemination materials produced.
Keywords: computers in mathematics education, problem solving, investigations, teachers'
beliefs, innovation, new information technologies
The MINERVA Project - Node DEFCUL
The experience described in this study was carried in one of the schools in the MINERVA
Project. This national Project, in operation since 1985, was designed for the introduction of
New Information Technologies (NIT) in primary, middle and secondary schools, both as tOpIc
of interest in itself and related to the teaching of all school subjects. The Node DEFCUL is one
of its groups, based at the Department of Education of the College of Sciences of the University
of Lisbon. It carries out development, research and extensive teacher training activities. A
detailed description of the work of this Pole of the Project was given elsewhere [2] and here we
will just review some aspects of its educational approach, based in the principles of active
learning and in a concern with broad cultural and educational changes.
The concept of New Information Technologies (NIT) implies far reaching ideas than the
mere use of electronic hardware. These technologies assume an important role in many social
activities, bringing about significant structural changes in ways of working and thinking. They
have influenced scientific research, economic planning, goods production, management,
communication media, etc. The NIT mean the possibility of automatic information processing
and communication. They represent new ways of looking at information and at the process of
knowledge development and dissemination, and touch on important values. In practice, most of
the work carried out in the schools has developed around the computer (with or without
peripherals), but the emphasis is in the general concept of NIT.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-311-320.jpg)
![302
One major pedagogical idea about the educational use of computers is that they are
essentially regarded as something for the students to work with. Of course, teachers need also
to be able to deal with computers in many ways, in particular for demonstrations. But the most
interesting and innovative aspects of the educational practices that can be expected to develop in
schools may have origin on the intensive interactions between students and computers.
The computer is also regarded as a working tool. There are other roles that it may fulfil,
and they may be of significant educational value. However, it is through its use as a general or
subject specific tool, for carrying out investigations and undertaking sophisticated projects, or
to perform quite simple tasks, that the computer is expected to find an important place as a
cultural artifact influencing human thinking and interpersonal relationships.
In an information-driven society, it becomes necessary to be able to use with ease the
computer and other equipment associated with NIT. New educational objectives need to be
considered. The youngsters need to develop, since quite early, the ability of knowing where
and how to locate information, how to select, interpret, process it and evaluate the results. It is
also necessary to reflect about what is its role in social, economical, and cultural processes and
what may be misuses and unethical procedures. Quite specially, it is important to be aware of its
role in scientific thinking, project design, and more generally, in the development of
knowledge.
It should not be surprising, therefore, that project work, usually undertaken in groups,
has been one of the most important activities within the schools. Students get an opportunity to
establish working goals, set strategies and methodologies, collect data and analyse it, and get
used to present their results and ideas to large audiences. In mathematics, a common activity is
also carrying out investigations, in which students explore the possible relationships between
concepts and try to get specific understandings or arrive at broad generalizations (for another
example of this kind of activity, see [3]).
The NIT, as a new subject, rise the issue of curricular articulation. It has been felt that
instead of setting a new discipline apart to deal with these matters, they should inform deeply in
the teaching of all existing school subjects, being integrated with them.
These technologies, besides their curricular implications, are also, and more generally, an
important factor of transformation of the school, yielding the setting of new objectives, new
learning situations, new activities, and values. They point towards the reorganization of the
spaces, working methods and teacher/student relationships.
The NIT put new demands on the teachers. They need to know how to use these
technologies in an effective and confident way. Teachers need to develop new attitudes and
abilities, have access to specialised information, receive formal training, and have the possibility
of informal interaction with other teachers with similar interests and concerns.
The role of the teacher is changing in a number of aspects. Knowledge is becoming more
and more of a dynamic nature, and the active involvement of the learner is critical in the process
of its acquisition and development. Teachers have to become experts in supporting this kind of](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-312-320.jpg)
![304
carry out. They had very different personal characters and also distinct opinions on education
matters, but they respected each others' ideas.
The four practising teachers and the supervisor decided to get involved in a project of
introducing calculators in this mathematics class. In a subsequent stage of the work computers
were also included. As they reponed, the group was, from the first moment, interested in
establishing a setting which could help to promote a different son of mathematical experience to
the students. The calculator and then the computer were viewed as good carriers of the
innovative strategies that they were looking for.
The work of this group was supported by a training programme on spreadsheets and
calculators in mathematics education. Within this programme they had several meetings with
practical activities, discussions and exchanges of experiences with teachers of other schools and
were visited several times by the project coordinator, often at their request [4].
The practising teachers felt that their lack of teaching experience was a good reason to be
gladly open to new ideas and teaching methodologies. They were much concerned with the
picture they conceived of the current situation in the teaching and learning of mathematics. They
were also very enthusiastic about doing problem solving activities and enhancing students
ability to investigate mathematical questions in rich content situations.
They have taken these concerns in their university studies. especially in the previous year,
in the mathematics methods course. Other important motivations and inspirations were received
through their participation in a mathematics teachers national meeting (PROFMAT). The
supervisor always supponed their initiatives and was ready to help them to go forward with
their projects. She had the caution of making the group reflect upon their actions and understand
the possible advantages and risks involved. but she never wanted to dictate their decisions.
The practising teachers emphasized the need for many changes in mathematics education.
They believed that the teachers' practices are too often a straight routine consisting in teacher
presentation of mathematical concepts. followed by a series of standard exercises where
students are supposed to acquire procedural behaviors intended to make them apply the
previous information presented. With this kind of exhaustive practice, they found very plausible
that students would be deprived of the real opportunity to understand the meaning of
mathematical ideas and would possibly perform many calculations without figuring out the
reasons for them.
With this in mind and the conviction that plain recepies were not available to create a rich
educational environment, the group of teachers embraced some pedagogical guidelines and
started out the search for new challenges to themselves and their students.
They wished to give students a less statical view of mathematics and to stress that many
mathematical questions demand creative strategies and can be explored in many different ways.
They began with problem solving activities which were supposed to encourage students'
work and discussion in groups. This and the opportunities for students to engage in
mathematical investigations created the general framework for the introduction of curricular
topics.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-314-320.jpg)
![Examining Effects of Heuristic Processes on the
Problem-Solving Education of Preservice Mathematics
Teachers
Domingos Fernandesl
Instituto de ｉｮｯｶ｡ｾｯ＠ Educacional, Trav. Terms de Sant'Ana 15, 1260 Lisboa, Portugal
Abstract: This eight-week study analyzed effects of two heuristic models of instruction in
mathematical problem solving on preservice teachers' problem-solving performance, on their
awareness of the problem-solving strategies they employ, and on their perceptions about
specific problem-solving issues. Both models taught four problem-solving strategies and
employed P6lya's four-step model of problem-solving; each subject solved or saw the solutions
to the same 24 experimenter-selected process problems. Study findings suggest that both
models of instruction significantly improved preservice teachers' problem-solving performance;
the explicit model appeared to be more effective in promoting organization of problem solutions
and description of problem-solving procedures. Study results and conclusions yielded
recommendations on the use of the focused holistic scoring, on the study design, and on teacher
education in mathematical problem solving.
Keywords: mathematical problem solving, problem-solving performance, problem-solving
strategies, problem-solving behavior, teacher awareness, teacher perceptions, heuristic models
of instruction, focused holistic scoring
It has been suggested that preservice teachers should be exposed to formal instruction in
mathematical problem solving. This recommendation has been made based upon the assumption
that if teachers are to teach problem solving then they must be exposed to formal instruction
which develops their abilities as problem solvers, their knowledge about problem solving, and
their skills to plan, implement, and evaluate problem-solving activities. Mathematics educators
have proposed models to meet these goals, and recommendations have been made concerning
the need to investigate the efficiency and effectiveness of those models.
Most research efforts on mathematical problem solving have focused at the precollege level
and have sought answers to questions such as (a) What kind of strategies do students use while
they are engaged in problem solving? [e.g., 12, 30]; (b) What methods of instruction are
appropriate to mathematical problem solving? [e.g., 4, 29, 23, 30]; c) What is the influence of
the processes children use on their problem-solving performance? [e.g., 12, 29]. These
research efforts identified strategies and processes used in problem solving as well as
IOn leave from Escola Superior de ｅ､ｵ｣｡ｾｯ＠ de Viana do Castelo.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-323-320.jpg)
![314
instructional procedures that seem to be effective in teaching problem solving to precollege
students.
As a consequence of demands to implement problem-solving programs in precollege
mathematics instruction, mathematics educators and professional associations have proposed
programs with goals of better teacher preparation and greater instructional effectiveness in
promoting problem solving in the classroom [4, 7, 10, 11, 14, 15,24,26]. However, there is
no research evidence in direct support of conjectures that the use of instructional approaches
which implicitly or explicitly emphasize problem solving will contribute to the improvement of
(a) teacher's performance in problem solving; (b) teachers' awareness about the problem-
solving skills that they employ; or (c) teachers' effective use of models of instruction in problem
solving.
Teacher education in mathematical problem solving is a natural consequence of the problem-
solving emphasis that has been recommended. The vision for the improvement of the
precollege mathematics curriculum set forth by the National Council ofTeachers of Mathematics
(NCTM) is based upon a teaching method which models and encourages problem-solving
behavior. In fact. NCTM's An Agenda for Action [25], Guidelines for the Preparation of
Teachers of Mathematics [26], and its Curriculum and Evaluation Standards for School
Mathematics [27] specify that (a) teacher training programs must be re-designed to meet the
problem-solving emphasis; (b) teachers must be competent problem solvers; and (c) teachers
must be able to teach problem-solving techniques. These goals have widespread support within
the mathematics education community. A practical result of this support has been the
development and implementation of models aimed at educating teachers in mathematical problem
solving [4, 11, 14, 15, 16].
The present investigation was based on the premise that teachers, if they are to become
teachers of problem solving, must be exposed to instruction which is directed toward improving
their abilities as problem solvers, their knowledge about problem solving, and their skills to
plan, implement, and assess problem-solving activities (R.I. Charles, personal communication,
October 25, 1986).
The major goal of this study was to examine the effects of two heuristic models of
instruction in mathematical problem solving on preservice teachers' problem-solving
performance, on their awareness of the problem-solving strategies they employ, and on their
perceptions about specific problem-solving issues
Method
SUbjects. The subjects were 68 preservice teachers enrolled in two classes of a three-
semester hour Mathematics Methods Course. More than 75 percent were in the 20-22 year old
age span and had no formal teaching experience. About halfof them claimed that they had taken
four or fewer mathematics courses in high school whereas the remaining half had taken five or
more. About 65 percent took four or fewer mathematics courses at the college level. All study
subjects were females.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-324-320.jpg)
![315
Treatments. Subjects in each class were randomly assigned to one of two treatments.
Two Treatment 1 subgroups and two Treatment 2 subgroups were obtained. Treatment 1
subjects were exposed to an implicit method of instruction in problem solving which made
organized use of specific problem-solving strategies to solve problems but did not overtly
identify or reflect upon the selection or application of those strategies. Treatment 2 subjects
were exposed to an explicit model of instruction which named, discussed, purposefully applied,
and reflected upon the organized use of specific problem-solving strategies to solve problems.
Both treatments taught four problem-solving strategies (Check Some Guesses, Make an
Organized List or Table, Find and Test a Pattern, and Make and/or Use a Drawing or Other
Model) and employed P6lya's four-step model of problem solving; each subject solved or saw
the solutions to the same 24 experimenter-selected process problems.(A sample of the problems
is displayed in Appendix 1.)
Procedure. Both experimental treatments were implemented in one 6O-minute laboratory
session per week during a period of eight weeks. Subjects were assigned at random to small
groups of three or four. The groups' composition remained constant throughout the treatment
sessions.
For each problem-solving strategy, six related problems were solved during two 6O-minute
sessions. In the first of the two sessions the instructor demonstrated the solution to an
introductory problem which elicited utilization of the targeted problem-solving strategy. This
problem was solved by following P6lya's four phases of understanding the problem, devising a
plan, implementing the plan, and looking back [28]. Afterwards, subjects working in small
groups solved a similar problem. Two homework problems were distributed at the end of the
first session; one was similar to the one solved in class, and the other was dissimilar but could
be solved using the same problem-solving strategy. In the second session, the same small
groups of subjects discussed their solutions to the homework problems until a consensus was
reached. Instructor-selected students then presented and discussed their groups' solutions to the
problems. Two more problems were distributed and solved. Their discussion was conducted
at the chalkboard by instructor-selected students.
Instruments. In order to assess the effects of the two treatments, three investigator-
developed instruments were used. The Problem-Solving Test (PST), a paper-and pencil
instrument, was used to gather pretest and posttest data on subjects' problem-solving
performance and on their awareness of the strategies they used. The Problem-Solving
Observation Guide (PSOG) was utilized to conduct 48 systematic observations of subjects'
small-group problem-solving activities in each treatment group; these 96 observations focused
on nine selected problem-solving behaviors. The PSOG also served to collect anecdotal data on
students' problem-solving activities. The Strategies Interview Protocol (SIP) structured the
collection of data on three issues: 1) subjects' perceptions of their ability to solve process
problems; 2) subjects' professed willingness to teach problem solving; and 3) subjects' ability
to identify problem-solving skills that should be taught to precollege students. The SIP also
permitted validation of subjects' self-reported uses of problem-solving strategies on the PST.
SIP data were gathered through interviews with random samples of eighteen treatment 1](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-325-320.jpg)
![318
solving to precollege students [17,30]. It supports assertions that, despite current limitations on
knowledge of how problem solving is learned, it is possible to design and implement instruction
that has positive effects on students' problem-solving perfonnance [18, 19,20,21].
Study conditions did not pennit a design which included a control group in which subjects
would be exposed to the problems of treatments I and 2 but would not be given implicit or
explicit instruction in problem-solving procedures. This model of instruction is titled Osmosis
by Kilpatrick [13] Thus, while the design supports the conclusion that treatments 1 and 2 are
equally effective, it does not address the significance of the pretest-posttest gains of either
group. Did the subjects show significant improvement in their problem-solving perfonnance as
measured by the PST? If so, can that improvement be attributed to the experimental treatments
(i.e., to instruction in problem solving)?
The pretest-posttest gains of the subjects, separated by treatment, are shown in Table 1.
Two-tailed l-tests for paired measures show that these gains are statistically significant (p <
.0005) for treatment I, treatment 2, and the combined group of 68 students. Since all pretest to
posttest difference scores were nonnegative, it is reasonable to infer that the preservice teachers'
problem-solving performance did improve during the eight-week period. While the absence of
a control group makes it impossible to attribute the gains recorded in this study to instruction,
studies which have included such a control group suggest that growth in problem-solving
performance requires direct instruction [30,31]. Thus, there is reason to believe that the large
gains in problem-solving performance observed in this study are a consequence of instruction.
Based upon the t-test results summarized in Table I, it is concluded that preservice teachers
can learn to solve process problems in mathematics. It is further suggested that such teachers'
problem-solving perfonnance can be improved substantially and equally well by instruction
which employs either an implicit or explicit model of mathematical problem solving. The data
gathered call for the note that success in applying problem-solving procedures seems to
begreater when a new problem is similar to the problems employed during instruction (Le., is a
near-transfer problem); perfonnance on far-transfer problems is less marked.
Subjects' explanatory work on the PST problems revealed parallel growth in the number of
problems solved and in the evidence that problem-solving strategies were used. Analysis of
treatment 1 subjects' work on corresponding items of the pretest and posttest illustrated that
overall organization and clarity of the explanatory work improved greatly from pretest to
Table 1. Paired t-Test Summary of Problem-Solvin& Test Gains by Treatment Groups and All
Subjects
Pretest Posttest
Group M* SD M* SD D <f t
Treatment 1 10.889 5.476 17.556 6.801 6.667 35 9.77-
Treatment 2 10.719 5.299 18.375 5.912 7.656 31 10.97-
All Subjects 10.809 5.354 17.941 6.364 7.132 67 14.61-
*Maximum possible value=32; ..p<.OOO5](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-328-320.jpg)
![319
posttest. Moreover, the organization and clarity of the work of treatment 2 subjects was
generally superior to that of treatment 1 subjects. Generally speaking, use of the language of
problem solving and systematic application of P6lya's four-step model differentiated treatment 1
explanations from treatment 2 explanations. It is concluded that explicit naming of strategies
and explicit reference to problem-solving procedures does improve the problem-solving
organization and explanatory power of preservice teachers.
Awareness and Utilization of Problem.Solving Strategies
Validation of subjects' self-reported uses of strategies in solving the problems of the posttest led
to the finding that preservice teachers exposed to either model of instruction were aware of the
four basic problem-solving strategies taught. The validation process also revealed that the
subjects frequently failed to indicate strategies that they had used. During the validation phase
(Le., administration of the SIP after the problem-solving posttest), subjects could justify almost
every strategy marked during the testing. Hence, the validation process found that subjects did
not mark strategies that they did not use. However, several subjects responded to the SIP
question "Did you use any other strategy on this problem that you would like to check now?" by
indicating strategies that they had overlooked while taking the test. Some had used a model
(diagram or sketch) but said that they were not certain that what they had used could be called a
'model'; others had made an organized list of guesses but questioned whether it could be called
a list or table since it did not have headings. Such uncertainties in identifying strategies were
frequent among treatment 1 students; lack of a common language of problem solving appeared
to be one source of their failure to indicate all of the strategies used on the test. A second
probable source was the subjects' test-taking priorities. Since the problem-solving test had a
time limit, the subjects' attention was focused upon solving problems; reflection on the
problem-solving process during testing received less attention. Posttest interviews are, then,
necessary to the accurate assessment of problem-solving strategies use during a test.
Subjects' written work on the posttest and their verification of strategy use during the SIP
interviews showed that they purposefully applied problem-solving strategies during the test.
This behavior was typical of problem solvers identified as "expert" by Shoenfeld [32].
Confidence in the efficacy of the strategies Table and Pattern are evident in the written work of
posttest papers. Whereas students abandoned such strategies after testing three or fewer cases
on the pretest, perseverance in testing seven or more cases produced solutions on the posttest.
Subjects' response to the SIP request to list several specific things to be taught to a student
in Grades 1-9 during a unit on problem solving provided another measure of their awareness of
the strategies and processes of problem solving. Although neither treatment included explicit
discussion of the problem-solving skills that should be taught to precollege students, subjects in
both treatments were able to identify those skills. Such identification indicates not only an
awareness of the skills but also a belief that they are understandable to elementary and middle](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-329-320.jpg)
![321
Shoenfeld [33] even when metacognition is not taught explicitly. Furthennore, the results of
the use of small-group activities in this study support recommendations to employ instructional
models which focus on the problem-solving process [3] and help students to recognize that
problem solving requires time and perseverance [8].
Teachers' Perceptions: Problem.Solving Ability and Willingness to Teach
Problem Solving
Analysis of data from the SIP led to the conclusion that the majority of preservice teachers
exposed to either treatment perceived themselves as able to solve process problems of the kind
presented in the experimental sessions. Subjects' experiences in solving the problems presented
during the study is one possible source of their volunteered profession of an increase in
confidence. A second source may be found in the fact that both treatments sought to maintain
an open and relaxed atmosphere during the problem-solving sessions. These results support the
recommendation of Stacey and Southwell [34] that students should be put at ease and that
teachers should acknowledge that problem solving is a time-consuming activity.
Based upon data from the SIP, it was concluded that subjects in either treatment were of the
opinion that it was very important to teach problem solving to precollege students; they also
professed a willingness to do such teaching. Related comments recorded on the SIP and PSOG
suggest that the subjects' willingness to teach problem solving is linked to their perceived ability
to solve problems. During early problem-solving sessions of the study, subjects expressed
doubt that the kind of process problems by which they were confronted could be presented to
their students. This position was supported by the subjects' observations that they experienced
difficulties in solving the problems. However, this assessment reversed as the subjects
successfully solved problems and became more confident of their problem-solving ability. This
supports an assumption common to models for educating teachers in mathematical problem
solving: teachers must be taught to be problem solvers if they are to be expected to teach
problem solving to precollege students.
Summary
The preceding discussion sought to link the main findings of this study to the results of related
studies at the college and precollege levels and to the recommendations of mathematics
educators who have proposed models for instructing teachers in problem solving. In making
those connections, it has made use of a post hoc analysis of subjects' gain scores on the PST
and of qualitative data provided by subjects' responses to open-ended items of the SIP,
observers' comments recorded on the PSOG. and subjects written explanatory discourse on the](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-331-320.jpg)
![323
Procedural Recommendations: Focused Holistic Scoring
The scoring of the paper-and-pencil instrument used to measure problem-solving performance
employed a five-point focused holistic scoring scale based upon the one developed by Charles,
Lester, & ODaffer [5]. This scale was chosen because it focused on the process of problem
solving rather than its end product. In the word of its developers, such a scale is appropriate
when the evaluator's interest is "in a general rating of the processes used and explicit criteria are
needed or wanted to guide the assigning of points" [5, p. 38]. This focus on process was
consistent with the goals of the study and is judged to be an important tool in the assessment of
problem-solving performance. However, experience in using the scale to score pilot versions
of the PST dictated that the scale's criteria be modified in order to make it applicable to all items
of the test. In particular, the test items for the strategy Check Some Guesses were difficult to
evaluate with the original criteria.
Refinement of the criteria of the focused holistic scoring scale in order to evaluate a
particular set of process problems is not viewed as a violation of the scoring procedure. In fact,
Charles, Lester, and ODaffer speak not of the focused holistic scoring scale but rather of a
focused holistic scoring scale. They point out that it is likely that revisions will be called for and
that the need for such revisions will become apparent only when the scale is put to work. This
assertion of a need to customize the scale was confirmed by this study. This study also
conftrmed the adaptability of the scoring procedure to the process problems involved.
The scoring of the several versions of the PST also amplified the issue of scoring
consistency. Experience confirmed the need to identify "anchor papers" for each point category
of the scale [5, p. 38]. In fact, it is necessary to identify anchor papers which are both item-
specific and criterion-specific. That is, for each item of the test and for each criterion of each
point category of the scale it is necessary to identify a paper which exemplifies satisfaction of
that particular criterion for that particular point category on that particular item. Thus, it is
recommended that pilot testing or prereading of all responses to a given item take place before
data collection begins. Such a scoring procedure will guide the refinement of the scale and will
support its consistent application.
Finally, it is recommended that each item be scored by at least two persons. Even when
operating with the modified scale and exemplary anchor papers, the two evaluators of the tests
used in this study found it necessary to negotiate the point assignment of approximately 20
percent of the items. In several cases, differences in the scoring of an item were revealed to be
due to an evaluator's overlooking critical evidence which was written on the back of a page or in
an area other than that designated as the work space. In other cases, an unusual method of
solution judged as "not understandable" by one evaluator was deciphered by the other evaluator.
In still other cases, the scoring disagreement was the result of a mismatching of a student's
work with the appropriate anchor paper. While each of these scoring conflicts was easily
resolved by the evaluators, it is clear that a single evaluator would have improperly scored at
least 10 percent of the items.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-333-320.jpg)
![324
Study Design Recommendations
Broad guidelines for needed research in mathematical problem solving have been set forth by,
among other researchers, Hatfield [9], Lester [18, 21], and Shoenfeld [31]. The dimensions of
needed research regarding the teaching of mathematical problem solving have been defined by
LeBlanc [16], Kilpatrick [13], and others. The specific research recommendations which
follow are directly related to the findings and limitations of this study.
1. The assertion that the subjects of this study improved their problem-solving performance
under either treatment is based upon the results of precollege studies and upon the large jump in
mean scores on the PST. There is a need to test that assertion with a design that includes a
control group that is exposed to the same set of process problems but is not exposed (implicitly
or explicitly) to instruction in problem-solving procedures.
2. Subjects' performances on the PST suggest that their pretest to posttest gains were
grounded in the five near-transfer items on the tests; gain scores on the far-transfer items are
less impressive. Since extended application of generic strategies is a goal of instruction in
problem solving, there is a need to focus on the study of far-transfer performance.
3. Failure to detect differences between the two treatment groups with respect to problem-
solving or strategy-use awareness could be a function of the limited time frame of the study.
Studies are needed that extend instruction over a longer period of time. Such studies might
address additional strategies as well.
4. This study assessed, among other things, preservice school teachers' problem-solving
performance and strategy-use awareness immediately following eight weeks of instruction and
concludes that such awareness/performance can be taught. Studies are needed to determine if it
is retained.
5. Subjects in this study professed a willingness to teach problem solving and exhibited an
ability to identify problem-solving skills that should be taught to precollege students. Clinical
studies of such preservice teachers should be conducted during their student-teaching experience
to determine whether that willingness is expressed in teaching which reflects the problem-
solving procedures that the teacher had been taught.
6. The principal assumption of this study was that teaching teachers to solve problems will
improve the problem-solving performance of their students. Since the findings of this study
suggest that teachers can be taught to solve problems, a next study is needed to measure the
effectiveness of such teachers in teaching problem solving to precollege students.
7. During the execution of this study, both the investigator and the observers noticed a
change in classroom atmosphere. Subject anxiety evident during the first two weeks quickly
gave way to confidence and optimism. A clinical study should be conducted to systematically
assess students' beliefs and attitudes during instruction in problem solving. Such research
could reveal the effects of models of instruction upon student's belief systems regarding
mathematics and problem solving.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-334-320.jpg)
![325
Classroom Practice Recommendations: Teacher Education
The review of the literature on precollege problem solving identified research-based
recommendations for precollege classroom practices [1,2,3,6, 16,20,34,35,36]. While the
search also identified recommendations for classroom practice in the instruction of preservice
teachers in problem solving, proposals at this level were not based on findings of studies
involving preservice teachers. The following recommendations are related to the findings of
this single study.
1. Preservice teachers are mathematically anxious and have little confidence in their ability
to do mathematics. Data gathered through formal and informal interviews during this study
indicate that anxieties appear to diminish and confidence appears to increase when a non-
threatening atmosphere for problem solving is established. It is recommended that such an
atmosphere be pursued by emphasizing that problem solving takes time and that perseverance
rather than innate ability is the key to the solution of the process problems presented.
2. Preservice teachers appeared to enjoy solving problems of the kind presented in this
study. They found the problems challenging (but within their grasp); they recognized the
problems as being adaptable for use with precollege students. Since the latter two conditions
were design criteria of the problems, it is recommended that problems used in the preparation of
teachers meet those criteria.
3. Students who entered this study expressing low confidence in their ability to do
mathematics reported that the use of small groups both reduced their anxiety and contributed to
their growth in problem-solving performance. Therefore, it is recommended that small-group
work be the primary mode of classroom problem-solving activity.
4. Chalkboard presentation of problem solutions by preservice teachers placed them in the
role ofdemonstrating solution procedures; it generated discussion both of the solution presented
and of alternative solution procedures. Presentation techniques improved and students became
aware that there truly was more than one way to solve a process problem. It is recommended
that such chalkboard presentations be a component of instruction model for teaching problem
solving to teachers.
5. Both treatments of this study were based upon heuristic procedures, procedures found to
be effective in teaching precollege students [17, 23, 29, 30]. That is, instructional features
common to the two methods of instruction were dialogue, insistence upon active participation
by the learners, and the systematic use of learners' suggestions (even when the instructor was
aware that the suggestion would not lead to a solution). Based upon the findings, it is
recommended that instruction of preservice teachers in problem solving should utilize heuristic
procedures.
6. Although this study found that the problem-solving performance of preservice teachers
exposed to the implicit model of instruction (treatment 1) did not differ from that of preservice
teachers exposed to the explicit method of instruction (treatment 2), the data from small-group
observations and the written work of students revealed that treatment 2 subjects did realize an](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-335-320.jpg)
![330
A Mathematics Problem Solving Course
There are two reasons for sharing a description of a problem-solving course. First, a great deal
of experience with this course suggests that it has potential to bring about change in the
classroom teaching of problem solving, and second, clear descriptions of successful teacher
education efforts may lead to the identification of questions amenable to research.
I begin by briefly discussing goals for teacher education and problem solving. Then I
discuss the general organization of the course. For the remainder of this section I focus on two
aspects of the course, the school curriculum and instruction.
Goals for Teacher Education vis-a-vis Problem Solving
Shulman [17] suggests that a teacher education program needs to construct answers to two
questions: (1) What do we want teachers to know?, and (2) What do we want them to be able to
do? Drawing on work with teachers and research on students' problem-solving processes and
effective teaching, the following goals were established for the problem solving course.
1. Teachers should be able to solve problems of at least the same level of
difficulty that they will use with their students.
2. Teacher's content and curricular knowledge and abilities.
2.1 Content knowledge - A teacher should know:
2.1.1 the meaning of a "problem" and "problem solving,"
2.1.2 why problem solving is important,
2.1.3 what factors influence success, and
2.1.4 the thinking processes involved in solving problems.
2.2 Curricular knowledge - A teacher should know:
2.2.1 types of problems and problem-solving experiences appropriate for
different grade/age levels, and
2.2.2 the roles problem solving has played in the mathematics
curriculum.
2.3 Curricular abilities - A teacher should be able to:
2.3.1 select and create problem-solving experiences appropriate for a
given population of students, and
2.3.2 evaluate problem-solving curriculum material.
3. Teachers' pedagogical knowledge and abilities:
3.1 Pedagogical knowledge - A teacher should know:
3.1.1 different roles for the teacher in the classroom,
3.1.2 teaching actions that facilitate problem solving,
3.1.3 ways to manage instruction,
3.1.4 assessment altematives,and
3.1.5 factors that influence the classroom climate.
3.2 Pedagogical abilities - A teacher should be able to:
3.2.1 implement a model for teaching problem solving,
3.2.2 implement classroom management practices,
3.2.3 develop and use assessment techniques, and
3.2.4 build a classroom climate conducive for problem solving.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-340-320.jpg)
![331
General Organization of the Course
This course has been taught in many configurations. The most common is a I-semester 15-
week course through a university. Also, it has been presented as a series of workshops over
time. Regardless of the configuration or time available, the course moves through three phases
with approximately one-third of the total amount of time devoted to each phase. Phase One
focuses on developing teachers' problem-solving abilities and shaping their beliefs and attitudes
about themselves as problem solvers and about problem solving (see Goal 1). Teachers need
some level of competence as problem solvers before they not only teach problem solving but
before they begin to learn about problem solving. Also, the importance of addressing affect at
the beginning of a teacher education program on mathematics problem solving cannot be
overemphasized. Many prospective and inservice teachers have considerable anxiety related to
problem solving. Beginning the course by solving problems and discussing feelings, attitudes,
and beliefs has been very successful; it is one way to give teachers permission to admit their
anxieties and begin to deal with them. In Phase Two teachers continue to solve problems to
further improve their problem-solving abilities and develop helpful attitudes and beliefs.
However, the emphasis here is on developing the teachers' content and curricular knowledge
related to mathematics problem solving (see Goal 2). Phase Three focuses on developing the
teachers' pedagogical knowledge and abilities (see Goal 3). Models for teaching problem
solving, techniques for assessing progress in problem solving, and classroom management
issues are explored.
School Curriculum and Instruction
There are many topics addressed in this course related to a teacher's knowledge about problem
solving, a teacher's knowledge and abilities related to curriculum, and a teacher's knowledge
and abilities related to instruction. In Phase 2, particular emphasis is given to examining types
of problem-solving experiences that can be used in school curriculum and the purpose or
purposes each serves in developing student thinking. In Phase 3, emphasis is given to the
teacher's role in facilitating the development of thinking. Because these topics are emphasized
in the course, they are discussed below in some detail.
Curriculum
Identifying the kinds of thinking experiences appropriate and useful for an elementary-middle
grades mathematics curriculum is a task partially amenable to research (see [18], for a historical](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-341-320.jpg)
![332
look at problem solving in the curriculum). Unfortunately, beyond the work of the
Mathematical Problem Solving Project at Indiana University in the 1970s, the nature of the
curriculum as it relates to problem solving has not been the subject of much research; it seems
that most suggestions for curriculum have emerged from popular opinion.
Two aspects of the school mathematics curriculum vis-a-vis problem solving are examined
in the course:
• problem-solving skills and strategies
• ways to integrate thinking to develop concepts, operations, and skills.
Problem Solving Skills and Strategies: A Stand Alone Approach
Extensive space in the elementary and middle school grades mathematics curriculum is devoted
to experiences aimed at developing students' use of strategies and skills for solving problems.
Explicit instruction is given to helping students learn how to make organized lists, how to
search for patterns, how to simplify problems, how to decide if an answer seems reasonable,
how to use deductive reasoning processes and so on. These experiences reflect a stand alone
approach to developing thinking [12]. In a stand alone approach to developing thinking,
emphasis is given to the thinking processes elicited in the experiences used; few mathematics
content demands are made of the student in completing them and no new mathematics concepts
typically emerge from working with them.
Experiences in existing curriculum that reflect the stand-alone approach usually appear as
special problem solving or thinking skills lessons such as can be seen in [9]. They appear as
lessons separate from those that focus on developing mathematics operations, concepts, and
skills. Much work over the past 10 years has been devoted to stand-alone lessons and their use
is now quite common in most classrooms. However, for many teachers these lessons are the
"extra ones" taught (if there's sufficient time) after they teach "the regular math."
Integrating Thinking Experiences: An Immersion Approach
The immersion approach for developing thinking is an emerging one that Prawat [12] suggests
does not have great support. However, in mathematics education many calls for reform in the
teaching of mathematics, most notably the Curriculum and Evaluation Standards from the
National Council ofTeachers of Mathematics [11], strongly encourage an immersion approach.
The immersion approach has its roots in a constructivist view of learning (see [3] for a
summary of the implications of constructivism for teaching). In an immersion approach,](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-342-320.jpg)
![334
How many different rectangular solids one layer thick can be made where each
consists of 60 same-size cubes? What are the dimensions of the rectangles that can be
traced? What ifthere are 25 cubes? 17 cubes?
You can trace a 5 by 4 rectangle from this solid.
Figure 2: A sample experience that can be used with an immersion approach.
Instruction
The stand alone and immersion approaches to developing thinking have implications for
instruction as well as curriculum. A teacher implementing stand alone instructional activities
such as those shown above could direct all questions and discussions to the thinking skills used
in an experience, not to the mathematics content. For the immersion approach the reverse could
be true. The questions and discussion could emphasize the mathematics understandings
students might gain from an experience with little attention given to the thinking skills exhibited
during the experience. In the problem solving class, discussions are held as to the merits and
limitations of these approaches. However, an argument is presented that the instructional
implications of the stand alone and immersion approaches when carried to the extreme are
inappropriate. Instead, an embedded thinking skills approach is encouraged [12].
In an embedded thinking skills approach, explicit attention is given both to the thinking
processes used in completing a task and to the mathematics content involved; a balance between
the two is attempted. It uses modeling and coaching by both the teacher and students. The
reciprocal teaching strategy of Palinscar and Brown developed initially for reading instruction
and later extended to mathematics [4] and Schoenfeld's [15] method for teaching problem
solving are two prominent examples of this approach discussed in the course.
The reciprocal teaching model as adapted to mathematics and Schoenfeld's method for
teaching problem solving emphasize the instructional milieu. In the reciprocal teaching model,
students work on three boards; Figure 3 shows a sample [4, p. 104]. In this approach,
"learning leaders [teacher or students] guide the group in working on three successive
chalkboards designed to help students proceed systematically...these procedures generate an
external record of the group's problem solving which can be monitored, evaluated, and
reflected upon" (p. 103).](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-344-320.jpg)
![335
Text ofProblem: Harry ate a hamburger and drank a glass of milk which totaled 495
calories. The m1lk contained half as many calories as the sandwich. How many
calories were in the sandwich and how many were in the milk?
PLANNING BOARD DRAWING BOARD DOING BOARD
A hamburger aNS a gla" 495
I I
of milk total.d 495 caloriu. H + H =495
Tlw milk contaiMd halt a, I I I I
tNlny caloriu than tlw L-.-.J
hamburger. H H
H =caloriu of hamburger
H =calori.. of milk
Figure 3: Sample boards from a reciprocal teaching approach.
Schoenfeld's [15] approach is based on a list of questions displayed in the classroom that the
teacher initially draws to the students' attention. With experience, students should learn to ask
themselves these questions as they solve problems. Schoenfeld's approach is aimed at
developing the student's metacognitive abilities for solving problems. Figure 4 shows the
questions Schoenfeld would display.
Questions to help you control your work.
• What (exactly) are you doing? (Can you describe it precisely?)
• Why are you doing it ? (How does it fit into the solution ?)
• How does it help you? (What will you do with the outcome when you attain it ?)
Figure 4: Schoenfeld's questions for student work.
The reciprocal teaching method and Schoenfeld's method are ones that reflect an embedded
thinking skills approach because the discussions of student work can address both the
mathematics content of the task and the thinking used to complete it. Another method that's
based on an embedded thinking skills approach also explored in the course is one developed by](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-345-320.jpg)
![336
Charles and Lester [5]. In this approach, a bulletin board is displayed in the classroom. Figure
5 shows a middle grades version of the bulletin board. Similar to Schoenfeld's method, the
teacher can initially draw students' attention to elements of the board and later, with experience,
students can utilize the board on their own.
UNDERSTANDING THE PROBLEM
• Read the problem.
• Decide what you are trying to find.
• Find the important data.
SOLVING THE PROBLEM
• Look for a pattem. • Draw a picture.
• Guess & check • Make an organized list.
• Write an equation • Make a tabie.
• Use logical reasoning • Use objects or act out.
• Work backwards • Simplify the problem.
ANSWERING THE PROBLEM & EVALUATING THE ANSWER
• Be sure you used all of the important information.
• Check your work.
• Decide whether the answer makes sense.
• Write the answer in a complete sentence.
Figure 5: A problem-solving bulletin board for the Charles and Lester method.
For the reciprocal teaching method and Schoenfeld's method, little specific direction is provided
for the teacher beyond suggesting that he or she facilitate student thinking. Work with the
Charles and Lester bulletin board suggests that most teachers interested in teaching problem
solving need specific suggestions for their roles during instruction beyond a bulletin board and
beyond being told that they should act as a facilitator. Based on work with teachers and
drawing from research on students' problem-solving processes and effective teaching, a
teaching strategy for problem solving was developed.](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-346-320.jpg)
![337
The Charles and Lester teaching strategy was designed to reflect four major roles a teacher
might play in facilitating student thinking and a sequence in which those roles are typically
played out during a lesson (see Figure 6).
Promote Student
Understanding
Monitor & Assess Promote Student
.. Student ThinkIng .. Reflection
and Work
Extend
.. Understanding
Figure 6: Some major roles a teacher can play in facilitating student thinking.
These four roles were translated into "teaching actions" and presented in a lesson plan format
(see Figure 7). The three phases of the lesson (Before, During, and After) were selected to
reflect the four roles a teacher might play to facilitate thinking. (Two of the roles are contained
in the After part of the plan.)
A complete discussion of the teaching actions can be found elsewhere (see [5]). Also,
recommendations for research on this teaching strategy can be found elsewhere [7]. The
remainder of this section contains the rationale for this teaching strategy.
The idea of a lesson plan was conceived as a vehicle for inviting teachers to begin teaching
problem solving. There may be concern when a teaching strategy is given in the form of a
lesson plan. One possible concern is that the strategy becomes a rigid sequence of behaviors
that does not allow for individual differences of either teachers or students. Another possible
concern is that a lesson plan may promote the false belief that an algorithm exists for teaching
problem solving. Although these concerns are legitimate, experience with this lesson plan and
research on teachers' planning behaviors and thought processes suggest that a problem-solving
lesson plan is not only helpful in getting teachers started teaching problem solving but may
indeed be a necessary ingredient in changing teachers' instructional behaviors. In a study by
Charles and Lester [6], interviews with teachers who used this lesson plan revealed that all of
the teachers found it helpful in getting started teaching problem solving. Also, the teaching
strategy was considered flexible enough to allow teachers to adapt it to their own teaching styles
but structured enough to help less confident teachers feel comfortable teaching problem solving.
Research on teachers' planning behaviors and thought processes provides an explanation
for the value of the teaching actions. A major component of teachers' planning is the selection
and sequencing of instructional activities [16]. In general, teachers do not begin their planning
for a lesson by thinking about specific instructional techniques they will use to achieve an
objective [8]. Rather, they begin by establishing activity chunks sequenced in a particular way.
Each activity chunk may contain several sub-activities. A sequence of activity chunks, provides
an agenda [10] or a script [I, 14] for the lesson. The value of the agenda is that it structures the](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-347-320.jpg)
![Teachino Actions Before
1. Read the problem.
2. Ask questions for
undemanding the
problem.
3. Discuss possible
solution strategies.
ram. Understanding
Teachino Actions During
4. ObselVe studelis.
5. Give hints as needed.
6. Require students to
check back and
a/lSl/erthe problem.
7. Give an extension
as needed.
Idon.r and Assess
338
[Insert problem statement here]
Questions for Understanding the Problem (fA 2)
(Write questions here related to
understanding the problem.]
Possible Hints for Solving the Problem (fA 5)
[Write clusters of hints related to
particular solution approaches]
Teacbino Actions After Possible Solution (fA 8)
8. Discuss solutions.
Name strategies. [ShOW one or more possible solutions here.]
9. Discuss related
problems and the
extension.
1O. Discuss special
features as needed.
11. Connect math
conteli.
Promote Rellection
E*nd Understandinos
Related Problems: (Reference similar problems solved previously.]
Problem Extension: (Write here an extension of the original problem.]
Figure 7: A lesson plan fonn for teaching problem solving.
activities of a lesson, making both the teacher's and students' actions predictable and reducing
the complexity of the information teachers encounter during instruction. With experience, the](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-348-320.jpg)
![339
agenda becomes internalized or routinized relieving the teacher from having to consciously
develop a mental road map for each lesson. Leinhardt [10] suggests that, "The use of routines
[agendas] also reduces the cognitive processing for the teacher and provides them with the
intellectual and temporal room needed to handle the dynamic portions of the lesson" (p. 27-8).
Experience with this lesson plan shows that teachers initially follow the teaching actions
quite closely. With experience, most teachers internalize the teaching actions and refer to the
lesson plan only on occasion. Observations and interviews with teachers also showed that only
after teachers internalized the teaching actions were they (cognitively) ready to focus on the
"dynamic portions" of teaching problem solving (e.g., ways to improve their skills giving hints
or ways to improve their discussions with students about problem solutions). Until teachers
internalized the teaching actions, they did not feel in control of the lesson and they were not
interested in exploring ways to improve individual teaching actions.
The Charles and Lester method described here is another example of an embedded thinking
skills approach to developing thinking. It is similar to the reciprocal teaching method and
Schoenfeld's method in that a board is used to draw attention to thinking processes and
discussions about student work solving a problem can focus both on the mathematics content
involved and on the thinking processes exhibited. However, unlike the reciprocal teaching
method and Schoenfeld's method, Charles and Lester's method provides specific ideas for
ways the teacher can facilitate student thinking and work.
Needed Research: Curriculum, Instruction, and Teacher Education
Reflecting on this course and the many instances in which it has been implemented suggests
several issues and questions that need investigation. These issues and questions can be grouped
into three areas: curriculum, instruction, and teacher education.
Curriculum
Building a school curriculum with a focus on problem solving or thinking is a challenging
activity; an activity for which there are many unanswered questions. For example, there are
unanswered questions about the nature of the instructional tasks that might be used in a
thinking-based curriculum. Which instructional tasks contribute most toward the development
of thinking abilities, and how do we evaluate tasks for inclusion in the curriculum? Much
attention in the 1980s was given to teaching problem-solving skills and strategies. Should these
continue as elements of the curriculum? Furthermore, are the problem-solving strategies
typically introduced in the elementary school grades the ones that should be included? A look at](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-349-320.jpg)
![340
the research on critical thinking makes it unclear as to which strategies should receive explicit
attention [13].
Curriculum tasks can be presented in a variety of environments. Many of the papers in this
book explore the nuances of computer environments, but there are indeed other environments
that should be considered (e.g., manipulatives). Should there be concern about the balance
among these environments in building curriculum?
Scope and sequence have always been key building blocks for school curriculum. Are
issues of scope and sequence no longer relevant when problem solving is the foundation of the
curriculum? How should a problem-solving based curriculum evolve across the grades? Where
should the curriculum fallon a fixed-dynamic dimension; fixed meaning the traditional scope
and sequence models and dynamic meaning that scope and sequence evolve situationally?
Perhaps the time has come for drastically different definitions and conceptualizations of
curriculum.
Finally, an issue seldom discussed is how much higher-level thinking can students and
teachers tolerate? Much human activity is aimed at reducing complexity to routine. Is there value
in routine? Does it provide time for mental and physical resources to rebuild themselves for the
next complex task to be encountered?
Instruction
An important issue related to teaching is how to invite teachers to begin making
thinking/problem solving the focus of instruction. Knowing how we want teachers to think and
act as facilitators of student thinking and knowing what advice will help them move toward that
vision are separate issues. Boyle e Ponder [2] identified four conditions necessary for teachers
to act on advice. The advice must (i) have administrative support, (ii) not conflict with the
teacher's role definition, (iii) be cost effective in terms of the teacher's time and energy, and (iv)
suggest specific behaviors. All of these need to be considered as ways to invite teachers to
become facilitators of thinking are explored.
Finally, four roles for a facilitator were identified in this paper; are they appropriate? Are
these roles generic? That is, do they apply to all tasks and for all task environments?
Teacher Education
It was mentioned at the beginning of this paper that a massive number of teacher education
activities related to problem solving are taking place throughout the United States. As issues of
curriculum and instruction continue to unfold and be resolved, issues of teacher education need
to be explored. One of the most important is articulating goals for teacher education. Several](https://image.slidesharecdn.com/aframeworkforresearchonproblem-solvinginstruction-230807162854-0c6e96f4/85/A-Framework-for-Research-on-Problem-Solving-Instruction-pdf-350-320.jpg)
More Related Content
Similar to A Framework for Research on Problem-Solving Instruction.pdf (20)
A Framework for Research on Problem-Solving Instruction.pdf
- 1. NATO ASI Series Advanced Science Institutes Series A series presenting the results of activities sponsored by the NATO Science Committee, which aims at the dissemination ofadvanced scientific and technological knowledge, with a view to strengthening links between scientific communities. The Series is published by an international board of publishers in conjunction with the NATO Scientific Affairs Division A Life Sciences B Physics C Mathematical and Physical Sciences o Behavioural and Social Sciences E Applied Sciences F Computer and Systems Sciences G Ecological Sciences H Cell Biology I Global Environmental Change NATo-pea DATABASE Plenum Publishing Corporation London and New York Kluwer AcademiCl Publishers Dordrecht, Boston and London Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest The electronic index to the NATO ASI Series provides full bibliographical references (with keywords and/or abstracts) to more than 30000 contributions from international scientists published in all sections of the NATO ASI Series. Access to the NATO-PCO DATABASE compiled by the NATO Publication Coordination Office is possible in two ways: - via online FILE 128 (NATO-PCO DATABASE) hosted by ESRIN, Via Galileo Galilei, 1-00044 Frascati, Italy. - via CD-ROM "NATO-PCO DATABASE" with user-friendly retrieval software in English, French and German (© WTV GmbH and DATAWARE Technologies Inc. 1989). The CD-ROM can be ordered through any member of the Board of Publishers or through NATO-PCO, Overijse, Belgium. Series F: Computer and Systems Sciences Vol. 89
- 2. The ASI Series Books Published as a Result of Activities of the Special Programme on ADVANCED EDUCATIONAL TECHNOLOGY This book contains the proceedings of a NATO Advanced Research Workshop held within the activities of the NATO Special Programme on Advanced Educational Technology, running from 1988 to 1993 under the auspices of the NATO Science Committee. The books published so far as a result of the activities of the Special Programme are: Vol. F 67: Designing Hypermedia for Learning. Edited by D. H. Jonassen and H. Mandl. 1990. Vol. F76: Multimedia Interface Design in Education. Edited by A. D. N. Edwards and S. Holland. 1992. Vol. F 78: Integrating Advanced Technology into Technology Education. Edited by M. Hacker, A. Gordon, and M. de Vries. 1991. Vol. F80: Intelligent Tutoring Systems for Foreign Language Learning. The Bridge to International Communication. Edited by M. L Swartz and M. Yazdani. 1992. Vol. F81: Cognitive Tools for Learning. Edited by PAM. Kommers, D.H. Jonassen, and J.T. Mayes. 1992. Vol. F84: Computer-Based Learning Environments and Problem Solving. Edited by E. De Corte, M. C. Linn, H. Mandl, and L. Verschaffel. 1992. Vol. F85: Adaptive Learning Environments. Foundations and Frontiers. Edited by M. Jones and P. H. Winne. 1992. Vol. F86: Intelligent Learning Environments and Knowledge Acquisition in Physics. Edited by A. Tiberghien and H. Mandl. 1992. Vol. F87: Cognitive Modelling and Interactive Environments in Language Learning. Edited by F. L. Engel, D. G. Bouwhuis, T. Basser, and G. d'Ydewaile. 1992. Vol. F89: Mathematical Problem Solving and New Information Technologies. Edited by J. P. Ponte, J. F. Matos, J. M. Matos, and D. Fernandes. 1992. Vol. F90: Collaborative Learning Through Computer Conferencing. Edited by A. R. Kaye. 1992.
- 3. Mathematical Problem Solving and New Information Technologies Research in Contexts of Practice Edited by Joao Pedro Ponte Joao Filipe Matos Departamento de Educac;ao, Faculdade de Ciâlcias Universidade de Lisboa, Av. 24 de Julho, 134-4° P-1300 Lisboa, Portugal Jose Manuel Matos Sec'1ao de Ciâlcias da Educac;ao, Faculdade de Ciâlcias e Tecnologia Universidade Nova de Lisboa P-2825 Monte da Caparica, Portugal Domingos Fernandes Instituto de Inova'1ao Educacional, Travessa Terras de Sant'Ana-15 P-1200 Lisboa, Portugal Springer-Verlag Berlin Heidelberg GmbH
- 4. Proceedings of the NATO Advanced Research Wor1<.shop on Advances in Mathematical Problem SoIving Research, held in Viana do Castelo, Portugal, 27-30 ApOI, 1991 CR Subiect Classification (1991): K.3.1 ISBN 978-3-642-83483-3 ISBN 978-3-642-58142-7 (eBook) DOI 10.1007/978-3-642-58142-7 This work is subject to copyright. M rights are reservoo.wh!!ther the whole Of" part 01 tha material is concemed. spacitically the rightsol translation, reprinting, re!.lse 01 illustrations. recitation, broadcasting, reproduction on mierolilms or in any OIher way. and sto(aga in data banks. Duptication 01 this publication Oi' parts thereo/ is parmittad ony undei" the provisioos 01 tho German COpyright Law 0/ S6ptember 9. 1965. in ｾｳ＠ cun&rll version. and p8fmission /o( usa m.Jst atway$ ba obtained lrom Springer-Variag. VlOIations ara liabie lor prosecution undar the German COpyright Law. C Springar-Variag Berlin He1de1barg 1992 Originally published by Springer-Verfag Berlin Heidelberg NfI'N Yor( in 1992 Soflcover reprint of !ha har<loover 1si edition 1992 Typasatting: C!mefa f88.dy by aultlors 45(.3140 - 5 4 3 2 1 O- Plinted on acid-rrea papar
- 5. Preface A strong and fluent competency in mathematics is a necessary condition for scientific, technological and economic progress. However, it is widely recognized that problem solving, reasoning, and thinking processes are critical areas in which students' performance lags far behind what should be expected and desired. Mathematics is indeed an important subject, but is also important to be able to use it in extra-mathematical contexts. Thinking strictly in terms of mathematics or thinking in terms of its relations with the real world involve quite different processes and issues. This book includes the revised papers presented at the NATO ARW "Information Technology and Mathematical Problem Solving Research", held in April 1991, in Viana do Castelo, Portugal, which focused on the implications of computerized learning environments and cognitive psychology research for these mathematical activities. In recent years, several committees, professional associations, and distinguished individuals throughout the world have put forward proposals to renew mathematics curricula, all emphasizing the importance of problem solving. In order to be successful, these reforming intentions require a theory-driven research base. But mathematics problem solving may be considered a "chaotic field" in which progress has been quite slow. There are many questions still to be resolved, including: - The purpose of problem solving: what is the problem solving activity? What are suitable problems for teaching purposes? How are they best explored in class? How can teachers develop their awareness about it? - Its curricular status: how are problems integrated with the remaining classroom activities? What instructional strategies can be used to improve students' competency? How should they be assessed? - The nature of the necessary research: what is the kind of knowledge that should be striven for? What are the most appropriate research methodologies? Are there critical variables to control, manipulate, and measure, and what are they? There are four main questions running through the papers presented in this book: (a) What is a problem-solving activity and how can it be assessed? (b) What psychological theories can be used to explain and improve students' problem-solving ability? (c) What are the implications of new information technologies for mathematical problem solving? (d) How can these issues
- 6. VI be carried out to practice, namely to teacher education and to the instructional environments? The papers are sequenced according to their focus on one of those themes; however, most of these aspects pervade back and forth through the book. A simple glance at the abstracts and keywords will tell the reader the main concerns ofeach author. Indeed, the nature of problem-solving as a mathematical activity is a critical issue. Problem solving has become a very popular expression, meaning quite different things for different people. It can be seen as an add-on to the existing practices, or more radically as a replacement. Alternatively, it may be seen as a new perspective which enables us to see new and old things in a different way. In some views, problem solving mostly consists of well- defined activities, which may have a peripheral or central role in mathematics learning. For others, problem solving is just an aspect of the more general concept of mathematical experience. Looking beyond well-defined and clearly formulated problems, one can devise other highly significant kinds of activity such as open-ended investigations, making and testing conjectures, and problem formulation, which may extend the flavour of creative mathematical thinking into the mathematics classroom. A problem formulated as such by a researcher or by a teacher may not have the same meaning (in fact any meaning at all) for a student. But the purpose of education is not that the students should just be expected to be able to solve the problems and tasks proposed by the teachers, but also to be able to raise, solve, and evaluate their own significant questions. A closely related issue concerns students' assessment. Without proper assessment the teacher does not have the possibility of ascertaining if the pupils are making any progress at all. Without assessment instruments and procedures there is no way of conducting research. However, assessing problem solving has been a major difficulty both for research and practice. If one is not just concerned with the number of exact solutions but also with the strategies and thought processes (successful or unsuccessful), and these are not overtly shown, then how does one assess them? Analytic scoring schemes, drawing upon Polya's four-stage model of problem solving, have been proposed as a frame for getting a composite picture of students' work. Alternatively, problem solving can be assessed in a holistic way, given the complex nature of the mental processes involved and the fact that this activity is always both socially and culturally situated. The tasks proposed may range from well-defined mathematical questions to open-ended real-life situations. However, the difficulties do not lie just in the researchers' models and in the teachers' capability to use new approaches. They have to do also with the students who are used to being tested on content knowledge rather on thinking processes, reasoning, or strategies arising in complex problems, and simply do not understand the new demands that are required from them. In a word, significant problem-solving experiences and new assessment tasks may imply not just new classroom procedures but also a significant cultural change.
- 7. VII Psychological research has been a major theoretical influence on mathematical problem solving. In the past, most research efforts in this field followed an established tradition in psychology: students are presented with ready-made, well-defmed, tasks in laboratory settings. Much valuable information can be gathered from this kind of research, but current educational thinking stresses the importance of activities of a very different kind, in which (a) students participate in the process of defining the nature and goal of the task, (b) things are not so well- defined from the very beginning but follow processes which may bear successive reformulations, and (c) the setting is much more complex in terms of roles and interactions than the usually straightforward researcher-subject relationship. Problems presented to the students in previous research include, among others, puzzles, word problems, simple and complex real- world applications, concept recognition tasks, strategy games, and questions relying on the knowledge of established formal mathematics. It is quite unlikely that general theories can be drawn up to encompass all these activities. And it is certainly questionable whether all of them have equivalent educational usefulness and value. However, one can recognize important steps that have been accomplished more recently. A general agreement has been reached around the need to take into consideration students' thought processes (especially metacognitive processes) and not just their problem solving competency measured by success rates in given kinds of problems. The need for theoretical orientations guiding research efforts and the value of the contributions of cognitive psychology to study those processes have been widely recognized. At this meeting the notions of conceptual field, didactic contract, cognitive conflict, sense making, and noticing were proposed to further extend the theoretical frameworks to study mathematical problem solving. Further progress is also dependent on refining the definition of key concepts and methodologies for research in this area. The availability of modern information technologies with its possibility to empower mathematical thinking and extend the range and scope of applications of mathematics poses new challenges for mathematical problem-solving research and practice. The computer can be viewed as a powerful intellectual tool, providing the means to automate routine processes and concentrate strategic thinking. Alternatively, it can be seen as a base to construct microworlds deliberately designed to foster specific kinds of knowledge. Attempts are being made to apply artificial intelligence techniques to software design in mathematics. But several contributions at the meeting made it clear that the software never works just by itself, and a critical role is played by the teacher in setting up the learning situation. With the computer, mathematics can become a much more experimental activity. However, the possibility of conducting easily a large number ofexperiences may prevent the most appropriate thinking from taking place - especially if students are not properly encouraged to develop critical and metacognitive processes. Problem solving is related to skills learning, concept and principle development, and reasoning processes. The way it is integrated into the classroom practices bears close connections with the teachers' confidence, feeling, educational agenda and espoused
- 8. VIII philosophy of mathematics. In this meeting, there was a special concern with the issues of classroom practice. In the countries in which problem solving is part of the curriculum, the activities that are performed in schools, if they exist, tend to not go much further than simple applications of content knowledge. How will teachers acquire confidence as problem solvers? How will they acquire the competence to conduct problem solving activities? What sort of support will be required? It is necessary to study the role of the teacher and the processes of teacher development, both through formal training programmes and through their participation in innovation processes. Formal pre-service and in-service courses are needed but they cannot replace the initiative of the teachers themselves giving rise to grass-roots innovative experiences. These, in tum, to be fruitful, consistent, and lasting should have support from research and teacher education institutions. The perspective of the computer as an intellectual tool, enabling explorations, and empowering the students, dominates current thinking concerning the use of this instrument in mathematical problem solving activities. However, the generalization of its use imposes stricter demands both on software development and teacher education. The growing interest of researchers in what happens in the classrooms may hopefully represent a turning point in this field. The concern with investigations, conjecturing, problem posing, and seeing problem solving in a wider context of mathematical activities may also represent a very important new direction of thinking. If that is the case, the computer with its strong invitation to experimentation may well have been a decisive factor in this evolution. Lisbon, May 1992 Jo3:o Pedro Ponte JOOo Filipe Matos Jose Manuel Matos Domingos Fernandes
- 9. Contributors and Participants Canada Joel Hillel Department of Mathematics and Statistics. Conconlia University Loyola Campus. 7141 Sherbrooke Street West. Montreal. Quebec H4B lR6. CANADA E_mail: [email protected] France Colette Laborde Laboratoire LSD2 - IMAG. Univ. Joseph Fourier-CNRS BP 53 X. 38041 Grenoble Cedex. FRANCE Jean Marie Laborde Laboratoire LSD2 - IMAG. Univ. Joseph Fourier-CNRS BP 53 X. 38041 Grenoble Cedex. FRANCE E_mail: [email protected] Jean ｆｾｩｳ＠ Nicaud CNRS University ofParis-Sud. Laboratoire de Recherche en Informatique Batiment 490. 91405 Orsay Cedex. FRANCE E_mail: [email protected] Greece Chronis Kynigos University of Athens 19 Kleomenous St.• 10675 Athens. GREECE E_mail: [email protected] Israel Tommy Dreyfus Center for Technological Education P.O. Box 305. Holon 58102. ISRAEL Fax: 972-1-5028967 Italy Ferdinanda Anarello Dipartimento di Matematica. University of Torino Via Carlo Alberto. 10. 10123 Torino. ITALY Fax: 39-11-534497
- 10. PaoloBoero Dipartimento di Matematica,University ofGenova Via L.B. Alberti,4, 16132 Genova, ITALY Fax: 39-10-3538769 Pier Luigi Ferrari University of Genova, Dipartimento di Matematica Via L.B. Alberti,4, 16132 Genova, ITALY E_mail: grandis@igecuniv Fax:39 10 3538763 Fulvia Furinghetti Dipartimento di Matematica, University ofGenova Via L.B. Alberti,4, 16132 Genova, ITALY Fax: 39-10-3538769 The Netherlands Henk van der Kooij ow&OC, Utrecht University Tiberdreef4, 3561 GG Utrecht, THE NETHERLANDS E_mail: [email protected] Fax: 31-30-660403 Portugal Domingos Fernandes Instituto de ｉｮｯｶｾｮｯ＠ Educacional x Travessa Terras de Sant'Ana - IS, 1200 LISBOA, PORTUGAL Fax: 351-1- 690731 Henrique Manuel Guirnaries Departamento de ｅ､ｾＬ＠ Faculdade de Cimcias, Universidade de Lisboa Av. 24 Julho - 134 - 4°, 1300 LISBOA, PORTUGAL E_mail: [email protected] Fax: 351-1-604546 Maria Cristina Loureiro Escola Superior de ｅ､ｾ＠ de Lisboa Av. Carolina Micaelis, 1700 LISBOA, PORTUGAL Joao Filipe Matos Departamento de ｅ､ｾｮｯＬ＠ Faculdade de Cimcias, Universidade de Lisboa Av. 24 Julho - 134 - 4°,1300 LISBOA, PORTUGAL E_mail: [email protected] Fax: 351-1-604546 Jose Manuel Matos ｓｾ＠ de Cimcias da ｅ､ｾＬ＠ Faculdade de Ciencias e Tecnologia, Universidade Nova de Lisboa 2825 MONTE DA CAPARICA, PORTUGAL
- 11. XI Jolo Pedro Ponte Departamento de ｅ､ｾＬ＠ Faculdade de Ci&1cias, Universidade de Lisboa Av. 24 Julho - 134 - 4°, 1300 LlSBOA, PORTUGAL E_mail: [email protected] ｆ｡ｸＺＳＵＱＭＱｾＵＴＶ＠ Jaime Carvalho e Silva Departamentode Matem4Iica. Universidade de Coimbra Apartado 3008 COIMBRA, PORnJGAL E_mail: [email protected] Fax: 351-039-32568 MariaGraciosa Ve1oso Departamento ､･ｾＬ＠ Faculdade de Ci&1cias, Universidade de Lisboa Av. 24 Juiho - 134 - 4°, 1300 LISBOA, PORTUGAL E_mail: [email protected] Fax: ＳＵＱＭＱｾＵＴＶ＠ Spain Juan Dfaz Godino Dep. Didactica de la Matematica, Escuela Universitaria del Professorado Campus de Cartuja. 18071 GRANADA, SPAIN E_mail: [email protected] Fax: 58-203561 United Kingdom Alan Bell Shell Centre for Mathematical Education University Park, Nottingham, NG7 2RD, UK Fax: 0602-420825 PaulEmest School of Education, University ofExeter St Luke's, Heavitree Road, Exeter, EXI 2LU, UK E_mail: (Janet)[email protected] Fax: 0392-264857 John Mason Centre for Mathematics Education,The Open University Walton Hall, Milton Keynes, MK76AA, UK E_mail: [email protected] USA Patricia A. Alexander Department ofEDCI/EPSY, College of Education, Texas A&M University College Station, TX 77843, USA E_mail: [email protected] Fax:409-845-6129
- 12. XII Randall I. Charles Dep. Mathematics and Computer Science, San Jose State University San Jose, CA 95192-0103, USA Fax: 408-924-4815 Robert Kansky Mathematical Sciences Education Board 818 Connecticut Avenue, NW, Suite 500, Washington, DC 20006, USA E_mail: [email protected] Fax:202-334-1453 Jeremy Kilpatrick Department of Mathematics Education, University of Georgia 105 Aderhold Hall Athens, Georgia 30602, USA E_mail: [email protected] Fax:I-404-542-SOIO or 4551 Frank Lester, Jr. Mathematics Education Development Center, School of Education, Indiana University W.W. Wright Education Building, Suite 309, Bloomington, IN 47405, USA E_mail: [email protected] Fax: 812-855-3044 Judah Schwartz Educational Technology Center, Harvard Graduate School of Education Nichols House, Appian Way, Cambridge MA 02138, USA E_mail: [email protected] Fax: 617-495-0540 Lora J. Shapiro Learning Research and Development Center, University of Pittsburgh 3939 O'Hara Street, Pittsburgh, PA 15260, USA E_mail:Shapiro@Pittvms Fax: 412-6249149
- 13. Table of Contents A Framework for Research on Problem-Solving Instruction. . . . . . . . . . . . . . . . . . .. 1 Frank K. Lester, Jr., Randalll. Charles Researching Problem Solving from the Inside . . . . . . . . . . . ; . . . . . . . . . . . . . . . . . 17 John Mason Some Issues in the Assessment of Mathematical Problem Solving. . . . . . . . . . . . . .. 37 Jeremy Kilpatrick Assessment of Mathematical Modelling and Applications ......................45 Henkvan tier Kooij A Cognitive Perspective on Mathematics: Issues of Perception, Instruction, and Assessment ................................................. 61 Patricia A. Alexander The Crucial Role of Semantic Fields in the Development of Problem Solving Skills in the School Environment ......................................... 77 Paolo Boero Cognitive Models in Geometry Learning ................................ 93 Jose Manuel Matos Examinations of Situation-Based Reasoning and Sense-Making in Students' Interpretations of Solutions to a Mathematics Story Problem .................. 113 Edward A. Silver, Lora J. Shapiro Aspects of Hypothetical Reasoning in Problem Solving ..................... 125 PierLuigi Fe"ari
- 14. XIV Problem Solving, Mathematical Activity and Learning: The Place of Reflection and Cognitive Conflict. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 137 Alan Bell Pre-Algebraic Problem Solving ................................... " 155 Ferdinando Anarello Can We Solve the Problem Solving Problem Without Posing the Problem Posing Problem? ......................................... 167 JudahL. Schwartz Problem Solving in Geometry: From Microworlds to Intelligent Computer Environments .......................................... 177 Colette Laborde, Jean-Marie Laborde Task Variables in Statistical Problem Solving Using Computers ................ 193 J. Dfaz Godino, M. C. Batanero Bernabeu, A. Estepa Castro The Computer as a Problem Solving Tool; It Gets a Job Done, but Is It Always Appropriate? .................................................. 205 Joel Hillel Insights into Pupils' and Teachers' Activities in Pupil-Controlled Problem-Solving Situations: A Longitudinally Developing Use for Programming by All in a Primary School ...................................................... 219 Chronis Kynigos Cognitive Processes and Social Interactions in Mathematical Investigations. . . . . . .. 239 Jotio Pedro Ponte, Jotio Filipe Matos Aspects of Computerized Learning Environments Which Support Problem Solving ... 255 Tommy Dreyfus A General Model of Algebraic Problem Solving for the Design of Interactive Learning Environments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 267 Jean-Franfois Nicaud
- 15. xv Problem Solving: Its Assimilation to the Teacher's Perspective. . . . . . . . . . . . . . .. 287 Paul Ernest Computer Spreadsheet and Investigative Activities: A Case Study of an Innovative Experience ............................................ 301 Jodo Pedro Ponte, Susana Carreira Examining Effects of Heuristic Processes on the Problem-Solving Education of Preservice Mathematics Teachers .................................... 313 Domingos Fernandes Mathematics Problem Solving: Some Issues Related to Teacher Education, School Curriculum, and Instruction ........................................ 329 RandallI. Charles Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 343
- 16. A Framework for Research on Problem-Solving Instruction Frank K. Lester, Jr.1, Randall!. Charles2 ISchool ofEducation, Indiana University, Bloomington, Indiana 47405, USA 2San Jose State University, San Jose, California 95192-0103, USA Abstract: Research on mathematical problem solving has provided little specific infonnation about problem-solving instruction. There appear to be four reasons for this unfortunate state of affairs: (1) relatively little attention has been given to the role of the teacher in instruction; (2) there has been little concern for what happens in real classrooms; (3) there has been a focus on individuals rather than small groups or whole classes; and (4) much of the research has been largely atheoretical in nature. This paper discusses each of these reasons and presents a framework for designing research on mathematics problem-solving instruction. Four major components of the framework are discussed: extra-instruction considerations, teacher planning, classroom processes, and instructional outcomes. Special attention is given to factors that may be particularly fruitful as the focal points of future research. Keywords: problem-solving instruction, teacher education, teaching, classroom processes, affects, instructional outcomes Despite the popUlarity of problem solving as a topic of research for mathematics educators during the past several years, there has been growing dissatisfaction with the slow pace at which our knowledge has increased about problem-solving instruction. Research has demonstrated that an individual must solve many problems over a prolonged period of time in order to become a better problem solver. Unfortunately, beyond this there is little specific advice for teachers that can be gleaned from research. In this paper we provide a brief discussion of some of the reasons for this unfortunate condition, we propose a step toward a solution, and we discuss some important considerations for future research concerned with mathematical problem-solving instruction. Why Has Progress Been Slow? Among the reasons why the research conducted thus far has provided so little direction for problem-solving instruction, four stand out as especially prominent: (a) lack of attention to the role of the teacher in instruction, (b) the absence of concern for what actually happens in real
- 17. 2 classrooms, (c) a focus on individuals rather than groups or whole classes, and (d) the atheoretical nature of much of the research. The Role of the Teacher Silver [35] has pointed out that the typical research report might describe in a general way the instructional method employed, but rarely is any mention made of the teacher's specific role. A step toward structuring descriptions of what teachers do during instruction was taken by the Mathematical Problem Solving Project (MPSP) at Indiana University with the identification of several teaching actions for problem solving [38]. Subsequent to the work of the MPSP, we identified ten teaching actions for problem solving [5, 6]. These teaching actions were selected by listing the thinking processes and behaviors that the research literature and other sources suggested as desirable outcomes of problem-solving instruction. We then identified teaching behaviors that seemed likely to promote these behaviors. Our task was made especially difficult by the fact that none of the literature on mathematical problem-solving instruction discussed the specifics of the teacher's role and very little of the research literature on teaching dealt with problem solving. As reasonable as our effort seemed at the time, we now view it as incomplete. What is needed is to consider teaching behavior, not simply as an agent to effect certain student outcomes, but rather as one dimension of a dynamic interaction among several dimensions. Observations of Real Classrooms A few years ago we conducted a large-scale study of the effectiveness of an approach to problem-solving instruction based on the ten teaching actions mentioned above. The research involved several hundred fifth and seventh grade students in more than 40 classrooms [6]. The results were gratifying: students receiving the instruction over the course of a year benefited tremendously with respect to several key components of the problem-solving process. However, despite the promise of our instructional approach, the conditions under which the study was conducted did not allow us to make extensive systematic observations of classrooms. Ours is not an isolated instance. Good and Biddle [13], Grouws [14], and Silver [35] have noticed an absence of adequate descriptions of what actually happens in the classroom. In particular, there has been a lack of descriptions of teachers' behaviors, teacher- student and student-student interactions, and the type of classroom atmosphere that exists. It is vital that such descriptions be compiled if there is to be any hope of deriving sound prescriptions for teaching problem solving.
- 18. 3 Focus on Individuals Rather than Groups or Whole Classes Much of the research in mathematical problem solving has focused on the thinking processes used by individuals as they solve problems or as they reflect back on their work solving problems. When the goal of research is to characterize the thinking involved in a process like problem solving, a microanalysis of individual performance seems appropriate. However, when our concerns are with classroom instruction. we should give attention to groups and whole classes. We agree with the argument of Shavelson and his colleagues [32] that small groups can serve as an appropriate environment for research on teaching problem solving. But. the research on problem-solving instruction cannot be limited to the study of small groups. Lester [25] suggests that. in order for the field to move forward. research on teaching ｰｲｯ｢ｬ･ｾ＠ solving needs to examine teaching and learning processes for individuals, small groups, and whole classes. Atheoretical Nature of the Research The absence of any widely-accepted theories to guide the conduct of research is a serious problem [1, 17,20]. The adoption of a theory orientation toward research in this area is crucial if progress is to be made toward establishing a stable body of knowledge about problem- solving instruction. In particular, we would like to see the development of theories of problem- solving instruction become a top priority for mathematics education research. Toward a Solution An important ingredient in making research on problem-solving instruction more fruitful is the clear description of the factorsl involved. In this section we provide an analysis of the factors relevant to the study of problem-solving instruction. This analysis is intended as a structure within which to design research. Categories of Factors Related to Problem-solving Instruction Our first attempt to provide an analysis of the factors that might be related to problem-solving instruction was made by Lester [24] in his general discussion of methodological issues lWe are aware that the word "factors" is commonly associated with experimental and quasi-experimental research methods. Our use of this word should not be interpreted as indicating that we have a preference for this sort of research to study problem-solving instruction. We have no such preference. Indeed, we believe that progress in this area will bemade primarily by means ofcarefully conducted ethnographic studies, longitudinal case studies of classrooms, observations of individuals and small groups, and clinical interviews. The word "factors" is used here exclusively to indicate what we consider to be key ingredients in the success or failure of problem-solving instruction.
- 19. 4 associated with research in this area. Lester's analysis was based on the earlier efforts of Dunkin and Biddle [9] and Kilpatrick [16]. Dunkin and Biddle's work was a broad view of the important categories of factors associated with research on teaching and was not specifically concerned with either mathematics or problem solving. Kilpatrick's analysis was restricted to research on teaching problem-solving heuristics in mathematics. After considering research on teaching behavior and teacher-student interaction (see e.g., [22, 23, 30, 31, 41, 42]), Charles [5] developed a refinement of Lester's original analysis. The primary improvement was that Charles' analysis incorporated two factors which were not present in Lester's work: a teacher planning category and interactions among categories of classroom processes. The structure presented below is a further refinement of our previous conceptualizations and other more recent conceptualizations of teaching. As is true of Charles's categorization, we have chosen to classify factors on the basis of three broad categories: Extra-classroom Considerations, Classroom Processes, and Instructional Outcomes. We have also identified a category that cuts across each of these categories: Teacher Planning. The four categories are discussed in the following sections. Category 1: Extra·classroom Considerations. What goes on in a classroom is influenced by many things. For example, the teacher's and students' knowledge, beliefs, attitudes, emotions, and dispositions all play a part in determining what happens during instruction. Furthermore, the nature of the tasks (i.e., the activities in which students and teachers engage during instruction) included during instruction, as well as the contextual conditions present also affect instruction. We have identified six types of considerations: teacher presage characteristics, student presage characteristics, teacher knowledge and affects, student knowledge and affects, tasks features, contextual (situational) conditions. Teacher and Student Presa&<; Characteristics. These are characteristics of the teacher and students that are not amenable to change but which may be examined for their effects on classroom processes. In addition, presage characteristics serve to describe the individuals involved. Typically, in experimental research these characteristics have potential for control by the researcher. But, awareness of these characteristics can be useful in non-experimental research as well help researchers make sense of what they are observing. Among the more prominent presage characteristics are age, sex, and previous experience (e.g., teaching experience, previous experience with the topic of instruction). Factors such as previous experience may indeed be of great importance as we learn more about the ways knowledge teachers glean from experience influences practice (see [33]). Teacher and Student Affects. Cognitions. and Metacognitions. The teacher's and students' knowledge (both cognitive and metacognitive) and affects (including beliefs) can strongly influence both the nature and effectiveness of instruction. As a category, these teacher and student traits are similar to, but quite different from, presage characteristics. The similarity lies in the potential for providing clear descriptions of the teacher and students. The difference between the two is that affects, cognitions, and metacognitions may change, in particular as a result of instruction, whereas presage characteristics cannot.
- 20. 5 Much recent attention has been given to the role of the teacher's knowledge as it relates to planning and classroom instruction (see e.g.,[33]). Doyle [8] suggests that teaching can be viewed as a problem-solving activity where the knowledge a teacher brings to the teaching situation is craft knowledge [34], that is, knowledge gleaned from experience. Doyle suggests that "teaching is, in other words, fundamentally a cognitive activity based on knowledge of the probable trajectory of events in classrooms and the way specific actions affect situations" [8, p. 355]. Thus, there is growing evidence that the knowledge a teacher brings to the classroom has a significant impact on the events that follow, and that there are a variety ofkinds of teacher knowledge that need to be considered in research on teaching (e.g., content knowledge, classroom knowledge). Task Features. Task features are the characteristics of the problems used in instruction and for assessment. At least five types of features serve to describe tasks: syntax, content, context, structure, and process (see [12]). Syntax features refer to the "arrangement of and relationships among words and symbols in a problem" [18, p. 16]. Content features deal with the mathematical meanings in the problem. Two important categories of content features are the mathematical content area (e.g., geometry, probability) and linguistic content features (e.g., terms having special mathematical meanings such as "less than," "function," "squared"). Context features are the non-mathematical meanings in the problem statement. Furthermore, context features describe the problem embodiment (representation), verbal setting, and the format of the information given in the problem statement. Structurefeatures can be described as the logical-mathematical properties of a problem representation. It is important to note that structure features are determined by the particular representation that is chosen for a problem. For example, one problem solver may choose to represent a problem in terms of a system of equations, while another problem solver may represent the same problem in terms of some sort of guessing process (see for example, Harik's discussion of "guessing moves" [15]). Finally, process features represent something of an interaction between task and problem solver. That is, although problem-solving processes (e.g., heuristic reasoning) typically are considered characteristics of the problem solver, it is reasonable to suggest that a problem may lend itself to solution via particular processes. A consideration of task process features can be very informative to the researcher in selecting tasks for both instruction and assessment. Contextual Conditions. These factors concern the conditions external to the teacher and students that may affect the nature of instruction. For example, class size is a condition that may directly influence the instructional process and with which both teacher and students must contend. Other obvious contextual conditions include textbooks used, community ethnicity, type of administrative support, economic and political forces, and assessment programs. Also, since instructional method provides a context within which teacher and student behaviors and interactions take place, it too can at times be considered a factor within this category. The six types of extra-classroom considerations are displayed in Table 1. It is important to add that we do not regard these six areas of consideration as comprehensive. It is likely that there are other influences that may be at least as important as the ones we have discussed.
- 21. Table 1: Key Extra-classroom Considerations Teacber ｐｲ･ｳ｡ｾ＠ Characteristics Examples: 6 Task Features In particular: • age, sex, ... • syntax • teacher education experiences • content • teaching experience • context • teacher traits (IQ, personality, teaching • mathematical and logical structure skills) • processes (e.g., "inherent" heuristics) Teacher Affects. Co&nitions & MetaCOKDitions Contextual Conditions In particular: Examples: • knowledge of content, curriculum, & • classroom contexts (e.g., class size, pedagogy textbook used) • affects about self, students, mathematics, • schooVcommunity contexts (e.g., ethnicity, teaching administrative support) • beliefs about self, students, mathematics, • sociaVeconomic/political forces problem solving, and teaching • mathematics content to be leamed Student PreS" Characteristics Examples: • age, sex, ... • instructional history • student traits (IQ, personality, ...) Student Affects. Cognitions & MetacoKDitions In particular: • mathematical knowledge, "world" knowledge • affects about self, teacher, mathematics, & problem solving • beliefs about self, teacher, mathematics, & problem solving Rather, our intent is to point out the importance ofpaying heed to the wide range of factors that can have an impact on what takes place during instruction. Category 2: Teacher Planning. Teacher planning is not clearly distinct from the other categories; in fact, it overlaps each of them in various ways. Of particular interest for research are the various decisions made before, during and as a result of instruction about student presage characteristics, instructional materials, teaching methods, classroom management procedures, evaluation of student performance, and amount of time to devote to particular activities and topics. Unfortunately, teacher planning has been largely ignored as a factor of importance in problem-solving instruction research. Indeed, in most studies teacher planning has not even been considered because the teachers in these studies have simply implemented a plan that had been predetermined by the researchers, not the teachers. Furthermore, it is no longer warranted to assume that the planning decisions teachers make are driven totally by the
- 22. 7 content and organization of the textbooks used and, therefore, need not be considered as an object ofresearch [10]. Category 3: Classroom Processes. Classroom processes include the host of teacher and student actions and interactions that take place during instruction. We have identified four dimensions of classroom processes: teacher affects, cognitions, and metacognitions; teacher behaviors; student affects, cognitions, and metacognitions; and student behaviors. Table 2 provides additional details on each of these dimensions. Table 2: Oassroom Processes Teacher Affects. Cognitions & Metacognitions Student Affects. Cognitions & Metacognitions • with respect to problem-solving phases • with respect to problem-solving (understanding, planning, carrying out plans, (understanding, planning, carrying out plans, & looking back) & looking back) • attitudes about self, students, mathematics, • attitudes about self, teachers, mathematics, & problem solving, & teaching problem solving • beliefs about self, students, mathematics, • beliefs about self, teachers, mathematics, & problem solving, & teaching problem solving Teacher Behaviors Student Behaviors Examples: questioning, clarifying, guiding, Examples: identifying information needed to monitoring, modeling, evaluating, solve a problem, selecting strategies for diagnosing, aiding generalizing, ... solving problems, assessing extent of progress made during a solution attempt, implementing a chosen strategy, determining the reasonableness ofresults Both the teacher's and the students' thinking processes and behaviors during instruction are almost always directed toward achieving a number of different goals, sometimes simultaneously. For example, during a lesson the teacher may be assessing the appropriateness of the small-group arrangement that was established prior to the lesson, while at the same time trying to guide the students' thinking toward the solution to a problem. Similarly, a student may be thinking about what her classmates will think if she never contributes to discussions and at the same time be trying to understand what the activity confronting her is all about. For convenience, we have restricted our discussion of classroom processes to the thinking processes and behaviors of the teacher and students that are directed toward activities (both mental and physical) associated with P6lya's [29] four phases of solving problems: understanding, planning, carrying out the plan, and looking back. That is, we have restricted our consideration to what the teacher thinks about and does to facilitate the student's thinking
- 23. 8 and what the student thinks about and does to solve a problem. We have not attempted to include a complete menu ofobjects or goals a teacher might think about during instruction. Teacher Affects. CoWitions. and MetacQWitions. These processes include those attitudes, beliefs, emotions, cognitions and metacognitions that influence, and are influenced by, the multitude of teacher and student behaviors that occur in the classroom during instruction. In particular, this dimension is concerned with the teacher's thinking and affects while facilitating a student's attempt to understand a problem, develop a plan for solving the problem, carry out the plan to obtain an answer, and look back over the solution effort. Teacher Bebayiors. A teacher's affects, cognitions, and metacognitions that operate during instruction give rise to the teacher's behaviors, the overt actions taken by the teacher during problem-solving instruction. Specific teacher behaviors such as those shown in Table 2 can be studied with regard to use (or non-use) as well as quality. The quality of a teacher behavior can include, among other things, the correctness of the behavior (e.g., correct mathematically or correct given the conditions of the problem), the clarity of the action (e.g., a clear question or hint), and the manner in which the behavior was delivered (e.g., the verbal and nonverbal communication style ofthe teacher). Student Affects. Coe;nitions. and Metacoe;nitions. Similar to the teacher, this subcategory refers to the affects and the cognitive and metacognitive processes that interact with teacher and student behaviors. The concern here is with how students interpret the behavior of the teacher and how the students' thinking about a problem, their affects, and their work on the problem affects their own behavior. Also of concern here is how instructional influences such as task features or contextual conditions directly affect a student's affects, cognitions, metacognitions, and behaviors. Student Behaviors. These behaviors include the overt actions of the student during a problem-solving episode. By restricting our attention to the problem-solving phases mentioned earlier, we can identify several behaviors students might exhibit as they solve problems. Sample behaviors are shown in Table 2. Category 4: Instructional Outcomes. The fourth category of factors consists of three types ofoutcomes ofinstruction: student outcomes, teacher outcomes, and incidental outcomes. Most instruction-related research has been concerned with short-term effects only. Furthermore, transfer effects, effects on attitudes, beliefs, and emotions, and changes in teacher behavior have been considered only rarely. Table 3 provides a list of the key outcomes associated with each of the three types. Student Outcomes. Both immediate and long-term effects on student learning are included, as are transfer effects (both near and far tranSfer). Illustrative of a student outcome, either immediate or long-term, is a change in a student's skill in implementing a particular problem- solving strategy (e.g., guess and check, working backwards). An example of a transfer effect is a change in students' performance in solving non-mathematics problems as a result of solving only mathematics problems. Also, of special importance is the consideration ofchanges in students' affects about problem solving or about themselves as problem solvers and the
- 24. 9 Table 3: Instructional Outcomes Student Outcomes 1. Immediate effects on student leaming with respect to: a. problem-solving skills b. perfonnance c. general problem-solving ability d. affects e. beliefs f. mathematical content knowledge 2. Long-tenn effects on student learning with respect to: a. problem-solving skills b. perfonnance c. general problem-solving ability d. affects e. beliefs f. mathematical content knowledge 3. Near and far transfer effects Teacher Outcomes 1. Effects on the nature of teacher planning 2. Effects on teacher behavior during subsequent instruction 3. Effects on teacher affects and beliefs about: a. effectiveness of instruction b. "worthwhileness" of instructional methods c. "ease" of use of instructional methods Incidental Outcomes Examples: student perfonnance on achievement tests, influence of instruction on student/teacher behavior in non-mathematics areas, affective changes with respect to mathematics in general, schooling, ... effect of problem-solving instruction on mathematical skill and concept leaming. For example, how is computational skill affected by increased emphasis on the thinking processes involved in solving problems? Teacher Outcomes. Teachers, of course, also change as a result of their instructional efforts. In particular, their attitudes and beliefs, the nature and extent of their planning, as well as their classroom behavior during subsequent instruction are all subject to change. Each problem-solving episode a teacher participates in changes the craft knowledge of the teacher [34]. Thus, it's reasonable to expect that experience affects the teacher's planning, thinking, affects, and actions in future situations. Incidental Outcomes. Increased perfonnance in science (or some other subject area) and heightened parental interest in their children's school work are two examples of possible incidental outcomes. Although it is not possible to predetermine the relevant incidental effects of instruction, it is important to be mindful of the potential for unexpected side effects.
- 25. 10 Discussion There are several factors in the structure described in this paper that deserve particular attention. In the following paragraphs we discuss these factors. Extra-clagroom Comidemdom: Teacher's Affects, Cogniliom and Metacogniliom Research on teaching in general points to the important role a teacher's knowledge and affects play in instruction. However, they have received relatively little attention in research on the teaching of problem solving [3, 7, 39, 40]. Future research should consider all of these factors when real teachers are involved. Questions such as the following need to be investigated: What knowledge (in particular, content, pedagogical, and curriculum knowledge) do teachers need to be effective as teachers of problem solving? How is that knowledge best structured to be useful to teachers? How do teachers' beliefs about themselves, their students, teaching mathematics, and problem solving influence the decisions they make prior to and during instruction? Extra-classroom Considerations: Task Features Although there has been considerable research on task variables (see [12]), it may be time to consider the very nature of the tasks used for instruction relative to problem solving. In most research, tasks have been relatively brief problem statements presented to students in a printed format Thus, research on task variables has considered variables like problem statement length and grammatical complexity. But, very little attention has been given to the identification of task variables that should be considered when the problem-solving task is presented, for example, through a videotape episode from an Indiana Jones movie (see [2]). Or, suppose the real-world tasks used for instruction were selected from those used in an instructional approach modeled after the concept of an apprenticeship [21]. Would the task variables of importance be the same as those examined and discussed in the existing research literature? Teacher Planning A teacher's behavior while teaching problem solving is certainly influenced by the teacher's affects, cognitions and metacognitions during instruction. However, some of this behavior during instruction is likely to be determined by the kinds of decisions the teacher makes prior to entering the classroom. For example, a teacher may have planned to follow a specific sequence of teaching actions for delivering a particular problem-solving lesson knowing that the exact ways in which these teaching actions are implemented evolve situationally during the lesson. Or, if the knowledge teachers use to plan instruction is case knowledge, that is knowledge of previous instructional episodes [8], then we would search for those cases that significantly
- 26. 11 shape the craft knowledge teachers use as a basis for planning and action. Future research should consider how teachers go about planning for problem-solving instruction and how the decisions made during planning influence actions during instruction. Classroom Processes: Teacher Affects, Cognitions and Metacognitions Students' cognitive and metacognitive behavior during mathematical problem solving have received a great deal of attention in recent years (see, for example, [7, 11, 36]). And, there is an indication that students' attitudes, beliefs, and emotions also are beginning to be the object of study [27, 28]. Research is needed that describes the thinking a teacher does during the teaching of problem solving. What does the teacher attend to during instruction? How do teachers interact with students during instruction? What drives the teacher's decisions at those times? Are teachers aware of their actions during instruction? Do they consciously assess their behaviors during instruction? Rich descriptive studies are seriously needed in this much neglected area. The work ofLampert [19] provides a promising model for consideration. Classroom Processes: Teacher Behaviors It is especially important that future research pay closer heed to what teachers actually do during instruction (i.e., teacher actions). In particular, there is a need to document teacher behaviors as they relate to the process of solving problems. It would be a significant contribution to our understanding of how to teach mathematical problem solving if careful, rich descriptions were prepared of a teacher's actions during each of P6lya's four phases (refer to Table 2). Such descriptions would help develop a clearer picture of what teachers should do to aid students to understand problem statements, to plan methods of solution, to carry out their plans, and to look back (evaluate, reflect, generalize) at their efforts. Classroom Processes: Student Affects, Cognitions, and Metacognitions As mentioned earlier, these factors have received substantial attention in the mathematical problem-solving literature. It would be valuable to know more about the conditions under which students' cognitions and metacognitions change and when these changes begin to occur. In a study we conducted [6] continual improvement in students' understanding, planning, and implementing behaviors (cognitions) occurred during the course of 23 weeks of problem- solving instruction. However, the design of the study did not allow us to make conjectures about the aspects of the program that were responsible for the improvement. The effects of a student's attitudes, beliefs, and emotions during problem solving on problem-solving performance is another area in need of serious attention. Recent work by Lester, Garofalo, and
- 27. 12 Kroll [26, 27] illustrates the often strong interaction among affects, cognitions, and metacognitions. McLeod and Adams [28] provide numerous questions for additional research in this area. Classroom Processes: Student Behaviors As important as it is to begin to study carefully the behavior of teachers during instruction, it is just as important to observe the behavior of students during problem-solving sessions. In particular, it would be valuable to know more about the way in which students' interactions affect their thinking processes and actions. For example, to what extent are students' choices and use of a particular problem-solving heuristic or strategy influenced by their peers' decisions? What group processes promote or inhibit successful problem solving? How are individual differences exhibited during problem solving? What kinds of individual differences influence success in solving problems and how can these differences be handled by the teacher? How are students' classroom behaviors influenced not only by their cognitions and metacognitions during instruction, but how are their behaviors influenced by the attitudes and beliefs they bring to the lessons? What kinds of student behaviors are attended to by teachers? How do student-student and teacher-student communications shape affects, beliefs, and cognitions demonstrated while solving problems and the knowledge students extract from the experience? Which instructional influences have the greatest effect on a student's actions? Instructional Outcomes: Students Both immediate and long-term effects on student learning should be considered with respect to cognitions and affects. In addition, possible transfer effects (both near and far) should be investigated. For example, does instruction involving geometry problems have any effect on students' solutions to non-geometry problems? Does instruction in solving routine story problems influence students' solutions to non-routine problems? Furthermore, it is not enough to be content with some vague, general improvement in the ability to get more correct answers (cf. [36, 37]). Instead, it is essential that researchers look to see if their instructional interventions actually develop the kinds of student behaviors they were designed to promote. Finally, how does problem-solving instruction influence the amount and nature of students' mathematics content knowledge? As an illustration of this last question, consider the situation in which a third grade teacher has emphasized during the course of an entire school year the development of various problem-solving strategies. To maintain this emphasis she has found it necessary to devote less attention than in the past to basic fact drill activities. Has the students' mastery of these basic facts been affected by this change in emphasis?
- 28. 13 Instructional Outcomes: Teachers Lampert [19] illustrates how reflecting on instructional practice influences subsequent planning and classroom behavior. It would be a useful contribution to document how systematic problem-solving instruction affects teachers' planning, affects and beliefs. Furthermore it would be valuable to know how such instruction influences teacher behavior during subsequent instruction. A Final Comment Our analysis of factors to be considered for research on problem-solving instruction is intended as a general framework for designing investigations of what actually happens in the classroom during instruction. As mentioned earlier, we recognize that there may be other important factors to be included in this framework and that certain of the factors may prove to be relatively unimportant. Notwithstanding these possible shortcomings, the framework we have presented can serve as a step in the direction of making research in the area more fruitful and relevant. Finally, at the beginning of this paper we recommended that the development of theories of problem-solving instruction should become a top priority for mathematics educators. We also believe that the type of theory development that is likely to have the greatest relevance should be "...grounded in data gathered from extensive observations of 'real' teachers, teaching 'real' students, 'real' mathematics, in 'real' classrooms" [24, p. 56]. It is only by adopting such a perspective that we are likely to make any significant progress in the foreseeable future. References 1. Balacheff, N.: Towards a problematique for research on mathematics teaching. Journal for Research in Mathematics Education 21, 259-272 (1990) 2. Bransford, J., Hasselbring, T., Barron, B., Kulewicz, S., Littlefield, J., & Goin, L.: Use of macro-contexts to facilitate mathematical thinking.. In: The teaching and assessing of mathematical problem solving (R. Charles & E. Silver, eds.), pp. 125-147. Reston, VA: LEA & NCTM 1988 3. Brown, C., Brown, S., Cooney, T., & Smith, D.: The pursuit of mathematics teachers' beliefs. In: Proceedings of the Fourth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (S. Wagner, ed.), pp. 203-215. Athens, GA: University of Georgia, Department of Mathematics Education 1982 4. Charles, R.: The development of a teaching strategy for problem solving. Paper presented at the Research Presession of the Annual NCTM Meeting, San Antonio, TX, April 1985 5. Charles, R., & Lester, F.: Teaching problem solving: What, why and how. Palo Alto, CA: Dale Seymour Publications 1982 6. Charles, R., & Lester, F.: An evaluation of a process-oriented mathematical problem-solving program in grades five and seven. Journal for Research in Mathematics Education 15, 15-34 (1984) 7. Charles, R., & Silver, E. (eds.): The teaching and assessing of mathematical problem solving. Reston, VA: LEA & NCTM 1988
- 29. 14 8. Doyle, W.: Classroom knowledge as a foundation for teaching. Teachers College Record 91(3), 347-360 (1990) 9. Dunkin, M., & Biddle, B.: The study of teaching. New York: Holt, Rinehart & Winston 1974 10. Freeman, D., & Porter, A.: Do textbooks dictate the content of mathematics instruction in elementary schools? American Educational Research Journal 26(3), 403-421 (1989) 11. Garofalo, J., & Lester, F.: Metacognition, cognitive monitoring, and mathematical performance. Journal for Research in Mathematics Education 16, 163-176 (1985) 12. Goldin, G., & McClintock, C. (eds.): Task variables in mathematical problem solving. Philadelphia: Franklin Institute Press 1984 13. Good. T. & Biddle, B.: Research and the improvement of mathematics instruction: The need for observational resources. In: Perspectives on research on effective mathematics teaching (0. Grouws, T. Cooney & D. Jones, eds.), pp. 114-142. Reston, VA: LEA & NCTM 1988 14. Grouws, D.: The teacher and classroom instruction: Neglected themes in problem-solving research. In: Teaching and learning mathematical problem solving: Multiple research perspectives (E. Silver, ed.), pp. 295- 308. Hillsdale, NJ: LEA 1985 15. Harik, F. Heuristic behaviors associated with problem tasks. In: Task variables in mathematical problem solving (G. Goldin & C. McClintock, eds.), pp. 327-352. Philadelphia: Franklin Institute Press 1984 16. Kilpatrick, J.: Variables and methodologies in research on problem solving. In: Mathematical problem solving: Papers from a research workshop (L. Hatfield & D. Bradbard, eds.), pp. 7-20. Columbus, OH: Eric Clearinghouse for Science, Mathematics, and Environmental Education 1978 17. Kilpatrick, J.: Research on mathematical learning and thinking in the United States. Recherches en Didactique des MatMmatiques 2,363-379 (1981) 18. Kulm, G.: The classification of problem-solving research variables. In: Task variables in mathematical problem solving (G. Goldin & C. McClintock, eds.), pp. 1-2l. Philadelphia: Franklin Institute Press 1984 19. Lampert, M.: When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research JoumaI27(1), 29-64 (1990) 20. Larkin, J.H.: Eight reasons for explicit theories in mathematics education. In: Research issues in the learning and teaching of algebra (S. Wagner & C. Kieren, eds.), pp. 275-277. Reston, VA: LEA & NCTM 1989 2l. Lave, J., Smith, S., & Butler, M.: Problem solving as everyday practice. In: The teaching and assessing of mathematical problem solving (R. Charles & E. Silver, eds.), pp. 61-8l. Reston, VA: LEA & NCTM 1988 22. Leinhardt, G.: Routines in expert math teachers' thoughts and actions. Paper presented at the Annual Meeting of AERA, Montreal, April 1983 23. Leinhardt, G.: Student cognitions during instruction. Paper presented at the Annual Meeting of AERA, Montreal, April 1983 24. Lester, F.: Methodological considerations in research on mathematical problem solving. In: Teaching and learning mathematical problem solving: Multiple research perspectives (E. Silver, ed.), pp. 41-69. Hillsdale, NJ: LEA 1985 25. Lester, F.: Reflections about mathematical prOblem-solving research. In: The teaching and assessing of mathematical problem solving (R. Charles & E. Silver, eds.), pp. 115-124. Reston, VA: LEA & NCTM 1988 26. Lester, F., Garofalo, J., & Kroll, D.: Self-confidence, interest, beliefs, and metacognition: Key influences on problem-solving behavior. In: Affect and mathematical problem solving: A new perspective (D. McLeod & V. Adams, eds.), pp. 75-88. New York: Springer-Verlag 1989 27. Lester, F., Garofalo, J., & Kroll, D.: The role of metacognition in mathematical problem solving: A study of two seventh grade classes. Final Report. Bloomington, In: Mathematics Education Development Center, National Science Foundation Grant MDR 85-50346. (ERIC # 314-255) 1989 28. McLeod, D., & Adams, V. (eds.).: Affect and mathematical problem solving: A new perspective. New York: Springer-Verlag 1989 29. P61ya, G.: How to solve it, Princeton University Press, Cambridge 1945
- 30. 15 30. Shavelson, R.: Review of research on teachers' pedagogical judgments, plans, and decisions. Elementary School1oumal83, 392-413 (1983) 31. Shavelson, R., & Stem, P.: Research on teachers' pedagogical thoughts, judgments, decisions and behavior. Review of Educational Research 51(4),455-498 (1981) 32. Shavelson, R., Webb, N., Stasz, C., & McArthur, D.: Teaching mathematical problem solving: Insights from teachers and tutors. In: The teaching and assessing of mathematical problem solving (R. Charles & E. Silver, eds.), pp. 203-231. Reston, VA: LEA & NCTM 1988 33. Shulman, L. S.: Those who understand: Knowledge growth and teaching. Educational Researcher 15(2),4-14 (1986) 34. Shulman, L.: The wisdom of practice: Managing complexity in medicine and teaching. In: Talks to teachers: A festschrift for N. L. Gage (D. Berliner & B. Rosenshine, eds.), pp. 369-386. New York: Random House 1987 35. Silver, E.: Research on teaching mathematical problem solving: Some underrepresented themes and needed directions. In: Teaching and earning mathematical problem solving: Multiple research perspectives (E. Silver, ed.), pp. 247-266. Hillsdale, NJ: LEA 1985 36. Silver, E.: Teaching and learning mathematical problem solving: Multiple research perspectives. Hillsdale, NJ: LEA 1985 37. Silver, E., & Kilpatrick, J.: Testing mathematical problem solving. In: The teaching and assessing of mathematical problem solving (R. Charles & E. Silver, eds.), pp. 178-186. Reston, VA: LEA & NCTM 1988 38. Stengel, A., LeBlanc, J., Jacobson, M., & Lester, F.: Learning to solve problems by solving problems: A report of a preliminary investigation. (Tech. Rep. 11.0. of the Mathematical Problem Solving Project). Bloomington, IN: Mathematics Education Development Center 1977 39. Thompson, A.: Changes in teachers' conceptions of mathematical problem solving. Paper presented at the annual meeting of the American Educational Research Association, Chicago, April 1985 40. Thompson, A.: Learning to teach mathematical problem solving: Changes in teachers' conceptions and beliefs. In: The teaching and assessing of mathematical problem solving (R. Charles & E. Silver, eds.), pp. 232-243. Reston, VA: LEA & NCTM 1988 41.Winnie, P. H., & Marx, R. W.: Reconceptualizing research on teaching. Journal of Educational Psychology 69(6),668-678 (1977) 42.Winnie, P. H., & Marx, R. W.: Matching students' cognitive responses to teaching skills. Journal of Educational Psychology 72, 257-264 (1980)
- 31. Researching Problem Solving! from the Inside 10hnMason Centre for Mathematics Education, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK Abstract: A methodology is offered for studying problem solving from the inside. This approach is first set in the context ofmillenia of educational thinkers, and then in the context of providing support for the professional development of teachers of mathematics. The approach is based on the development and strengthening of each teacher's own awareness of their own mathematical thinking so that they can more readily enter and appreciate the thinking of their pupils. It is described and illustrated, through offering experience of working on mathematics, through offering challenging assenions, and through suggesting ways of working on mathematics teaching. Observations are made about validity and validation through this methodology. The heart of the methodology is based on the observation that what really matters in teaching is to be awake to possibilities in the moment, as a lesson unfolds; moments when the teacher experiences the true freedom of a conscious choice. Three extracts from different styles ofpresentation of mathematics for students at a distance are offered for consideration. Keywords: awareness, metacognition, frameworks, noticing, conjectures, investigations, insights, generalizing, specializing, teaching Introduction What are we doing when we engage in research on problem solving? What is the enterprise? Is it to understand and appreciate the processes involved? Is it to locate a framework, a programme for "teaching" problem solving? The first aim already begs many questions in referring to the processes, implying as it does that processes have independent existence and that the researcher's job is to locate them in some Platonic world of forms. Modern epistemological views suggest that such processes are constructed by the observer as part of the process of distinguishing features and making distinctions on the way to understanding. Though they may "exist" for one researcher, they may not for others. Ifdistinctions are to be useful to others, then it is necessary to assist others to construct and enter the same reality. The second aim assumes that direct instruction is possible and even desirable. The trouble with direct approaches is that they may not leave room for contingent or collateral learning. 1 I take the word problem to refer to a person's state of being in question, and problem solving to refer to seeking to resolve or reformulate unstructured questions for which no specific technique comes readily to mind
- 32. 18 What would useful research conclusions look like? Who would try to use them? There is a long history of thoughtful advice about problem-solving, from such eminent names as Plato, Aristotle, Pappus, Bacon, Descartes, Fermat, Euler, Bolzano, Boole, Hadamard, and P6lya [29]2. They sought to organize thinking, and some of them even sought to mechanize it, as well as to assist novices in learning mathematics. The impulse to mechanize thought has appeared in every generation. The availability of sophisticated computers has stimulated a burgeoning of activity by numerous workers in the fields of Artificial Intelligence and Cognitive Psychology, who have likewise contributed to the mechanization of problem solving in mathematics3 in the idiom of their times. When research findings are translated into practice, they turn from observation into rules, from heuristics into content. Attempts to pass on insights become attempts to teach patterns of thought. Once the "patterns of thought", the heuristics, become content to be learned, instruction in problem-solving takes over and thinking tends to come to a halt. Dewey put it nicely, when he said Perhaps the greatest of all pedagogical fallacies is the notion that a person learns only the particular thing he is studying at the time. Collateral learning ... may be and often is much more important than the actual lesson. Approaching the "content" too directly may not always be wise. In an age noted for its pragmatism which begins to rival that of Dewey's times, collateral learning may need to be re- emphasized. To counteract the functionalism of modern educational curriculum specification and assessment, Bill Brookes, in a seminar, suggested that We are wise to create systems for spin-offs rather than for pay-offs. Mathematicians too have been very slow to acknowledge and accept distinctions made by researchers, fearful perhaps that they will lose touch with their own creativity. People do not, on the whole, relish being told how they think or how they might think differently. In order to influence teaching, researchers have traditionally moved to programme development, almost always text based, with or without training for teachers in the use of the materials. Even where teachers are supported, it is usually to help them to "deliver the programme", like some mail-order firm. My approach is to work from the inside, from and on my own experience. This is not as solipsistic and idiosyncratic as it sounds, for it draws on results of outer research which speaks 2 See particularly George POlya [29] for a study of the use and teaching of heuristics. For recent surveys, see also Alan Schoenfeld [32] and H. Burkhardt, S. Groves, A. Schoenfeld, and K. Stacey [2]. 3 Particularly notable is the work of Edmund Furse [7]. Furse has managed to build a program that will read standard mathematical texts (translated into a suitable formal language), and then solve the end-of-chapter exercises. Its solutions look like those of an expert. Furthermore, there is no mathematics embedded in the program, only a syntactic pattern recognition engine.
- 33. 19 to my experience. Furthennore, there is a long history of people working effectively in this way, particularly in the phenomenological and hermeneutic traditions, building as they do on psychological insight of ancient peoples in India, China and the Middle East. Recently there has been a notable increase in interest from academics, with attention being drawn to metacognition and reflection4• Humboldt observed that the essence of thinking lies in abstraction, Dewey spoke of turning a subject over in the mind, Piaget stressed the role of reflective abstraction, Vygotsky drew attention to the internalization ofhigherpsychologicalprocesses through being in the presence of more expert thinkers, Skemp lays great stress on the development of reflective as well as intuitive intelligence, and Kilpatrick [19] made an explicit call for the development of self-awareness in mathematics education. Apart from practical considerations such as that teacher-proof materials have never succeeded, and that there is no evidence for a single way to teach mathematics, there are three principle justifications for an approach from the inside: to appreciate the struggles of another, it is necessary to struggle oneself, and to be able to re-enter that struggle; to learn to struggle and to make best use of available resources, it is useful to be in the presence of someone who themselves is in question, who is struggling to know; teaching mathematics is ultimately about being mathematical oneself, in front of and with pupils, and so in order to develop one's teaching, to work at being mathematical, it is necessary to develop one's mathematical being. I do not mean to imply that it is best for teachers to be struggling with the same mathematics as their pupils, although often it is the case that a teacher who does not know answers can be of more help than a teacher who does. More precisely, knowing the/an answer can make it harder to appreciate pupils' thinking, and more likely that you will direct them to your own solution. Instead of dwelling on mathematical answers, teachers can be more helpful by dwelling in their own questions about what the pupils are thinking, how their powers can be evoked, what advice might be helpful, and leave the actual mathematical thinking to the pupils. To do this requires awareness of their own thinking, which comes from inner research. To influence pupils in how to come into question, to be in question, and to resolve questions it is helpful if the teachers are themselves in question, either about some other mathematics, or about their teaching of mathematics, or both. I concur with Brown, Collins, and Duguid [1] who advocate a cognitive-apprenticeship vocabulary for describing one form of effective teaching, and this is consistent with current social-constructivist thinking about the role and significance of peers and experts in learning. The medieval Guild system, when functioning effectively, had much to recommend it. 4 Some would attribute this to the dawning of the age of Aquarius. Others. such as Julian Jaynes. [17] would claim that we are experiencing an evolutionary change in the structure ofconsciousness.
- 34. 20 The difficulty as always is to translate description into action. I do not advocate trying to describe effective teaching in behavioural tenns so that others can then (supposedly) "do it". All attempts at this of which I am aware, have failed. What I advocate is establishing a collegial atmosphere in which teachers work on and develop their awareness of their own mathematical thinking, and of themselves as teachers. I stress the collegial, because it is not enough to be a hennit, operating alone behind the closed door of the classroom. It is essential to test out insights and conjectures with others, to participate in a wider community of fellow seekers. Such peer-checking is an essential part of the larger methodology ofthe Discipline of Noticing (see Mason and Davis [28] and Davis [3]). In this paper I want to illustrate rather than derme what I mean by researching/rom the inside, and to make some remarks about what I have learned over the past twenty-five years. I shall offer some mathematical questions5 to illustrate a little bit of what I mean by working from the inside, and to indicate that it is fundamental to me to be consistent in my approach. To speak about working fonn the inside is to monger words. To appreciate what I have learned, you must experience it from the inside. If my words speak to your experience, then all well and good. Ifnot, then we have both wasted time! When working with teachers I often suggest at the beginning that I shall be working on three levels: on Mathematics, on Teaching and Learning Mathematics, and on Ways of Working: - on Mathematics, - on Teaching and Learning Mathematics. Having come to mathematics education from mathematics research, I find much that is similar between working on mathematics and working on teaching and learning of mathematics, particularly in the need for particular examples, and for attention to be devoted to how generalizations are drawn from or seen in the particular. I draw attention to these three levels because I neither claim to know, nor wish to show or tell people, how to teach. Rather, I wish to draw out from their experience for conscious attention the powers that they already possess, and through that awareness, try to help them be able to do the same for their pupils. Socrates' image of midwife for ideas in Plato's dialogue the Thaetetus has much to recommend it, though care is needed in isolating the useful features of the much invoked but frequently misunderstood Socratic Dialogue (see for example [12, 13]). Instruction, that term so beloved of American educators, is much more subtle than directing pupils through the stages of some institutional regime. It involves more than being sensitive to pupils' needs, more than actualizing a few slogans. It is about being yourself, being aware of 5 Mindful that a question is ink on paper, and a problem is a state which arises in a person in the presence of a question, in a context.
- 35. 21 yourself and your powers, and evoking those same powers in others. Thus teaching and learning are for me almost synonymous. They are about entering the potential space (D. Winnicot, quoted in [18]) between you and another, and inviting others to join you in that space. Sometimes direct instruction, that is, giving pupils specific instructions as to what to do, is appropriate. Telling pupils things can indeed be effective (despite the current bad press which exposition receives at present), especially if pupils are prepared to hear what is being said because of recent experience which has raised doubts or questions. Sometimes a less direct prompt to recall prior advice or instruction is enough, and at other times being verbally silent but physically and mathematically present, or even being physically absent is sufficient. Pupil dependency on teacher intervention can easily and unintentionally be fostered. Tasks are sometimes directive, sometimes indicatively prompting investigation, sometimes spontaneously arising from pupil or group. Samples of attempts to manifest these in print, purely as support materials, are given at the end of the paper. Examples of Approaching from the Inside In this section I offer some mathematical questions, together with a few remarks about what often emerges when using them. If the questions are too easy, then nothing will be learned from them. If you get stuck, then there is a chance something can be learned. If you simply scan or skim through them, then nothing will be gained. One Sum Take any two numbers that sum to one. Which will be the larger sum: the square of the larger added to the smaller, or the square of the smaller added to the larger? Everyone naturally specializes. Some use simple fractions, some use decimals, and some use algebraic symbols. I refer to all of these as specializing, since people naturally tum to something specific which they find confidence inspiring. Pointing this out can help to ease a transition from number to algebra. Square Sums 32 + 42 = 52 102 + 112 + 122 = 132 + 142 212 + 222+ 23 2 + 242 = 252 + 262 + 272 Most people find the first statement a well known and unexceptional fact. Juxtaposed with the second, which is somewhat surprising, interest mounts. When the third is offered, there is
- 36. 22 conviction that there must be a pattern which continues. Locating that pattern evokes powers of pattern spotting and expression which lie at the heart of algebraic generalization and thinking. The assumption of pattern is endemic in mathematics, but not always born out: 32 + 42 = 52 33 + 43 + 53 = 63 ... ? Map Scaling Imagine that you are sitting at home with a map of your country, and you wish to scale it down by a factor of a half. To do this, you put a transparent piece of paper over the map, and using your own home-town as centre, scale every other feature of the map halfway along the line joining it to your centre. Now imagine that a friend uses the same map, but using a different centre. Will your two scaled maps look the same or different, and why? Many people have incomplete intuitions about scaling. There is something paradoxical about being able to use any point as centre, and still get the same map. Even experienced mathematicians can be caught out in a conflict between their educated intuition and their gut response. A computer programme such as Cabri-geometre [20] enables a simultaneous scaling of the same line or circle from two different points, and watching the two images develop is often a surprise even when you know what to expect. The following has a similar effect, particularly on astronomers: The Half Moon Inn In England you often see pub signs showing a vertical half-moon (the straight diameter is vertical) in a black sky with a few stars shining nearby. When, if ever, can you see a vertical half-moon? These questions are useful for opening up discussion about the extent to which intuition can be educated, and the extent to which early intuitions are always present, if suppressed. Fischbein [6] takes the view that early intuitions are robust against teaching, and Di Sessa [4] develops the notion ofpsychologicalprimitives as basic building blocks in constructing stories to account for things that happen in the world. The development of much of mathematics can be seen as the gradual refinement of intuitions: for example from discontinuity as a jump in the graph, to a more complex phenomenon; from a function having a derivative of 0 at a point being flat nearby, to the possibility that it could have arbitrary slope arbitrarily close to such a point.
- 37. 23 Planets [33] Consider a collection of identical spherical planets in space. From some places on the the surface of some of the planets, it is possible to see one or more other planets in the collection. What is the total·area on all the planets of these places? This question has proved fruitful because, after successfully specializing to two-dimensions, most people develop an argument which generalizes only with great difficulty to three dimensions. After some initial success, it is usually necessary to prompt people to go back to the two-dimensional case and seek an argument which will generalize. Remarks about my Approach Inner research begins with self-observation. This is a constant and on-going process, guided by the concerns of the people I am trying to assist, and by my own propensities. Teachers often remark to me that it is very rare that they have the time to work on mathematics themselves, either alone or with colleagues. I take this as resonance with the notion that it would be better if they did engage in mathematical thinking, but that conditions are not supportive. Many are the programmes of in-service support for mathematics teachers; the most significant benefit to participants of any programme is rarely the content, and more often the opportunity and stimulus to "indulge" in mathematics, and to compare reflective notes with colleagues. Where teachers are transfixed by "picking up some idea they can use tomorrow" the benefit is short- lived, as evidenced by their return next time for another "fix". Give someone a fish, and you feed them for a day; Teach them to fish, and you feed them for a lifetime.6 After many years of inviting people to engage in mathematical thinking, it became clear that on the whole it was much more effective to begin with simply stated tasks. As with any useful observation about teaching, this is not meant to be an unbroken rule; rather it is meant to inform choices made in the moment. I am always tempted to start with something "interesting", which means "interesting to me", and because of the complexity, people find it difficult to separate off a little bit of their attention to observe themselves. Starting with apparently very simple questions which invoke the type of thinking which is needed when you get stuck - specializing and generalizing, conjecturing and convincing, animating the static and freezing the dynamic, working backwards and working forwards - enables participants to attend to their experience as well as to the task, and thereby to find discussion and elaboration fruitful. More important than the completion of tasks is awareness of what you are doing. But attention is usually drawn into the mathematics. For example, when teachers watch videotaped episodes from classrooms, 6 Purportedly a Chinese proverb, source unknown.
- 38. 24 their attention is drawn initially to the task posed to the pupils, and whether they, the viewers, can do it. Only when they feel confident about and familiar with the mathematics can they attend to the acts of teaching. I struggled for a long time with Gattegno's memorable observation that Only Awareness is Educable [9]. He has a quite specific meaning which I only partially appreciate, connected with his Science of Education. This is an approach which has much in common with mine, in that it seeks the roots of mathematics and of thinking in actions and experience, although Gattegno emphasizes particularly the actions and experience of the neonate in its cot. I do not pretend to have mastered Gattegno's ideas, nor to be carrying them further. I simply note similarities. I found increased meaning in his observation when I augmented it by twoother assertions, informed by that most ancient of metaphors for the psychological make-up of human beings, the image of the Chariot? The chariot is drawn by a horse, and has a driver as well as an owner. One way to read it is as follows: the chariot represents the body, which needs maintenance; the horse represents the emotions (the source of motion) which need grooming and feeding and sometimes require blinkering; the driver represents the intellect which needs challenging tasks; the reins represent the power of mental imagery to direct, which need to be kept subtle and loosely-taut; the shafts represent the psychosomatic interplay between emotion and the musculature (see [31]); the owner represents a different form of awareness that is possible, but only when the chariot is functioning correctly will the owner deign to use it. The two extra observations suggested by this image are that Only behaviour is trainable, and Only emotion is hamessable. I capitalize the only in each case, because it is often the only which causes the most strife when people encounter and contemplate these assertions. The definitiveness often generates interest and thought, whether in favour or in opposition. The three onlys also act to inform an approach to the construction of mathematical tasks for pupils, outlined in Griffith and Gates [10]. The positive role of frameworks, and for slogans too [14], is that they can serve to heighten awareness, to remind you in the midst of preparation or lesson, of some alternatives to your standard, automatic reaction. The negative side is that instead of in-forming, they come to pro- form, even de-form awareness. The education of awareness, and the integration of 7 Katha Upanishad 1.3.3-5, see for example S. Rhadakrishnar [30]. A more modem version of the image can be found in Gurdjieff [111.
- 39. 25 observations through subordination8 are part of the development of being. but the metaphor of growth suggests a monotonic progression. and this does not conform with my experience. Often what once seemed sorted out, needs to be re-evaluated and re-questioned. and stimulus to do this comes through working with colleagues in order to seek resonance with their experience. Mathematical questions. and the recall of teaching or learning incidents through anecdotal re-telling or through video-tape are devices for focusing attention. In recalling an event in retrospect. there is a temptation to judge. to berate oneself for failing to do something. This is a waste of energy. To notice something about yourself is like suddenly taking a breath of clean air. like suddenly coming upon an unexpected vista. The energy which accompanies the insight has to go somewhere. and it is all too natural to throw it away on judgement and negativity. To make use of what is observed. it is better simply to note. to "in-spire" the in-sight into oneself. to accept. By re-calling and re-entering specific salient moments. moments that come readily and vividly to mind soon after an event. it is possible to take a three-centred approach to developing one's being. The horse. carriage. reins. shafts and driver can be employed in a balanced and intentional storing of experience which can serve to inform practice in the future. This is the theory behind the presentation and recommendations in Mason. Burton and Stacey [24]. By vividly re-entering a significant moment through the power of mental imagery. you contribute to a rich store ofexperience to be resonated at a later date. To stimulate the growth of an inner monitor. it is necessary to direct some of the energy released from noticing. for that purpose. What tends to happen is that you become aware of missed opportunity. After the lesson it suddenly comes to you what you could have done; a few moments after a pupil asks a question. you find yourself answering in your usual automatic fashion; despite intending to listen to what pupils have to say. you find yourself telling them what you think; while working on a question you gradually become aware that you are not getting anywhere. These moments of awareness come at fll'St in retrospect. The challenge is to move them into the moment. so that you become aware of opportunities when they are relevant. Strategies and techniques for doing this are part of the Discipline of Noticing. and have been described elsewhere. Noticing Action in the Moment To become aware of your own thinking. and to be able to make use of personal insights when teaching. it is necessary to be able to catch action in the moment that it happens. But action is very hard to catch in yourself when you are being successful. It is likely. for example. that you found at least one of the tasks presented earlier so easy that you simply "did it". The trouble 8 A reference to Gattegno's memorable title. What we owe children: The subordination ofteaching to learning. Gattegno [8].
- 40. 26 with success is that although it breeds psychological confidence, it offers no support if and when things start to get sticky. It is relatively easy to catch yourself when you are stuck, because attention naturally drifts from the task at hand to awareness of being stuck. Unfortunately, being stuck does not involve much action, and again, this state is not terribly helpful in itself. It is an honourable state, full of potential, and it can be turned to positive advantage, if you recognize and acknowledge it, and cast around intentionally for advice and assistance. A passing teacher can at this point say many things to you, some of which you can hear, while others will go right past you. If the teacher always provides the same sort of help, then you may come to recognize that constant help, or you may simply come to depend on it Let me take one class of examples: teacher interventions when pupils get stuck. There is considerable interest at present in scaffolded instruction in mathematics, in which the teacher, with Vygotsky's zone ofproximal development in mind, asks questions to locate just where the pupil's current "zone" lies, and then tries to "operate in that zone". But the sorts of questions that teachers ask in these circumstances have been described over and over again over the years. The new metaphors make little difference to what happens, just different ways of reading them. Edwards and Mercer [5] concluded after extensive study of transcripts that Despite the fact that the lessons were organized in terms of practical actions and small-group joint activity between pupils, the sort of learning that took place was not essentially a matter of experiential learning and communication between pupils.... While maintaining a tight control over activity and discourse, the teacher nevertheless espoused and attempted to act upon the educational principle of pupil-centred experiential learning ... (p.156) John Holt [15] has a most succinct version in an interaction with Ruth: We had been doing math, and I was pleased with myself because, instead of telling her answers and showing her how to do problems, I was "making her think" by asking her questions. It was slow work. Question after question met only silence. She said nothing, did nothing, just sat and looked at me through those glasses, and waited. Each time, I had to think of a question easier and more pointed than the last, until I found one so easy that she would feel safe in answering it. So we inched our way along until suddenly, looking at her as I waited for an answer to a question, I saw with a start that she was not at all puzzled by what I had asked her. In fact, she was not thinking about it. She was coolly appraising me, weighing my patience, waiting for that next, sure-to-be-easier question. I thought, "I've been had!" The girl had learned to make all her previous teachers do the same thing. If I wouldn't tell her the answers, very well, she would just let me question her right up to them. Bauersfeld calls this thefunnel effect, as teacher and pupil are together drawn down a funnel of increasing detail and special cases. The outcome is usually a question so trivial that the pupil is
- 41. 27 in wonder that the teacher could ask it, but their attention is absorbed by each successive question and is drawn away from the larger task. There are alternatives. When you recognize that you have something specific that you are trying to get pupils to "see" or "say", you can, for example, stop your current questioning, acknowledge what you are doing, and simply tell them the idea that you have in mind. This requires more presence and strength than might appear, and that presence and strength are built up over a period of time through noticing missed opportunities. You can then suggest that they reconstruct it in their own language, perhaps by repeating it to a colleague. Then they can explore variations of the idea, and pose themselves more general questions which succumb to the same approach. That works if they see the relevance of what is said to what they are trying to do. Unfortunately, when a teacher leaves a group of pupils after an intervention it is often the case that the pupils act as if the teacher had never been. They carry on as before. I conjecture that one reason is that teachers remain caught in giving direct advice, much of which does not relate to the pupils' experience. More fruitful is a gradual movement from direct advice, though indirect prompts which evoke recent memories of similar situations, with the overall aim of withdrawing prompts altogether and detecting pupils using the suggestions spontaneously themselves. Brown, Collins and Duguid [1] referred to this asfading. I have found it useful to keep in mind a transition in intervention from directed, to prompted, to spontaneous use of technical terms and heuristic advice by pupils. When you are stuck on a problem, you tend to be deeply involved in that problem. When you seek advice, you immediately try to apply what you are told to the problem, and so tend not to be aware of the shape or form of that advice. In order to assist pupils to become aware of the existence of useful advice, metacognitive shifts can be effective. Instead of asking Can you give me an example?, the question What question am I going to ask you? can draw attention away from the particular to the general, prompting a shift in the structure of attention [23, 26]. The pupil may not know the first few times, so you can resort to your original question. But at least you are signalling that the advice itself is of some, indeed more importance than success on the particular task. In this way, pupils can move from requiring direct advice, to prompted advice, and eventually, to giving themselves that same advice. In the process they will have internalized their own mathematical monitor, and intentionally contributed to the growth of their inner teacher ([16, 27]). In terms of evaluating the effectiveness of some framework such as directed, prompted, spontaneous, advice on what to do when you are stuck, or any other proposed to pupils and/or teachers, I want to know whether people find the ideas informing their practice. But I distrust asking them directly. Even observing lessons to see if there is some manifestation of the framework is a form of prompting probe which is likely to distort, and assumes that what is of value is observable. In seeking to evaluate it is all too tempting to be caught in a cause-and-
- 42. 28 effect paradigm, in which treatment is administered and pupil reactions noted. But teaching and learning do not follow cause-and-effect. They are concurrent not consequential. I consider that teaching takes place in time, through specific acts, while learning as taking place over time, through maturation. Evidence for "learning" based on direct instructions and prompts does not approach what pupils have internalized and use spontaneously. Another way to express this is by the following assertion Wounds are to patients, as assessment is to students. The value of such an assertion is in the challenge that it induces, and the multiplicity of interpretation, including contradictory views. Just as health is defined by evaluators as "healthy response to illness", so learning is defined as "appropriate response to assessment". But health is much more than response to illness, and learning much more than response to examination questions. Indeed the medical metaphor is deeply embedded in educational rhetoric. In seeking evidence, I prefer to wait for spontaneous reference and use of related ideas by teachers when talking or writing to colleagues about what they do. The language they use may not be the same as that used by me, indeed I would be wary if it were. And when the language varies, I cannot assert that the expression they come to has been stimulated, influenced or informed by me. Similarly with pupils, I distrust direct questions and oblique prompts which try to ascertain whether pupils have integrated advice into their own thinking. or about the value they have found for themselves [25]. Searching their work for evidence of using a particular framework ignores unmanifested influence and behaviour that is informed by exposure to that advice, while use of the terms of some framework may simply be the effect of expecting that that is what I want to see. I am more impressed by spontaneous use in later courses or in other work. This approach renders statistical analysis powerless, but when spontaneous manifestation occurs, it provides evidence ofreal value and significant use. This sort of external validation is therefore rather unreliable for evaluative purposes. Validation of the effectiveness of inner research lies with the researcher in seeking resonance with pupils and colleagues, not just once, but frequently and repeatedly. Re- questioning becomes part of established practice. Progress is not achieved by building on certainty, because insights are constantly in need of verification in fresh times and places, and of restatement in current idioms and language so that people can recognize them in their own experience. I conclude with three examples of types of written materials, designed to assist teachers to sharpen their awareness of their own thinking, and perhaps then to use some modified version of the tasks with their pupils. The examples are merely ink on paper. What is important is to
- 43. 29 draw out observations made while working on them, and to integrate those insights into thinking so that they can draw upon them in the future, to inform future action. Example A: Exposition with Exploration This extract is from notes prepared for a weekend of problem-solving by undergraduates, graduates and tutors on the theme of Fractals and Chaos. Dragon Curves The name Dragon Curve is derived from the appearance of a particular curve which arises in several contexts: as a fractal curve, and as the result of folding paper. A - Paper Dragons. Take a long (thin?) strip of paper, stretch it out from left to right in front of you, and fold it in half by placing the right-hand end on top of the left, so that the strip is now half as long. Make a crease. Fold the result in half again (right over left), and repeat several times. Now open it out so that each crease forms a right angle, and place the paper on its edge on a blank piece of paper. The edge of the strip forms a paper dragon curve with degree = the number of fold operations you made. The unopened folded strip comes in layers, and when opened, there are creases. Number the segments between creases from left to right as 1,2, ... . Now fold the paper back up, and observe the sequence of segments as you go up the layers. The layer-sequence of a dragon curve is the sequence of segment numbers which appear when it is folded in layers starting with segment 1 in the bottom layer. For example, for d = 2, there are four layers, which appear in the order 2,3,4,1 Before opening the paper out, how many layers will it have? In general? How many creases? Make sure that segment 1 is at the bottom. Find a way to predict the layer sequence of a degree n dragon curve, in terms of the layer sequence of the degree n-I curve.
- 44. 30 With an unfolded dragon curve, start at segment 1 and traverse the paper strip recording an R or an L for right or left turns as you traverse the curve. This is the turn-sequence. One way to find a description is by generalizing from several systematically developed examples. Another is to work from what happens to a turn sequence when a new fold is made. A Dragon Curve is the curve obtained as the limit of a sequence of more "folded" paper dragon curves. B - General Dragons. The original paper dragon was made by always folding the right-hand end over the left to make the folds and creases. But at any stage you could fold the right-hand under the left instead. Afold-sequence is an infinite sequence of O's and U's, and is to be interpreted as instructions to fold the right-hand end of the strip Over or Under what is already there. There is a 1 to I correspondence between the fold-sequences and the real numbers on [0, I]. A fold-sequence gives rise to a generalized paper dragon curve, by opening out the strip of paper and making the creases all form right-angles. One version is that the dth Dragon Curve is formed from the d-I th by inserting in front of, between each letter of, and at the end of the tum- sequence for the d-Ith curve, chosen in tum from L, R , L, ... or the letters R, L, R, ... depending on whether the dth term of the fold-sequence is 0 or U. Work out two different ways of determining the turn- sequence of a dragon curve of degree d in terms of the previous sequence for degree d-J. Which points of a paper dragon curve are also points of the limit Dragon Curve? Show that any sequence beginning (0, ...J has the same effect as some other sequence (U, ...). Show that two different sequences both beginning with 0 give different results. Think binary. Specify how to work out the tum-sequence corresponding to a given fold-sequence. Why does this work?
- 45. 31 Another version is that the next Dragon Curve is fonned from the last by inserting an L or an R as appropriate in between two copies of the previous sequence, but transfonned according to whether the letter in the fold-sequence is 0 or U. For an 0, the sequence stays as it was. For a U, it is reversed and the letters interchanged. THEOREM: Dragon curves are (almost) simple. That is, they never cross themselves, though they do touch at various points. Example B: Exploration Why does this work? Suggestion: to cross itself, the curve would have to turn through 360 degrees between two creases. This extract is taken from Mason [22] which are materials designed to stimulate teacher's own mathematical thinking in a context of topics taught in the classroom. There is very little exposition or even assistance, since it depends on teachers or groups of pupils to work together puzzling them out with the assistance of a tutor when necessary. "Readers" are expected to select from the multitude of activities presented, and to explore around them as they see fit. What is the Problem? Adding together two numbers is a basic arithmetic operation, as is multiplication. The idea of what constitutes a number grew, historically, from whole numbers through fractions to decimals (though the full story is immensely complex as one might imagine). At each stage, a new notation for numbers was suggested in order to make thinking or computation easier and, with the notation, the new kinds of number came (slowly) to be accepted. Each time the old idea had to be integrated into the new, and the new seen as an extension of the old. Thus, from whole numbers, the move to fractions is seen as encompassing the whole numbers and so the arithmetic of fractions must reduce to or extend the arithmetic of whole numbers. Similarly, since fractions can also be represented using decimal-names, the arithmetic of decimals must reduce to or extend the arithmetic of fractions. This process turns out to be more problematic than might be expected! ｾ＠ WHY NOT? Why is 1.5 + 2.5 not 3.10? Why is 0.3 x 0.2 not 0.6? Why is 1.2 x 100 not 1.200? Why is 3.0/10 not 3? Why is 3.05/10 not 3.5?
- 46. 32 Why is 1.23 + 3.4 not .157? In each case, construct a plausible reason (based on some memorized but perhaps incorrectly applied rule of thumb) for a pupil to believe that the answer given is correct. ＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭｾ＠ Comments There are usually sound, but perhaps misapplied, reasons for the ideas that pupils pick up! Rules such as "to divide by ten, drop a zero" make sense in context but can easily be misapplied if the rule is not your own summary of what is already understood. ｾ＠ MOVING DECIMAL POINTS Rules such as the following are often found in textbooks: • to multiply 3 by ten, add a nought to get 30; • to multiply 3.45 by ten, move the decimal point one place to the right to get 34.5. What connections are there between these two rules, and how might a pupil be expected to encounter that connection? A pupil, trying to apply the corresponding rule for division, said You can't move the decimal pOint in 2.3 to the left two places because there is nowhere for it to go! Try to see the decimal point as fixed, and the number-name moving past it. What differences in perception, if any, are involved in seeing the decimal point as moving, and the decimal point as fixed but the digits as moving, and how does this relate to multiplication by ten? ＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭｾ＠ ｾ＠ DECIMAL SUMS Perform the following additions, using them to generate a rule or technique for adding two decimal numbers. • 1.2 + 0.7, 1.22 + 0.77, 1.222 + 0.777, 1.2,' + 0.7: ＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭｾ＠ ｾ＠ MORE DECIMAL SUMS Can you predict the length of the period of the sum of two periodic decimals? The following sums might help - use a calculator and your knowledge offractions to see what is happening. 1.1: 23: + 2.9: 87:, 1.1: 2345: + 2.3: 47:, 1.1: 23456: + 2.9:876:. 1.01: 2345: + 2.888: 765: ＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭｾ＠ Comments The periodic decimals in this activity were chosen to illustrate several aspects of adding periodic decimals that might othelWise be overlooked. The invitation is to produce a
- 47. 33 complete theory or description ofthe addition of such decimal numbers. To predict the length of the period of the sum of two periodic decimals, specialize systematically in order to see what is going on. (Question: Can you have a periodic decimal-name in which the period is infinite?) You can also convert to fractions and then convert back again of course. ｾ＠ SAME AND DIFFERENT What is the same, and what is different, about the products in each of the following rows, and between the rows? • three 2s, thirty 20s • thirty-four 12s, three 20s, thirty-four 1.2s, three point four 1.2s ＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭｾ＠ Comments What "rules" are pupils expected to deduce from such patterns? How are they helped to see connections between the rules? When a decimal-name is the only or major name of a number available, multiplication of decimals turns out to be much more complicated even than addition. ｾ＠ DECIMAL PRODUCTS Begin to carry out each of the following products by long multiplication, in order to experience the uncertainties inherent in multiplying infinite decimal-names. • 0.3 x 1.5, 0.33 x 1.5, 0.333 x 1.5, • 0.3 x 0.3, 0.33 x 0.33, 0.333 x 0.333, ·1.2 x 0.8, 1.22 x 0.81, 1.222 x 0.818, 0.3: x 1.5 0.3: x 0.3: 1.2: x 0.8:1: Use your knowledge of fractions to verify what is suggested by the calculator. ＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭｾ＠ Comments Using long multiplication, at what point in the calculation of 1.2,' x 0.8,"" can you be sure of even the first digit in the product? Just finding the first non-zero digit in a product of two periodic decimals is not always easy if you confine yourself to some rule for multiplying the decimal-names witlwUl converting to fractions. Example C: Worked Examples One difficulty with neatly recorded solutions by experts is that many pupils never realize the struggles which even an expert may have been through before reaching a conclusion. That is why I stress being mathematical with and in front of pupils. Mason, Burton, and Stacey [24] presented a kind of"resolution with commentary" which has proved popular with students. The attempt is made to give evidence of dead ends and fresh starts, mistakes and corrections, conjectures which tum out to need modifying, and so on. In a similar vein, Open University tutors are encouraged to run technique bashing sessions, in which the tutor works on questions
- 48. 34 on an overhead-projector, trying to expose as much of their inner chatter and thoughts as possible while they work. Students are encouraged not to take notes, but to try to enter the screen, as if the tutor's words were inside their own head. Providing examples is always tricky, for the recipient may not attend to the features which the presenter is mentally stressing and which make the example exemplary. ALL ONES [21] Which of the numbers 1, 11, 111, 1111, 11111, ... can be a perfect square? STUCK? Try them on your calculator. Make a conjecture. Which digits do perfect squares end in? Resolution Trying the numbers on a calculator seems a good place to start. The calculator shows that only 1 seems likely to be a perfect square, but it is far from clear why this may be. The terms 11, 1111, 111 111, ... are divisible by 11, which suggests looking at the quotients 1, 101, 10101, ... obtained by removing the factor of 11. These numbers would have to have a factor of 11 as well, if the original number was to be a perfect square - a lot of energy can be spent going this route! Start again. The square roots on the calculator (to two decimal places) are 1, 3.33, 10.54, 33.33, 105.41, 333.33, .... I could look for a pattern here, trying to make use of the appearance of those threes. I stop and ask myself which digits can be the last digits of a perfect square. I find that numbers ending in 1,2,3,4,5,6,7,8,9,0, when squared, have as their last digits 1, 4, 9, 6, 5, 9, 4, 1, O. Only numbers ending in 9 or 1 could possibly be square roots of numbers consisting solely of Is. Dead halt - what now? Carry the same idea further - what two-digit numbers can be the last two digits of a perfect square? Careful- that looks like a lot of work. I really want to know whether 11 can be the last two digits of a perfect square. Trying all the cases as I did for single digits will get an answer, but so will a little algebra. Suppose lOa + 9 or lOa + 1 were square roots of an all 1's number. Their squares are 100a2 + 180a + 81 and lOOa2 + 20a + 1.
- 49. 35 What do I want? I want the second-last digit to be a 1. But the second-last digit comes from the last digit of 18a + 8 or from 2a, which in both cases is even! None of the other all 1's numbers can be perfect squares apart from 1 itself. Checking back over the argument, I wonder where the fact that 1 is a perfect square shows up? I have also shown a great deal more than I wanted originally, because the argument looks only at the last two digits. It never occurred to me at the beginning that only the last two digits mattered - perhaps I was blinded by a plethora of 1s. References 1. Brown, J., Collins, A., & Duguid, P.: Situated cognition and the culture of learning. Educational Researcher 18 (1) 3242 (1989) 2. Burkhardt, H., Groves, S., Schoenfeld, A., & Stacey, K. (eds.): Problem solving:A world view. Shell Centre, University of Nottingham 1988 3. Davis, J.: The role of the participant observer in the discipline of noticing. In: Proceedings of the Fourth Conference on Systematic Cooperation Between Theory and Practice in Mathematics Education (H. Steinbring & F. Seeger, eds.), University of Bielefeld 1990 4. Di Sessa, A.: Phenomenology and the evolution of intuition. In: Problems of representation in the teaching and learning of mathematics (C. Janvier, ed.), pp. 83-96. Hillsdale: Erlbaum, 1987 5. Edwards, D. & Mercer, N.: Common knowledge: The development of understanding in the classroom. London: Methuen 1987 6. Fischbein, E.: Intuition in science and mathematics: An educational approach. Dordrecht: Reidel 1987 7. Furse, E: Understanding mathematics: From task to cognitive architecture, version 15, transcript of a talk given Feb 28, available from him at the Polytechnic of Wales, Pontypridd 1990 8. Gattegno, C.: What we owe children: The subordination of teaching to learning. London: Routledge & Kegan Paul 1978 9. Gattegno, c.: The science of education, Part 1: Theoretical considerations. New York: Educational Solutions 1987 10. Griffith, P. & Gates, P.: Project Mathematics UPDATE: PM753A,B,C,D, Preparing to teach angle, equations, ratio and probability. Milton Keynes: Open University 1989 11. Gurdjieff, G.: All and everything, First Series, pp. 1193-1199. London: Routledge & Kegan Paul 1950 12. Hansen, D. Was Socrates a "Socratic teacher"? Educational Theory 38(2), 213-224 (1988) 13. Haroutunian-Gordon, S.: Teaching in an "ill-structured" situation: The case of Socrates. Educational Theory 38(2), 225-237 (1988) 14. Harris, I.: Forms of discourse and their possibilities for guiding practice: Towards an effective rhetoric. Journal of Curriculum Studies 15(1),2742 (1983) 15. Holt, J.: How children fail, London: Pitman 1964 16. Hyabashi, I. & Shigematsu, K.: Metacognilion: the role of the inner teacher (3). In: Proceedings of PME XII (A. Borbas, ed.) Vol 2, pp. 410416. Vezprem, Hungary 1988 17. Jaynes, J.: The origins of consciousness in the breakdown of the bicameral mind, University of Toronto, 1976 18. Khan, M.: Hidden selves: Between theory and practice in psychoanalysis, London: Maresfield Library 1989 19. Kilpatrick, J.: Reflection and recursion. In: Proceedings of the Fifth International Congress on Mathematical Education (M. Carss, ed.), Birkhauser, Boston, pp. 7-29, 1984 20. Laborde, J.-M.: Cabri-goometre, Laboratoire des Structures et de Didactique, Ins!. IMAG, Grenoble, 1988
- 50. 36 21. Mason, J.: Learning and doing mathematics, Macmillan, London, pp. 25-26, 19899 22. Mason, J.: Project Mathematics UPDATE, PM75ID, Dealing with decimals, Open University, Milton Keynes, 1989 23. Mason, J.: Mathematical abstraction as the result of a delicate shift of attention. For the Learning of Mathematics 9(2),2-8 (1989) 24. Mason, J. , Burton, L., & Stacey, K.: Thinking mathematically. London: Addison Wesley, London, 1984 25. Mason, J. & Davis, J.: The use of explicitly introduced vocabulary in helping students to learn, and teachers to teach in mathematics. In: Proceedings of PME XI (C. Bergeron, N. Herscovics, & C. Kieran, eds.), Vol. 3, pp. 275-281, Montreal 1987 26. Mason, J.& Davis, J.: Cognitive and metacognitive shifts. In: Proceedings of PME XII (A. Borb4s, ed.), Vol. 2, pp. 487-494. Vezprem, Hungary 1988 27. Mason, J. & Davis, J.: The inner teacher, the didactic tension, and shifts of attention. In: Proceedings of PME XIII, Vol. 2, pp. 274-281. Paris, France 1989 28. Mason, J. & Davis, J.: Notes on a radical constructivist epistemethodology applied to didactic situations. Journal of Structural Learning 10, 157-176 (1989) 29. P61ya, G.: How to solve it. Princeton: Princeton University Press 1945 30. Rhadakrishnar, S.: The principle upanishads. London: George Allen & Unwin 195310 31. Rolf,l.: Structural integration. San Francisco: Guild of Structural Integration 1963 32. Schoenfeld, A.: Mathematical problem solving. Academic Press: New York 1985 33. Soviet Olympiad. Crux Mathematicorum 7(8),237 (1981) ｾｨｩｳ＠ is an edited version of undergraduate course material for the Open University M101 Foundation Course in Mathematics, which has been running since 1978. lOA more modern version of the same image can be found in [111.
- 51. Some Issues in the Assessment of Mathematical Problem Solving! Jeremy Kilpatrick Department of Mathematics Education, University of Georgia, Athens, GA 30602, USA Abstract: To assess problem solving in mathematics adequately one must address the narrowing effects of current testing practice and of the continued pressure for efficient measurement. A new psychometrics is needed. Also, solutions need to be communicated and to be assessed as communication. A central issue is to make a valid assessment that sees problem solving as situated and mental processes as multiple and nonlinear. Keywords: assessment, mathematical problem solving. task validity, process scoring, measurement theory, communicating solutions To the extent that problem solving has become a common shibboleth if not necessarily a central feature of the school mathematics curriculum in recent years, it has raised serious issues of assessment. Most of these issues concern how problem-solving performance might be measured, but several relate to assessment in general and others concern how problem solving is to be understood. This paper addresses some issues of each type as they touch on possibilities for further thought and research in mathematics education. Validity of Tasks In a paper prepared for this conference, John Mason [10] argues for spontaneous evidence that pupils have learned to solve problems. He claims that asking pupils directly is likely to distort what they can do and are willing to do. Recent research on situated cognition underlines his concern. Researchers have raised the question of how reasonable it is to assume that a task is the "same task" when it is set for pupils to solve as it is when they define and set the task for themselves. Newman, Griffin, and Cole [14] tried to make the same task happen in two different settings. They gave a group of fourth graders (9- to 1O-year-olds) a one-on-one tutorial in which the child was given four stacks of cards, each stack being of a different color and bearing 1I am grateful to Vema Adams for suggesting the idea of problem solving as a composition task.
- 52. 38 the picture of a different movie star. The child was asked to find all the ways that pairs of stars could be friends. Children who did not invent a systematic procedure for checking whether they had all pairs were given hints to lead them to such a procedure. As a check on whether they had developed some system, a fifth stack was added and the task repeated. Children who did not arrive at a system were explicitly shown one. Then in a laboratory setting the children were given four chemicals and asked to find all pairs. Even though the teacher posed the problem in the laboratory, the children did not take solving that problem as their goal until they had formed some pairs and wanted to form more. Although none of the children started out by doing the task, those who finally did were much more successful than on the first task because they had discovered the task on their own. The children were not all doing the same task in the laboratory setting, and that led the researchers to reconsider what had happened in the movie-star tutorial. It became clear that many of the children may not have been doing the movie-star task at all in the first two trials. Newman and his colleagues observed that in the usual division of labor, the researcher sets the task and the subject is expected to do it. They concluded: When experimenters present a well-defined task to subjects in a standardized way, they have little chance to observe the subjects' formation of new goals or their application of a procedure to new situations. [14, p. 173] What applies to researchers giving laboratory tasks to subjects would seem to apply equally well to teachers giving assessment tasks to pupils (see also [14]). Spontaneous evidence of problem formulation and solution activities is valuable and ought to be built into continuous-assessment programs in mathematics. How planned assessment might be put into a context that permits an examination of how pupils are formulating goals and how they are applying procedures to new situations remains as a major challenge. Perhaps the overarching assessment issue stimulated by this line of research is how to make a valid assessment 0/what students know and can do in/ormulating and solving mathematicalproblems. Objective Assessment The assessment of intellectual performance has a long history. Over 4000 years ago in China, the emperor was testing his officials every three years [3]. Civil service examinations that included arithmetic were in place in China by 1115 B.C. and were being widely copied in Europe and the United States by the 19th century. England underwent an examination fever in the mid-19th century in which much of the examining changed from oral to written form. Mathematics always seemed to playa prominent role in innovative examinations (see [7], for details).
- 53. 39 The movement to new-type, or objective, testing that occurred in several countries during the ftrst half of this century tended to stress the measurement of a pupil's knowledge as the prime index of that pupil's achievement in a school subject. At the concluding conference of the International Examinations Inquiry [12, pp. 240-252], for example, E. L. Thorndike argued that for many purposes examinations should be as psychometrically "pure" as possible and therefore that examinations in mathematics, say, should not be needlessly confounded with measures of verbal ability or of one's general knowledge of the world. The emphasis on pure measures and objective scoring, coupled with the behaviorist view that complex thinking can be decomposed into simpler bits to be learned ftrst, has led to the multiple-choice and short-answer tests so common today. Such tests emphasize what pupils do not know: A high score in this kind of test does not infallibly demonstrate the attainment of what we call a liberal education; but a low score does infallibly demonstrate a lack of liberal education, because it reveals the absence of the foundation upon which a liberal education must stand.... One may have a flourishing tree without fruit, but one cannot have fruit without a tree; knowledge - ample and accurate knowledge - is the tree on which the fruit we call culture must grow. [M. McConn, quoted in 6, p. 140] One of the problems we face in assessing problem solving is that most current assessment techniques are directed toward grading the lumber of the tree of content knowledge rather than tasting and judging the fruit of problem solving. Pupils have become so accustomed to tests that look only for knowledge that they are not sure how to respond to requests that they solve complex problems either as part of class work or an assessment. An issue that has arisen from the extensive use of objective knowledge-based tests is how to contend with the effects of current assessmentpractice. Assessing Products and Processes Another source of difftculties in assessing problem solving is a failure to distinguish clearly between different types of problems. Historically, problems appear to have been included in the school curriculum primarily to provide a means to introduce and practice standard solution techniques [16]. Consequently, most problems in the curriculum are rather straightforward and routine. Sections in standardized tests labeled "problem solving" tend to consist of simple applications of content knowledge to well-structured, well-rehearsed, and stereotypical problem situations. Such tests may capture the kind of problem solving (or exercise answering) that is typical of many textbooks and much instruction, but they do not assess problem solving of a more open and original kind. Efforts to study the "processes" used in answering such exercises rather than simply assessing their "products" may be somewhat misguided.
- 54. 40 For example, some researchers have developed "analytic scoring schemes" [9, pp. 63--64] based on "phases" of problem solving similar to the phases used by Polya [15] in How to Solve It. To the extent that such schemes are applied to the processes pupils appear to use when they are solving problems by "thinking aloud," the schemes may be helpful to teachers and researchers in understanding how a pupil is tackling a problem. It is difficult to see the value of such schemes, however, when the problems are routine and the evaluator is sifting through the pupils' written work for evidence of "understanding the problem," or "devising a plan." Any inference about the process a pupil is using is bound to be somewhat shaky and needs to be checked against other information. Furthermore, the point of solving a routine problem is to get the official solution that everyone knows is there. It matters not how that solution has been found; the point is to get it. In the first U.S. National Assessment of Education Progress that dealt with mathematics, some of the assessment was conducted by way of interviews. The test constructors thought that a "real-life" exercise in balancing a checkbook might yield interesting problem-solving processes. Perhaps it did, but in the context of such an assessment, all that people were really interested in after the exercise was scored was whether or not the checkbook had been balanced. The number of examinees who did it one way versus the number who did it another way may intrigue some researcher but seemed to have essentially no practical value. The same thing seems to be true of various procedures for solving simple word problems. When such procedures are studied as part of a research project or to provide a teacher with information about his pupils, they can be informative. When they yield only descriptive information related to an assessment, however, they seem relatively barren. The issue for assessment is not so much what the pupil was thinking while she or he devised a solution as what solution the pupil offers. Unless they are rephrased, simple word problems typically ask for only a word, number, or sentence in response. They do not challenge the pupil to compose an explanation or justification. Perhaps, for example, the National Assessment checkbook exercise would have been more illuminating if the respondents had been asked not merely to balance the checkbook but to explain how balancing is done - after they had done it and not while they were attempting it. The relevant issue is how to assess problem solution in the sense of how and why the problem was solved the way it was rather than in the sense of what answer was obtained. New Approaches to Assessment Attempts to include more challenging problems in assessment instruments have run into several complications. Such problems tend to be difficult for pupils to respond to, especially in the
- 55. 41 context of a timed test. Unless previous instruction has familiarized pupils with similar problems, they may not know how to begin or what sort of answer is expected. The process of grading their answers may yield scores whose meaning is uncertain or whose reliability is shaky. The California Assessment Program [2] tried out a set of five open-ended questions in its 1987-1988 Survey of Academic Skills at Grade 12. The questions turned out to be rather difficult: the responses rated as showing "demonstrated competence" were never more than 20% of the total, and from 50% to almost 70% were no response or an "inadequate" response. The Mathematics Assessment Advisory Committee conjectured that the poor performance had resulted from the pupils' lack of experience in expressing mathematical ideas in writing. Several generations of short answer testing in the U.S. appear to have taken their toll on pupils' facility with open-ended questions. In the Netherlands, the Hewet Project [8] appears to have had relatively more success than the California Assessment Project in using alternatives to timed written tests. The Hewet Project tried a variety of alternatives. One consisted of two-stage tasks in which the pupils took a timed written test consisting of open-ended and essay questions, the responses were scored, and the pupils had the opportunity to retake the test at home over several weeks to obtain a second score. Other alternatives consisted of take-home tasks, essays, and oral tests. Perhaps because of the different pattern of mathematics instruction and testing in Dutch schools compared with California schools, but even more likely because of the special instruction provided in the Hewet Project, the level of performance was relatively high. The 28 pupils (from 2 teachers) participating in an internal examination earned scores that averaged from 6 to 8 out of 10 on a timed written test, a take-home task, and an oral test. The project did not let problems of intersubjectivity in grading control the selection of tasks. The criteria, or principles, for developing alternative assessment tasks were that the tasks should (a) improve learning, (b) allow candidates to show what they know, (c) operationalize curriculum goals, (d) not be selected primarily to allow objective scoring, and (e) fit into usual school practice [8, p. 183]. The demand for objective scoring and high reliability are part of the fetish of efficiency that shapes and directs much assessment today. As Norman Frederiksen [4] observed, however, efficient tests tend to drive out less efficient tests, leaving many important abilities untested - and untaught. An important task for educators and psychologists is to develop instruments that will better reflect the whole domain of educational goals and to fmd ways to use them in improving the educational process. (p.201) In a call for "useful assessment," Wolf, Bixby, Glenn, and Gardner [17, p. 35] argue that such assessment should (a) be multidimensional, (b) capture knowledge and skill as exercised in
- 56. 42 context, (c) be longitudinal enough to permIt mquiry into the procedures by which understanding developed, and (d) offer information about the pupil's ability to amplify his or her thought through connection to tools, resources, and other thinkers. This view of useful assessment is in harmony with many current efforts by mathematics educators to construct new techniques and materials for assessment. The difficulty does not appear to be a lack of creativity but rather our tradition-bound views of how measurement must be conducted. A key issue is how to deal with pressuresfor efficiency and reliability ofmeasurement. Measurement Theory Psychologists appear to be searching not only for new instruments but also new theories to support the construction of such instruments. "It is only a slight exaggeration to describe the test theory that dominates educational measurement today as the application of twentieth century statistics to nineteenth century psychology" [11]. Current test theory treats problem-solving ability as a single, continuous variable, whereas current cognitive psychology conceives of problem solving as entailing a variety of processes, including the restructuring of knowledge, the development of internal representations, and the use of sophisticated strategies to construct as well as to monitor and critique solutions. Pressures to create efficient test instruments may have inhibited the development of measurement models to handle the new view of cognition as nonlinear, dynamic, and context-bound. One recent framework replaces the linear structure of abilities with a lattice model [5], but much further effort is needed. The issue is how to develop a new psychometrics. Communicating Problem Solving Mathematics educators and researchers concerned with mathematics education need not wait for new measurement models before they begin investigating new approaches to the assessment of problem solving. One of the most promising of these approaches is to treat the solving of a problem as a task in composition. That is, just as in writing a composition, one can distinguish knowledge telling from knowledge transforming [1], so in solving a problem, one can observe that some thinking is an almost mechanical execution of a well-practiced procedure, whereas other thinking operates at several levels to yield an understanding of the problem through various transformations that eventually produce a solution. When a written account of a solution to a mathematical problem is required from a pupil, the pupil engages in an activity much like writing a composition. The pupil needs to plan how the argument will be organized, what the reader needs to know, and how the various ideas are related. The written solution can
- 57. 43 be evaluated in much the same way that an essay is judged, and one can see whether the solution involves only mechanical performance or some deeper level of understanding. Such compositions have long been used in written examinations in many countries, but they have fallen out of favor in recent years. Their restoration would benefit instruction in problem solving as well as its assessment. Teachers may not have given sufficient emphasis to the creation of a composition as a means of reporting on one's problem solving. They have tended to tell pupils to show their work, which emphasizes the thinking process of the moment, rather than asking them to write up their solution in a coherent fashion, which would emphasize the need to look back at what one has done and construct a clear communication about it. When one couples a nonroutine mathematical problem to the task of writing an ･ｳｾ｡ｹ＠ about its solution, one opens up many opportunities for assessment. Pupils can keep journals that chronicle their evolving thoughts about the problem. They can submit drafts of their essay responses to the teacher or to other pupils for editorial comment (not so much about the mechanics of the prose as about its communicative power). They can work on the problem - and the construction of its solution in essay form - in collaboration with other pupils. In this fashion, problem solving becomes an opportunity to learn and practice both mathematical and communication skills. Researchers are likely to continue to attempt to break problem solving into components so that those components can be assessed separately and reliably. Efforts along that line are not to be disparaged, but they should not be accepted uncritically either. One needs continually to ask whether the tasks in a proposed instrument qualify as mathematical problems of any significance. Along with these analytic efforts, some efforts should be made to explore a more holistic approach to problem-solving assessment in which a single significant problem is taken as the unit of interest and the pupil's task is not only to find a solution that is personally satisfying but to write up a solution that would satisfy a reader. Mathematics is about communication. The pupil who cannot communicate what he or she has done with a problem has not truly solved it. The communication may be oral, it may be written, it may take a variety of forms. But the assessment of mathematical problem solving should concentrate on how the pupil expresses a solution. The issue: how to reconstruct problem-solving assessment as communication. Emerging Views To address the issues raised in this paper will require much effort by many people. Mathematics teachers need not only to be informed about the progress being made toward the resolution of these issues but also to contribute to that resolution. The spirit behind the issues can be captured in the following views, which have begun to find support in the literature but
- 58. 44 are not widely enough known and understood today in the mathematics education community: Mind is both multiple and nonlinear. Cognition is situated both socially and culturally. A task to assess pupils' cognition is an activity situated in a context. Consequently, assessment needs to be multi-faceted, developmental, context-sensitive, and holistic. References 1. Bereiter, C., & Scardarnalia, M.:The psychology of written composition. Hillsdale, NI: Erlbaum 1987 2. California Assessment Program:A question of thinking: A flTst look at students' performance on open-ended questions in mathematics. Sacramento: California State Department of Education 1989 3. DuBois, P. H.: A test-dominated society: China, 1115 B.C.--1905 A.D. In: Proceedings of the 1964 Invitational Conference on Testing Problems (C. W. Harris, ed.), pp. 3-11. Princeton, NI: Educational Testing Service 1965 4. Frederiksen, N.: The real test bias: Influences of testing on teaching and learning. American Psychologist, 39, 193-202 (1984) 5. Haertel, E. H., & Wiley, D. E.: Representations of ability structures: Implications for testing. Unpublished manuscript 1991 6. Kandel, I. L.: Examinations and their substitutes in the United States (Bulletin No. 28). New York: Carnegie Foundation for the Advancement ofTeaching 1936 7. Kilpatrick, I.: The chain and the arrow: From the history of mathematics assessment. Paper presented at the ICMI Study on Assessment in Mathematics Education and Its Effects, Calonge, Spain, April 1991. 8. Lange, I. de.: Mathematics, insight and meaning. Utrecht: Rijksuniversiteit Utrecht, Vakgroep Onderzoek Wiskunde Onderwijs en Onderwijscomputercentrum 1987 9. Lester, F. K., Ir. & Kroll, D. L.: Assessing student growth in mathematical problem solving. In Assessing higher order thinking in mathematics (G. Kulm, ed.), pp. 53-70. Washington, DC: American Association for the Advancement of Science 1990 10. Mason, I.: Researching problem solving from the inside (In this volume), 1991 11. Mislevy, R. I.: Foundations of a new test theory (Report No. RR-89-52-0NR). Princeton, NI: Educational Testing Service 1989 12. Monroe, P. (Ed.): Conference on examinations (Dinard, France, September 16-19, 1938). New York: Columbia University, Teachers College, Bureau of Publications 1939 13. Newman, D., Griffin, P., & Cole, M.: Social constraints in laboratory and classroom tasks. In: Everyday cognition: Its development in social contex (B. Rogoff & I. Lave, eds.), pp. 172-193. Cambridge, MA: Harvard University Press 1984 14. Newman, D., Griffin, P., & Cole, M.: The construction zone: Working for cognitive change in school. Cambridge: Cambridge University Press 1989 15. Polya, G.: How to solve it (2nd ed.). Princeton, NI: Princeton University Press 1945 16. Stanic, G. M. A., & Kilpatrick, 1.: Historical perspectives on problem solving in the mathematics curriculum. In R. I. Charles & E. A. Silver (Eds.), The teaching and assessing of mathematical problem solving (Research Agenda for Mathematics Education, Vol. 3, pp. 1-22. Hillsdale, NI: Erlbaum; Reston, VA: National Council of Teachers of Mathematics 1988 17. Wolf, D., Bixby, I., Glenn, I., & Gardner, H.: To use their minds well: Investigating new forms of student assessment. Review of Research in Education, 17,31-74 (1991)
- 59. Assessment of Mathematical Modelling and Applications Henk van der Kooij OW&OC. Tiberdreef 4.3561 GG Utrecht, The Netherlands Abstract: In The Netherlands mathematics education is changing from learning about structures and mechanic manipulation of formulas into learning how to formulate, how to structure. The so called realistic approach to mathematics education is strongly process oriented in stead of product oriented.Students (and teachers) are active in (re)inventing mathematics for themselves.ln this kind of education important skills to be tested are: choosing suitable strategies for solving problems, modelling real problems and critical analyzing given models and given solutions. The HEWET and HAWEX-project have shown that there are appropriate ways for assessment of mathematical modelling and applications, even in time restricted written tests (like the final examination). Keywords: realistic mathematics education, mathematical modelling, applications, HEWET project, HAWEX project, final examination The HEWET-project resulted in a new curriculum for the upper grades (age 16-18) of the pre- university level of secondary education in The Netherlands: mathematics-A. The curriculum is aiming at those students who are preparing for a study in social sciences at university. Since 1987 nationwide the final examinations are based on this curriculum. The HAWEX-project led to new curricula (mathematics-A and -B) for the upper grades of Havo (higher general education, a middle level of Dutch secondary education, see Appendix 1). Mathematics-B is aiming at the students who are preparing for any technical study in higher vocational education. Mathematics-A is aiming at those students who are preparing for any social study, but also at those students who go to work after finishing school. The first experimental examinations were held in 1989 and 1990. Although the curricula are different because of the different groups of students they are aiming at, all three of them are based on the same idea: the realistic approach of mathematics education. In this approach modelling, applications and problem solving are very important ingredients. Consequently the assessment of this kind of education must contain aspects of modelling and applications. Before turning to the assessment we will first look at some aspects of the education itself. The general ideas of the realistic approach of mathematics education and the consequences for the HEWET mathematics-A curriculum are described extensively, respectively by Treffers [3]
- 60. 46 and De Lange [2]. We will look at the way the ideas of the realistic approach work out in the classroom. Mathematization in the Classroom In the realistic approach mathematization plays a very imponant role. Real world problems are explored intuitively, resulting in mathematizing that real situation.Imponant activities in this pan of the process of mathematization are: Organizing, structuring, schematizing and visualizing. As soon as the problem has been transformed to a mathematical problem it can be attacked and treated with (more or less advanced) mathematical tools. Keywords of this pan of mathematization are: representing relations in formulas, proving, refining and adjusting models, combining and integrating models, formulating a new mathematical concept, generalizing. A beautiful example of a problem that asks for a number of these activities from students is the 'Rat-problem', taken from one of the HEWET-booklets. The problem stans with a text from 'Rats', a book written by a well known Dutch novelist Maarten 't Han: GROWTH OF RAT POPULAnONS. As regards the progeny of one pair of rats during one year the numbers given vary considerably. In the next chapter I shall discuss the scanty information supplied by research into the fenility of rats in nature, but at this point it might be interesting to estimate the number of offspring produced by one pair under ideal conditions. My estimate will be based on the following data. The average number of young produced at a binh is six; three out of those six are females. The period of gestation is twenty one days; lactation also lasts twenty one days. However, a female may already conceive again during lactation, she may even conceive again on the very day she dropped her young. To simplify matters, let the number of days between one litter and the next be fony. If then a female drops six young on the first of january, she will be able to produce another six fony days later. The females from the first litter of six will be able to produce offsprings themselves after a hundred and twenty days. Assuming there will always be three females in every litter of six, the total number of rats will be 1808 by the next first ofjanuary, the original pair included. This number is of course entire fictitious. There will be deaths; mothers may reject their young; sometimes females are not in heat for a long time. Nevenheless, this number gives us some idea of the host of rats that may come into being in one single year. The simple question on this text: Is the conclusion that there will be 1808 rats at the end ofthe year correct? Although the question is simple, the problem turns out to be very difficult.Given as a homework task no more than 5 or 6 students out of a group of 30 are successful in solving it. They have to demonstrate and explain their solutions to the other students.
- 61. 47 Four examples ofwell structured schema: (A) (B) (C) t -1 0 1 2 3 4 5 6 7 8 9 0 2 3 4 5 6 7 8 9 N(t) 2 2 6 2 6 6 2 2 2 4 4 total number 8 2 8 2 4 8 146 278 536 974 1808 N(t) _ 2 ｻＬｾｾＮ＠ ＭｲｬＭＫｉｾｉｾｉＭＭｾｉ＠ ｾｉＭＫｉｾｉＭＭｉｲＭｲｬＭｲｬ＠ t - -1 0 1 2 3 4 5 6 7 8 9 adults = 4 7 10 22 43 73 139 (D) adults young] youngO t number offemales 1 0 0 -1 1 0 3 0 1 + 3 = 1 3 3 1 4 + 3 4 3 3 2 7 + 3 = 7 3 12 3 10 + 3x4 = 10 12 21 4 22 + 3x7 = 22 21 30 5 43 + 3x1O = 43 30 66 6 73 + 3x22 = 73 66 129 7 139 + 3x43 = 139 129 219 8 268 + 3x73 = 268 219 417 9 487 + 3x139 = Figure 1 I 4 7 10 22 43 73 139 268 487 904
- 62. 48 When all solutions are put on the blackboard the solutions themselves become subject of study. It is obvious that only those students who found some kind of structuring, are successful in finding the complete solution. Another surprising fact is that no two solutions are similar (or at least look the same). Schema like (C) and (D) are very transparent, because they neglect the male rats. Each of these two structures is leading to a further going schematizing. Schema (C) can be translated into a graph and/or a matrix: from Yo Y1 a Yo to :: C: ｾ＠ ) Figure 2 Refinement of the idealized model (see the text of Maarten 't Hart) is possible by changing the weights in the graph and in the matrix. Schema (D) leads to a recurrence relation: N(t+1) =N(t) + 3 x N(t-2), with initial values: N(-1) = 1, N(O) = 4, N( I) = 7 This relation can easily be put into a computer, with the possibility to look at the growth of the population for a much longer period. The growth turns out to be exponential. The Rats problem is not an easy one. It appears at the end of the two years students are working on mathematics A (pre-university level). But throughout the course there are many opportunities to become familiar with mathematization activities on (more or less) complex problems taken from the (more or less) real world. Mathematization activities are not reserved to real world problems only. The next problem, taken from a booklet on mathematics B, is a purely mathematical one: Given the straight lines 1: y =2x - 10 and m: y =10 - O.5x A is on line 1 and B on line m, so that AB is horizontal and AB = 6 Calculate the coordinates of A and B. (Hint: let xB = x; express YB, xA and YA in x) The students who tried to solve the problem using the hint, did not succeed (see figure 3). They did not understand our formal, algebraic approach, typical for mathematicians! Those who looked for a solution in their own way, did a very good job. Look at the four following solutions. Each one is based on knowledge the students already have from earlier lessons.
- 63. 5 (a) Algebraic Approach x A =0.5y + 5 xB =20 - 2y 49 Figure 3 ｾＭ xB = 2.5y - 15 = 6 ,soJA = 8.4 and x A = 9.2 ;xB = 3.2 Figure 4 (b) Dynamical approach. Translation of line lover vector (-6 ,0) gives line 1': y =2x + 2. The intersection of I' and m is B: 2x + 2 =10 - O.Sx , so it follows that x =3.2 ,etc. (c) Using the meaning of slope ｂｾｾＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭｾ＠ Using the slopes I :y=2x -10,soifAy=1thenAx =0.5 m :y = 10 - 0.5 x . If Ay = 1 then Ax =-2 So a vertical step AY = 1 causes a horizontal widening Ax = 2.5 From AB = 6 = Ax it follows that AY = ＲｾＵ＠ = 2.4 Conclusion: yA =y B =8.4 and xB =8 - 2x2.4 = 3.2 and xA =8 + 0.5x2.4 = 9.2 Figure 5
- 64. (d) Geometrical solution h 6 SO· h 36 2 4 "6=T5' . =T5= . so y = 8.4; ｾ＠ and xB follow by substitution of y = 8.4 in the formulas 50 6 Figure 6 The HEWET and the HAWEX-project have shown that by the realistic approach of mathematics education students become good (and sometimes very professional) problem solvers, who are fully aware of what they are doing and for what purpose. They are critical of the way they are attacking problems, and also of the way other people do. Very often they surprise their teachers with more elegant solving strategies than the ones the teachers have in mind. In order to enable them to arrive at that level , it is very important that every student is given the opportunity to create and to use his/her own solving strategies, at least until other strategies (of other students or the teacher's one) are accepted as being more suitable. A more extensive discussion about these ideas can be found in Gravemeijer [1]. The Assessment It may be clear that students who have learned mathematics this way, may not be "punished" by examinations in which only technical and algorithmic skills are tested. In The Netherlands the final examinations are time restricted written tests. Traditionally the stated problems are of a very technical kind. The students have to show that they are well trained in using standard algorithms. This kind of testing only shows what a student does not know and he is punished for that. The new curricula are asking for tests in which students can demonstrate they are able - to choose an appropriate strategy for solving a problem, - to criticize a given model, - to integrate different mathematical models. Part of the HAWEX-project was the description of general goals, knowledge and skills that can be tested in the final examinations (see Appendix 2).This was not done in the HEWET- project. It appeared that many teachers did not take these higher goals for granted Of course very open problems, like the Rat-problem, don't fit in written tests like the final examinations. Three examples of problems are given to show some possibilities for testing the stated goals within the final examinations.
- 65. 51 Example 1. Mathematics-A, Pre-University Level. In a thesis about juvenile criminality a researcher assumes that 30% of the students at secondary school occasionally have committed shop-lifting. A headmaster wants to know whether the percentage of 30 is also true for the 1200 students of his school. Assume that indeed 30% of the students of this school have shop-lifted. A random sample of 15 students is taken. »1. Calculate the probability that at least 5 students in the sample have ever committed shop-lifting. Assume that 6 out of a class of 20 students have ever committed shop-lifting. »2. Calculate the probability that in a random sample of 10 students out of this class, less than 3 students have ever committed shop-lifting. A teacher decides to make a thorough investigation by questioning all students of the school. He knows that in such a study not everybody will tell the truth, so a method must be used in which it is not always necessary to answer truthfully. He makes use of the following method: - every student is asked: "have you ever shop-lifted?"; - before answering the student has to throw a die; the result of the throw remains unknown to the teacher; - the question now must be answered in the following way: ifyou have thrown: 1,2,3 or 4 5 6 then your answer must be: the truth "yes" or "no" always "yes" always "no" The student is the only one who knows whether the given answer is given by chance or according to the truth. This method of questioning, known as 'randomized response technique', makes it possible to draw conclusions by studying all the given answers. Of the 1200 given answers, 416 were "yes". The teacher estimates that the number of students that ever committed shop-lifting is 324. »3. Explain how he arrives at this estimation. The number of 324 students is obviously smaller than the 360 students you could expect according to the thesis. Of course the students of this school are not a random sample of all students in secondary education. Therefore the assumption of the researcher may not be rejected because of this sample. >>4. Verify whether or not the assumption ofthe researcher should be rejected in case of a random sample where exactly 324 students did ever commit shop-lifting. Use a significance levelof5%. The randomized response technique is discussed in the mathematics lesson. One student proposes a much simpler method with the following instruction:
- 66. 52 ifyou have thrown: then your answermust be: 1,2 or 3 the truth 4, 5 or 6 the opposite to the truth In this case the privacy ofevery person is also guaranteed, he says. »5. Is this variant ofthe randomized response technique a useful one? The students are accustomed to questions like 1,2 and 4. Questions 3 and 5 are highly original. They have never seen something like that before. These questions really ask for skills that surpass the level of algorithms and techniques. Just try to solve it for yourself, before looking at the solutions of students! Students (and teachers) who never are given room to develop their own strategies for solving problems, will fail on answering this kind of questioning.Three solutions of students: (a) P(5) = 116, so 200 persons (=1200/6) always answer "yes". P(I,2,3,4) = 4/6 so there are 800 answers according to the truth. 416 - 200 = 216 and therefore 216 out of 800 answer "yes", according to the truth. Conclusion: P ("yes") = 216/800 and so the expected number is 1200 x 216/800=324. (b) Let X be the number of students who have ever.committed shop-lifting, then: (5/6)X + (116)(1200 - X) = 416 and that gives X = 324. (c) A visualisation of the problem in a tree: 1200 ＱＮＲｾ＠ 800 200 200 yeV'.no Iyes lno m 200 200 Figure 7 The total number of "yes" is 416, so ? = 216. That means P("yes") = 216/800. The expected number of 'criminals' will be about 1200 x 216/800 = 324. Example 2. Mathematics-A, Higher General Education. One of the exhaust gasses emitted by a car, is carbon-monoxide (CO). The amount of CO (the so called CO emission) depends on the temperature of the engine and on the driving speed. That appears from an article in the magazine Verkeerskunde. The article was illustrated by the graph on figure 8: The CO emission for a wann engine is given by the formula: (1) e = 4.4 + 196.0/v e in g/km v in km/h »1 The emission decreases when the speed increases. How can you see this in the formula?
- 67. 53 I. II 0---11......" • • _WiII....... ,,, ! . ｾＢ＠ ｝ｾﾷＮＭＭＭＭ］ＭＭＧＭﾷＭﾷＭＭＮＭｉ＠ - - " .....11,1_1 Figure 8 Assume fonnula (1) may be used for a speed of60 km/h. »2. Calculate the emission (in glkm)for this speed. The CO emission for a cold engine is given by the formula: (2) e =6.9 + 298.5/.. At a certain speed the emission ofa car with a cold engine was 14 gIkm. »3. Calculate the speed ofthat car. There also are fonnulas in which the CO emission is given depending on the drive length and the duration of the drive. For a warm engine this fonnula is: (3) E = 4.4L + O.054T E = amount of CO in g, emitted during the drive L = drive length in km T = duration of the drive in sec. »4. Calculate the total CO-emission (in g) for a drive of5 km in 8 minutes with a warm engine. »5. Calculate the total CO emissionfor this drive also withformula (1). »6. Find a formula for the total CO emission E depending on drive length Land duration T for a cold engine. Mostly the students on the mathematics-A program of HAVO are very poor 'algebraists'. Therefore the last question was only meant for the best students, to demonstrate they could do more than they had leamed in the classroom. We thought the solution had to be something like this: formula (2): e.. 6.9 + 298.5 v . L " =3600 xT" (from hours to seconds!) substitution in formula .(2) gives: .. 6.9 + 298.5 T I =6.9 + 0.083 x r: 'f 3600 E =e xL - 6.9 xL + .0.083 xT Figure 9
- 68. 54 Only one student answered the question this way, but there were many solutions to the problem based on analogy: (a) wann engine e =4.4 + 196.0/v gives E =4.4L + 0.054T, so cold engine e =6.9 + 298.5/v gives E =6.9L + 0.082T "0.082 because 196/0.054 = 3629.6296 and 298.5/3629.6296 = 0.0822398. I've taken four decimals just like is done in the 'wann' fonnula." (b) a short answer that shows insight in the structures of the fonnulas: The fonnula is E = 6.9 L +0.083T 0.083, because you have to divide 298.5 by 3600 (60 x 60 from sec to hours) (c) a very fine solution, using some information from earlier questions: (4) E = 6.9L + x With a cold engine you get e = 6.9 + 298.5/37.5 = 14.86 (he takes L = 5 km, T = 8 min and v = 37.5 km/h from »4) so: E = 5 x 14.86 = 74.3. Substitution in (4): 74.3 = 6.9 x 5 + x so x = 39.8 39.8/480 = 0.0829 (because 8 x 60 = 480 sec) Conclusion: E = 6.9L + 0.0829T. Example 3. Mathematics-B, Higher General Education. The church tower on the photo will be examined. The plan of this tower is a square of 6 by 6 meters. The roof consists of four identical rhombus. The lowest corners of the rhombus are 18 m above the ground. The top of the tower is 26 m above the ground. The remaining corners of the rhombus are 22 m above the ground, each on the line of symmetry of the four side walls. Figure 10 On the worksheet ( see Appendix 3) you see a start of a drawing of the tower in a parallel projection. »1. Complete the drawing o/the tower.
- 69. 55 The quality of the bell ringing depends on the volume of the room in which the bell is hanging. The floor of this room is 12 m. above the ground. The ceiling of that room can be built on a height of 20, 22 or 24 m. »2. Draw theform ofeach ofthese three possible ceilings (scale 1:100). The ceiling is built on a height of 22 m. »3. Calculate the volume ofthe room where the bell is hanging. Space geometry is one of the two main subjects of the mathematics-B curriculum. The students have to 'use their spatial imagination in an effective way' ( see Appendix 2). They need that imagination for each of the three questions. Although the questions look very 'closed', the students haven't learned a standard way to answer them. Consequently there is a large variety of solutions. Three examples of answers to question »3: (a) skelch, n Ｔｾ＠ Figure 11 Four pyramids are cut off. The volume of one pyramid is G x h/3 = 4 x ...J18 x ...J4.5 = 12 The volume is V =4 x 12 + 216 =264 (He oversees the volume of the 'square': 18 x 4 = 72). (b) 4 Figure 12 A nice idea: The four pyramids seem to fill up some space, but the student didn't notice that the inner space isn't filled up!
- 70. 56 (c) Figure 13 Very clear and simple: The volume of one pyramid is (0.5 x 3 x 3) x 4/3 = 6 So the volume is: V = 6 x 6 x 10 - 4 x 6 = 336 Alternative Tests Final examinations are not the most appropriate tool to test the so called higher goals of the curriculum.In the HEWET-project several ways of alternative testing are developed [2]. In the HAWEX-project a new phenomena is introduced: a practical examination. Sixty mathematics-B students did a practical job on Space geometry,working in couples for three hours. The concrete object was a lamp, built of 8 half cubes. Some of these half cubes are rotative, so there are many different shapes possible. Each couple had 10 cardboard half cubes to use, if necessary. Fourteen problems, starting simple and very complex in the end, had to be solved. Photos and drawings of the different shapes of the lamp were the source for the problems they had to solve (figure 14). Of course the couples were allowed to deliberate. They also were allowed to consult the teacher.At the end of the three hours each couple had to hand in one set of solutions to the 14 problems. This way of testing offers some advantages over 'normal' tests. The students have to cooperate. They must convince each other, for they had to hand in only one solution.Of course this skill can never be tested in a written test. The problems can be more complex than the ones in the written test, because the students can deliberate, while the teacher can lend a helping hand, if necessary.
- 71. 57 Figure 14 The students did like this way of testing very much: "It's fine that you can talk to somebody when you feel uncertain about how to attack a problem", "There is no trouble with stress, the atmosphere is quite calm". The results were very good: only one of the thirty couples failed. Some Final Remarks The realistic approach of mathematics education is strongly process oriented in stead of product oriented. Therefore the traditional written test is not the best instrument to measure the achievements of a student. Alternative tests are fitting better to the goals of the curricula of the HEWET and HAWEX-project. The composition of such tests is not an easy job. Many teachers in The Netherlands feel incapable of doing it, while others won't take (or have) the time for it. Therefore this task should be done by professional teams. From both projects it has become clear that the way students do attack problems differ from the way most mathematicians do. A mathematician often starts solving a problem by translating the problem to some algebraic form.Doing this, the problem becomes static. Most students keep working within the context of the problem, only using (some) algebra when it is not avoidable. This way of solving problems is much more dynamic. That's why students' solving strategies often look more elegant, much more simple and straight to the point than the solution of mathematics teachers. The student solves the stated problem, while the mathematician often escapes from the uncertainty of the problem into the safety of the mathematical techniques.
- 72. 58 Appendix 1: A Global Picture of the Dutch School System, and the Place of the Two Projects within it university (4·6years) 18 -1••-,••••..••••,-••-•••••-,••1 vwo (high level) HDO higbervocatiooal educalioo (4years) (middle level) MDO intennediate vocational education (4years) mavo/tbo (lower level) secondary education ＱＲＫＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭｾ＠ primary education (age4·12) Figure 15
- 73. 59 Appendix 2: Goals, Knowledge and Skills for the Upper Grades of Havo Goals: Mathematics-B: The curriculum is mainly focused on the use of mathematics in exact sciences.The students have to have to their disposal algorithmic and geometric skills and have to be able to use them in mathematical contexts and in geometric, natural scientific, technical and other situations. Mathematics-A: The curriculum has a general educational value and is focused on the use of mathematics in society. The goal of mathematics-A is that students can understand and solve problems from reality with use of mathematical tools. Knowledge and Skills: To achieve the mentioned goal, the students have to learn: - to analyze problems and to show logical relations between data, statements and results.(A,B)*) - to choose a suitable mathematical method to solve a problem and to use algorithms when solving.these problems.(A,B) - to use calculators and computer programs when solving problems.(A,B) - to analyze critically articles from news media with mathematical presentations, reasoning or calculations.(A) - to interpret mathematical solutions within the given context.(A) - to recognize and extract mathematical essentials out of texts.(A) - to use their spatial imagination in an effective way.(B) - to combine and to integrate different solving methods.{B) - to use new concepts or measures in new situations after a short description.{A,B) - to present the choice of a method, the process of solving and the results conveniently arranged in words or by use of suitable other representation fonns.{A,B) *) (A): only for mathematics-A (B): only for mathematics-B (A,B): for both mathematics -A and-B Main subjects Mathematics B: - Applied Calculus - Space Geometry Mathematics A: - Tables,Graphs and Fonnulas - Discrete Mathematics - Statistics and Probability
- 74. 60 Appendix 3: Worksheet Figure 16 References 1. Gravemeijer, K.,van den Heuvel, M., & Streefland, L. : Contexts, Free Productions, Tests and Geometry in Realistic Mathematics Education. Utrecht: OW&OC 1990. 2. de Lange, J. :Mathematics, insight and meaning. Utrecht: OW&OC 1987 3. Treffers, A. & Goffree, F. : Rational Analysis of Realistic Mathematics Education. In: Proceedings of the Ninth International Conference for the Psychology of Mathematics Education (L. Streefland, ed.), pp 79-122. Utrecht: OW&OC 1985
- 75. A Cognitive Perspective on Mathematics: Issues of Perception, Instruction, and Assessment Patricia A. Alexander College of Education, Texas A&M University, College Station, Texas 77843, USA Abstract: In this paper, concerns systemic to educational theory and practice are discussed in relation to the current state of mathematics education research and mathematics instruction. These concerns are synthesized into three general issues addressing the perception of the discipline, the nature of schooling, and measurement models and assessment practices. Potential solutions to these issues are offered. Keywords: cognitive psychology, knowledge, classroom culture, context, metacognition, misconceptions, assessment, measurement theory, anchored instruction Society is presently faced with a staggering number of life-threatening and planet-threatening problems, such as AIDS, economic stagnation, and environmental abuse. Any solution to these enormous problems will require logical, rational, creative thought, and systematic investigation; that is, any solution requires mathematical problem solving. Yet, where will we find those individuals capable of the formulating the solutions to these and to other problems that have still to be detected? Finding these individuals is, perhaps, even a greater problem facing our society today. As Paulos [41] so aptly stated in his bestselling book on innumeracy: I'm distressed by a society which depends so completely on mathematics and science and yet seems so indifferent to the innumeracy and scientific illiteracy of so many of its citizens...(p. 134). Despite the decades of research activity and aggressive instructional reforms [e.g., 35, 36], more and more American students seem to be exiting educational programs with inadequate mathematics knowledge, with a distaste for or a fear of mathematics, or with a cache of mathematical formulae and terminology that they seem unwilling or incapable of transferring to unfamiliar, complex, or "real world" situations. These are conditions that underlie the issues to which the title of this presentation refers. My presence at this international conference further suggests that these questions of importance to mathematics education are of broader concern to cognitive researchers
- 76. 62 investigating human intellectual processing. For the past five years, I have been engaged in research on the interaction of domain knowledge and strategic processing. This research has afforded me the opportunity to examine theory and practice in mathematics, as well as in the domains of human biology/immunology, physics, and social studies. Not surprisingly, this enterprise has led me to see the current state of mathematics education research and mathematics instruction in terms of underlying issues; issues that have relevance to a number of academic disciplines outside of mathematics. Indeed, it would not be inconceivable to say that almost every academic discipline is currently exhibiting the same general symptoms frequently attributed to mathematics: declining achievement and rising incompetence among students, instruction that is decontextualized, routinized, and focused on lower-order cognitive processing, student disenchantment, and teacher frustration. Rather than diminish the importance of these issues for mathematics, these shared symptoms hint at the systemic nature of the affliction, a point I will return to in the conclusion of this presentation. I concur with Schoenfeld [59] that the issue for research is to understand and abstract the common symptoms from isolated cases and examples and to use that understanding to create environments that foster intellectual growth and academic achievement, thereby treating the affliction. Thus, for the purpose of this presentation, I will merge a number of concerns into three global issues with relevance to mathematics. Specifically, I will consider how perceptions of the discipline, the nature of schooling, and measurement models and assessment practices may negatively affect mathematics learning. Even though I treat these three issues as if they were discrete, I fully acknowledge that they are highly interrelated topics. That is, one cannot completely separate questions about the discipline of mathematics from questions about how mathematics is taught or how it is assessed. The separation, however, does permit me to identify some global concems that can more readily be discussed here. Perceptions of the Discipline of Mathematics Mathematics is a discipline that consists of many related domains. Disciplines can be distinguished by the fact that they encompass extensive bodies of knowledge that are oriented toward fundamental principles or generalizations [45, 67]. These fundamental principles form the cruxes around which domain-specific knowledge is internalized [2]. Domains, by definition, are fields of study that, like disciplines, consist of declarative knowledge (knowing that), procedural knowledge (knowing how) and conditional knowledge (knowing when and where) [39, 55]. Further, from a within-individual perspective, disciplines (and domains) have both tacit and explicit dimensions, as well as formal and informal components [4]. The formal component refers to that knowledge which is specifically taught, while the informal pertains to that knowledge which is acquired from everyday situations or experiences. Despite these unifying characteristics, disciplines are not all alike [2, 23, 52, 64]. Some can be characterized as rather principle-driven, procedurally-rich, or well-structured (e.g.,
- 77. 63 physics). Others can more accurately be described as composite (e.g., social studies) or ill- structured (e.g., reading) in nature [19, 46]. Such differences are nontrivial and should be considered important both to practice and research in mathematics. Relatively speaking, mathematics is a fairly well-structured discipline that is procedurally-rich and which can be represented by many well-defined tasks or problems [29,71]. These general distinctions, however, should not obscure the structural variability that exists between the discipline of mathematics and other disciplines or within mathematical domains. Geometry, for example, appears to be a domain that is highly visual and spatial in nature, syllogistic in reasoning, and centered around postulates, theorems, and proofs. Algebra, by comparison, has a much more sequential and computational character, and is oriented toward determining functional relationships. Not only are there valid distinctions to be made between domains, but also within them. That is, while mathematics domains may fall toward one end of the structuredness continuum or another, there remains variability within each. Let me again use geometry to illustrate how a mathematics domain can be at once both well-structured and ill-structured. As a well- structured domain, geometry may be part of one of two broad families: Euclidean and non- Euclidean. In both families, we can systematically discuss the relationships between dimensions, coordinates, angles, and so on, by postulates, theorems, and proofs. In Euclidean geometry, the sum of the angles of every triangle equals 180 degrees. In a non-Euclidean geometry, such as that of Lobachevsky, the sum of the angles of every triangle is not equal to 180 degrees. Geometry emerges as a more ill-structured domain when we debate which of the two families is better applied to investigate variables that describe our world. If we consider space and time on a scale in which straight rather than curved dimensions are appropriate, then we will adopt a Euclidean perspective. Yet, if we consider scales in which space and time are best represented as curved or nonlinear dimensions, then we will adopt a non-Euclidean perspective. In truth, any mathematics domain can be approached in a more or less well- structured manner. Introducing uncertainty within any given problem, leaves more room for diversity, novelty, and creativity in solution, thus making the problem more ill-structured. The reason for discussing the nature of mathematics here is that, regrettably, there are numerous indicators that teachers and researchers tend to be short-sighted in their perceptions of mathematics, tending to see only the well-structured dimensions. Likewise, as I will discuss later, there is evidence that the structure of the discipline itself has not always been weighed in generalizing the results of certain research programs. If we were to deduce the nature of mathematics from the instructional practices that are evidenced in American classrooms, we would surmise that mathematics knowledge is fixed, static, and complete. In other words, we would lose sight of the fact that mathematics is, in truth, developing, dynamic, and incomplete. Based on school practices, we would also infer that mathematics is not just relatively but entirely a well-structured discipline composed completely of well-defined problems [52]. As Resnick [52, p. 32] states:
- 78. 64 Educators typically treat mathematics as a field with no open questions and no arguments, at least none that young children or those not particularly talented in mathematics can appreciate. We would further conclude that mathematical understanding is achieved by the acquisition of basic mathematics facts and algorithmic procedures and is demonstrated by one's ability to identify the correct formula and execute it in an automatized and effortless manner. In other words, mathematics as practiced in the classroom is generally arithmetic; arithmetic that stresses lower-order cognitive processing [17]. According to Peterson [44],47% of instructional time is devoted to unguided paper-and-pencil seatwork and 43% to whole class instruction. She further reports that for the vast majority of this instructional time (85%), students are engaged in lower-level cognitive processing or not cognitively engaged at all. While this pattern of unguided practice and low-level cognitive processing occurs in other areas (e.g., reading/language arts [18]), there seems even less active involvement or verbal engagement during mathematics instruction than in these other curricular areas [52]. Unless teachers possess an understanding and appreciation of the complexity and diversity of mathematics, unless teachers learn to see the ill-structured aspects of even the most well-structured of mathematics domains, and unless they identify the opportunities for higher-order mental processing, there remains little hope that they will engender these same understandings and perceptions within their students. How does the nature of mathematics influence research activity? Judging from the research in expertise and misconceptions, it would seem that with few exceptions [e.g., 37, 60] the choice of domain or the selection of domain-related tasks is of little or no significance to reported outcomes. Quite to the contrary, I would argue that the dissimilarity among domains or domain-specific tasks should be carefully evaluated in discussions of expertise and misconceptions [2]. Before we accept the "strong domain knowledge - weak general strategy" arguments espoused in the expert-novice research, for instance, we should remember that much of this research not only involved more well-defined domains (e.g., physics, computer science) but also well-structured problems (e.g., solving quadratic equations). As the research in expertise has reached into less structured fields or tasks [e.g., 40, 73], it has been more difficult to distinguish novices from experts and it has been harder to overlook the role played by general cognitive and metacognitive strategies [28]. The research on misconceptions exhibits some of these same problems. Is it coincidence that the research on misconceptions has occurred primarily in mathematics and science? Are the patterns in students' mis-understandings in other disciplines as evident, as resilient, or as detrimental? For one, the selection of well-structured domains may have been influenced by the availability of tasks for which outcomes are verifiable or refutable. If there is no agreed-upon "correct" concept, then how can there be a mis-conception to investigate? In educational psychology, for example, one can argue that cognitivism is a more appropriate theory than behaviorism to explain human learning. However, while debatable, neither theoretical
- 79. 65 perspective is truly refutable. The use of well-defined tasks in mathematics and science makes this verification much easier and pennits one to trace the source of the error to some underlying misconception. Nonetheless, I concur with Spiro et al. [64] that the differences in domains probably give rise to differences in domain-related errors. To achieve a greater understanding of misconceptions, therefore, we must study conceptual errors within domains and with tasks that are ill-structured as well as well-structured. Further, we must seek to uncover whether the cause ofcertain misconceptions relates more to the lack ofcontent-specific knowledge, the lack ofproblem-solving abilities, or some combination thereof. In summary, understanding of the discipline of mathematics is central to the health of mathematics research and education. Research agenda and interpretations of findings, as well as decisions about what should be taught and how it should be taught in schools, should be consistent with our understanding of the discipline. In this way, we may avoid the overgeneralizing or trivializing ofthe effects such inherent differences may have on mathematics learning and instruction. The Nature of Schooling and Knowledge Acquisition There is no doubt that schooling helps to shape students' conceptions of mathematics. Although I do not support Lave, Smith, and Butler's [33] contention that classroom practices, more than conceptual limitations or strategic deficits, are the primary source of problems in learning, there is little question that such practices significantly contribute to the lack of mathematics advancement. My own observations of classroom behavior and that of others (e.g., [50]) have shown that teachers are frequently content if their students can complete stereotypical problems using the routinized arithmetic procedures they were taught. That their students understand the concepts upon which these procedures are based or recognize their purpose or value seems of little consequence. That students may perceive the beauty of mathematics is beyond consideration. What is it about the nature of schooling and the culture of classrooms that inhibits mathematics learning? The practice of mathematics in American schools has certain salient attributes that may provide some clues. First, school mathematics consists of a series of classes or courses isolated from other academic disciplines. That is, teachers and students understand "mathematics" to be content-specific knowledge that can be taught and learned without regard to history, reading, or any other school subjects. Second, not only is mathematics separated from other academic disciplines but the various fields of study within mathematics are similarly taught in this disassociated manner. Thus, the high school student comes to believe that algebra, geometry, and trigonometry are discrete areas of study rather than mathematical domains related on the basis of shared principles. Finally, students are given little opportunity to discuss mathematical concepts or externalize their thinking during problem solving [31, 32],
- 80. 66 nor are students encouraged to construct meaning within the social context of the classroom [10, 15,25]. The consequence of these instructional rituals are that students equate "doing" mathematics (i.e., arithmetic computation) with "knowing" mathematics [17, 26]. Further, because the "doing" of mathematics primarily entails the completion of routine, well-defined, teacher- prescribed problems, students do not readily see the correspondence between mathematics and complex problem solving. Since "school" problems do not frequently demand it and since teachers do not seem to stimulate or reward it, higher-level, strategic processes may fail to become fully developed. Yet, competence in mathematics, to say nothing of expertise, requires not only declarative "facts," and domain-specific procedures but also general cognitive and metacognitive strategies [58]. Without the ability and the motivation to apply general cognitive or metacognitive strategies, every problem that is the least bit novel or complex would become an insurmountable obstacle. Despite this realization, the focus on domain-specific facts and procedures at the expense of strategic processing remains characteristic of classroom practices. As I [3] and others [43, 49] have argued, it is of little value to persist in the theoretical or practical separation of domain knowledge and strategic processing. The very fact that a conference on mathematics and problem solving is needed, however, suggests that the integration of these two essential knowledges in theory, as well as in practice, has yet to be achieved. The concerns about the interaction of domain knowledge and strategic processing are not as simple as they may first appear, however. More than just realizing that both are essential to mathematics performance, we must come to understand how the development of one form of knowledge influences the other. That is, how does the acquisition of domain knowledge change the amount and type of strategic processing? Are those who know more in less need of general strategies, for instance? Likewise, how do general cognitive and metacognitive strategies contribute to the acquisition and transfer of domain knowledge? Are general strategies, such as analogical reasoning, basic to knowledge transfer and knowledge restructuring, as has been suggested? [e.g., 3,72]. We also need to investigate how the nature of the tasks (e.g., well-defined or ill-defined) impacts strategic processing. For instance, are general strategies more critical with tasks that are ill-defined or novel than they are for tasks that are well-defined? Such questions about the interaction of domain knowledge and strategic processing can only be addressed through extended research programs. Beyond compartmentalizing mathematics by discipline and by domain, and by fostering the separation of domain knowledge from strategic processing, school mathematics is a weak representation of the discipline from which it is drawn. Descartes, the 17th century philosopher, held that mathematics was basic to all knowledge, since it permitted humankind to view their world in logical and reasoned ways. Schools would also hold mathematics to be basic, although in school venacular "basic" has unfortunately come to mean "minimal" or "simplistic." Schools have come to equate mathematics with the procedures of addition, subtraction, multiplication, and division. What they fail to see is that proficiency in executing
- 81. 67 these "basic" procedures is only a rudimentary stage toward thinking mathematically, just as the ability to decode and to recognize words are basic to reading but do not, themselves, constitute reading. Even the problems that students practice in school are poor approximations of the complex and ill-structured problems (e.g., predicting earthquakes, forecasting the economy) that are tackled by contemporary mathematicians [65]. As a result of these and other factors, school mathematics remains a poor sampling of the discipline that it is intended to represent [30, 50]. Mathematics, ofcourse, is not unique in this regard. Most of us would be hard-pressed to name any class or course of study that currently represents a "good" sampling of the discipline from which it arises. The abstraction of mathematics from realistic contexts is another hallmark of schooling. "School" problems, for instance, bear little resemblance to the real-world dilemmas that students encounter; that is, the problems that are the mainstay of students' drill-and-practice have little in common with students' out-of-school knowledge and experiences [25, 59]. Lave et al. [33] argue that this deliberate abstraction and decontextualization of mathematics came about because of the misconception that such abstraction facilitated knowledge acquisition and transfer. To this theoretical explanation, I would add a pragmatic one. I would contend that the abstraction of mathematics also occurs because few teachers are capable of or willing to deal with the inherent complexities ofmathematics. Data would suggest, for instance, that there are individuals teaching mathematics who do not have the certification nor the competencies to do so [35]. In addition, teachers of mathematics are handed instructional materials (Le., curriculum guides, texts) that are highly abstracted in that they are only skeletal, simplistic representations of the discipline. Through this abstraction, mathematics is converted to a hierarchy of basics that can be symbolically represented, algorithmically manipulated, and easily verified [30]. Unfortunately, teachers limited understanding of the discipline does not permit teachers to elaborate or extend those instructional materials adequately; in essence, they are unable to put any meat on the skeleton they are given. As a result, instruction remains abstracted and students acquire only a piecemeal knowledge of mathematics. To paraphrase a line from Shulman [62], teachers teach what they know--and students learn what teachers teach. I do not wish to suggest that teachers deliberately set out to "dummy-down" the mathematics curriculum or to inhibit students' learning. To the contrary, these teachers are performing their perceived role in the culture of the classroom. Certain social and cultural theorists might well suggest that the culture of the classroom is merely a reproduction, if not reflection, of the culture of society which is intended to perpetuate a class system [24, 34]. Even from a more conservative perspective, it would seem that teachers are modeling the mathematical pedagogy, as well as communicating the content-specific knowledge, that they have internalized as a consequence of their educational experiences. What that culture of the classroom has taught them is that teacher and textbooks are the authorities and that the task for the student is to passively absorb what is communicated to them verbally or in print [31]. As Lampert (p. 32) writes:
- 82. 68 Even when teachers give an explanation rather than simply stating a rule to be followed, they do not invite students to examine the mathematical assumptions behind the explanation, and it is unlikely that they do so themselves... Even though this practice conflicts with cognitive theory that tells us that one learns best when actively involved [e.g., 74, 76, 77], this same passivity can be witnessed in other curricular areas. Instructional approaches such as reciprocal teaching [38], cooperative learning [16], and cognitive apprenticeships [15] represent attempts to alter the culture of the classroom. There is yet another explanation for why the simplication or abstraction of mathematics occurs in school. There seems to exist the misperception that mathematical literacy can only occur with the onset offonnal instruction, and that prior to formal instruction children have little conceptual understanding about mathematics. As has occurred with language literacy, mathematics educators must come to realize that many valuable lessons are learned outside of the classroom, and that formal instruction can be enhanced by activating students' informal knowledge base. There also appears to be a serious underestimation of the mathematical capabilities of young children [22, 53]. The research of my colleagues and I [e.g., 7, 8, 75] has reinforced that point. Specifically, in our research on analogical reasoning, the majority of preschoolers we tested evidenced reasoning abilities assumed to be beyond their developmental capabilities [e.g., 47] when they were presented with a motivating task performed within an appropriate and motivating setting. Researchers in mathematics have reported similar findings [11, 12]. Consequently, the more we build upon students' preexisting conceptual and strategic knowledge in classroom instruction, the more likely we are to foster subsequent advancement. Measurement Models and Assessment Practices As I see it, there are two problems facing mathematics and mathematics education with regard to assessment. The first is the mismatch between current theories of learning and instruction and models of assessment [14, 50]. The second pertains to the way that assessment is practiced in schools [30]. The more I investigate human intellectual processing, the more I come to appreciate its complexity. My recent efforts to test the interaction of domain and strategy knowledge [e.g., 5], to propose a framework for relating various forms of knowledge [6], and to articulate a theory of creativity [4] have made me acutely aware of the dynamic, interactive, and multidimensional nature of processing that occurs within the individual. Just the ability to read and comprehend a passage in a book or solve a word problem requires an intricate blend of domain-specific knowledge, and strategic processing. I have also come to recognize that all individuals, as social beings, must operate within a sociocultural milieu that significantly influences their actions. Further, as psychological beings, humans dohot frequently operate in ways that are coldly rational but are greatly affected by their beliefs, goals, interests, and by their self-perceptions.
- 83. 69 When you map human complexity onto current models of assessment, you begin to realize that what we are presently able to test is a pale reflection of what we need to know about human performance. It is likely that we will never be able to match our understanding of human processing with measures that are equally dynamic or complex. To do so may not even be desirable. However, I would argue that the simplistic notions of what constitutes mathematics, as operationally defined by the tests that we currently use, is far from satisfactory. Certainly there is much more to be known about mathematics than that which appears on standardized achievement tests [e.g., 54]. What I suggest is that the mismatch between theories of learning and instruction and models of measurement can be lessened in three ways. Specifically, I recommend that we: (a) take new looks at traditional mathematical tasks; (b) develop novel mathematical tasks or tests; and, (c) devise new, cognitive-oriented measurement models. As far back as 1978, in their research on "debugging," Brown and Burton alerted us to the fact that even the most commonplace of mathematical procedures, like addition, could be reexamined from a cognitive perspective. By means of such reexamination, researchers can become aware of the underlying procedural deficits that result in performance errors. Brown and Burton, for instance, determined that the errors students made when performing basic mathematics procedures (e.g., addition/subtraction) were systematic and traceable to particular procedural misconceptions. VanLehn's [71] research on repair theory and Sleeman's [63] study of students' misunderstandings in basic algebra are additional examples of researchers who have undertaken detailed cognitive analyses of traditional mathematical tasks. Still other researchers have seen fit to devise mathematical problem-solving tasks that vary greatly from the traditional paper-and-pencil ones we discussed earlier in this paper. Bransford and the Cognition and Technology Group at Vanderbilt [13], for example, have created innovative computer-based problem-solving tasks they refer to as "anchored instruction." The objective of these videodisc activities is to immerse students in a stimulating environment that relates to their out-of-school experiences, permits sustained exploration, and encourages problem-solving from multiple perspectives. Our work on the interaction of domain-specific and strategy knowledge led our research group at Texas A&M to develop a series of domain knowledge, strategy knowledge, and interactive knowledge measures that permitted evaluation of subjects' academic performance both unidimensionally and multidimensionally [5]. Research on realistic or authentic assessment in science and in reading are additional examples of the efforts to devise innovative tasks that permit richer cognitive analysis of student leaming. What is clear from these various efforts is that the concern for higher-order problem solving in the research community and in educational reforms [e.g., 35] is not yet matched by a large array of assessment tasks that measure problem solving in valid and reliable ways. Perhaps the lack of a clear or consistent definition of what constitutes "problem-solving," "higher-order thinking," or "strategic processing" is one barrier to the development of cognitive-based mathematical assessments [6]. Considering the diversity of interpretations that exist in the literature [e.g., 3], arriving at such a consistent definition will certainly be no simple matter.
- 84. 70 Still another way to infuse new life into assessment is with the development of measurement models that deviate from more traditional psychometric approaches. Item- response theory or IRT, for example, differs from classical true score theory in that IRT is not sample dependent and considers the entire response pattern of individuals instead of a composite score. This means that IRT is more robust than classical true score theory with regard to estimates of item difficulty and discrimination. Although IRT has not been widely applied to mathematics, it remains a promising measurement model. Building on IRT, Tatsuoka [68, 69] has devised rule-space analysis, a procedure in which various erroneous rule patterns are mapped into multidimensional space. Tatsuoka has used the rule-space computer program to identify "bugs" in students' solution of signed-number addition and subtraction problems. Multidimensional scaling (MDS) is yet another measurement approach that could provide a more in-depth analysis of mathematical performance [61]. Pellegrino, Mumaw, and Shutes [42] have used MDS to compare the spatial abilities of experts and novices. In my own research, MDS has been useful in determining whether certain types of errors were more associated with deficits in domain-specific knowledge or strategic processing [1]. Despite these three promising avenues for lessening the mismatch between theories of learning and instruction and measurement models, the activities being undertaken in the research arena have had limited effect on school assessment practices. Several factors contribute to this condition. First, the cognitive techniques I have been describing are still very much in the developmental stages [23, 77]. Second, the procedures, tasks, and tests described require a high level ofexpertise that is not typically available among school personnel. Third, even those problem-solving tasks that seem more appropriate to classroom use, like Bransford's anchored instruction, are not ready for widespread application. Fourth, many of the measurement models and error detection tasks require sophisticated technology to implement. Given the current state of schooling, then, what can be done to improve assessment practices in mathematics? To this question, I offer several responses. First, educators need to look beyond the correct score to the thinking and knowledge that is evident within student responses, whether that response is correct or incorrect. As the research of Brown and Burton, VanLehn, and my own research [I, 7, 29] has repeatedly shown, students' errors are nonrandom, often predictable, and frequently informative. In the same regard, the research my colleagues and I have conducted has made it apparent to us that errors are not all equivalent. In that research, we have found that error patterns appear to represent either higher or lower levels of domain or strategy knowledge. Thus, these error patterns become clues to the organization and accessibility ofstudents' domain-specific knowledge and of their problem solving abilities. If students' error patterns can be identified, then more effective intervention programs can be planned. Yet another way that school assessment practices can be improved is by better preparing teachers to deal with test construction and test interpretation [27,66]. The tests that teachers devise will continue to playa critical role in mathematics education. If nothing else, these tests signal to students what teachers value in the way of knowledge. If teachers talk about the
- 85. 71 importance of problem solving but continue to measure only lower-level declarative or procedural knowledge, students will focus on the "to-he-evaluated" content [56]. It would seem, therefore, that teacher training programs should be organized to provide preservice teachers with the knowledge and skills they need to construct tests that are not only valid but also informative, related to out-of-school experiences, and oriented toward problem solving. Earlier I mentioned the value of looking at student errors. Based on the research, it should also be possible to teach teachers ways to construct tests that permit them to perform simple error analyses. By constructing tests that would identify procedural "bugs," teachers should be better equipped to diagnose student misconceptions and to address those in instruction. Training programs should also ensure that teachers are knowledgeable about various forms of assessment (e.g., norm-referenced, criterion-referenced), so that they are able to interpret test results accurately and communicate those results effectively [66]. More importantly, an effective training program should work to ensure that teachers are capable of making appropriate use of test data in planning and implementing instruction. Perhaps if they felt more secure about devising their own tests and were more knowledgeable about the limitations of current standardized measures, then teachers would assume greater responsiblity for student assessment and feel less pressured to rely on standardized test scores [27]. Potential Solutions Throughout this paper, I have been describing global issues that I believe to be systemic to education and which seem to be important to the future of mathematics research and practice. I would be remiss, however, if my discussion did not conclude with some consideration of ways that these issues can he positively addressed. If the issues are, as I have stated, systemic to educational research and instructional practice, then the potential solutions to those problems can also be described as systemic. These potential solutions, then, speak not only to mathematics but also to other disciplines. Further, they speak to the SOCiological and psychological dimensions of learning and instruction, as well as the cognitive. For the purposes of this paper, I will only briefly describe these potential solutions. More in-depth consideration must be reserved for some other time and place. Proposal I: Explore Learning and Instruction as Sociocultural Events As the research on situated cognition [e.g., 25, 59] suggests, learning cannot be separated from the sociocultural context in which it occurs. As a microcosm of the society in which it operates, classrooms have their own languages, value systems, and cultures that impact learning. If we are to foster the acquisition and utilization of intricate mathematic concepts and reasoning
- 86. 72 abilities, then we must make mathematics thinking and problem solving a natural aspect of the classroom culture. We must encourage students to engage in discourse about mathematics. As Lampert [31, 32] suggests, discourse in the classroom as it relates to mathematics should appear like the discourse of argument and conjecture, with teachers and students working together for the purpose of "sense-making." In this situation, the engagement in problem solving is more important than quick resolution. Students come to see mathematics as a process more than as a set offacts and fonnulae to memorize [57]. Social and cultural theorists might argue that such a drastic reconceptualization of mathematics education will require a genuine demand within society for change; a demand that has yet to be made [48]. Proposal II: Conceive of Instruction as a Partnership Requiring the Active Participation of Students and Teachers Alike As the writings ofDewey, Bruner, and, more recently, Wittrock, Resnick, and Bransford have shown us, students who are actively engaged learn better than those who are not. If we persist in endowing teachers with full responsibility for "telling" students what they need to know, while students passively sit by absorbing the content, then mathematics learning will remain limited. Several techniques have been suggested for making students more active and willing participants in learning. As noted earlier, Collins et aJ.'s [15] cognitive apprenticeship is one instructional model, as is Palincsar and Brown's [38] reciprocal teaching approach. These instructional approaches provide for teacher modeling, thoughtful guidance, and a gradual transfer of instructional responsibility to the student. Along with these approaches, I would also recommend that there be some explicit teaching of general problem solving strategies with the choice of instructional approach being made on the basis of the task and the competency of the students. Proposal III: Build on the Interests and Experiences of the Students It is hard to overestimate the impact that interest and motivation play in students' learning [20, 21]. The success of Bransford's videodisc-based "anchored instruction" is largely attributable to the fact that programs like the Sherlock Holmes series are visually inviting productions. As a result, students are eager to participate and able to maintain attention. Yet, so much of what we ask students to do in classrooms is of limited appeal to them and is removed from their out-of- school experiences. What we need to do is to seek ways to lessen this mismatch [51]. This can be accomplished by engaging students in problem solving that is contextualized, concrete, and realistic, and which speaks to issues and topics that interest them.
- 87. 73 Proposal IV: Show Students the Relatedness of Knowledge Just as we want to build connections between students lives in and out of school, we want to help them learn to see the relatedness of knowledge. One of the concepts that was evident in the literatures on the integration of domain and strategy knowledge [3] and on creative problem solving [4] was that expertise involves the ability to see relationships across tasks, domains, and disciplines. For the truly expert or the truly creative, there are no artificial boundaries of knowledge. What these individuals possess is the flexibility of thought that permits them to draw connections between concepts that are not apparent. In schools, therefore, we must make efforts to show students how what they learn in mathematics has relevance to what they learn in history, music, reading, or science. Likewise, we need to help them see how the various domains of mathematics (e.g., algebra, calculus, trigonometry) are related. If integrated structures of knowledge are the hallmarks of expertise and creativity, then we should work toward these ends in our classrooms. Concluding Remarks Despite the somewhat somber tone of this paper, I believe that there is great hope for the future of education. Now, more than at any other time in our history, we have a deep understanding of the nature of learning and instruction, we are sensitive to the problems that exist, and we are willing, even anxious, to bring about change. By integrating this burgeoning body of knowledge with the content of the various disciplines, we can not only continue to learn more, but assure that future generations will have the ability and the desire to face challenges that are currently beyond imagination. References 1. Alexander, P. A.: Categorizing learner responses on domain-specific analogy tests: A case for error analysis. Paper presented at the annual meeting of the American Educational Research Association, Boston, MA April 1990 2. Alexander, P. A.: Domain knowledge: Evolving themes and emerging concerns. Educational Psychologist in press 3. Alexander, P. A., & Judy, J. E.: The interaction of domain-specific and strategic knowledge in academic performance. Review of Educational Research, 58, 375-404 (1988) 4. Alexander, P. A., Parsons, J. L., & Nash, W. R.:. Toward a theory of creativity. Manuscript submitted for publication 1991. 5. Alexander, P. A., Pate, P. E., Kulikowich, J. M., Farrell, D. M., & Wright, N. L.: Domain-specific and strategic knowledge: Effects of training on students of differing ages or competence levels. Learning and Individual Differences, 1,283-325 (1989) 6. Alexander, P. A., Schallert, V. L., & Hare, V. C.: Coming to terms: How researchers in literacy and learning talk about knowledge. Review of Educational Research in press.
- 88. 74 7. Alexander, P. A., Willson, V. L., White, C. S., & Fuqua, J. D.: Analogical reasoning in young children. Journa1 of Educational Psychology, 79, 401-408 (1987) 8. Alexander, P. A., Willson, V., White, C. S., Fuqua, J. D., Clark, G. D., Wilson, A. F., & Kulikowich, J.: Development of analogical reasoning in four- and five-year-olds. Cognitive Development, 4, 65-88 (1989) 9. Brown, J. S. & Burton, R. R.: Diagnostic models for procedural bugs in basic mathematical skills. Cognitive Science, 2,155-192 (1978) 10. Brown, J. S., Collins, A., & Duguid, P.: Situated cognition and the culture of learning. Educational Researcher, 18,32-42 (1989) 11. Carpenter, T. P. & Moser, J. M.: The acquisition of addition and subtraction concepts. In: Acquisition of mathematics concepts and processes (R. Lesh & M. Landau, eds.), pp. 7-44. Orlando, FL: Academic Press 1983 12. Carraher, T. N., Carraher, D. W., & Schliemann, A. D.: Mathematics in the streets and in schools. British Journa1 of Developmental Psychology, 3, 21-29 (1983) 13. Cognitive and Technology Group at Vanderbilt: Anchored instruction and its relationship to situated cognition. Educational Researcher, 19,2-10 (1990) 14. Cole, N. S.: Conceptions of educational achievement. Educational Researcher, 19,2-7 (1990) 15. Collins, A., Brown, J. S., & Newman, S. E.: Cognitive apprenticeship: Teaching the craft of reading, writing, and mathematics. In: Knowing, learning, and instruction: Essays in honor of Robert Glaser (L. B. Resnick, ed.),pp. 453-494). Hillsdale, NJ: Erlbaurn 1989 16. Dansereau, D. F.: Cooperative learning strategies. In: Learning and study strategies: Issues in assessment, instruction, and evaluation (C. E. Weinstein, E. T. Goetz, & P. A. Alexander, eds.), pp. 103-120. San Diego: Academic Press 1988 17. Davis, R. B.: Conceptual and procedural knowledge in mathematics: A summary analysis. In: Conceptual and procedural knowledge: The case of mathematics (J. Hiebert, ed.), pp. 265-300, 1986 18. Durkin, D.: What classroom observations reveal about reading comprehension instruction. Reading Research Quarterly, 14,481-533 (1978-79) 19. Frederiksen, N.: Implications of cognitive theory for instruction in problem solving. Review of Educational Research, 54,363-407 (1984) 20. Gamer, R., & Alexander, P. A.: Skill, will, and thrill: Factors in adults' text comprehension. Paper presented at the annual meeting of the American Educational Research Association, Chicago, April 1991 21. Gamer, R., Alexander, P. A., Gillingham, M. G., Kulikowich, J. M., & Brown, R.: Interest and learning from text. American Educational Research Journal in press 22. Gelman, S. A. & Markman, E. M.: Categories and induction in young children. Cognition, 23, 183-209 (1986) 23. Glaser, R., Lesgold, A., & Lajoie, S.: Toward a cognitive theory for the measurement of achievement. In: The influence of cognitive psychology on testing (R. R. Romig, 1. A. Glover, J. C. Conoley, & J. W. Whitt, eds.), pp. 41-85. Hillsdale, NJ: Erlbaum 1987 24. Goetz, J. P., & Grant, L.: Conceptual approaches to studying gender in education. Anthropology Quarterly, 19, 182-196 (1988) 25. Greeno, J. G.: A perspective on thinking. American Psychologist, 44, 134-141 (1989) 26. Greer, B., & Mulhern, G.: New directions in mathematics education. London: Routledge 1989 27. Hills, J. R.: Apathy concerning grading and testing. Phi Delta Kappan, 72, 540-545 (1991) 28. Hofstadter, D. R.: Metamagical themas: Questions for the essence of mind and pattern. New York: Basic Books 1983 29. Kulikowich, J. M., & Alexander, P. A.: The effects of gender, ability, and grade on analogy performance. Contemporary Educational Psychology, 15,364-377 (1990) 30. Kulm, G.: Assessing higher order mathematical thinking: What we need to know and be able to do. In: Assessing higher order thinking in mathematics (G. Kulm, ed.), pp. 1-4. Washington, DC: American Association for the Advancement of Science 1990
- 89. 75 31. Lampert. M. How do teachers manage to teach? Harvard Educational Review. 55.178-194 (1985) 32. Lampert. M.: When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Association, 27. 29-63 (1990) 33. Lave. J.• Smith. S.• & Butler. M.: Problem solving as everyday practice. In: The teaching and assessing of mathematical problem solving (R. I. Charles & E. Silver. eds.). pp. 61-81. Reston. VA: The National Council ofTeachers of Mathematics 1988 34. McDade. L A.: Knowing the "right stuff": Attrition. gender. and scientific literacy. Anthropology Quarterly. 19.93-114 (1988) 35. National Council of Teachers of Mathematics: Curriculum and evaluation standards for school mathematics. Reston. VA: Author 1989 36. National Research Council: Everybody counts: A report to the nation on the future of mathematics education. Washington. DC: National Academy Press 1989 37. Owen. E.• & Sweller. 1.: Whata do students learn while solving mathematics problems. Iournal of Educational Psychology. 77. 272-284 (1985) 38. Palincsar, A. S., & Brown, A. L.: Reciprocal teaching of comprehension-fostering and comprehension- monitoring activities. Cognition and Instruction, I, 117-175 (1984) 39. Paris. S. G.• Lipson. M. Y.• & Wixson. K. K.: Becoming a strategic reader. Contemporary Educational Psychology, 8.293-316 (1983) 40. Pate. P. E., Alexander, P. A.• & Kulikowich. 1. M.: Assessing the effects of training social studies content and analogical reasoning processes on sixth graders' domain-specific and strategic knowledge. In: Middle school research: Selected studies (D. B. Strahan, ed.), pp. 19-29. Columbus,OH: Research Committee of the National Middle School Association 1989 41. Paulos. 1. A.: Innumeracy: Mathematical illiteracy and its consequences. New York: Hill and Wang 1988 42. Pellegrino, 1. W.• Mumaw, R. J.• & Shutes, N. 1.: Analyses of spatial aptitude and expertise. In: Test design: Developments in psychology and psychometrics (S. E. Embretson, ed.), pp. 45-76. Orlando. FL: Academic Press 1985 43. Perkins. D. N., & Salomon, G.: Are cognitive skills context-bound? Educational Researcher, 18, 16-25 (1989) 44. Peterson. P. L.: Teaching for higher-order thinking in mathematics: The challenge for the next decade. In: Perspectives on research on effective mathematics teaching (D. A. Grouws, T. J. Cooney. & D. Jones. eds.). pp. 2-26. Reston. VA: The National Council of Teachers of Mathematics 1988 45. Phenix, P. P.: The use of disciplines as curriculum content. In: The sugject of curriculum (F. L. Steeves. ed.), pp. 1-16. New York: Odyssey Press 1968 46. Phillips. D. C.: Philosophy. science. and social inquiry. Oxford: Pergamon Press 1987 47. Piaget, J.: Piaget's theory. In: Handbook of child psychology (4th ed.). (P. Mussen. ed.), Vol I, pp. 103- 128. New York: Wiley 1983 48. Pitman, M. A., & Eisenhart. M. A.: Experiences of gender: Studies of women and gender in schools and society. Anthropology & Education Quarterly. 19.67-69 (1988) 49. Prawat. R. S.: Promoting access to knowledge, strategy, and disposition in students: A research synthesis. Review of Educational Research, 59,1-41 (1989) 50. Putnam. R. T.• Lampert. M., & Peterson. P. L.: Alternative perspectives on knowing mathematics in elementary schools. In: Review of Research in Education (C. B. Cazden. ed.). Vol. 16, pp. 57-150. Washington. DC: American Educational Research Association 1990 51. Resnick. L. B.: Learning in school and out Educational Researcher. 16.13-20 (1987) 52. Resnick. L. B.: Treating mathematics as an ill-structured discipline. In: The teaching and assessing of mathematical problem solving (R. I. Charles & E. A. Silver. eds.), pp. 32-60. Reston. VA: The National Council of Teachers of Mathematics 1988 53. Resnick, L. B.: Developing mathematic knowledge. American Psychologist. 44.162-169 (1989)
- 90. 76 54. Romberg. T. A.• & Carpenter. T. A.: Research on teaching and learning mathematics: Two disciplines of scientific inquiry. In: Handbook of research on teaching (3rd ed.). (M. C. Wittrock. ed.). pp. 850-873. New York: Macmillan 1986 55. Ryle. G.: The concept of the mind. London: Hutchinson 1949 56. Schallert. D. L.• Alexander. P. A., & Goetz, E. T.: Implicit instruction of strategies for learning from text. In: Learning and study strategies: Issues in assessment. instruction. and evaluation (C. E. Weinstein, E. T. Goetz. & P. A. Alexander. eds.). pp. 193-214. San Diego: Academic Press 1988 57. Schoenfeld. A. H.: Mathematical problem solving. New York: Academic Press 1985 58. Schoenfeld. A. H.: What's all the fuss about metacognition? In: Cognitive science and mathematics education (A. H. Schoenfeld, ed.). pp. 189-215. Hillsdale, NJ: Erlbaum 1987 59. Schoenfeld. A. H.: Problem solving in context. In: The teaching and assessing of mathematical problem solving (R. I. Charles & E. A. Silver, eds.). pp. 82-92. Reston. VA: The National Council of Teachers of Mathematics 1988. 60. Schoenfeld. A.• & Herrmann, D. 1.: Problem perception and knowledge structure in expert and novice mathematical problem solvers. Journal of Experimental Psychology: Learning, Memory. and Cognition. 8, 484-494 (1982) 61. Shoben, E. J., & Ross. B. H.: Structure and process in cognitive psychology using multidimensional scaling and related techniques. In: The influence of cognitive psychology on testing (R. R. Ronning. J. A. Glover. & J. C. Witt, eds.). pp. 229-266. Hillsdale. NJ: Erlbaum 1987 62. Shulman. L. S.: Those who understand: Knowledge growth in teaching. Educational Researcher, 15.4-14, (1986) 63. Sleeman. D.: An attempt to understand students' understanding of basic algebra. Cognitive Science. 8. 387- 412 (1984) 64. Spiro, R. J., Feltovich, P. J., Coulson, R. L., & Anderson, D. K.: Multiple analogies for complex concepts: Antidotes for analogy-induced misconceptions. In: Similarity and analogical reasoning S. Vosniadou & A. Ortony. eds.). pp. 498-531. Cambridge, England: Cambridge University Press 1989 65. Stewart, I. The problems of mathematics. New York: Oxford University Press 1987 66. Stiggins. R. J.: Assessment literacy. Phi Delta Kappan. 72, 534-539 (1991) 67. Schwab. J. J.: Structure of disciplines: Meanings and significances. In: The structure of knowledge and the curriculum (G. W. Ford & L. Pugno, eds.), pp. 6-30. Chicago: Rand McNally 1964 68. Tatsuoka. K. K.: Rule space: An approach for dealing with misconceptions based on item response theory. Journal of Educatiooal Measurement. 20, 345-354 (1983) 69. Tatsuoka. K. K., & Tatsuoka, M. M.: Spotting erroneous rules of operation by the individual consistency index. Journal of Educatiooal Measurement, 20, 221-230 (1983) 70. VanLehn. K.: On the representation of procedures in repair theory. In: The development of mathematical thinking (H. P. Ginsburg, ed.). New York: Academic Press 1983 71. VanLehn. K.: Mind bugs: The origins of procedural misconceptions. Cambridge, MA: MIT Press 1990 72. Vosniadou. S., & Brewer, W. F.: Theories of knowledge restructuring in development. Review of Educational Research, 57, 51-67 (1987) 73. Voss, J. F., Greene, T. R.• Post, T. A.• & Penner, B. C.: Problem solving skill in the social sciences. In: The psychology of learning and motivation: Advances in research theory (G. H. Bower, ed.), Vol. 17, pp. 205-232. New York: Academic Press 1983 74. Weinstein. C. E.: Training students to use elaboration learning strategies. Contemporary Educational Psychology. 7. 301-311 (1982) 75. White, C. S., & Alexander, P. A.: Effects of training on four-year-olds' ability to solve geometric analogy problems. Cognition and Instruction. 3.261-268 (1986) 76. Wittrock. M. C.: Learning as a generative process. Educational Psychologist, 11.87-95 (1974) 77. Wittrock. M. C.• & Baker, E. L.: Testing and cognition. Englewood Cliffs, NJ: Prentice Hall 1991
- 91. The Crucial Role of Semantic Fields in the Development of Problem Solving Skills in the School Environment PaoloBoero Dipartimento eli Marematica, Universita eli Genova, Via L. B. Alberti,4, 16132 Genova. Italy Abstract: I will try to frame from a theoretical standpoint, and clarify, the following points: Which aspects of problem solving are more context sensitive, on the short term (in relation to the single problem situation), and on the long term (in relation to the development of the problem solving skills)? Which elements of the context can have more significant effects on problem solving processes? Which are the most important differences between the "contextualized" problem solving processes and the "non-contextualized" problem solving processes? Which teaching actions associated with the chosen contexts can the teacher perform in order to enhance pupils' results in problem solving? Keywords: problem-solving skills, fields of experience, context sensitivity, semantic fields, meaning, representations, theoretical frame 1. Introduction Extensive research in the past 15 years has emphasized the importance of the "context" for problem solving processes. In particular, I would like to recall: - the research in ethnomathematics concerning the development of applied mathematical problem solving skills outside of the school (see [3, 12]); - the research on "context sensitivity" in problem solving. In several cases it concerns basic research of the influences that the "context", evoked in the problem, may exert on the choice ofproblem solving strategies. In other cases it concerns comparative research on the effects of curricular teaching choices which are aimed at pressing for the development, through the contextualization of problems, of problem solving skills [11]. In both cases the work is based on the assumption that the "context" can have an effect on the cognitive behavior during problem solving activities, in the first case, as the element which has a role (along with others) in determining a single performance and, in the second case, as the element which has an effect upon the development ofthe problem solving skills.
- 92. 78 In this regard I would also like to recall the research work carried out in psycholinguistics which (see [10)) reveals the sensitivity to the context in the acquisition of connectives, not only as the element with which to influence a single performance, but also as an important factor for the growth oflinguistic skills. More generally, referring to Brown's, Donaldson's, Gelman's, Nelson's researches, French [10] writes: "Despite the fact that these investigators study quite different domains within cognitive development, make different assumptions about the origin of particular cognitive abilities, and describe their results in different terms, they are in basic agreement about the importance of contextual factors in both the acquisition and display of cognitive abilities. Their essential premise is that cognitive competence initially arises within, is embedded within, and is practiced within particular contexts..... Last but not least, there is empirical evidence, to which little consideration is given in current research on problem solving, which concerns the generally good results achieved in problem solving with teaching projects based on the teaching "by problems" referred to "areas of interest" that may be identified by, and stimulating for the pupils (refer to various projects carried out in Great Britain [2], The Netherlands [8, 13,], USA [14], etc.). I think that the foregoing justifies the interest for research on the role of the "context" in problem solving, that goes beyond the simple observations available and the results of the research acquired thus far and provides elements to clarify the following points (the first three, mainly oftheoretical interest and the fourth ofparticular interest to teaching): - which aspects of problem solving are more context sensitive, on the short term (in relation to the single problem situation), and on the long term (in relation to the development of the problem solving skills)? - which elements of the context can have more significant effects on problem solving processes? - which are the most imponant differences between the "contextualized" problem solving processes and the "non- contextualized" problem solving processes? - which teaching actions associated with the chosen contexts can the teacher perform in order to enhance pupils' results in problem solving? To study the above problems further I believe that it is necessary: - to construct a theoretical reference frame that may be used to speak with sufficient accuracy of the context from the standpoint of its intrinsic characteristics and from pupil's and teacher's perspective (Refer to paragraph 2). - to organize extensive long term experimental activities (with many pupils and teachers involved) which have common characteristics, namely: the presence of"contexts" which require considerable teaching, gradually proposing problem situations whereby an increasing command
- 93. 79 of the subjects discussed may be acquired; the possibility to carefully observe what is performed in the classroom and to systematically collect the material produced by the pupils. In practice, in my case, this is accomplished through primary and comprehensive school projects developed by the group coordinated by myself since 1976. These projects: - concern many disciplines: all of the major primary school disciplines and mathematics and experimental sciences in the comprehensive school; - are experimented by about 250 teachers; 43 teachers actively collaborate with researchers to elaborate teaching proposals and analyze work produced in the classroom; - require a strong commitment to develop verbal competence and verbalization processes (even in problem solving). The choice is made in line with the assumption that the verbal language is a thought construction tool (and not only of communication). It also provides the possibility to be able to "monitor" the pupils' mental work (in particular, in several classes, it is possible to collect all the written material produced by a pupil over periods of 3 to 5 years); - require (for pedagogical and generally cultural choices, regardless of research on problem solving) that the construction of subject-related and problem solving competences takes place through activities regarding important questions for the knowledge of the natural and social reality. The "contextualisation" of most of the problems put forward in our projects is radical, in the sense that not only does one refer to contexts which are familiar to children but these contexts are effectively "performed" in class, and in many cases the results achieved with the solution to a problem are effectively used to carry on with the work. 2. Experience Fields and Semantic Fields In contextualized problem solving work the pupil can refer to hers or his experiences (intellectual or material) regarding the context (see [1D. On the other hand, in the solution to the problem the context may be present directly (with its "material" constraints) or by means of (external) representations. Finally, the pupil's work is usually subject to the effects of suggestions, the manner in which the problem is presented, etc. by the teachers who, in tum, are influenced by the conceptions that they have of the context and of the manners by which the pupils refer to it. There appears, hence, to be three components of the context; a component within the pupil ("internal component of the pupil"), an "external component" and a component within the teacher ("internal component ofthe teacher"). These three components are all subject to evolve with time, even if the evolution takes place with differing speeds, extension and characteristics. In particular, as far as the external component is concerned, a "history" related evolution (the modem day motor cars are not those of 50 years ago) and a much more rapid and closer to the
- 94. 80 pupil evolution, associated to problem situations presented by the teacher and to the objects and (external) representations immediately available to the pupil, overlap. As regards the internal context of the pupil, the evolution is continuous and intense and concerns: the ability of the pupil to perceive (becoming more complete and ordered with time) the external context and related, intrinsic constraints; the command of systems (increasingly more complex) and of relations that tie the specific variables ofeach context together; the invariants and schemes [15] steadily constructed to adapt to problem situations, inside and outside the school, related to a given context. As far as the teacher is concerned, his internal context evolves too in relation mainly to the experience which the teacher gradually acquires in the ways in which the pupils refer to the external context and, at times, even to their knowledge of the external context and relationships present in it. This dynamic insight into the context, in particular, takes account of the process of cultural growth and "maturation" of the pupils, provided under the guidance of the teacher in the school environment. We will refer to the "field ofexperience" as the set of three components ofthe context referred to a sector ofhuman experience (social, material, intellectual...) that may be identified by the teachers and pupils as one and sufficiently homogeneous from the standpoint of the "scripts" (actually or potentially) involved; for example, if we speak of "orientation" or of "industrial production" or of the "measure of time", we delimit "fields of experience" in which the pupils (at least, from a certain age on, different for each field of experience) can refer to their experiences, conceptions, etc. In this paper I will consider only "real world" fields of experience; but the same definition might be applied also to "purely mathematical" fields of experience (for instance: "numbers", or "geometric figures", or "algebra"). Obviously, "numbers" or "geometric figures" or "algebra" become veritable "fields of experience" only after a suitable experience (at the beginning of compulsory education, pupils' "internal context" is already rich in numerical experiences, but rather poor in algebraic experiences). For a purely mathematical field of experience, the external context contains only external representations (formulas, verbal definitions and statements, diagrams, geometric figures,...). We may observe that the constraints depending on the teacher and, more generally, classroom social interactions are not included in my definition of "field of experience"; this observation suggests possible integrations of the concept of "field of experience", as a concept referring to the construction of"meanings", in other didactical or psychological theories. We note that two or more different fields of experience can have parts in common. For instance, both in the "field of experience" of hand-craft productions and in that of industrial productions the problem of the formation of production costs is very important. Similarly, the "sun shadow" phenomenon is referred to when dealing with the aspects of orientation and the measure of time (with elementary methods that are transparent and easily grasped by 10-11 years old pupils).
- 95. 81 It is therefore possible to find limited environments (sometimes fields of experience. at times not easily recognized as such by the pupils) within the "fields ofexperience" that may not be broken down further and integrate many problem situations, rich of disciplinary implications. The problem of the "formation of production costs" and the "sun shadow" constitute two examples in this regard. We will call these basic, irreducible and meaningful environments "semantic fields" (and here again. as researchers. we may consider, with their evolution in time. the "external component", the "internal component" of the pupil and the "internal component" of the teacher). What are the purposes of these concepts? As for the purposes of the teaching planning, they may be used to determine sufficiently homogeneous "environments" in which to develop teaching activities aimed at the construction of concepts and specific abilities in various disciplines. In particular, the "fields ofexperience" provide the possibility to organize a flexible curriculum, in accordance with the subject matters identified by the pupils, with reference to their extra-mural experiences, whereas the "semantic field" may be used to identify the basic core of concept construction and problem solving processes, that may be included in different fields of experience according to the particular circumstance (interests of pupils, particular social-cultural situations, etc.); for each semantic field precise teaching itineraries may be developed and enhanced through experience. The concept of "conceptual field" discussed by Vergnaud [15] may be utilized as a further instrument with which to identify and organize the elements of the disciplinary curriculum into an ordered structure, the same elements which are to be pursued through "context-related" work in the fields of experience. In practice, in our work associated with our projects, we realized the importance of taking into consideration, within the teaching planning, both the standpoint of the pupils' cognitive involvement (referred to the fields of experience) and the standpoint of the disciplinary organization of knowledge (represented by the conceptual fields). Furthermore, the concepts of "field of experience" and of "semantic field" may, for research purposes, enable a reasonably accurate analysis of various aspects of context-related concept construction and problem solving processes to be carried out. And it is strictly in this sense that we will use them in this paper. 3. Method The elements upon which we will base our analysis are: episodes (that is, moments of work in the classroom, carefully documented, usually concerning one problem situation. in which it is possible to follow, for a sufficiently long period of time - at least half an hour - the behaviors of one or more children); comparative assessments of classes which work differently within the framework of the same project, or of different projects; cases of pupils followed
- 96. 82 carefully for extended periods (ranging from one to five years); clues of (for example) coherent indications coming from teachers who work without any interaction between themselves. In particular we are striving to balance out the advantages of classroom situations (or of individual children) followed diligently in their problem solving activities with the advantages of statistical results, more superficial and random, which relate to an high number of pupils (this year there are over 600 pupils for each age group of classes of 6 to 14 years old pupils, subjected to controlled experiments, even through common periodic tests, in our projects, for a total of well over 5000 pupils). In this paper, I will consider "contextualized problems": "contextualized" means "clearly related to some experience field and integrated in a consistent, long term classroom work in that experience field". Problem situations refer to the external context of the experience field directly accessible to pupils during classroom activities, or also to an external context which may be partly evoked on the basis of common life direct or indirect experience (like in the following example d». 4. Specific Problem Solving Aspects Sensitive to Semantic Fields We can identify numerous problem solving aspects which, to a greater or lesser extent, appear to be subject to the effects of the peculiar "semantic field" in which the problem solving situations given to the pupils are integrated: Conceptual aspects. For instance let us consider the concept of angle. The epistemological analysis of this concept shows the need for emphasizing the dynamic meaning (amplitude of rotation) and the invariance in relation to the variation of the lengths of the sides. On the other hand the only meaningful example of rotation of an object in which the length of the object varies when undergoing rotation is that of the rotation of the shadow cast by the sun. Afterwards, it is possible to verify (as we have in effect seen in comparative investigations concerning several hundred lO-years old pupils) that the conceptual characterization of the dynamic aspect of the angle, through the work on the shadow of the sun, constitutes the concept of angle with a reasonably high percentage of success among the pupils, as regards the discrimination of the concept and the possibility to transfer it to other environments. If we consider the following problem situations (in the field of experience "The Earth and the Sun") we may understand better how the concept of angle and other important concepts and skills are involved: (a) (9 and 11 years old pupils, after 15 hours of work in the field): from the shadowfan ofa stick registered on a large sheet ofpaper, to the shadowfan drawn on the personal copy-books: - form the "dynamical" conception of angles, to the "static" meaning; properties of scale reduction; ...
- 97. 83 Figure 1 (b) (9 and 11 years old pupils, after 25 hours of work in the field): to find the height ofa tower, knowing its shadow on the ground; (b') to trace the shadow on the ground (direction and length) ofa two meters stick, knowing the shadow of an 8 cm nail on a horizontal surface, etc. - these "classical" geometrical problems involve an increasingly deep mastery of "angles", "parallelism", etc. (c) (11 years old pupils, after 40 hours of work in the field) completion of the hourly shadow fan (if possible!) - observe how the axial symmetry plays an important role as a tool to solve the problem in a situation where the symmetry axe is not directly given; and how the angle as a rotation invariant is interwoven with the symmetry concept. !J Figure 2 (d) (11 years old pupils, after 70 hours of work in the field) at what latitude must fly a plane in order to see the sun just lying on the horizon line for many hours?- a good problem situation to force and deepen the mastery of complex modelization processes and of many mathematical concepts involved.
- 98. 84 "Theorems in action" [15]. the example which we have studied to some depth concerns the distributive property of multiplication in respect to addition and regards the widespread occurrence of this "theorem in action" in situations in which the 7 years old children must evaluate the overall cost of (for example) 4 objects costing 320 liras each. If we give this problem to children who already possess a reasonable amount of experience in working with money, in real or simulated purchases, many children spontaneously break the price down into 300 + 20 and then separately consider 4 times 300 liras and 4 times 20 liras. At the end, they add up the partial sums thus obtained. By proposing similar problems, in parallel, referred to lengths (for example, the problem of calculating the length of a corridor giving access to 4 rooms, each 320 cm long,) to classes which, moreover, are involved in the same project, the distributive property becomes apparent to a lesser extent. The semantic field of the "goods/money exchange" seems to be able (as a result of the material organization of the monetary values and of the social practice of price break-down, according to the money available during payments) to activate the "theorem in action" of distributivity in a relatively high percentage of children (over 70% of the seven years old). We have also observed that the "theorem in action" of distributivity does not occur frequently when we propose the problem situation of "four time 320 liras" in classes with a poor experience in working with money. From the developmental standpoint, the relatively early emergence of the distributive property in the semantic field of the "goods/money exchange" allows pupils to perform complex numerical strategies, thus contributing to the increasing of the mastery of numbers and the connected meanings of operations [7]. Procedural aspects. We have analysed various situations in which specific characteristics of the semantic field (material aspect of the external context, extra-mural experiences of the pupil, in particular referred to widespread social practices, etc....) seem to encourage the acquisition of complex procedures in problem situations that have been properly contextualized. For example, in class work situations in which the 8 years old children are asked "how many sheets 0[21 em paper need to be arranged next to each other on the walls o[the room" , it is very frequent to see them resort to a "covering strategy" (directly, on the wall, or graphically). This strategy frequently evolves (in further problems) towards a numerical strategy acting on the measurements. From the developmental standpoint, this allows children to explore an important meaning of division and to approach an efficient algorithm to perform divisions (see [7]). "Planning skills" in the solution to complex problems (which require a proper linkage of arithmetical or geometrical operations). Among the "planning skills", first and foremost, we believe that the development and handling of hypotheses are of particular importance. In research work conducted last year [6], regarding the first signs of these skills, on 65 children
- 99. 85 from 4 classes under supervision, from Class II to Class IV, in mathematical and non- mathematical, context-related and non context-related problem solving activities, I was able to ascertain how the development of hypothetical reasoning skills in problem solving is apparent in a large proportion of cases, in well integrated problems within the fields of experience of our primary school project and not in "decontextualized" problems. In particular, this concerns two types of hypotheses: The heuristic-exploration hypotheses necessary to iteratively approach the required solution. They involve, each time, an evaluation of the trials made and an adjustment of the solution strategy. In a problem concerning the division of a sum of money (for example 38000 liras), necessary for a classroom activity, between all 17 pupils of the class, given to 8 years old, when the children still do not know or understand the written calculation techniques of division, many children resort to strategies such as "ifeach childpays 1000 liras, it means that 17000 liras are collected ... too little ..., if every child pays 2000 liras then 34000 liras are collected ... still not enough ..., if each child pays 3000 liras... 51000 liras... too much, then /,11 try with 2100 liras..., etc." (see [7, 9]). The assumptions concerning constraints specific to problem situations (for instance, in a geometric problem in which an explanation about the general procedure to measure an angle, drawn on a sheet of paper with an assigned protractor, is required, it is necessary to take into account the following two possibilities: angle with sides of length greater than that of the protractor radius; angle with at least one side smaller than ..., and hence formulate a coherent measurement project for this.) Furthermore, as far as the complex problems are concerned, we have realized that certain semantic fields give the children the possibility to participate in the problem construction operation from within the problem situation (this occurs, for example, in problems in which one must construct and compare budgets for travel expenses with different means of transportation), or to make them solve complex problems, on the basis of the overall sense of the problem situation and extra-mural experiences. The comparisons carried out indicate that these problem solving experiences modify the capability to deal, productively, with the usual problems of standard complexity. Finally, we can consider the solution to complex arithmetical problems without given numeric data (achieved through the verbalization of the solution strategy, "in general", or even with the older children, by using algebraic expressions): this is a very important activity for the effects it has on the development of the design capability, on computer-aided problem solving and also on the construction of the meanings of the arithmetical operations; it appears practicable by a high percentage of pupils, only for heavily contextual-related problem situations in the fields of experience, familiar to the children. From a general standpoint, "complexity" in applied mathematical problems may concern the number of choices and steps that the pupil must keep under control (depending on his
- 100. 86 technical knowledge and previous experience), the coordination of different concepts and procedures (see problem situation (c), or the coordination of "additive" and "multiplicative" models in the semantic field of the formation of production costs), the selection of pertinent aspects in a "complex" reality. Thus the mastery of complexity involves complex mental processes, not reducible to the composition of simpler skills and concepts. Semantic fields offer the opportunity to "force" the mastery of complexity through the involvement in the problem situation, the reference to extra-mural experiences, the management of "space" and "time" familiar images and environments (this is evident in the before mentioned problem situations (c) and (d». Problem solving general logical-linguistic prerequisites. Many activities of considerable logical content relative to semantic fields (for example, at 8 years ofage, under the request to write down "how to measure one's own shadow alone" ... and then "why Mario's project doesn't work") appears "to force" the acquisition of competences which are important also to problem solving. 5. Elements of the Semantic Field which, if Appropriately Handled by the Teacher, Can Have an Effect on Problem Solving - Constraints and "social" or "material" relationships present in the external context of the semantic field: the structure of monetary values, the capability to break-down money into different monetary values, the relationships between the "height" of the sun and the "length" of the shadow, the relationships existing between "weights" and "distances" in the equilibrium of a beam balance of unequal arms, etc. may be effectively used by the teacher "to force" the pupils to acquire important concepts and abilities for problem solving purposes. However, it should be pointed out, in this regard, that by simply exposing the pupil (in hers or his extra-mural experiences) to these realities may not be sufficient to produce acquisitions that may be used in problem solving activities. In other words, various aspects (even very "real") of the external context may not be spontaneously taken into consideration by the child. For example, for the 9 years old, as for the 10-11 years old, we have been able to ascertain that over 50% of the children are convinced that the shadow of the sun is longer at 12 noon than at 9 am in the morning, simply "because at 12 noon the sun is more intense". - external representation (drawings, written records, etc.) as the mediating element of the problem solving process, both in the relation between thought development and internal representation and in the relation between internal representation and material constraints of the semantic field. In some cases the representation plays an essential role to enable the development of an efficient solution process, as occurs for the "echo problems": the 11/12 years old child who produces (or sees) a drawing of the type:
- 101. 87 ｹｾＭ ｾＺ＠ > Figure 3 Generally finds the correct solution quicldy to a problem in which she or he must determine the distance from the wall, knowing the time elapsed between the emission of the sound and the echo and the speed of sound. In other cases, certain forms of representations can constitute obstacles towards finding a correct solution to the given problem; for instance, in the problem: "a 60 cm long stick projects a shadow 0195 cm, how long is the shadow projected, at the same time, by an 80 cm long stick?" many 10/11 years old pupils (and even older) assume that the shadow projected by the 80 cm long stick is equal to 95+20 cm. Among the reasons for this, perhaps, there is also the presence of an additive relationship suggested by the manner in which it is verbally expressed: "they add 20 cm because as the length 01 the stick increases so the length 01 the shadow increases", (statement allowed in the Italian language but which suggests an additive model rather than a proportionality model). - conceptions referred to the field of experience: in several problem situations they may act in the sense of building obstacles to the solution of the problem, in other cases, on the other hand, they can guide the pupil in the construction of a correct solution strategy. An example of the first type is, for instance, that constituted by the conception of hereditary characters as the mix of the hereditary characters of the father and the mother in the son, which constitute an important factor of error when (at the end of a prolonged work on the probabilistic modelling of Mendelian standstill hereditary characteristics) the pupils are given a difficult problem such as that of tracing back the characteristics of the sons to the characteristics of their parents. - memory of extra-mural experiences and behaviors relative to the field of experience in which one works: let us consider, for example, a problem in which it is necessary to plan a train journey, with the times of many trains to be coordinated, costs to be evaluated, etc.; the 14 years old children show forms of reasoning which are evidently suggested by direct experiences carried out with their parents or with other adults; of interest is the fact that at times in these problems, even good problem solvers can be in difficulty due to the lack of these external references.
- 102. 88 - experiences and concepts of the teacher as regards the field experience in which one works: they act in various ways; in particular, we are able to identify their effects on the formulation of texts of the problems, on the interaction between the teacher and the pupils in difficulty and on the choice of certain problem situations with respect to others. 6. Differences Between Contextualized and Non-Contextualized Solving Activities The elements determined through the previous analysis suggest that there are important qualitative differences in the ways the pupils deal with problem solving activities, for the case of contextualized problems and for the case of non contextualized problems. These differences may be determined by indicators which, in tum, suggest assumptions on the nature of the differences and their long term effects. The differences can be explored by giving classes, which essentially follow the same course, identical and difficult problems (at the limits of the pupils' problem solving capability), in parallel, in one case as problems that are well integrated in the subject work as it is developed, and in the other case as fictional problems. Among the indicators the following can be listed: - the use of verbal language: for a given structural complexity of the problems, it is more widespread in strongly contextualized problem solving and, moreover, it is, for the same child, generally of a different nature in both cases (for many children in decontextualized problem solving a sequential language structure is often noted, index of the adherence to a scheme [15]; whereas the same children, in contextualized problem solving, produce solution texts rich in parallel time connections, assumptions, results); - heuristic strategies: their presence is anticipated and much more extensive in contextualized problem solving (as we have already shown for the conditions in which the development and handling of assumptions of the type "I try ..." - see [6] - appear for the first time); - behavior of good problem solvers: as already observed by Lesh [11], even in decontextualized problem solving situations, they refer extensively to the context evoked by the problem texts; - behaviour of weak pupils: for the case of a decontextualized problem, if they are able to solve it, this generally occurs by making use of schemes; otherwise there is a situation of total standstill. For the case of a "truly" contextualized problem, we can observe, in addition to the use of schemes, clever actions being attempted, for example by the drawing of the problem situation, by recording possible actions on the variable involved ("I take", "I put", "I move", "I measure", etc.).
- 103. 89 The mediating action of the teacher can be put to good use on such attempts. All this suggests the hypothesis that the inclusion of the problem situation within a field of experience, in relation to the overall work in progress, can favor: - the development of mental process to internally and externally represent problem situations related to perceptions of the context and of connections between the variables implemented practically during the work in the field ofexperience; - planning activities oriented by the awareness of the aims which the solution to the problem has in relation to the development of the school work in the field of experience. The mental work appears less "contracted", less oriented towards the research of acquired reproducible schemes, and evolving mental environments appear to be created in which images and hypotheses and reconstructed experiences are handled. 7. The Role of the Teacher in Contextualized Problem Solving in Semantic Fields We will take for granted the complexity of the questions concerning the choice of the fields of experience within which to propose and choose problem situations (refer to [4] and [5] in this regard). I would like to pause a while here to discuss the question of the handling of problem situations in the classroom. On the basis of what has been said in the previous paragraphs, the constraints and "concrete" aspects of the external context may not be perceived by the pupil; on the other hand, at times, the conceptions of the pupil or the external representations operate in a way of forcing the pupil to reach incorrect solutions. It is therefore necessary that the teacher does not limit herself or himself to propose "stimulating" problem situations for the pupils' strategy construction processes, but rather to intervene directly to act as go-between the external context and the internal context of the pupil and as mediator of the culture elaborated by humanity to date. For this purpose I do not believe it possible that in certain applied mathematical problem solving situations, referred to certain "scientific revolutions" (Talete's Theorem, Archimedes Principle for the equilibrium of the lever, Mendelian transmission of hereditary characters) the simple proposition of opportune problem solving situations has the effect of conducting towards the elaboration of correct solution strategies and associated conceptualizations. For example, for the case of the problem of the length of the shadow of the 80 cm long stick, knowing the length of the shadow cast by the 60 cm long stick, it is true that the pupil may realize, on account of the results of crucial experiences proposed by the teacher, that the additive model does not work; it is nonetheless difficult that she or he alone is able to understand that a multiplication model is better suited to the reality! The role that the teacher must perform for the productivity of her or his work therefore appears to depend upon the field of experience chosen and upon the particular problem
- 104. 90 situations offered. The analysis of passed experiences (one's own and of others) and, within certain limits, the historical-epistemological analysis of the problem situations can provide useful references in order to decide, each time, whether to simply propose stimulating problem situations, or also be a vehicle by which to directly transmit cultural aspects. Then there is the problem of the pupils who have considerable learning difficulty and, more in general, the problem of the relationship between autonomous and individual work carried out by the pupils, and the work perfonned under the individualized guidance of the teacher, and collective work. As far as problem solving is concerned, we do not believe (in relation to the experiments conducted to date) that the collective work for the solution to a problem can contribute to the development of the personal problem solving ability, whereas we have many positive experiences (episodes, case analyses, comparative investigations between different classes and teachers who carry out the same project ...) regarding the effects of the teacher's individualized interventions in the weaker child's work (for example, through written questions which the teacher makes in relation to the strategy gradually developed by the pupil) and the effects of the work of comparing the strategies produced in the class (which in certain cases provides an effectively increasing wealth of strategies which the children are able to produce and, at any rate, always stimulates them to look for additional strategies). What are the functions ofthe semantic field in all this? The semantic field in which one operates can offer the teacher an important occasion: - to support the work of the pupil in difficulty by referring to hers or his school and extra-mural experience; - to evaluate and compare the strategies produced by the pupils, not by themselves, but rather in relation to the purposes that are to be achieved with the solution to the problem; - to construct a higher and precise conscience, in the class, of what "solving a problem" means, it also associated to the realism of the problem situations given in relation to the works in progress in the field of experience. Finally, we must ask whether the use of the computer in the classroom can modify the points of view expressed in this paper. It seems to me that the use of the computer as an instrument for solving problems can provide the teacher with the occasions to substantiate the proposition of complex problems without explicit numerical data, the request to verbally specify the reasoning followed for the solution to a problem, the comparison among various strategies (more or less apt to be implemented on the computer). It also appears that the computer used as a simulation instrument can enhance the external context of certain fields of experience (for example with moving images, graphics, etc.). I nonetheless fear that, with the computer in the classroom, there is the risk (for the teachers and pupils alike) of losing sight of, or underestimating the acquisition of those crucial problem solving skills, which seem to stem from an intense relationship between the pupil and the external context ("cultural" or "material") ofthe various fields of experience.
- 105. 91 References 1. Bauersfeld, H., Krummerheur, G., & Voigt, 1.: Interactional theory of learning and teaching mathematics and related microetbnographical Studies, Proceedings ofTME-II, IDM Bielefeld, 1988 2. Binns, B. et al.: Mathematical modelling in the school classroom: developing effective support for representative teachers. In: Modelling, applications and applied problem solving - Teaching mathematics in a real context (W. Blum, M. Niss, & I. Huntley, eds.). Chichester: Horwood 1989 3. Bishop, A.: Mathematics enculturation. Dordrecht: Kluwer 1988 4. Boero, P.: Mathematics literacy for all: Experiences and problems. In: Proceedings of the 13th International Conference for the Psychology ofMathematics Education, Paris, 1989 5. Booro, P.: Semantic fields suggested by history. Zentralblat fUr Didaktik der Mathematik 89(4), 128-132, 1989 6. Boero, P.: On long term development of some general skills of problem solving. In: Proceedings of the 14th International Conference for the Psychology of Mathematics Education, 1990 7. Booro, P., Ferrari, P. L. & Ferrero, E.: Division problems: Meanings and procedures in the transition to a wrillen algorithm. For the Learning of Mathematics 9 (1989) 8. De Lange, J.: Mathematics - Insight and meaning. Utrecht: OW & OC 1987 9. Ferrari, P.: Hypothetical reasoning in the resolution of applied mathematical problem solving at the age of 8- 10. In: Proceedings of the 13th International Conference for the Psychology of Mathematics Education, 1989 10. French, L.: Acquiring and using words to express logical relationships. In: The development of word meaning (S. Kuczaj & M. Barret, eds.). Berlin: Springer-Verlag 1985 11. Lesh, R.: Conceptual analysis of mathematical ideas. In: Proceedings of the 9th International Conference for the Psychology of Mathematics Education. Utrecht, Holland 1985 12. Solomon, Y.: The practice of mathematics. London: Routledge 1989 13. Treffers, A. & Goffree, F.: Rational analysis of realistic mathematics education. In: Proceedings of the 9th International Conference for the Psychology of Mathematics Education. Utrecht: OW & OC 1985 14. Usiskin, Z.: The sequencing of applications and modelling in the UCSMP 7-12 curriculum. In: Blum et aI (eds.), pp. 176-181, 1989 15. Vergnaud, G.: La theorie des champs conceptuels, Recherches en Didactique des Mathematiques 10 (1990)
- 106. Cognitive Models in Geometry Learning Jose Manuel Matosl ｳｾ＠ de Ciencias de ｅ､ｾＬ＠ Faculdade de Ciancias e Tecnologia, 2825 Monte da Caparlca, Portugal Abstract: Recent studies on the ways in which we fonn categories of objects indicate that we admit that there are special elements (prototypes) in most classes. Moreover, these classes of objects seem to be organized in such a way that apparently it exists a special hierarchical level, the basic level. that is generally used in communication. Researchers have identified several types of cognitive models that seem to play an important role in the ways in which we fonn and organize these categories. The explanatory power of these models is tested in the re- interpretation of findings from several investigations. Keywords: cognitive models, scripts. schemata. metaphors. metonymy. mental images. language and mathematics. geometry learning. van Hiele theory. social cognition One of the concerns of psychologists and educators has been the ways in which students abstract ideas and concepts from their experiences. These abstractions help us organize our experiential world. Gestalt theory conceived abstraction as a reorganization of the field of perception. Piaget viewed it as the formation of schemas. and cognitive scientists incorporated in it the mechanisms of generalization. differentiation. and the recognition of patterns. Abstraction involves. among other things. the formation of categories of objects. Traditional psychological studies considered that a small set of simple properties is necessary and sufficient to establish membership of a category. These categories have defining or critical attributes which determine which elements are members or are not members of a category. Take for example the set ofred flowers. Boundaries of this category are assumed to be sharp and not fuzzy. i.e.• given any flower. either it is red or not [14]. In Piaget's studies. for example, equivalence relationships determine membership to categories. Moreover it assumed that we use necessary and sufficient properties of each category when perfonning inferences. deductions. and when we construct a taxonomy of categories. These two assumptions are usually taken to represent what is called the classical view of categorization. lThis article is based in part of the author's doctoral dissertation in process of completion at the University of Georgia and was partially supported by a grant from the Institute of International Education, New York and another grant from the Ministry of Education of Portugal.
- 107. 94 It is the purpose of this paper to review current psychological perspectives on categorization and to apply these perspectives to the discussion of research findings on the learning of some geometric concepts. Category Theory The classical view of categories was first challenged by Wittgenstein's work. He pointed out that the category of games does not have a set of common properties shared by all the members. Some games share amusement, others luck, competition, skills, or a mix of these. Wittgenstein conjectured that game was a cluster concept, held together by a variety of attributes, but no instance contains all the attributes. The category of games has ajamily resemblance structure. Like family members, games are similar to one another in some, but not in every, ways. Some categories, like games or numbers, have no fixed boundaries and can be extended depending on one's purposes. Some categories, like numbers or polyhedra, have central members. Any defmition of numbers must include the integers, as any definition of polyhedra must include the cube [14, 19]. The Study of Prototype Effects Experiments later performed by Rosch and her colleagues empirically confirmed Wittgenstein's philosophical investigations [32, 33, 34, 35, 36, 37, 38]. Their research proceeded in two directions: horizontally it looked for asymmetries among members of one category; vertically, it looked at asymmetries within nested categories [36]. Their work produced evidence of two phenomena: prototype effects and basic-level effects. The term prototype effects refers to the experimental finding that in some categories not all members have an equal condition. Rosch's initial research with color [32] showed that there are colors (focal colors) that have a special cognitive status. On the one hand, they were preferred by her subjects as best examples. On the other hand, subjects learned them easier than the other colors. This finding run contrary to the previous scientific belief that colors are arbitrarily named, i.e., that the specific colors chosen to be named are determined by language alone. She called these focal colors cognitive reference points, and prototypes [33]. She later extended her research to other categories, usually of physical objects, and in each case, she found prototype effects, i.e., subjects judged that certain members of each category were more representative of the category than others. She showed, for example, that her subjects believed robins to be very typical birds, whereas chickens were less typical. These effects, however, were not discovered in every category. Rosch, for example, was unable to find prototype effects in categories of actions like walking, eating, etc. [9].
- 108. 95 Rosch's research also tried to find other consequences of these prototype effects. She showed that they predict perfonnance on several tasks focusing on the ways in which central members of a category are related with peripheral members and with the category itself. She found that: (a) less typical members of a category are less associated with that category; (b) typical members appear to have an advantage in perceptual recognition; (c) when people think of a category member, they generally think of typical instances of that category. She also showed one asymmetry in the ways in which members of some categories are related to others: (d) subjects considered less typical members to be more similar to more representative examples than the converse. Finally, she showed that: (e) the categories studied had a structure of family resemblances [2, 19,43]. Basic Level Effects Investigations about the ways in which people nest the classification of objects have been the object of research. Brown has been credited as being the first to present the problem: We ordinarily speak of the name of a thing as if there were just one, but in fact, of course, every referent has many names. The dime in my pocket is not only a dime. It is also money, a metal object, a thing, and, moving to subordinates, it is a 1952 dime, in fact a particular 1952 dime with a unique pattern of scratches, discolorations, and smooth places. [5, p. 14] He pointed out that we have the feeling that some of these names are the "real names," the others being achievements of the imagination. Although we know that Brown's dime is a "coin," or a "thing," we are compelled to think that its real name is "dime". Moreover these special names seem to be frequently linked with non-linguistic actions. Brown's ideas prompted cognitive anthropologists to search for folk taxonomies-the ways in which cultures use the fonn "A is a kind of B." Berlin and his coworkers (referred in [19]) examined folk classification of plants and animals of speakers of Tzeltalliving in a region of Mexico. They found out that, although their infonnants could name animals and plants in a variety of ways, they tended to use a single level of classification. Berlin called this level of classification the folk-generic level (or basic level). This level was in the middle of folk classification hierarchy. Further research on the Tzeltallanguage discovered that most children initially learn names at this folk-generic level, and later they find out simultaneously how to differentiate and generalize these tenns [19]. Rosch and her colleagues developed a series of experiments that confinned most of Berlin's findings [38]. They found that the psychological most basic level was in the middle of taxonomic hierarchies. Basic-level categories are basic in perception, function, communication, and knowledge organization. Essentially they found that the basic level is:
- 109. 96 - The highest level at which category members have similarly perceived overall shapes. - The highest level at which a single mental image can reflect the entire category. - The highest level at which a person uses similar motor actions for interacting with category members. - The level at which subjects are faster at identifying category members. - The level with the most commonly used labels for category members. - The first level named and understood by children. (...) - The level at which terms are used in neutral contexts. (...) - The level at which most of our knowledge is organized. [19, p. 46] Other researchers [9] also found that, unlike scientific taxonomies, classification in folk taxonomies is not always exclusive, i.e., instances can share several taxonomic levels. Moreover, the implicit categorization criteria may vary. Sometimes, for example, we use the function as the categorization criterion for superordinate category ("clothing"). Other times we use the unity of place ("furniture"). In other cases, we form categories ("groceries") from a composition of criteria. There is also evidence that folk taxonomies are not very extensive [29]. Implications for Cognition Prototype and basic level effects destroyed the notion that concepts are organized by sets of necessary and sufficient conditions, and have prompted the development of new cognitive models that can accommodate the experimental findings. At the core of this theoretical effort is the notion of mental representations-"a set of constructs that can be invoked for the explanation of cognitive phenomena, ranging from visual perception to story comprehension" [14, p. 383]. This section will analyze the ways in which several psychological theories account for the effects described previously. The holistic perspective provides the simplest account for prototype effects. This theory maintains that, for example, a term like "dog" refers to the mental category dog, which is in itself an unanalyzable gestalt. It assumes that mental categories are composed of templates, usually imagistic, that are isomorphic to the object they represent, are unanalyzable, and that implicitly show the relations between the several features or dimensions of the object. An object belongs to a certain class if it provides a holistic match to the template of the class. Computer scientists working in pattern recognition have been using this theory. The theory, however, seems to be limited to the categorization of concrete objects-it is difficult to talk about templates for categories like "furniture", or for more abstract entities like "justice" [44]. Another theory, the featural approach, supports the idea that human minds use more elementary categories and only a few of the words that we manipulate code unanalyzable
- 110. 97 concepts. Rather, most words are labels for mental categories which are themselves sets of simpler mental categories, usually called features, properties, or attributes. Each concept is represented by groups of features that have a substantial probability of occurring in instances of the concept. Some of the proponents of this theory developed the notion of a group of weighted features. An object is an instance of a category if the sum of its values for each feature is greater than a given threshold. Each member of a category is further removed from the prototype the more it differs in highly weighted features [19,44]. Some researchers [23, 24, 31] have designed a similar model using dimensions instead of features to represent concepts. This view departs from the featural view especially in the treatment of continuous dimensions like size. Specific instances depart from the prototype in continuous degrees. If the relevant dimensions of birds were thought to be animacy, size, and ferocity, a robin, for example, might have a 1 in the feature of animacy, .7 in size, and .4 in ferocity. A key concept that is a consequence of this approach is the notion of semantic metric spaces, which are thought to be multidimensional Euclidean real spaces. The vector (1, .7, .4) in R3 would represent a robin [44], providing a literal meaning to the notion of semantic distance which is interpreted as the Euclidean distance on a semantic space. Typically, researchers in this area attempt to determine relevant dimensions, discuss the meaning of clusters of concepts, or discuss meanings of translations in the semantic metric space. The semantic distance between an instance and the prototype corresponds to the degree of category membership. Critics to these two last perspectives have pointed out that: (a) the knowledge represented in a concept includes more than a list offeatures, namely the relationships among the features; (b) the model does not provide for contextual or background effects. The dimensional perspective also raises additional problems because of its use of semantic metric spaces. The requirement of orthogonal dimensions, the necessity of the isotropy of the semantic space, the very possibility of coexistence of concepts and their members in the same space are some examples of the difficulties of these theories [19,44]. Global theories of cognition have provided ways to accommodate prototype effects. Rumelhart, for example, proposes that knowledge of each concept is represented by a schema (he uses the plural as schemata). A schema is "a data structure for representing the generic concepts stored in memory" [39, p. 34] and contains the relationships among the components of the concept in question. A very similar construct, a/rame, is proposed by Minsky [25]. Rumelhart describes the features of schemata using four analogies. First, each schema is a type of informal, private, unarticulated theory about the nature of events, objects, or situations that we face. The total set of our schemata constitutes our private theory about the nature of reality and represent knowledge in all levels of abstraction ("Schemata are our knowledge" [39, p. 41]). This means that we are constantly testing this
- 111. 98 theory. and that we use it to make predictions about unobserved events. According to Minsky [25]. this is accomplished by an i,gormation retrieval network. Second. schemata are active processes like procedures or computer programs. As such they are able to detennine the extent in which they account for the pattern of observations. and are capable of invoking other subprocedures (or other subframes). Third. schemata are like parsers that work with conceptual elements. On the one hand. we are able to find and verify the appropriate schemata. On the other hand. schemata enable us to find constituents and subconstituents in our observations. Finally. the internal structure of schemata is like scripts of plays that can be played with different actors. The scripts correspond to prototypes of the concepts. They have several variables that can be "associated with (bound to) different aspects of the environment on different instantiations of the schema" [39. p. 35]. Minsky talks about terminals. which are "slots that must be filled by specific instances or data" [25. p. 96]. We are aware of the typical values of these variables and of their interrelationships. Both schemata and frames provide a similar explanation of prototype effects. Rumelhart claims that the meaning of a concept is encoded in terms of the prototypical situations or events that instantiate that concept. and Minsky defines a frame as the representation of a stereotyped situation. In more specific explanations, both researchers stipulate that each schema's variables have default values that are responsible for our expectations and other kinds of presumptions. These default values are "attached loose to their terminals" [25, p. 97], which allow their replacement by new items that better fit our experiences. Cultural Models The previous models we have been discussing still do not account for contextual or background effects. nor do they provide any explanation for basic level effects. From an educational point of view, contextual effects are very important to understand children's enculturation into school mathematics. It is a plausible conjecture that children's conceptualizations depend heavily on the social and cognitive context in which learning takes place. Students and their teachers live in a school culture and we can expect that they intersubjectively share mathematical concepts in some degree. Both students and teachers are also members of larger social groups and we may expect that they bring the mathematical knowledge of such groups into the school. A broader approach to mental representations was needed. one that would take into account the role played by the community of human minds upon the individual. To describe this common knowledge, cognitive anthropologists and linguists have developed the notion of a cultural model-"a cognitive schema that is intersubjectively shared by a social group" [9, p. 809] used by American researchers, or of social representations-system(s) of values. ideas
- 112. 99 and practices with a twofold function: first, to establish an order which will enable individuals to orient themselves in their material and social world and to master it; and secondly to enable communication to take place among the members of a community by providing them with a code for social exchange and a code for naming and classifying unambiguously the various aspects of their world and their individual and group history [10, 26, 41] used by investigators working in the sociological European tradition. A special example of these cultural models is the notion of scripts, developed by Schank and Abelson in the context of text comprehension, which are cultural models adapted to the study of events. A script is "a coherent sequence of events expected by the individual, involving him either as a participant or as an observer" [I, p. 33] and it can be interpreted as an extension of schemata to dynamic episodes [2, 14]. Scripts may be thought metaphorically as a cartoon strip. The departure of an instantiation of events from the expected prototypical episode gives rise to prototype effects. People, for example, have a "restaurant script" that is composed of a stereotyped set of events that they expect to happen under certain circumstances. It depends on a sociocultural institution, that is, the existence of a place that serves and sells food [9]. Idealized Cognitive Models Recently linguists and anthropologists have been converging in their study of cultural models. An example is Lakoff's work on cognitive models. Lakoff develops an idea of mental representations that borrows some of the features of Rumelhart's schemata, Minsky's frames, and Abelson's scripts, and adds linguistic and cultural components. He proposes that we organize our knowledge by means of idealized cognitive models, and category structures and prototype effects are by-products of that organization. To build his construct Lakoff departs from two assumptions shared by the featural and the dimensional approaches [19]. The first is that goodness of example is a direct reflection of degree of category membership, that is, subjects' willingness to say that a chicken is not a good bird implies that chickens do not have a high degree of members of the category of birds. Although the construct of a graded membership to mental categories can explain some prototype effects, others are not. A classical example of a category (first presented by Fillmore) that does not have a graded membership is the category of "bachelor", for which there are clear conditions for membership. Nevertheless persons like the pope, or Tarzan do not have a clear status of membership to this category. Lakoff proposes that prototype effects in this category are produced not because the category is graded, but because we have an idealized cognitive model of bachelor based in the context of a human society where there are certain expectations about marriage and marriageable age. The worse the fit between that idealized cognitive model and our knowledge of the background conditions, the less appropriate we feel that the concept
- 113. 100 should be used [19, 28]. Later we will discuss the category of even number which is not graded either. The assumption, shared by the featural and the dimensional approaches, that prototype effects mirror mental representations of categories, that is, categories are represented in the mind in terms of prototypes and degrees of category membership are determined by their degree of similarity to the prototype, does not capture the complexity of some categories. For example, the concept of mother is based on a complex aggregate of several models: - The birth model: The person who gives birth is the mother. C •••) - The genetic model: The female who contributes the genetic material is the mother. - The nurturance model: The female adult who nurtures and raises the child is the mother of that child - The marital model: The wife of the father is the mother. - The genealogical model: The closest female ancestor is the mother. [19, p. 74]. The concept of mother is not defined by necessary and sufficient conditions, and all those models converge in a prototypical ideal case. Prototype effects can be explained by tensions between these models in some situations (stepmother, surrogate mother, foster mother, etc.). Lakoff claims that a major source of prototype effects is associated with our use of metonymy-"a situation in which some subcategory or member or submodel is used (...) to comprehend the category as a whole" [19, p. 79]. Social stereotypes, where a subcategory has a socially recognized status as standing for the category as a whole, are examples of our use of metonymy. For example, in the United States, the category "working mother" is not a mother that happens to be working. Rather it is defined in contrast with the social stereotype of a "housewife-mother" which is defined by the nurturance model. Prototype effects in the case of a working mother arise from its comparison with only one of the models in the cluster and not against the whole category. Put in another way, the "housewife mother" usually stands for the whole category of "mothers". Consider an unwed mother who gives up her child for adoption and then goes out and gets a job. She is still a mother, by virtue of the birth model, and she is working-but she is not a working mother! The reason is that it is the nurturance model, not the birth model, that is relevant. Thus, a biological mother who is not responsible for nurturance cannot be a working mother, though an adoptive mother, of course, can be one [19, p. 80]. Other kinds of metonymic models include: typical examples, ideals, paragons, salient examples, and others that we will discuss later. Neither the featural and dimensional approaches, Rumelhart's schemata, Minsky's frames, nor Schank and Abelson's scripts account for prototype effects that result from metonymy. Another important source in the construction of all kinds of models is the use of analogy and metaphor. Each metaphor is based on a similarity between a source and a target domain,
- 114. 101 together with a source-to-target mapping [19]. Metaphors allow us to extend the similarities between the domains beyond their initial state and they structure most of our conceptual system. We use them to map structures usually on the physical and eventually on the mental world into other domains through imaginative processes [20]. The result of any such mapping, from physical experience in the source domain to social or psychological experience in the target domain, it that elements, properties, and relations that could not be conceptualized in image-schematic form without the metaphor can now be so expressed in the terms provided by the metaphor [28, p. 28]. Johnson later elaborated this notion, and proposed the construct of kinesthetic image schema- basic experiential structures that are a consequence of the nature of human biological capacities and the experience of functioning in a physical and social environment [17]. Reason for Johnson is no longer detached from human beings as functioning organisms. These image schemas significantly structure our experience prior to, and independent of, any concepts, and are responsible for many of the metaphors we use in abstract domains. Examples of these schemas include: the container schema that consists of a boundary distinguishing an interior from an exterior; the part-whole schema that involves the whole, the parts and a configuration; the link schema, where there are two entities and a link connecting them; the center-periphery schema where a central element is thought to be more important than the periphery; the source- path-goal schema that includes a source, a destination, a path, and a direction; the up-down schema; the front-back schema; and the linear order schema [19]. Quinn and Holland [28] argue that these imagetic schemas, from which metaphors are based, are not only predicated in our bodily experiences but may also be built upon elements shared by the cultural group. In summary, Lakoff proposes that the structure of thought in general, and the categorization in natural languages in particular is characterized by cognitive models that fall in four types [19]: (a) Propositional models that specify elements, their properties, and the relations holding among them. (b) Image-schematic models that specify schematic images. (c) Metaphoric models that are mappings from one of the above models in one domain to a corresponding structure in another domain. (d) Metonymic models that make use of the previous models and map one element of the model to another. By distinguishing among these types of cognitive models, Lakoff is able to propose a process of creation of complex cognitive models. He argues that there is a "significant level of human interaction with the external environment (the basic level), characterized by gestalt perception, mental imagery, and motor movements" [19, p. 269]. This is the level at which people function most efficiently and successfully using basic-level and image-schematic concepts.
- 115. 102 Categorization of Mathematical Objects Mathematical categories have been the object of research both by psychologists, linguists, and mathematics educators. For some of these researchers, mathematical knowledge is a field of certainty, bound by the laws of logic, and a clear example of analytic truth. Armstrong, L. Gleitman, and H. Gleitman's investigation on prototype effects [3] provides a typical example. In an attempt to prove that prototype effects were unrelated to the ways in which we categorize, they compared categorization of mathematical entities with the categorization of real world objects. The rationale for this approach was that if prototype effects could be found in mathematical categories, like "even number", then Rosch's prototype theory should be wrong, because these prototype effects are unrelated to membership gradience-the category of "even numbers" has a clear membership rule. Their implicit assumption was that mathematical entities have a clear, declarative membership rule, and that their subjects were applying it. This conception of mathematics runs contrary to recent developments in philosophy, history, and sociology of mathematics [4, 18, 30,47]. As a result of investigating the roots of mathematical knowledge, researchers in these fields have been proposing that mathematics knowledge is generated by social interactions and that mathematical truth is intersubjectivily shared by the community of mathematicians. Lakatos' work, in particular, shows how mathematicians themselves may not be in agreement over the meanings of mathematical entities, even when such meanings are provided by definitions. Although Lakatos's field is history and philosophy of mathematics, he does provide evidence of prototype effects in the category of polyhedra. In his historical account of the discussions over a precise definition of the concept of polyhedra, Lakatos shows how some mathematicians have come up with counterexamples of polyhedra that did not verify Euler's formula, and how other mathematicians would claim that they were presenting "monsters" and using "wrong" definitions of polyhedra. Moreover there are central examples of polyhedra-all mathematicians would agree that any definition of polyhedra should include prototypes like the five platonic solids. A second point can be made about Armstrong, L. Gleitman, and H. Gleitman's investigation. Even if we accept that mathematical categories are classical, we would still have to show that they are thought as such by the subjects themselves. As Gardner pointed out [14], their research may very well show that even mathematical categories display a structure similar to other categories. We will further discuss their research later in this paper. A strong case for the subjectivity inherent to mathematical entities is put forth by Fischbein. In his review of the role of intuition in thought he gives examples of what he terms analogic and paradigmatic models in mathematics and physics. [11]. Analogic models are similar to what we have termed here metaphoric models, and a paradigm, in Fischbein's terminology, is an instance of a category that is used to represent the whole category and is thought to be a particularly good example of the category. This last definition shares both the characteristics of
- 116. 103 a prototypical and a metonymic model. Fischbein also agrees that mathematical categories may not be classical. He proposes in his book about intuition [11] that when we define a concept we never do it as a pure logical construct The meaning subjectively attributed to it [the concept], its potential associations, implications and various usages are tacitly inspired and manipulated by some particular exemplar, accepted as a representative for the whole class (p. 143). Fischbein's point is as much about students as about mathematicians themselves. Fischbein compares these reasoning processes to Kuhn's paradigms in scientific thought, and calls this phenomenon "the paradigmatic nature of intuitive judgment" (p. 143). An Example: the Concept of Number Rosch [33] studied examples ofcognitive reference points that included vertical and horizontal lines, and numbers that are powers of 10. Part of her research looked for prototype effects using linguistic hedges-"terms referring to types of metaphorical distance" (p. 533), like "almost", "virtually", "essentially", "loosely speaking". She made use, for example of stimuli as "103 is essentially 100". She found that, within the decimal system, multiples of ten constitute reference points. Both 97 and 102 were judged essentially 100, but not vice versa, and both were considered closer to 100 than 100 was close to them. As a by-product these asymmetries questioned the isotropy of semantic spaces [33]. Analyzing these results from a linguistic perspective, Lakoff [19] adds that the natural numbers, for most people, are characterized by the words for the integers between zero and nine, plus addition and multiplication tables and rules of arithmetic. These digits are the central members of the category of natural numbers, from which the other members are generated. Any natural number can be written as a sequence of digits, the properties of large numbers are understood in terms of the properties of the single-digit numbers, and the computations with large numbers are understood in terms of computation with the single-digit numbers. Each of the single digits generates subcategories of its own when multiplied by 10, 100, etc. These results were actually predicted by Wertheimer [52]. He may be credited as being the first to draw attention to the special place multiples of ten have in our vocabulary: "He is a man in his thirties," or "X died in the twenties of last century." Natural numbers are an example of a category composed of some central members and some rules for generating the other members. Lakoff [19] claims this is a metonymic model, where the single-digit numbers stand for the whole category. He also claims that the category of natural numbers itself is a central category in more general categories of numbers. For example rational numbers are understood as quotients of natural numbers, real numbers as infinite sequences of single-digits, etc. These other categories of numbers are understood
- 117. 104 metonyrnically in terms of the natural numbers. Every axiomatic system involving numbers must include the natural numbers, so their centrality is reflected even in the work of mathematicians. Data from mathematics educators confirm this "dissolution of hierarchies" [11, p. 147]. Tall and Vinner [45] report that often students did not regard -{2 as a complex number although some of them defined real numbers as "complex numbers with imaginary part zero" (p. 154). The effects found by Rosch are explained because we use the powers of ten as a submodel to comprehend the relative size of the numbers, especially in the context of approximations and estimations. There are also other models that we use to comprehend numbers. For example, in the context of body temperature, 380 Celsius is a cognitive reference point where fever is involved, and where American money is concerned, a cognitive model often includes powers of five (nickels, dimes, quarters). Each of these models produces prototype effects. It is important to note these prototype effects are not equivalent to graded category membership. In fact subjects in Armstrong, L. Gleitman, and H. Gleitman's investigation [3] agreed that the categories of even and odd numbers are well defined. Nevertheless the researchers found prototype effects using reaction time and ratings. Lakoff [19] claims that these effects are the result of the superposition of all those models over the even-odd structure of the natural numbers. Another Example: Preferred Triangles Geometry, in particular, relies heavily on metaphors. Let us just look at the terms "altitude", "height", "base", "length", and "width". We talk about "the altitude of a triangle", "the altitude of a trapezoid", "the altitude of a parallelogram", but very rarely about "the altitude of a rectangle" ("length" and "width" are used instead), or "the altitude of a square" (we use "side"), and never about "the altitude of a rhombus" (altitudes of rhombuses are seldom used). The same can be said about the term "base". There is "the base of a triangle" but not "the base of a rhombus". "Base" and "altitude" are also used with solids in the same way. Virtually every textbook will say that to calculate the area of a rectangle we have to multiply the "length" and the "width", whereas the area of triangle is computed using the "base" and the "height". The term "the altitude of' is used in English common language mainly in relation with mountains. We would prefer the terms "height of a building" or "height of a person" but not "the altitude of a house". Although we would say that "the plane is at an altitude of 9 Km" we would not say "the altitude ofthe plane is 9 Km". The underlying message is that we are asking students to imagine triangles as mountains, whereas rectangles are thought to be like rooms or football fields (rhombuses apparently are thought to be diamonds or kites). We are in fact using
- 118. 105 a "mountain metaphor" to work with triangles and a "football field" or a "room metaphor" to compute the areas ofrectangles. Although such worldly terminology would be condemned by Hilbertian formalists, its use guarantees that students attribute meaning to their actions on mathematical objects. But mathematicians themselves also make an extensive use of metaphors. As Thom puts it, "the mathematician gives a meaning to every proposition" [46, p. 202].Terms like "manifold", "fiber bundle", "curvature", "projection", "kernel", "closure", and many others are all evidence ofmathematicians' concern for meaning. It is this concern for the attribution of meaning to mathematical entities that the Teachers' Edition of Merrill Mathematics, Grade 5 expresses when giving recommendations for the lessons about the computation of the areas of the rectangle and the triangle: Make sure each student understands the relationship between length times width for a rectangle, and base times height while computing the area of a right triangle. Otherwise, students will not be as apt to use what the already know to solve these new problems (p. 388). The attempt to use what students already know is exactly the purpose of the educational use of the mountain or the football field metaphors. The mountain metaphor may only be part of the picture. Several researchers have been reporting students' preference for the upright/horizontal position of geometric figures [8, 12, 13, 51, 54], and in one case [8] one of the informants (Bud) distinguished among several triangles by the directions they were pointing. Some researchers [13] have interpreted these phenomena as "perceptual difficulties" (p. 137) but this description does not provide specific information. We would argue instead that it is a cognitive, not perceptual, problem produced by the interaction of several cognitive models. The up-down schema in Johnson's terminology [19] may account for the preferred orientation, and a metaphor mapping the human act of pointing to some of Bud's triangles may help to explain his answers. We will discuss these points later. The use of such processes is a necessary and unavoidable characteristic of thought. It facilitates students' identification of the relevant elements and their relationships, and permits their integration with previous knowledge [27]. There is however an unwanted side effect to it. It is hard to imagine into which direction is an obtuse triangle pointing. Moreover, in the culture of school mathematics it is irrelevant where triangles are pointing. Students also tend to have problems when they attempt to apply the mountain metaphor to triangles that are not in the "mountain position" (no side horizontal), or that do not look like mountains (obtuse triangles with a horizontal side other than the larger side). A non-obtuse triangle in a "mountain position" seems to be the cognitive reference point [19] employed by students. Prototype effects are
- 119. 106 likely to occur with different triangles, as shown by Vinner and Hershkowitz [51] and Wilson's [54] investigations that included the concept of the altitude of a triangle. Concept Images and Concept Definitions The construct of prototypes has also been used by researchers in geometry learning. Some researchers have reported that students' choice of examples of geometric concepts and their defmitions of the same concepts do not match [7, 13,21,55]. The construct of concept images has been proposed by Vinner and some of his colleagues [45,48, 51] as an explanation for those findings. A concept image is "the total cognitive structure that is associated with a given concept" [45, p. 152] and is composed of the images associated with that concept together with a set of properties and processes. For example, the concept image of a function may include a picture of the graph, a picture of the algebraic expression that defines the function, together with the students' defmition of function. Concept image is contrasted with concept definition, which is a verbal defmition that accurately explains the concept [48] and may differ from the mathematical definition [51]. Vinner distinguishes between formal and informal learning, and claims that in the later we need a concept image and not a concept definition. Concept definitions introduced by means of a definition will remain inactive or eventually be forgotten. In a specific intellectual task only portions of the concept image are actually evoked (temporary or evoked concept image). These portions might be contradictory and produce conflict in one person's mind when these opposing portions of the concept image are used simultaneously [45,48]. These researchers have been using this construct to interpret the finding that often visual identifications and drawings made by students do not match their definitions [15, 49, 50, 51]. Although Vinner and his colleagues did not perform these investigations within the framework of categorization theory, Hershkowitz [15] recently attempted a reinterpretation of their findings consonant with categorization theory and van Hiele theory. We will discuss her proposals elsewhere (see.[22]). Applying the construct of concept images to the problem of the mismatch between the choice of examples and the definitions we could say that the students' concept images and concept definitions were not matching, and the concept image was taking precedence in identification or production tasks. However, we would still fail to explain the incompleteness of the definitions, the absence of the distinction between necessary and sufficient conditions, the ambiguity of the terms, and how a contradiction between an imagetic and a propositional representation of concepts could occur in students' minds.
- 120. 107 Learning of Some Geometric Concepts In this section we will analyze four areas that have been the focus of recent research in geometric learning. In each of them researchers have detected evidence of students' non- standard mathematical knowledge. For each of these areas we will attempt to look at them through the eyes of category theory and provide an explanation of the sources of that knowledge. This analysis will permit us to uncover prototype effects with roots in distinct cognitive models, namely image schematic, metaphoric, metonymic models, and scripts. The Influence of Visual Prototypes and Metaphoric Models Prototype effects caused by image schematic models, characterized by a gestalt of a geometrical figure, are well known by researchers [13, 15,21,40,42,51,53]. They are the most simple cases ofprototype effects. The characteristics of these prototypes can be summarized by: (1) a preferred position; namely triangles, squares, rectangles, and parallelograms must have a horizontal base [13, 21, 51, 53]; (2) symmetry; for example, obtuse triangles with their bases in a smaller side are not recognized, or a right triangle is thought as a half-triangle [6,13,51]; (3) an overall balanced shape; namely students do not recognize "skinny" triangles, "pointy" triangles, or extremely small squares [6, 13, 15]. These characteristics provide a good description of the prototypical geometrical figures. Moreover, these gestalts do not require some characteristics that are significant from a standard mathematical point of view. For example, sides may be curved or "crooked" [6, 13,40]. There is also some evidence showing that students form such image schematic models even when only a verbal definition is given [51]. Students also show a substantial agreement about these models. In fact it may happen that in some cases these image schematic models are intermix with some metonymic models. In the case of triangles, Vinner and Hershkowitz [51] present evidence suggesting that the "overall balanced" isosceles triangles are taken metonymically as best examples of the whole category of triangles. This idea can be extended to other categories. It is reasonable to conjecture that a whole set of"overall balanced" rectangles may stand for the whole category of rectangles. There is an overall agreement about the sources of these models. As we have seen previously, upon entering school, children are able to identify balls, cans, boxes, and other shapes. Usually in school they learn the names of two-dimensional geometric figures. The first type of objects is composed of real world objects. Children learn them by manipulation or observation, and they are capable of identifying them regardless of their position. Objects of the second kind tend to be learned and used mainly in school. Research has shown [13] that
- 121. 108 geometric figures are usually presented in pictures that match the three characteristics of students' mental images described above: preferred position, symmetry, and balanced shape. In summary, children's prototypes are heavily influenced by the best exemplars shown to them in the school environment. Image schematic models, however, do not present a complete picture. There is evidence that there are also perceptual problems involved in the identification of geometrical figures, namely in the perception of right angles. Vinner and Hershkowitz [51] have shown some evidence suggesting that isolated right angles or right angles included in right triangles are more difficult to identify when none of the sides is horizontal. Some of their subjects used the strategy to turn the figure so that they could accomplish a better identification of the right angles. Other models may also be involved in the identification of geometrical figures. Previously we mentioned a metaphoric model used (or implicitly used) by the mathematical community when dealing with triangles, the "mountain metaphor". Here we will analyze two examples of students' metaphoric models. Burger [6], for example, reports that Bud (one of his subjects) explained that some of his triangles were different from others because they were "pointing that way [to the right, or down]" (p. 52). The idea that triangles point to a direction is what Johnson [17] would call a metaphoric model based on our kinesthetic image schema of pointing. For Bud, triangles (at least some triangles) are embodied, i.e., some of their properties "are a consequence of the nature of human biological capacities and of the experience of functioning in a physical and social environment" [19, p. 12]. This way to think about triangles is not Bud's particular model. Rather it is a social model that we intersubjectively share, because we are all able to understand Bud's point. In some contexts we ourselves would be willing to say that a triangle is pointing to a direction. Fuys, Geddes, and Tischler [13] report an example of another metaphoric model. One of their subjects (Gene), when asked if a square was a rectangle, answered "Na, that's a box" (p. 83). Of course Gene knew that, literally, a square is not a box. He was using a box as a metaphoric model of a square. Gene also thought that "the sides of a rectangle" referred to the vertical sides, whereas the horizontal sides were not "sides" but "top" and "bottom". Again he was using a metaphoric model. When we use English words to denote objects in the world that look like rectangles we may make this linguistic distinction. Both these metaphoric models were based on his experiences with objects on his environment. Effects Due to Prototypical Actions Investigations focusing on students' difficulties in drawing elements on a figure have also found prototype effects. Previous explanations have interpreted these phenomena as prototype
- 122. 109 effects produced by comparisons with a prototypical image of the expected drawing. In this section we will argue for a more dynamic interpretation that takes into account the actions that students are expecting to perform when asked to draw elements of geometric figures. We will focus on two important cases: drawing the altitude of a triangle and drawing the diagonals of a polygon. Researches on students' drawing of the altitude of triangles have focused on the determination of the characteristics of the triangle with which students experience more problems. Vinner and Hershkowitz [51] asked students to draw an altitude of 14 triangles. The triangles varied on their orientation, their type (isosceles, right, and obtuse), and whether the altitude to be drawn was going to be inside or outside the triangle. Their results show that the orientation has almost no effect on students' ability to draw the altitude. However, altitudes that fell on the side or outside the triangle, and triangles that deviate from isosceles triangle had a negative impact on students' performance. Vinner and Hershkowitz were able to produce a statistically significant sequence of increasingly difficult triangles on which to draw an altitude, as a consequence of their research. From the easiest to the more difficult, the sequence is: (a) isosceles triangle (non-equilateral) with altitude falling on the side that has different length, (b) scalene triangle with altitude falling inside the triangle, (c) obtuse triangle with altitude falling outside the triangle, and (d) right triangle. These researchers conducted a similar investigation using the diagonals of a polygon [15, 16]. It showed that in the case of concave polygons, only the diagonals inside the polygon, and that did not contain any side, were drawn. The models involved in these investigations are distinct from the ones previously described, because they involve action. The participants were not using image schematic models exclusively. They were expecting to perform a sequence of actions familiar in a certain context. This sequence of familiar actions fits exactly the definition of script mentioned previously [1]. An interpretation of Vinner and Hershkowitz's research using scripts may say that the typical script for drawing the altitude of a triangle occurs in the context of isosceles (non- equilateral) triangles. Students then seem to attempt to adapt this script to the other cases. When given an isosceles triangle the student draws the altitude of the triangle so that it falls perpendicularly on the middle of the side that has different length (74% of the students were able to do it). When the triangle is quasi-isosceles this script is still maintained, but it breaks down for many students producing prototype effects when the triangle is considerably non- symmetric (only 40% of the students answered correctly). In the case of the isosceles triangle the altitude coincides with the median and with the perpendicular bisector. This is no longer the case when the triangle does not resemble an isosceles triangle. The original script is changed by the students into two incompatible scripts. A considerable number of students choose to draw the median (20%) whereas a smaller number (7%) draws a perpendicular bisector to the side. The original script breaks down for an even greater number of students in the last two cases
- 123. 110 (only 32% and 30.5% of the students answered correctly in the last two cases respectively), and again some students choose the median (21 % and 20%) others the perpendicular bisector (7% and 9%). A similar interpretation can be produced in the case of the diagonals of polygons. Consequences for Research The previous theoretical discussion produces some consequences for research. The first consequence may be drawn at a theoretical level: researchers need to improve on the theoretical models of geometric thinking currently available, namely van Hiele theory. Although this theory needs to be reformulated in some ways (in its implicit cognitive model and in its definition of the levels), it is still very successfully used in research. A second consequence lays at a methodological level: there is a need to design research programs that complement the analysis of students' individual productions with observations of the contexts in which such productions are developed, namely the social interactions that occur in the classroom. A final consequence stands at a practical level and it is still in its interrogative form: what should teachers, textbook authors, and curriculum developers conclude from the previous discussion? The role played by these mental models seems to indicate that they are at the very heart of cognition. But it also seems to indicate that they are the very source of some of the students' difficulties with mathematical knowledge. References 1. Abelson, R.: Script processing in attitude formation and decision making. In: Cognition and social behavior (1. Carroll, & J. Payne, eds.), pp. 33-45. Hillsdale, New Jersey: Lawrence Erlbaum 1976 2. Anderson, J.: Cognitive psychology and its implications. San Francisco: W. H. Freeman 1980 3. Armstrong, S., Gleitrnan, L., & Gleitrnan, H.: What some concepts might not be. Cognition 13, 263-308 (1983) 4. Bloor, D.: Knowledge and social imagery. London: Routledge & Kegan Paul 1976 5. Brown, R. How shall a thing be called. Psychological Review 65, 14-21 (1958) 6. Burger, W.: Geometry. Arithmetic Teacher 32(6), 52-56 (1985) 7. Burger, W.: Thought levels in geometry. Interim Report of the study "Assessing Children's Development in Geometry". Oregon State University 1985 8. Burger, W., & Shaughnessy, J.: Characterizing the Van Hiele levels of development in geometry. Journal for Research in Mathematics Education 17,3148 (1986) 9. D'Andrade, R. : Cultural cognition. In: Foundations of cognitive science (M. Posner, ed.), pp. 795-830. Cambridge, Massachusetts: MIT Press 1989 10. Duveen, G., & Lloyd, B.: Introduction. In: Social reprsentations and the development of knowledge (G. Duveen, & B. Lloyd eds.), pp. 1-10.. Cambridge, UK: Cambridge University Press 1990 11. Fischbein, E.: Intuition in science and mathematics. An educational approach. Dordrecht D. Reidel 1987 12. Fisher, N.: Visual influences of figure orientation on concept formation in geometry. In: Recent research concerning the development of spatial and geometric concepts (R. Lesh & D. Mierkiewicz, eds.), pp. 307- 321. Columbus, Ohio: ERIC Clearinghouse for Science, Mathematics, and Environmental Education 1978
- 124. 111 13. Fuys, D., Geddes, D., & Tischler, R.: An investigation of the van Hiele model of thinking in geometry among adolescents. Brooklyn, New York: Brooklyn College, School of Education 1985 14. Gardner, H.: The mind's new science: A history of the cognitive revolution. New York: Basic Books 1985 15. Hershkowitz, R., Bruckheimer, M., & Vinner, S.: Activities with teachers based on cognitive research. In: Learning and teaching geometry, K-12, 1987 Yearbook (M. Lindquist & A. Schulte, eds.), pp. 222-235. Reston, Virginia: National Council of Teachers of Mathematics 1987 16. Hershkowitz, R. & Vinner, S.: Children's concept in elementary geometry - A reflection of teacher's concepts? In: Proceedings of the Eighth International Conference for the Psychology of Mathematics Education (B. Southwell, R. Eyland, M. Cooper, J. Conroy, & K. Collis, eds.), pp. 63-69. Darlinghurst, Australia: Mathematical Association of New Sout Wales 1984 17. Johnson, M.: The body in the mind. The bodily basis of meaning, imagination, and reason. Chicago: University of Chicago Press 1987 18. Lakatos, I.: Proofs and refutations: The logic of mathematical discovery. New York: Cambridge University Press 1976 19. Lakoff, G.: Women, fire, and dangerous things. What categories reveal about the mind. Chicago: University of Chicago Press 1987 20. Lakoff, G., & Johnson, M.: Metaphors we live by. Chicago: University of Chicago Press 1980 21. Mason, M.: The Van Hiele model of geometric understanding and geometric misconceptions in gifted sixth through eighth graders. In: Proceedings of the Eleventh Annual Meeting. North American Chapter of the International Group for the Psychologhy of Mathematics Education (A. Maher, G. Goldin, & R. Davis, eds.), pp. 165-171. New Brunswick, New Jersey: Center for Mathematics, Science, and Computer Education 1989 22. Matos, J. M. Changes in the van Hiele theory to accommodate prototype effects originating from image- schematic, metaphoric, and metonymic models. Paper presented at the Fifth International Conference on Theory of Mathematics Education (TME-5), Paderno del Grappa, Italy 1991 23. McDonald, J.: Accuracy and stability of cognitive structures and retention of geometric content. Educational Studies in Mathematics 20,425448 (1989) 24. McDonald, J.: Cognitive development and the structuring of geometric content. Journal for Research in Mathematics Education 20, 76-94 (1989) 25. Minsky, M.: A framework for representing knowledge. In: Mind design. Philosophy, psychology, and artificial inteligence (J. Haugeland, ed.), pp. 95-128. Cambridge, Massachusetts: MIT Press 1981 26. Moscovici, S.: La psychologie des representations sociales. Revue ｅｵｲｯｰｾｮｮ･＠ des Sciences Sociales, Cahiers Vilfredo Pareto 14(38-39),409-416 (1976) 27. Petrie, H.: Metaphor and learning. In: Metaphor and thought (A. Ortony, ed.), pp. 438-461. Cambridge, England: Cambridge University Press 1979 28. Quinn, N., & Holland, D.: Culture and cognition. In: Cultural models in language and thought (D. Holland, & N. Quinn, eds.), pp. 3-40. Cambridge, England: Cambridge University Press 1987 29. Randall, R.: How tall is a taxonomic tree? Some evidence for dwarfism. American Ethnologist 3,543-553 (1976) 30. Restivo, S.: The social relations of physics, mysticism, and mathematics. Boston: D. Reidel 1983 31. Rips, L., Shoben, E., & Smith, E.: Semantic distance and the verification of semantic relations. Journal of Verbal Learning and Verbal Behavior 12, 1-20 (1973) 32. Rosch, E.: Natural categories. Cognitive Psychology 4, 328-350 (1973) 33. Rosch, E.: Cognitive reference points. Cognitive Psychology 7, 532-547 (1975) 34. Rosch, E.: Cognitive representations of semantic categories. Journal of Experimental Psychology: General 104(3), 192-233 (1975) 35. Rosch, E.: Human categorization. In: Studies in cross-cultural psychology (N. Warren, ed.). Vol. I, pp. 1- 49. London: Academic Press 1977
- 125. 112 36. Rosch, E.: Principles of categorization. In: Cognition and categorization (E. Rosch, ed.), pp. 27-48. Hillsdale, New Jersey: Lawrence Erlbaum 1978 37. Rosch, E., & Mervis, C.: Family resemblances: Studies in the internal structure of categories. Cognitive Psychology 7, 573-605 (1975) 38. Rosch, E., Mervis, C., Gray, W., Johnson, D., & Boyes-Braem, P.: Basic objects in natural categories. Cognitive Psychology 8, 382-439 (1976) 39. Rumelhart, D.: Schemata: The building blocks of cognition. In: Theoretical issues in reading comprehension. Perspectives from cognitive psychology, linguistics, artificial intelligence, and education (R. Spiro, B. Bruce, & W. Brewer, eds.), pp. 33-58). Hillsdale, New Jersey: Lawrence Erlbaum 1980 40. Scally, S.: The effects of learning Logo on ninth grade students' understanding of geometric relations. In: Proceedings of the Eleventh International Conference Psychology of Mathematics Education PME-XI O. Bergeron, N. Herscovics, & C. Kieran, eds.), Vol. II, pp. 46-52. Montreal: ｕｮｩｶ･ｲｳｩｴｾ＠ de Montreal 1987 41. Semin, G.: Prototypes et representations sociales. In: Les ｲ･ｰｲｾｮｴ｡ｴｩｯｮｳ＠ sociales (D. Jodelet, ed.), pp. 239- 251. Paris: Presses Universitaires de France 1989 42. Shaughnessy, J., & Burger, W.: Spadework prior to deduction in geometry. Mathematics Teacher 78, 419- 428 (1985) 43. Smith, E.: Concepts and induction. In: Foundations of cognitive science (M. Posner, ed.), pp. 501-526. Cambridge, Massachusetts: MIT Press 1989 44. Smith, E., & Medin, D.: Categories and concepts. Cambridge, Massachusetts: Harvard University Press 1981 45. Tall, D., & Vinner, S.: Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics 12, 151-169 (1981) 46. Thorn, R.: Modern mathematics, does it exist? In: Developments in mathematical education. Proceedings of the Second International Congress on Mathematics Education (A. Howson, ed.), pp. 194-209. Cambridge: Cambridge University Press 1973 47. Tymoczko, T. (ed.): New directions in the philosophy of mathematics. Boston: Birkhl1user 1986 48. Vinner, S.: Concept definition, concept image and the notion of function. International Journal of Education in Science and Technology 14(3),293-305 (1983) 49. Vinner, S., & Dreyfus, T.: Images and definitions for the concept of function. Journal for Research in Mathematics Education 20, 356-366 (1989) 50. Vinner, S., & Hershkowitz, R.: Concept images and common cognitive paths in the development of simple geometrical concepts. In: Proceedings of the Fourth International Conference for the Pshychology of Mathematics Education (R. Karplus, ed.), pp. 177-184. Berkeley, California: University of California 1980 51. Vinner, S., & Hershkowitz, R.: On concept formation in geometry. Zentralblatt filr Didaktik der Mathematik 15(1),20-25 (1983) 52. Wertheimer, M.: Numbers and numerical concepts in primitive peoples. In: A source book of gestalt psychology (W. Ellis, ed.) pp. 265-273. London: Routledge & Kegan Paul 1938 53. Wilson, P.: The relationship between children's definitions of rectangles and their choices of examples. In: Proceedings of the Eighth Annual Meeting PME-NA, North American Chapter of the International Group for the Psychology of Mathematics Education (G. Lappan, & R. Even, eds.), pp. 158-162. East Lansing, Michigan 1986 54. Wilson, P.: Feature frequency and the use of negative instances in a geometric task. Journal for Research in Mathematics Education 17, 130-139 (1986) 55. Wilson, P.: Investigating the relationships of student definitions and example choices in geometry. In: Qualitative research in education: Substance, methods, experience (J. Goetz, & J. Allen, eds.), pp. 54-69. Athens, Georgia: College of Education, University of Georgia 1988
- 126. Examinations of Situation-Based Reasoning and Sense-Making in Students' Interpretations of Solutions to a Mathematics Story Probleml Edward A. Silver, Lora J. Shapiro Learning Research and Development Center, University of Pittsburgh, Pittsburgh, PA 15260, USA Abstract: This paper discusses a series of five studies examining over 800 students' solutions to division-with-remainders story problems. The findings from the studies suggest that students' difficulty in solving these problems is due, in large part, to a failure to engage in situation-based reasoning and sense-making in interpreting computational results obtained when solving the problem. Data from recent studies not only has provided direct evidence to support this hypothesis but also suggested that students' dissociation of sense-making from school mathematics is a major barrier to their engaging in or displaying the reasoning that leads to correct solutions. Keywords: context sensivity, division story problems, remainders, semantic processing, sense-making, situation-based reasoning Over the last twenty or thirty years, considerable attention has been paid to understanding the nature of skilled problem solving, particularly in mathematics. Among the many significant findings from this research is the importance of semantic processing in the initial stages of mathematical problem solving [5, 7]. Relatively little attention, however, has been paid to semantic processing at later stages of problem solving. In this paper, we discuss a series of empirical research studies that have specifically examined such semantic processing, as it occurs in children's interpretations of problem solutions. The research described here has concentrated on children's situation-based reasoning and sense-making involved in their interpretations and solutions to division-with-remainders story problems. These problems are particularly rich contexts in which to study situation-based reasoning and solution interpretations because different problems can be represented with the same symbolic division expression despite that fact that they have different answers, the 1Some of the research described here was supported by a grant to the Learning Research and Development Center from the U. S. Department of Education for the Center for the Study of Learning. The opinions expressed here are those of the authors and do not necessarily represent the views of the center or the department
- 127. 114 determination of which depends on aspects of the situational context and the quantities involved in the problem. For example, each of the following problems can be represented by the same expression, 100+40, but each has a different answer. Mary has 100 brownies which she will put into containers that each hold 40 brownies. 1) How many containers can she fill? 2) How many containers will she use for all the brownies? 3) After she fill as many containers as she can, how many brownies will be left over? Unlike the case for most other story problems encountered by students in elementary school, sense-making is not an optional activity in solving these problems, since merely performing an accurate computation cannot ensure arriving at a correct solution. Divison-with-remainder problems are not only cognitively complex; they are also quite difficult for middle school students to solve. Children's difficulty in solving them has been well-documented in a number of national and state assessments. For example, findings from the Mathematics portion of the National Assessment of Educational Progress (NAEP) [11] showed that only 24% of a national sample of 13-year-olds was able to correctly solve the following problem: "An army bus holds 36 soldiers. If 1,128 soldiers are being bused to their training site, how many buses are needed?" To better understand the basis for the observed difficulty students have in solving division- with-remainder problems, a series of studies has been conducted with students in sixth, seventh and eighth grades. The overall findings suggest that a major factor in students' failure to solve division story problems with remainders is their failure to engage in semantic processing in the later stages of problem solving. In particular, students apparently fail to relate their computational results to the situation described in the problem; i.e., they fail to engage in situation-based reasoning about the problem and the solution. Each of the studies described here has contributed to a deeper understanding of the role of sense-making in students' problem solving, and each is discussed briefly in the remainder of this paper. More detailed reporting of the procedures and findings cilll be found in the cited papers by Silver and his colleagues. Early Studies In the earliest study of students' difficulty with division-with-remainder problems [14], it was hypothesized that students were failing to attend to relevant information implicitly represented in the problem situation but not explicitly stated in the story text (e. g., no one is to be left behind; on one bus there may be some ｾｰｴｹ＠ seats). Several multiple-choice problem variants were created that made this relevant information more salient, and the performance of about 160
- 128. 115 middle school students was examined on these variants. Unlike other research on the role of semantic processing in mathematical problem solving, the focus in this study was on enhancing students' ability to relate the problem solution and the story context rather than on enhancing initial mappings between the story text and a mathematical model. The results of this study demonstrated that students' performance could be significantly improved by making explicit certain implicit information in the problem or in the required solution. In subsequent research, Silver and his colleagues [15, 16] continued to use multiple-choice tasks to examine students' difficulties with division-with-remainders problems. In these studies, over 500 students' performances were examined on three division problems, similar to the "brownie problems" presented earlier. The results indicated that students' performance on each type of problem was improved by the solving of the related division problems. In general, the results were consistent with the explanation that enhanced performance was due to students' increased sensitivity and attention to the relevant semantic processing involved in the target problem solution. In particular, experience with the related problems may have drawn attention to the need for relating the computational result either to the story text or to the story situation in order to obtain a final solution to the problem. Taken together, these results and the NAEP assessment findings suggested that students' failure to solve the division story problems was due, perhaps in large part, to an incomplete mapping among referential systems that were relevant to the problem. In particular, it appeared that students might map successfully from the problem text to a mathematical model (in this case, probably a division computation), to obtain a computational answer within the domain of the mathematics model, but fail to return to the problem's story text or to the situation to which the story text referred in order to determine the correct solution. The lack of semantic processing after obtaining a computational result was hypothesized as a major factor in the incorrect solutions generated by many students. Figure I presents a schematic representation of this hypothesized version of a student's unsuccessful solution attempt for this type of problem. In the case of a successful solution, it was hypothesized that the student would move from the story text to the mathematical model, and then after obtaining a computational result, map back from the mathematical model to the story text or to the story situation in order to decide what solution would make sense, given the contextual constraints imposed by the problem situation. In these early studies, the validity of the hypothesized models of correct and incorrect solution processes was suggested on the basis of indirect evidence available from examinations of students' performance on multiple-choice items. In order to validate these models more directly and to extend the examination of children's solutions and interpretations of division- with-remainder problems, further investigations have been conducted using interviews with individual students and open-response written task formats with groups of students. Two of these more recent investigations are discussed in the next section.
- 129. 116 ... ---------- .. Figure 1. Schematic representation of hypothesized unsuccessful solution Recent Studies Using a structured individual interview, Smith and Silver [19] examined the problem-solving performance of 8 middle school students on the following problem: Students at Greenway Middle School will go by bus to their end-of-year picnic at Kennywood Park. There will be a total of 1,128 students and adults. Each bus holds 36 people. How many buses are needed? Subjects participated in a 15-30 minute individual interview during which time they were given the problem, asked to read it aloud and requested to think aloud while solving it. Upon completing their solution, students were asked to critique an alternative numerical solution to the problem. Students who gave the correct response of 32 buses, were told that some people think the answer is 31 1/3 and they were asked to evaluate that answer. Alternatively, if students' who gave an incorrect response (e.g., 31 1/3), were told that some people think the answer is 32, and they were asked to evaluate that answer. The interview protocols from this study brought to light some interesting facets of students' sense-making with respect to the division problem. Furthermore, the interviews revealed that some students who gave answers that would have been incorrect, if they had chosen them on a multiple-choice task were able to offer interesting and valid interpretations of the apparently incorrect numerical answers - interpretations which would have remained invisible in multiple-choice responses. The second study sought to expand the findings of the interview study by using a paper- and-pencil, open-response question with a larger sample of students. Although not providing the same level of direct access to students' thinking and reasoning about the problem that
- 130. 117 individual interviews could provide, the open-response question format was viewed as a desirable alternative to multiple-choice questions for use with a large sample of students. Silver, Shapiro and Deutsch [18] presented a similar task to a mixed ability sample of 195 students at a large urban middle school: The Oearview Little League is going to a Pirate game. There are 5402 people, including players, coaches and parents. They will travel by bus and each bus holds 40 people. How many buses will they need to get to the game? Examination of the interview protocols and written responses of the students in both studies revealed several similar findings and observations with respect to the nature of students' solution processes and interpretations, to possible factors contributing to students' lack of success in engaging in situation-based reasoning and semantic processing with respect to problem solutions and interpretations, and to the validity of the proposed hypothesized models ofsuccessful and unsuccessful problem solutions. Solution Processes and Interpretations Solution Processes. Prior research had not attended to the solution processes used by students as they solved division-with-remainder problems. Therefore, these studies provided an opportunity to examine directly which of the many possible solution processes (e.g., drawing pictures, forming sets, etc.) were used by students to solve the problems. Although many solution processes could have been used, it was found that all students used an algorithm or combination of algorithms to solve the problems. Moreover, the vast majority of students used the long division algorithm, despite the fact that it was more difficult. Approximately one-third of those using long division made a calculation error. There was, however, a significant minority who used other algorithmic procedures, such as repeated addition or repeated subtraction. These students tended to be better able to execute the procedures flawlessly, and were somewhat more likely to obtain a correct solution Interpretations. In the interview study, three of the eight students provided unprompted, spontaneous interpretations of their whole number solutions in terms of the number of buses needed. This result was similar to that found in the written study, where about one-third of the students presented appropriate interpretations of their numerical calculations. The interpretations in this category included expected explanations, such as "a whole number of buses was needed because you cannot have a fraction of a bus" or "you need an extra bus so everybody can go". Situation-based interpretations of alternative solutions did appear, however they were generated by a relatively small number of students. For example, one student in the 2Two other versions of this problem were used with dividends of 532 and 554.
- 131. 118 interview study obtained a numerical answer of 31 1/3, but gave 31 as her solution. When when asked what she thought about 32 as a possible solution, she responded that she didn't choose 32 because you could "squish the left over kids on the bus" since the remainder is less than one-half. Another student applied everyday knowledge to make sense of his numerical answer of 31 1/3, by suggesting the solution of 31 buses and a mini-van to transport the remaining people. When asked what he thought of the answer 32, the student replied that 32 buses would also be an acceptable solution, but he continued to assert that all you really needed was 31 buses and a mini-van. Two students in the paper-and-pencil study supplied a similar explanation about buses and a mini-van to explain their non-whole-number answers. Influences on Performance and Processing Task Format and Setting. In the interview study, five of the eight students provided interesting interpretations of their numerical calculations. Three students did so spontaneously, and two others did so when the interviewer presented them with an alternative solution. Explanations were less frequently provided in the paper-and-pencil study, with only about one- third of the students offering appropriate interpretations. The differential frequency of evident sense-making by students in these studies may be due, at least in part, to features of the task format used in each study. In the interview study, students were given the opportunity to provide explanations spontaneously and when prompted to do so by the interviewer; whereas in the written study they had only one opportunity to provide an interpretation. In addition to less frequent opportunities for sense-making, the paper-and-pencil task may also have elicited fewer interpretations because it required students to provide written rather than oral explanations Many students were clearly unfamiliar with providing written explanations of their solutions to problems. Some students expressed objections to being asked to provide written explanations, and many others simply left blank the interpretation section of their answer paper. Although the paper-and-pencil, open-response format was intended to capture students' solution processes and their situation-based thinking and reasoning, it appears that this task format failed to do so for many students. As we shall discuss below, there is evidence that more situation-based thinking and reasoning occurred than was communicated in writing on students' papers. Until written explanations become a more prevalent feature of mathematics assignments, students are likely to express discomfort and display a lack of facility in completing such tasks. Students' reluctance or inability to provide written explanations was compounded by an apparent dissociation for many students between formal mathematics and sense-making. Dissociation of Sense-making from School Mathematics. Since the written task was administered by their mathematics teacher, during the normal mathematics class period, the
- 132. 119 students probably viewed it as a formal classroom exercise and, therefore, responded in a manner which they believed to be both mathematically correct and acceptable to their teacher. A follow-up discussion with teachers of the students who participated in the paper-and-pencil study suggested that the children actually engaged in more sense-making than they were willing to reveal in writing on their papers. In particular, reporting their recollections of discussions which followed this problem-solving activity in their classrooms, the teachers noted that many of their students argued vigorously for alternative solutions using a variety of interpretations for the remainder and explanations of how to represent their interpretations numerically; yet almost none of these creative interpretations appeared in their written solutions. For example, the teachers reported that students suggested getting a mini-van, taking a cab or using a car, instead of hiring an extra bus. In addition, they indicated that some students argued that an extra bus was not needed because some students would be absent and would not attend the game; some other students said that if a parent would come along, the extra kids could walk to the baseball garne because the school was close to the stadium. The teachers also said they saw some students who worked on the problem on their desk tops or book covers or scrap paper, that these students solved the problem by using repeated addition or other alternative algorithms, but having done this, then wrote long division computations on the "official" paper that was to be collected for review. The impact of students' perceptions of the importance of engaging in teacher-sanctioned, mathematically acceptable behavior was also noted in the interview study. Even in that non- classroom setting, students' work generally reflected an overemphasis on the application of formal algorithms and an underemphasis on the exhibition of behaviors that included situation- based reasoning and interpretations - behaviors which students seemed to view as unacceptable mathematical behavior. For example, one student struggled to use a teacher- approved procedure (long division) rather than an alternate procedure (repeated addition) that made more sense to him, despite the fact that he repeatedly encountered difficulties in executing the long division algorithm. Another student exhibited great concern about doing what her teacher "told us yesterday" (i. e., how to write the remainder as a fraction); her concern with recalling this procedure was so great that she failed to consider the relationship between the numerical answer and the solution to the problem being solved. These students appear to have learned in their mathematics instructional experience that theform of procedures and answers is at least as (or even more) important than theirfunction in solving a problem. In the written responses, it was also evident that students' attention was focused on matters of form, particularly the manner in which the numerical computation was to be carried out and how the final answer should be expressed, rather than the relationship between the numerical answer and the problem being solved. This was most evident in the number of students (54% of sixth graders and 36% of all the students) who provided detailed, step-by-step narrative descriptions of the procedures they used to obtain their numerical answers. An excessive
- 133. 120 emphasis on particular calculation procedures or notational form is likely to impede students from correctly solving problems in which an interpretation of the numerical response is needed. If issues of mathematical formalism are paramount in the students' attention during problem solving, then a strong motivation for interpreting the numerical result is less likely to exist. Moreover, if a student is narrowly focused on issues ofrequired procedural or notational form, then the student may not even recognize the need to interpret the obtained solution. To engage in such processing, a student must perceive the need to interpret the numerical solution. The fact that many students were concerned more with fonn than function may be a reflection of the imbalance in the middle school curriculum and the predominance of instruction focusing on rote computational procedures. Students' tendencies to use memorized procedures rather than sense-making to solve the problem and their reluctance to share their creative interpretations and alternate solution processes, probably reflects their views of what is considered appropriate mathematics - prescribed algorithmic procedures that have little or no connection to real world, non-classroom considerations. Students' tendencies to dissociate formal, school mathematics from sense- making have also been noted by Cobb [8] and Schoenfeld [13]. These tendencies may also be indicative of an unwritten didactical contract [4] between students and their mathematics teachers - a contract which obliges teachers to provide for students procedures and knowledge that can be memorized and obliges students to apply these procedures without reference to any of their other non-classroom experience or knowledge. The existence of such a contract would explain the apparent dissociation of school mathematics from sense-making for many of the students sampled and the apparent dissociation of personal invention, creative thinking and situation-based reasoning from acceptable mathematical activity for many others. Until students see a richer connection between their situation-based reasoning and the kind of thinking that occurs in mathematics classrooms, they are likely to be inhibited from being successful in solving problems that require sense-making. Hypothesized Solution Models Based on the above findings, there is considerable direct evidence that students' failure to interpret their computational results is a major barrier to their obtaining a correct solution. The data thus provide general support for Silver's hypothesis that students' inabilities to successfully obtain a solution to the division-with-remainders problem was due to their failure to engage in semantic processing the later stages of the solution process. Similarly, the proposed model of problem-solving success was also generally supported by the responses of students who solved the problem correctly, since their solutions tended to reveal an appropriate interpretation of the numerical solution.
- 134. 121 The only data that were in apparent conflict with the hypothesized models involved some students who were able to obtain correct solutions without giving explicit evidence of sense- making following their numerical solution. These were students who did not use the long division algorithm but used instead an alternative algorithm, such as repeated addition or repeated subtraction. Although more mathematically primitive than the long division algorithm, these alternate procedures may have been more intuitively linked to the situation described in the problem. For example, adding up or subtracting down more naturally parallels the act of loading individuals onto a bus. Thus the ｾｴｵ､･ｮｴｳ＠ who used these algorithms, unlike those who used long division, may have utilized these procedures as a natural consequence of their situation-based reasoning about the problem. It is unlikely that their use of these procedures can be attributed to schoolleaming, since these procedures would not have been taught as a formal procedure for solving a division problem. Rather than suggesting counter-evidence for the correct solution model proposed earlier, the fact that this group of students did not offer a written explanation to accompany their solutions suggests not only that the procedures themselves (Le., repeated addition and repeated subtraction) implicitly contain a situation-based interpretative framework (Le., loading onto buses or subtracting from the group waiting to be loaded) but also that the answers obtained from such procedures require no further explanation or interpretation. Future Studies The series of studies reported here has contributed not only to our understanding of students' difficulties in solving division-with-remainder problems, but also to our understanding of the factors which influence semantic processing and problem-solving performance. Our findings on the nature and form of students' interpretations and explanations, the context sensitivity of students' responses, and the general dissociation of sense-making from school mathematics suggest that further consideration and investigation of these issues will be fruitful. In studies of problem solving across different contexts (e.g., [1)), and studies on the development of children's understanding of long division (e.g., [2)) other investigators have reported an apparent relationship between students' use of solution strategies and certain contextual features of the problem situation. These findings, like the results of the studies reported here, suggest the value of examining more closely both general and specific relationships between problem-solving procedures and situational contexts. In particular, it would be natural to examine different situational contexts which impact on students' sense- making. For example, problem contexts could be varied, as could the context in which the task is administered. Examples of the latter might include interview formats or paper-and-pencil tasks designed to require students to think about alternative solutions or interpretations through the use of prompts. For example, Curcio and DeFranco (F. Curcio, written communication,
- 135. 122 January 26, 1991) presented students with division-with-remainder problems using two different formats for their administration: in one format students critiqued the written work and solution of a hypothetical student, and in the second setting they acted out the actual ordering of buses for a school trip. Although limitations in the design of the study hinder interpretation and generalization of their results, the finding that students exhibited different kinds of reasoning on the two tasks are nevertheless intriguing. We are currently analyzing students' responses to alternative paper-and-pencil formats from another study specifically designed to elicit interpretations by having students consider and evaluate a proposed solution to the "bus" problem. Another potentially promising avenue of investigation would involve the extension of the hypothesized models ofcorrect and incorrect solutions to multi-step problems. These problems are difficult for children and, like the division-with-remainders problems, a fundamental source of the difficulty may be the semantic processing requirements imposed by the problems. In order to complete successfully a chain of computations necessary to produce a successful solution to a multi-step problem, the solver may need to interpret the result of each calculation with respect to the ultimate goal of the problem and/or even with respect to the situation described in the problem. Thus, it is likely that models proposed here can be generalized to multi-step problem solving. In addition to continued research related to the general issue of understanding how and when students connect mathematics to situations, the findings of these studies suggest the need for research on curricular and instructional changes in mathematics classrooms. Consistent with calls (e.g., National Council of Teachers of Mathematics, [12]) for increased instructional attention to mathematical reasoning and problem solving, as well as greater emphasis on communication and explanations, a number of efforts are underway to create mathematics programs and classroom environments in which these features are emphasized. For example, teachers and researchers involved in schools in the QUASAR (Quantitative Understanding: Amplifying Student Achievement and Reasoning) Project [17] and teachers involved in other efforts with similar goals (e.g., [3, 6, 9, 10]) are developing and implementing challenging instructional activities in which sense-making and communication are featured as an integral part of the curriculum. Findings from these research programs will serve to further our understanding of the complex relationship between mathematical problem solving, situation- based reasoning and sense-making - this understanding is critical to our meeting the goals of the broad-based reforms currently called for in mathematics programs and assessments. References 1. Baranes, R., Perry, M., & Stigler, J.: Activation of real-world knowledge in the solution of word problems. Cognition and Instruction 6(4), 287-318 (1989) 2. Boero, P., Ferrari, P., & Ferrero, E.: Division problems: Meanings and procedures in the transition to a written algorithm. For the Learning of Mathematics, 9(4),17-25 (1989)
- 136. 123 3. Bransford, J., Hasselbring, T., Barron, B., Kulewicz, S., Littlefield, J., & Goin, L.: Use of macro<ontexts to facilitate mathematical thinking.. In: The teaching and assessing of mathematical problem solving (R. Charles & E. Silver, eds.), pp. 125-147. Reston, VA: LEA & NCTM, 1989 4. Brousseau, G.: The crucial role of the didactical contract in the analysis and construction of situations in teaching and learning mathematics (Occasional Paper No. 54). Bielefeld, FRG: UniversiUlt Bielefeld, Institut fUr Didaktik der Mathematik, November 1984 5. Carpenter, T., Hiebert, J., & Moser, J.: Problem structure and first grade children's initial solution processes for simple addition and subtraction problems. Journal for Research in Mathematics Education 12,27-39 (1981) 6. Carpenter, T. P., Fennema, E., Peterson, P. L., Chaing, C., & Loef, M.: Using knowledge of children's mathematics thinking in classroom teaching: An experimental study. American Educational Research Journal 26(4),499-532 (1989) 7. Chi, M., Glaser, R., & Rees, E.: Expertise in problem solving. In: Advances in the psychology of human intelligence (R. Sternberg, ed.), pp. 7-75. Hillsdale, NJ: Lawrence Erlbaum Associates 1982 8. Cobb, P.: The tension between theories of learning and instruction in mathematics education. Educational Psychologist 23 (2), 87-103 (1988) 9. Cobb, P., Wood, T., & Yackel, E.: Constructivist approach to second grade mathematics. In: Constructivism in mathematics education (E. von Glasersfeld ed.). Dordrecht Kluwer 1991 10. Lampert, M.: Connecting mathematical teaching and learning. In: Integrating research on teaching and learning mathematics: Papers from the First Wisconsin Symposium for Research on Teaching and Learning Mathematics (E. Fennema, T. Carpenter, & S. Lamon, eds.), pp. 132-165. Madison: University of Wisconsin, Wisconsin Center for Education Research 1988 11. National Assessment of Educational Progress: The third national mathematics assessment Results, trends and issues. Denver, CO: Author 1985 12. National Council of Teachers of Mathematics: Curriculum and evaluation standards for school mathematics. Reston, VA: Author 1989 13. Schoenfeld, A.: When good teaching leads to bad results: The disasters of"well taught" mathematics courses. Educational Psychologist 23(2), 87-103 (1988) 14. Silver, E.: Using conceptual and procedural knowledge: A focus on relationships. In: Conceptual and procedural knowledge: The case of mathematics (1. Hiebert, ed.), pp. 181-189. Hillsdale, NJ: Lawrence Erlbaum 1986 15. Silver, E.: Solving story problems involving division with remainders: The importance of semantic processing and referential mapping. In: Proceedings of the Tenth Annual Meeting of PME-NA (M. Behr, C. Lacampagne & M. Wheeler, eds.), pp. 127-133. DeKalb, IL: Author 1988 16. Silver, E., Mukhopadhyay, S., & Gabriele, A.: Referential mapping and the solution of division story problems involving remainders. Paper presented at the annual meeting of the American Educational Research Association, San Francisco, Maich 1989 17. Silver, E., Smith, M., Lane, S., Salmon-Cox, L., & Stein, M.: QUASAR (Quantitative Understanding: Amplifying Student Achievement and Understanding) project summary. Learning Research and Development Center, University of Pittsburgh, Fall 1990 18. Silver, E., Shapiro, L., & Deutsch, A.: Sense-making and the solution of division problems involving remainders: An examination of students' solution processes and their interpretations of solutions. Paper presented at the annual meeting of the American Educational Research Association, Chicago, April 1991 19. Smith, M. & Silver, E.: Examination of middle school students' solutions and interpretations of a division story problem. Paper presented at the annual meeting of the American Educational Research Association, Chicago, April 1991
- 137. Aspects of Hypothetical Reasoning in Problem Solving Pier Luigi Ferrari Dipartimento di Matematica, UniversitA, Via L. B. Alberti, 4, 16132 Genova, Italia Abstract: In some problem situations, conditional fOnDs, as well as other complex syntactical fOnDs, have been widely found in children's reports of their own resolution procedures. This happens mainly in arithmetical problems (when children do not yet know algorithms to solve them) or in complex geometrical problems (with a crucial presence of figures). In most cases, these fOnDS may be related to hypotheses stated and checked by the pupil. The ability at producing conditional fOnDS in problem-solving environment seems to be correlated to problem-solving skills. The functions of hypothetical reasoning in mathematical problem- solving are investigated and the interplay between the production of hypothetical reasoning and the comprehension oflogical connectives is discussed briefly. Keywords: hypothetical reasoning, conditional fOnDS, verbal reports, logical connectives, problem solving strategies 1. Introduction 1.1. Verbal Language in the Resolution of Complex Problems It is widely accepted that, in primary school, pupils generally meet with difficulties when dealing with complex problems, i.e. problems requiring a strategy with more than one step. At this regard the research of the Genova Group [2, 5) has pointed out the importance of the mastery of verbal language for all the pupils related to: (i) the comprehension through the text and the representation of the problem situation; (ii) the design of a resolution strategy; (iii) the need of pupils to keep in touch with the problem situation while perfonDing a procedure. (iv) the representation, comparison and discussion in the classroom of the strategies produced by pupils. As regards (ii), verbal language seems to play a major function mainly in complex problems, or in problems pupils do not know any algorithm to solve; pupils do not rely on
- 138. 126 verbal language very much in simple problems (when a good representation, possibly iconic, of the problem situation, is often a good starting point to work out an answer) or when 'deterministic' algorithms are available which allow them to perform some calculation with no need of analyzing the data carefully before planning a strategy. In other words, verbal language is a powerful tool in order to design strategies in complex problem situations, as its application seems more general than methods based on iconic representations, which generally do not work, by themselves, in all the problem situations; some cases of pupils who can easily solve simple problems but come to a standstill when dealing with more complex ones may be a clue in this direction. Moreover, the systematic use of verbal language in the analysis of problem situations and the design of a strategy may prevent pupils from dangerous attitudes and behaviors such as performing any algorithm whatever without attending to the data and so on. On this account in our project pupils are forced to use verbal language in problem solving also by means of suitable problem situations (such as problems without numbers and so on). As for (iv), it is crucial the transition from a strategy built and performed step by step to a strategy described in verbal language, with the connections between the successive steps made explicit (in other words, from a strategy as a tool to a strategy as an object). The discussion and comparison of strategies has proved an important factor of improvement of problem solving performance, in particular as regards those pupils who generally meet with difficulties when dealing with new problem situations. The skills involved in this transition are also relevant as prerequisites for computer aided problem-solving. Actually, it has been observed [10] that pupils aged 13 to 14, in their first approach to informatics, meet with difficulties mainly related with the verbalization processes of their thinking, the mastery of connectives and specific linguistic structures and the management of different languages. In particular, the use of connectives to describe processes within situations logically organized according to constraints not depending upon the pupils and the teacher (such as the working of a machine) may be an intermediate step to a more formal use of them in programming. 1.2. Making Hypotheses and Problem-Solving Many clues suggest that, to plan a strategy as a whole, it is crucial the mastery of some connectives, in order to explicitly relate distinct statements, and, in particular, the possibility of making and managing hypotheses. This happens, for example, in the transition from children's trial and error strategies designed step by step to the mastery of a more effective, algorithm for division (such as the so-called 'Greenwood' - or 'canadian' algorithm) [5]. In this situation pupils when trying to plan their strategies are forced to take into account, for each trial, the possibility of a positive or negative remainder. This happens also when pupils need to change
- 139. 127 their strategy (because the first one they have tried to plan does not work) or to perform the same strategy on different data. In brief, there is a great deal of situations in which the mastery of advanced syntactical forms (in particular, conditional forms) is useful in order to design, manage, change, perform or compare a resolution procedure. Examples of this kind will be discussed in 3.2.. This is interesting enough for research, as in the last years we have observed some correlation between problem-solving skills and the ability at producing conditional forms in suitable settings. These aspects are discussed in § 4. 1.3. Goals of the Research The main purposes of the research I am reporting are: - to state the conditions under which the production of conditional forms is more frequent; - to analyse the functions of hypothetical reasoning in problem-solving; - to analyse the interplay between the mastery of complex syntactical forms (including conditional forms) and problem-solving skills; - to discuss the interplay between the production of hypothetical reasoning and the comprehension of logical connectives, with particular regard to implication (as studied, for example, by O'Brien at al. [12]. 2. The Framework of the Research 2.1. The Educational Context: the Stress on Verbal Language Acquisition In all the paper, until further notice, I will refer to pupils who have experienced the Genova Group's Project for primary school. Related to the object of this study, we point out the following characteristics of the Project (a further information on the educational context in which the research is carried out can be found in [1, 2, 5, 8]): -long-term planning of the educational work in all the subject areas (usually with the same teacher from grade 1 to grade 5); - familiarity with verbalization processes, due to activities such as reporting in written form the strategies used to solve a problem as well as a discussion or an experiment performed in class; - stress on the construction of linguistic competence to describe procedures or relationships among facts; in particular, the stress on the mastery of connectives (such as 'before', 'after', 'when', 'while', 'because', 'if...then', ...), focusing on the semantic point of
- 140. 128 view; for example, this is done, from the age of 6, by means of activities like verbal reporting in situations that are logically organized according to constraints not depending upon pupils or teachers (such as the working of a machine); - acquisition of the algorithms for addition, multiplication, subtraction and division at the end of processes of progressive schematization and generalization of the heuristic strategies built by the pupils. 2.2. Methodological Questions: Making Hypotheses Versus Using Conditional Forms From the beginning of the research we have dealt with the problem of the interplay between making hypotheses (as a mental process) and the linguistic forms through which hypotheses are expressed. It seems that they should not be mechanically connected. The educational context described in 2.1. (and in particular, the stress on verbalization processes and descriptions of processes and situations organized according to external constraints) may induce pupils to report their thought more accurately than in other classes and, from the other hand, may prevent them from a superficial or semantically unrelevant use of connectives. As regards the first issue, pupils' written reports are produced so that Ericsson and Simon's conditions on the reliability of verbal reports [7] are satisfied, since pupils are only requested to describe their procedure in general and not to perform selective, generative or inferential processes. Verbalizations are generally performed while pupils are dealing with a problem or immediately after they have completed the task. Anyway, we have found pupils who (very likely) make hypotheses but do not report them when verbalizing their procedures (what does not implies the unreliability of the reports - for details see § 5.) and others who write down conditional forms with another function than making hypotheses. For example, there are pupils who write down conditional forms even if, strictly speaking, they are not related to any hypothesis (see § 3.2.); there are also children (at the age of 11-13) who can use complex syntactical forms (and, in particular, conditional forms) when talking freely with some particular adult person, such as their grandmother and so on, but they cannot use them to describe any relationship or process. It is well known the influence of 'conversational styles' on the production and comprehension of linguistic forms, and in particular connectives (for example see [13]). These behaviors are much more frequent out of our project.
- 141. 129 2.3. Experimental Design Through all the paper I refer to research carried out within the Genova Group's Project, based on a lot of materials collected in the last years. These materials, produced by pupils from the age of 8 to 10, are: - children's verbal traces written down during the resolution process of applied mathematical problems, with or without numerical data; - children's written reports of their own procedures for applied mathematical problems, with or without numerical data; - children's written descriptions of everyday-life processes (such as the preparation of a cup of coffee and so on); - other texts produced in non mathematical situations; - recordings of interviews (related to how children think they solve problems). 3. Hypothetical Reasoning: Conditions and Functions 3.1. Conditions for the Production of Hypothetical Reasoning Through all the paper, by 'spontaneous' I mean freely expressed in an educational context, as sketched in 2.1., which deliberately guide pupils' behaviors to the production of complex linguistic forms. The spontaneous production of conditional forms in problem-solving seems to depend upon the following factors: - the context of the problem: if pupils regard the problem situations as artificial and do not master the meanings involved, they often produce stereotyped answers (if the problems are trivial) or come to a standstill (if they are difficult); in situations like these pupils hardly engage themselves in the endeavor of stating working hypotheses or analyzing data and conditions which are inherent to the situation. Very often pupils, even if they can find a solution, do not compare it in a critical way with the situation (see also [3]); - the nature of the task: for example, hypothetical reasoning hardly can be found when the task is to solve a problem which needs only recalling and applying a resolution procedure the pupil has already learn and used in analogous situations, or when the resolution procedure is constructed by a sequence of steps closely related each other without the arising of alternatives or difficulties, or when the numerical data distract the pupils from reflecting on the procedure; - the time of the performance: when the verbalization is performed while a problem is being solved, a wider presence of hypothetical reasoning and other syntactically complex forms has been observed; if it happens after a resolution procedure has been found, I have observed a
- 142. 130 great amount of sequentially-structured texts, with a wide usage of temporal connectives (such as 'then', 'afterwards' and so on), which are most likely referred to both temporal ordering and logical consequence. In other words, conditional forms have been found mainly in arithmetical problems (when children do not yet know algorithms to solve them) or in complex geometrical problems (with a crucial presence of figures). No conditional form has been observed in simple problems when children already know the algorithms to compute the arithmetical opemtions. The availability of a "deterministic" algorithm often prevents children from producing well organized verbalizations, with use of conditional or other complex syntactical forms. The need for explicit reasoning may emerge if the pupil realizes that no algorithm he knows works to get a solution (and he does not come to a standstill) or if he is comparing and discussing the procedures produced in class. More generally, the sort of cognitive detachment and the stress on communication purposes realized when comparing strategies seem to foster the production of complex syntactical forms. 3.2. Functions of Hypothetical Reasoning in Problem-Solving The following functions, which are not clear-cut and do not exclude each other, have been recognized: (A) individuation of conditions upon which depend, at least partially, the resolution strategies of a problem; this happens often when pupils deal with problems involving actual measurements or other 'strong' interactions with physical world; for example, in a problem involving the measurement of an angle (dmwn on the paper) with a goniometer, the pupils must deal with the length of the sides of the angle, which may be too short to perform the measurement immediately: "ifthe sides are too short, I must extend them..."; or when planning an experiment (related to the measurement of the shadows of a nail in different places) pupils must deal with a lot of problems (the thickness of the board, the length of the nail, the horizontality of the board,...) they take into account by means of conditional forms: " If the board is not as thick as the other, do not drive in it the nail by more than 2 cm ..." ; "If it happen that the board is lying on the gravel, then level the ground until it becomes ...."; "If you cannot go on levelling it..., then lay the board .. .". In situations like these pupils seem to use conditional forms to express hypotheses in order to become progressively aware of the data and the conditions involved in the problem. For details, see [8]. (B) explorative hypotheses, which are found often in the transition from trial-and-error strategies designed step by step to more effective algorithms or procedures. For instance, in the problem of sharing out the total cost of a trip out of the pupils of a class, some of them make hypotheses as follows:"The total cost of the trip is 74000 lire and there are 19 ofus. If any
- 143. 131 child pays 1'000 lire it makes 19'000, too little; then let us try with 2'000 lire: it makes 38.000. It is not yet enough. If any child pay 10'000 lire, it makes 190'000, which is too much, ..." ; so they go on, by trial and error, and come to "little less than 4'000 lire" and then, specifying "how much one has to take off" they come to "about 3'900 lire"). When building a strategy to compute how many sheets of 21 cm need to be arranged next to each other on the walls of the room in order to cover the length of the wall completely, good problem-solvers use hypothetical reasoning to manage the possibility of a positive or negative remainder: "/ try with 34 sheets 21 cm long; ifit is larger (than the wall), / try with a lower number, or, if/have spared some space on the wall, / try with a greater number ...tt [8]. Explorative hypotheses are crucial in the planning of a resolution strategy: the pupils can put themselves in a particular case (stating some particular hypothesis) and try to draw some conclusion; this may be useful as a step of a process of progressive approximation to the solution or just to make explicit some relationship involved in the problem situation. In the example at the beginning of this paragraph the first trial ("Ifany childpays 1'000 lire it makes 19'000") is not only the first step in an approximation process, but also helps the pupil to grasp the mathematical relationship involved in the problem. (C) "justification" hypotheses, which are made by some children after they have chosen and performed a strategy. For example, some children, related to the task of scale representations of the height of all pupils in the classroom on a sheet on a sheet 42 squares large, write reports like the following: "The tallest pupil is Roberto, who is 140 cm tall; if/ take 7 cm = 2 squares, Roberto wants 40 squares, all right; this worksfor all the other children as well,for if/consider another pupil. he is shorter than Roberto, and his height becomes less than 40 squares". The first 'if' is related to an 'explorative' hypothesis, whereas the second one has a function at least partially different, since the pupil has already found his answer and needs no further 'exploration' but only 'logically' justifying what he/she has done. The second 'if' selects a situation which is 'potentially' dangerous for the whole procedure and allows the pupil to focus on it. This use of conditional forms may be found also in 'working' mathematical proofs (not in formal deductions!). (D) Change of data; after the construction of a strategy to solve a given problem, conditional forms are used to deal with the same problem situation with different initial data: "we have made 20 biscuits and we have spent...; ifwe make 30 biscuits, we spend...". In this situations, pupils progressively become aware that they need not to build once again the whole procedure, but that it is enough to perform the same procedure on different data. They sometimes remark that it is not necessary "to clear all the blackboard", but it is enough to clear the old numbers, write down the new data and perform the calculations without modifying the procedure. This function of conditional forms is closely related to the emergence of the algorithm as an autonomous object, separated from the data on which it is performed. For details see [8].
- 144. 132 (E) 'pleonastic' hypotheses; in some situations, mainly in complex problems or in problems involving proportion or comparisons, there are children who regard the data as hypothetical even though they are given under no condition. For example, some pupils write down reports such as: "iftoday is April 28, from April 28 to may there are 8 days" or even "if a child must pay half the full fare, I multiply the childfare by the number ofchildren" and so on. There is some evidence supporting the relevance of complexity: for example, if the task is to state how much meat a dog eats in a month, knowing that he eats 3 hg every day, no pupil use pleonastic 'if' to manage the multiplication: "The dog eats 3 hg every day, so in a week he eats 3 x 7 = 21 hg, ..."; if the question is made more complex by introducing further questions (related to the cost of meat and so on) some of the same pupils write down reports such as "If the dogs eats..., in a week he...and ifmeat costs..." . This behavior has been found in a lot of different problem situations. More research is needed to explain all this. The 'selectivity' in the use of this fonn seems to exclude that this 'if' does not correspond to any mental process of the pupil or to any hypothesis. One may conjecture that pupils uses pleonastic 'if to insert the data in a sort of complex elaboration structure which allows them to focus on data and, at the same time, on some inferential step they want to point out. There is possibly a connection between this 'pleonastic' use and the use discussed in (D); nevertheless, pleonastic uses of conditional fonns have been found also before pupils have dealt with problems with modified data. (F) Comparison of strategies; sometimes, in our classes, children, after they have solved a problem individually, are requested to compare their strategies each other; in this situation, fonns of hypothetical reasoning as the following have been observed: "I have done this way... if I had done as Claudia I should have found:... Then the reasoning ofClaudia needs longer and more difficult calculations than mine.. .". This the only context in which even poor problem-solvers can produce conditional fonns spontaneously and frequently. 4. Verbal Reports, Hypothetical Reasoning and Problem Solving Skills I have observed a good correlation between problem-solving skills and the ability at producing conditional fonns when describing their problem-solving strategies [8]. In particular, all pupils classified as good problem-solvers can use conditional fonns (in problem solving or in other contexts), even they do not always use them when reporting a resolution procedure. I remark that by 'good problem-solvers' I mean pupils who are able to give acceptable solutions to most of the problems (either contextualized or not) they are administered during the year, not regarding too much the quality of the resolution processes or the reports. By 'poor problem solvers' I mean pupils who almost never are able to design some strategy to solve complex problems and often meet with difficulties even when solving simple problems.
- 145. 133 When describing everyday-life processes, good problem-solvers generally display a wide use of conditional or other advanced syntactical forms. Poor problem-solvers generally do not use any complex syntactical construction in order to connect different statements, but often just state some external constraints of the process. For example, to describe the preparation of some coffee, a lot of good problem-solvers write statements such as: "put water into the coffee-pot till when it isfull.", whereas poor problem-solvers use mainly statements such as "you must put enough ofwater into the coffee-pot, in order to prepare the right amount ofcoffee". Poor problem-solvers almost never use conditional forms in their resolution procedures, and hardly in descriptions of processes. I have found conditional forms spontaneously produced by poor problem-solvers mainly in the comparison of strategies. The transfer of these skills to problem solving has generally proved very difficult. This can strengthen the hypothesis that problem solving environment is not very suitable to force the acquisition of hypothetical reasoning, but offers a good opportunity to refine, extend and develop it. 5. Space, Time and Hypothetical Reasoning In pupils' written reports very often temporal connectives (such as 'when' 'till when' and so on) are used in place of proper conditional forms (such as 'if...then...'). The use of a temporal connective instead of a conditional one seems to depend upon the meaning of the situation. When describing the working of a slot-machine children uses either 'if... then.. .' or 'when' to express conditional controls; this seems not to be at random, because to test each coin 90% of children who insert a conditional control at that step use the form 'if. .. then...' ("if the coin is 'good', the machine .. ."), whereas to test the total amount of the coin already inserted some 80% of the children who insert at least a conditional control use 'when' ("When the amount of the coins inserted is 400 lire, the machine ..."). When describing other processes (e.g. the preparation of a cup of coffee) about 40% of the children uses almost once constructions such as "when P, Q" or "P till when Q" (e.g., "when the coffee-pot is ready, put it on the fire" or "put water into the coffee-pot till when it is full"). About 40% uses almost once 'if P then Q' (e.g. "if the water is not yet boiling, wait a bit"). About 15% of the children (all good problem-solvers) use both constructions. These children, when take into account the final amount of coffee use 'if...then.. .' ("if the coffee is not enough, I must put more water, if it is too much I must put less water"), whereas 'when' is mainly used related to more 'intrinsic' steps of the process, which are more difficult to master from the outside ("when water boils, I must put the fire out"). These examples show also that temporal and logical relationships are closely connected for children. There is no clear-cut distinction between them, even though in some situations seem to prevail logical aspects, and in others temporal ones.
- 146. 134 At this regard, a first analysis of pupils' written reports of their strategies, in a lot of situations including arithmetical and geometrical word problems (at the age of 9-10), has shown two main patterns of resolution procedures, with regard to the structure of the text. There are pupils who write down their procedure (while or after solving the problem) as a sequence of their own operations: "Before I compute, then I do ... and afterwards I find .. .". They hardly state some relationship concerning the problem situation explicitly and do not use complex syntactical forms often but organize their reports by means of only temporal connectives (such as 'before', 'afterwards', and so on). This temporal organization of the text seems to embody even the 'logical' organization of their procedure. Some of these pupils are very clever at solving complex problems and often succeed even in changing quickly their strategy when the first they have tried does not work, and it is likely that they actually make hypotheses when solving a complex problem even they do not report them. On the other hand, there are pupils who write down some relation among the 'objects' involved in the problem situation, and pay less attention to the temporal organization of their performance. Among 4th graders the first group is larger than the second and in either group there are good and poor problem solvers. The second style seems more useful in order to improve performance. Actually, all the poor problem solvers following the first style who improve their performance during the year change their style, for example introducing some statement (sometimes as an equation or a diagram, more often in verbal language) to describe the problem situation or to organize their procedure from the 'logical' - not only temporal - point of view. On account of this, the mastery of some syntactical construction, which allow children to state different kinds of relationship among facts, such as, for example, causal relationship, has proved to be a factor of improvement in problem-solving performance for either group. In geometrical problems we have found a wide production of conditional forms. Geometrical figures have emerged as crucial in order to mediate between the problem situation and the need to plan a strategy. One figure may embody a complex procedure (such as the iterative construction, square by square, of a polygon satisfying some given properties) or an 'action-proof' (such as the equivalence of 2 polygons). This function of spatial representations may allow even some poor problem-solvers to make some hypothesis even though they do not yet master the corresponding linguistic forms (for details, see [6]). 6. Hypothetical Reasoning and the Learning of Logical Connectives A thorough discussion about the learning oflogical connectives is perhaps out of the scope of this volume. A lot of studies has shown the difficulty of pupils of almost any age in understanding logical connectives, in particular implication (for example, [12]). These
- 147. 135 difficulties might be justified by the tension between the everyday-life meaning of connectives (in natura1language environment) and the 'logical' one, which are quite different. I want only to sketch one example in which the research I have reported and the research on the learning of logical connectives have different outcomes. Some results of Markovits [11] show that the use of drawings does not improve at all children's understanding of implication (in a 'logical' sense). On the other hand, in our project a significant number of pupils, at the age of 10, can manage conditional forms in order to organize a resolution procedure in a large set of problem situations, and the presence of drawings seems to improve their ability in making hypotheses, as happens in geometrical problems. This may be justified by the fact that from a (classical) logical point of view the semantics of an implication 'p->q' is a purely combinatorial function of the truth-values of the propositions p and q, whereas pupils seem to use spontaneously conditional forms with a much richer meaning, including temporal and spatial meaning. This meaning is partially conveyed by everyday-life verbal language, whose semantics are quite different from the semantics of the formal languages of mathematical logic. This may explain the 'curious' findings of Markovits, as far as most likely the use of 'statical' drawings may focus on the combinatorial, truth-functional meaning of implication, whereas verbal language alone may anyhow preserve some other kind of meaning more understandable by pupils. 7. Further Remarks From the data I have reported in this paper seem to emerge the following ideas, which are relevant from the didactical point of view: - processes of 'progressive schematization' from informal children's strategies to more effective algorithms, as described, for example, in [5] and [14], are very useful not only in order to achieve the conscious mastery of algorithms but also as a ground for the construction of crucial skills concerning the design and management ofprocedures and reasonings as well as programming; - activities inducing pupils to reflect upon their own actions and thoughts, by means of comparison in the classroom or other, are very useful as far as they allow pupils not only to perform their procedures and reasonings but to regard them as autonomous products; anyway, the possibility ofreflecting should be deliberately built, through a long term educational work; - the ability at 'spontaneously' producing procedures and reasonings seems to depend more upon the mastery of the specific meanings of the problem situation than upon its complexity itself; the awareness of the specific meanings is very important as regards the mastery oflinguistic connectives; for more details on the relevance of 'semantic fields' see [4]; - children's procedures and reasonings are strongly related to time; most pupils represent them as processes with a relevant temporal dimension; only few pupils can represent problem
- 148. 136 situations by means of 'static' relationships without a temporal dimension; on the other hand, the mastery of time as a basis for the representation of processes seems to be related to the mastery of time as an explicit variable of the problem; for more details on the function of time see [9]. References 1. Boero, P.: Acquisition of meanings and evolution of strategies in problem solving from the age of 7 to the age of 11 in a curricular environment. In: Proceedings of the 12th Conference of the International Group for the Psychology of Mathematics Education, vo1.l, 177-184, 1988. 2. Boero, P.: Mathematical literacy for all: experiences and problems. In: Proceedings of the 13th Conference of the International Group for the Psychology of Mathematics Education, vol.1, 62-76, 1989. 3. Boero, P.: On long term development of some general skills in problem solving: a longitudinal comparative study. In: Proceedings of the 14th Conference of the International Group for the Psychology of Mathematics Education, vo1.2, 169-176, 1990. 4. Boero, P.: The crucial role of semantic fields in the development of problem-solving skills in the school environment. (in this volume), 1991. 5. Boero, P., Ferrari, P. L., & Ferrero, E.: Division Problems: Meanings and procedures in the transition to a written algorithm. For the Learning of Mathematics 9(3), 17-25 (1989). 6. Bondesan, M. G, & Ferrari, P. L.: The active comparison of strategies in problem-solving: an exploratory study. In: Proceedings of the 15th Conference of the International Group for the Psychology of Mathematics Education 1991. 7. Ericsson, K. A., & Simon, H. A.: Verbal reports as data. Psychological Review,.87, 215-251 (1980). 8. Ferrari, P. L.: Hypothetical reasoning in the resolution of applied mathematical problems at the ages of 8-10. In: Proceedings of the 13th Conference of the International Group for the Psychology of Mathematics Education, VoU, 260-267, 1989. 9. Ferrari, P.L.: Time and hypothetical reasoning in problem solving. In: Proceedings of the 14th Conference of the International Group for the Psychology of Mathematics Education, vo1.2, 185-192, 1990. 10. Lemut, E. & Ferrero, E.: On the linguistic prerequisites of computing literacy. In: Proceedings of the IFIP Technical Committee, 3.139-144, North Holland 1988. 11. MarkovilS, H.: The curious effect of using drawings in conditional reasoning problems. Educational studies in mathematics,17, 81-87 (1986) 12. O'Brien, T.C., Shapiro, BJ. & Reali, N.C.: Logical thinking - language and context. Educational studies in mathematics, 4.201-219 (1971) 13. Rumain, B., Connell,I. & Braine, D.S.: Conversational comprehension processes are responsible for ...• Developmental Psychology, 19 (4),471-481 (1983). 14. Treffers, A.: Integrated Column Arithmetic According to Progressive Schematization. Educational Studies in Mathematics, 18, 125-145 (1987)
- 149. Problem Solving, Mathematical Activity and Learning: The Place of Reflection and Cognitive Conflict Alan Bell Shell Centre for Mathematical Education, University of Nottingham, Nottingham N67 2RD, England Abstract: In order for problem solving to take its place as a central activity of the mathematics curriculum, the notion must be extended to include broader kinds of mathematical activity, including the gaining of insights into phenomena through the application of mathematical ideas and processes. But beyond simply engaging in mathematical activity, efective learning (which leaves the students more competent at these activities afterwards than they were before) needs tasks which provoke reflection, cognitive conflict and discussion. A range of types of such tasks is discussed. Keywords: mathematical actIvIty, learning, reflection, cognitive conflict, discussion, classroom tasks, diagnostic teaching Introduction I am sure that we all welcome the moves towards establishing problem solving as the central activity of the mathematical curriculum, with the learning of specific skills and concepts providing the necessary basis of knowledge for use in this activity. There has been a long tradition of teaching skills out of relation to their use in meaningful activity; this is comparable with a curriculum in woodwork in which students learn how to use hammers, chisels, saws and planes but without ever designing and constructing an article of furniture. The teaching of what used to be called Crafts in UK schools has happily moved away from such practices, as indicated by its present title of Design Technology. In a similar way, foreign language teaching now emphasizes competence in communication, and uses authentic examples of writing, display and conversation in the target language. Nevertheless the notion of problem solving is too limited to provide a comprehensive aim for our teaching. The real aim must be to introduce students to authentic mathematical activity in such a way as to encourage and enable them to engage in such activity in their subsequent professional, personal and social lines. This paper is intended to contribute to this aim, first briefly by broadening the concept of problem solving into mathematical activity, and then by considering how such activities might become effective learning experiences. For it is not our aim simply that students should engage in problem solving or other mathematical activities; we wish them to acquire more or less permanent
- 150. 138 competencies, so that they actually become more effective perfonners on future occasions. Learning has taken place when the student possesses, after the activity, some ability, competence, knowledge or skill which he did not have before. Thus, this paper is addressing, first, the question what is to be learnt in mathematics lessons, and secondly, how it may be most effectively learnt. Applied and Pure Mathematical Processes Mathematics has two aspects, roughly fitting the traditional labels applied and pure. First, it is a means of gaining insight into some aspect of the environment. For example, the exponential or compound growth function gives us insight into the way in which a population with a given growth rate grows over time - first slowly and then with an increasingly rapid rate of increase. Some ofthese propenies are encapsulated in well known puzzles - such as that of the water lily, doubling its area each day, where will it be the day before it covers the whole pond? - or in the frequent exhortations of our financial salesman to consider how a modest investment might grow. Another related example is that of the decrease of the rate of inflation - which many people believe means that prices are coming down. In the home environment, a little knowledge of the symmetry group of the rectangular block will tell us when we have turned the mattress on the bed as many ways round as we can; and a modest knowledge of probability and statistics will help us to interpret advertising claims about what toothpaste seven out of ten movies stars use, and not to be excessively hopeful that our next child will be a boy if we have already produced three girls. These are all 'useful' aspects of mathematics - and note, by the way, that they all depend on the application of conceptual awareness, not on any technical skill; they are useful in the same way as is the knowledge gained in most of the subjects of the curriculum - history, geography, literature, science - that is, deriving from knowledge of some key facts and explanatory concepts. The second aspect of mathematics is somewhat less loudly commended in public nowadays. It is the pure mathematical aspect which it shares with art and music, the solution and construction of puzzles and problems, and the enjoyment of recognizing and making patterns. Mathematical problems in newspapers and magazines still attracts a following, and we might speculate that the capacity to appreciate mathematics as an art to enjoy is initially present in most people, though it often gets suppressed by distasteful school experiences. These two modes of interaction of people with mathematics, representing the applied and the pure mathematical approaches, have been identifiable throughout history as the mainsprings of mathematical activity. Freudenthal [5] also distinguishes applied and pure mathematical processes:
- 151. 139 "Arithmetic and geometry have sprung from mathematising part of reality. But soon, at least from the Greek antiquity onwards, mathematics itself has become the object of mathematising... What humans have to learn is not mathematics as a closed system, but rather as an activity, the process of mathematising reality and if possible even that of mathematising mathematics." More briefly, "Mathematics concerns the properties of the operations by which the individual orders, organizes and controls his environment." [6, p. 157] Components of Mathematical Competence The capacity to do mathematics involves several different kinds of acquisitions. We need to distinguish (a) between skills, conceptual structures and general strategies; (b) between symbols and meanings, and (c) between mathematical concepts or structures and the contexts which embody them; with the notion of transfer of a concept, learned in one context, to its application in another. Skills include all the common computational algorithms - for the 4 rules, for moving the decimal point, geometrical constructions, equation-solving rules. They are routines consisting of a number of steps and knowledge of them implies the ability to move from one step to the next fluently. Conceptual structures are interconnected sets ofrelationships. The concept of place value is such a structure; it includes the fact that the figure after the decimal point represents a tenth, that moving the point to the right multiplies the number by powers of ten; it governs the rules for carrying, and the rules for putting numbers in order of size. Strategies (including both strategic skills and strategic concepts) are plans which guide a person's choice of what skills to use or what knowledge to draw upon at each stage in an activity of problem solving or investigation. Identifying data and conclusion, trying easier numbers, proving, are examples of general strategies. Fuller treatment of the learning of strategies appears in Bell [I], and of all these topics in Bell, Costello and Kuchemann [2]. These acquisitions can be classified on three dimensions: first, as schemes governing actions or knowledge (skills, concepts); secondly, according to the type of connectivity they possess, skills being mainly linear, though with branches, concepts being more like interconnected networks; thirdly, according to the degree of generality, the main body of mathematical concepts and skills being at a lower level than strategies, which govern the way in which the mathematical concepts and skills are deployed.
- 152. 140 Symbols and Meanings The relation between symbol-manipulations and their meaning in terms of the concepts denoted is at the heart of mathematics. It is probably also responsible for much of the difficulty of teaching mathematics, since the symbol-manipulations are the most visible part of the activity, and it seems natural to teach them directly without realising that comprehension of the underlying conceptual transformations is important for effective permanent learning. A demonstration of the distinctiveness of these two aspects and also of the need to ask carefully "how does the pupil view the question", is given incidentally in an experiment on the learning of fractions by 9-10 year olds [7]. In one of their tests the following items appeared. and: Look at the square in the top picture. Four of the ten equal parts are shaded. Now look at the bottom picture. This square must have the same amount shaded. How many of the five equal parts should be shaded so that the sam& amount will be shaded in both squares? The squares are unit squares. /1/1/1/1/1/1 00000 00000 00000 If there are six triangles for'every fifteen circles, how many triangles would there be for five circle.? 00000 The later, corresponding items are: 4 • .Q and 10 5 15 5 Find the number that goes in the box. It seems likely that at least by some pupils the symbolic questions were treated as requests to manipulate the fraction symbol according to the rule "divide the top and bottom by the same number". It is also possible that in the diagrammatic case of the fraction item the answer was obtained by filling up the total area in the second square equal to that shaded in the first. If this is so, in neither case was the concept of a fraction necessarily involved in the thinking. The low correlation (about 0.37) between pupils' results on the two forms of test confirm that to the pupils the two types of problem are substantially different.
- 153. 141 The ability to be able to use symbolisms effectively, working within them when appropriate, but also to be able to re-establish the links with the underlying concepts when that is necessary, needs extensive experience in making the translations both ways. Structure and Context The most characteristic typical mathematical activity consists of the recognition of a relational structure in a context, and the use of known properties of that structure to effect a transformation which reveals some previously hidden aspect of the situation. For example, consider the problem of finding the number of litres of petrol equivalent to the 5.5 gallons held by the tank of a Mini, given that a litre is 0.22 gallons. One has to recognise the operation of division implied by the situation, and then use the relevant algorithm to reveal the answer, 25 litres. This is a quite typical, if elementary, fragment of mathematical activity. Another problem might require the original size of a poster, reduced by a scale factor of 0.22 to give a magazine picture 5.5 cm high. Again the operation required is division, but the type of structure is no longer that of quotition, nor of repeated subtraction but rather as the inverse of multiplying by a scale factor. The frrst problem might in fact be solved by a pupil by adding sufficient 0.22s together to give 5.5; if the conversion factor had been 2.2 instead of 0.22 this would be even more likely. The second is most unlikely to be solved by any such process. Thus the aspects of an operational problem, as perceived by the pupil, depend on structural properties of the context and on the particular numbers. For further discussion of these factors, in particular the effect of familiarity of context and the transferability of structural knowledge across contexts see Bell, Costello and Kiichemann [4, Ch. 2]. To secure the leaming of a articular skill, strategy or concept it is first necessary to present the leamer with tasks which bring them into play. There is thus a need in the curriculum both for broader tasks, such as extended investigations or practical problems or open situations, which bring general strategies into playas well as a possibly indetenninate range of particular mathematical concepts; and also more sharply focused tasks which require the use of a particular concept, such as place value, or rotational symmetry. In fact, most tasks call into play both general strategies and particular knowledge. What is learned from them depends on what is focused upon, what is attended to. It is at this point that the teacher's intervention is of crucial importance, in directing the students' attention to those aspects of the situation which they are intended to learn, making them aware of what they've done and how they have done it. What we are asserting here is that the two basic mechanisms of learning are (a) insight, and (b) repetition.
- 154. 142 Each new insight adds a further link to the conceptual structure; repetition (with appropriate variation) forges the link more firmly. Repetition develops fluency of response, but as Brownell [3, 4] showed in his research on drill in number facts, without attention to increasing insights, its effects fade quickly. New insights may be consonant with existing cognitive structure, or may conflict with it. In the former case, they are assimilated; in the latter, they may be rejected, or ignored, or deflected into something different which does conform - until the disequilibrium or cognitive coriflict is intolerable, and reorganization and renewal of the cognitive structure takes place, in which the new ideas and the old combine in a new synthesis. Piaget described this process as that by which an organism learns through interacting with its environment. We have used this as the basis of a teaching methodology which uses coriflict-discussion - the provocation of cognitive conflict focusing on known or likely points of misconception, followed by resolution through group or class discussion. A number of particular types of task have proved valuable for this purpose. We shall describe some of them, and report the results of experimental teaching using this methodology, which has become known as Diagnostic Teaching. Diagnostic Teaching This project was (and is) concerned in developing methods of providing cognitive conflict and discussion focussed on key conceptual obstacles revealed by research into pupils' understanding of several important topics. As the title implies, this method begins by conducting conversations, interviews and tests with students, looking carefully at their mistakes and using these to help to understand the ways of thinking that lie behind them. Next comes devising and trying out teaching ideas and methods aimed at overcoming particular misconceptions, and testing sometime afterwards to see how successful they are. Work has covered several known problem areas of the curriculum - directed numbers, decimal place-value, choice of operation with decimal numbers, algebra - but subsequently colleagues have been applying the same principles and method to a variety of other curriculum topics including graphical interpretation, shape and place-value with primary children, probability with sixth-formers. Other outcomes of the work have included some new insights into students' understanding in each of these fields. Here we want to discuss the kinds of classroom-task which have emerged as particularly helpful in generating the conflict and discussion needed to promote learning. Making up questions, 'marking homework' (using a real piece of homework or a specially constructed one), filling in tables, and games have all proved useful, and ways of combining group and whole-class discussion have been explored. But before describing these tasks, we shall indicate the conceptual field in which we were working and the particular misconceptions we found.
- 155. 143 Number Problems and Operations A by-product of the advent of the calculator has been the realization that the recognition of operations presents significant difficulties to many students. These have formerly often passed unnoticed in our preoccupation with the teaching of methods of calculating. (The operation required could often be inferred from the sizes of numbers from the chapter heading, or from cue words like 'times' or 'altogether'.) We gave the following question to a middling third-year secondary class: "Mushrooms cost 40 pence per pound. If I buy 25 pence worth, what weight will I get?" 90% gave a correct estimate for the answer (i.e. half pound, three quarter pound or similar), but asked what calculation was required, only 10% answered correctly 25 + 40, most of the remainder giving 40 + 25. Another question concerned meat price (The price of minced beef is shown as 88.2 pence for each kilogram; what is the cost ofa packet containing 0.58 kg of minced beef) and its result: only 29% of the 15 year olds correctly chose multiplication; 42% suggested division. These results indicate the widespread nature of the misconceptions that multiplying makes bigger, division makes smaller, and that division is necessarily of a bigger number by a smaller one. A further problem is that many pupils lack fll'Ill and correct understanding of the meanings of 3 + 24, 24 + 3 and Ｓｾ＠ and the relation between them. When asked how to find a number of apples each, when 24 are divided among 3 people, our pupils would generally write 31Mor 24 + 3; but when faced with 3 + 24 they would often identify this with ＳｾＮ＠ The degree of inconsistency here was frequently great. The faulty choices of operation being made seemed to derive from the numerical misconceptions. Hence if these could be remedied, the main pull towards faulty choices should disappear. We have noted too, that the misconception that 'small + big is impossible' is confounded with the understanding of the division notations, + and )_, so this also needs attention. Moreover it is reasonable to assume that suitable teaching of the correct quantity relations e.g. Cost = Amount x Unit Price, as applying whatever the size of the numbers involved, would contribute to improvement. The teaching material was aimed at these three aspects: notational misconceptions, numerical misconceptions and the invariance of quantity relations in problems in contexts such as price and speed. The materials were developed through several trials and also used comparatively. We shall mention the comparative experiments later concentrating here on the actual design of the material and how it was modified in the course of the trials.
- 156. 144 Predict and Check The first material on the division notation consisted of the sheet Thinking about division, from which an extract is shown. C.lculition M ••ning in wo..... Ans...,. C.lcullto,. Cheek 15 • 3 ｊＮＺｾＬＮＮ＠ J.vJJ 6,,: ｉｉｾ＠ ｾ＠ S .)To ｦＮＺＬｾ＠ .. ,..to ｾ＠ ,t 5 2 • 10 1.-<; ,1.,..W ｾ＠ 0'2 This was meant to expose the different interpretations of the division notation, and to be self correcting, in that the calculator check would show up any mistaken interpretation, which could then be corrected by the pupil. In the event. some pupils made no errors, others responded with the same working throughout irrespective of the differences from question to question. and very few actually used the check infonnation to correct their errors. Often they obtained the calculator answer first then repeated this for the answer column. The idea of a worksheet designed so as to enable you to check your ideas and correct them was too unfamiliar; it needed a substantial reorientation of the pupils' attitudes. which at this stage we had not achieved. In revising this sheet we moved away from the list of similar questions. and also turned the task into one involving group activity. Filling in Tables
- 157. 145 For this task, each group of four pupils had a set of small cards and a large, mostly blank, table as shown. The cards had to be used appropriately to fill all the blank cells. A typical lesson using this activity as recorded, and we report some episodes from it. The task looked easy at first, but disagreements soon arose about whether a particular card was correctly placed and some vigorous arguments took place. During the group activity, the teacher would, if appropriate, question the groups' placing of a card, and suggest they discuss it further. When the groups were all satisfied with their placings, a class discussion was held. To begin this, a representative of one of the groups filling in a blackboard copy of the table and other groups indicated where they agreed or disagreed. The teacher encouraged them to explain their decisions, taking a neural chairman' stance. The most heated discussion arose about whether, for 2 apples shared among 8 people, the cards 2 + 8 and 2)8 were correctly placed in the same line. Some pupils maintained that 2 + 8 was impossible so you had to do 8 + 2. But they agreed that sharing 2 apples among 8 girls was possible, and they they would each have 1/4. We quote part of the discussion. PI: In division you can't have a 1/4 left over. It would be nothing or a whole one. T: What about the example, then? (2 apples, 8 girls) PI: We've been told that with sharing you've got to do it with a whole number. If there's one left and you're dividing, say by 5, it's 1 remainder. T: I see. So what about the example then? PI: Well, if you give someone this question, 2 apples shared among 8 girls, they'll say a 1/4. But if you give them that (2 + 8) they'll say a different answer P2: They'll say 4 T: Why do you think they'll say 4? P2: 'Cos if it's numbers you can't chop them up. P3: You can chop them up ... 'cos there's subtraction, you can say 4 take I, then you're chopping up a number. The claim here by PI is that real situations are one thing, but number operations are in a different category and have their own different rules. At another point, pupil P3 again gets strongly moved to put the other right. P4: You can't have that (2 + 8) because 8 doesn't go into 2 - well, it does, but it's not 4, is it! P5: For 8 + 2 we've put how many 2s go into 8. T: So what have you put for 2 + 8? P5: We've put the same P3: (Bursting with protest) but it's not the same, you're changing it round, aren't you! ... (explodes) You'd be saying, like 100 divided by 2 and 2 divided by 100; there's a big difference, you know! P5: What's the difference then? P3: 'Cos 100 divided by 2 is 50! P5: Yes, and what's 2 divided by 100 then? P3: (laughs) ... I'm not sure
- 158. 146 Connict-Discussion This was the kind of conflict-discussion which we aimed to produce in each lesson, by giving a short start activity which would lead to the exposure of such misconceptions as are present in the pupils' schemes and ideas. So it could be said that we deliberately gave them questions which at least some pupils would get wrong; and without forewarning them of possible hazards. The principle was that if there is an underlying misconception, then it's 'better out than in'; it needs to be seen and subjected to critical peer group discussion. Of course, establishing a classroom atmosphere in which this is an accepted activity is not a trivial matter, and it may take some time. Challenges This lesson aired questions concerning both division notation and the numerical misconceptions. Another lesson aiming at the latter offered three number changes, 5 --+ 10, 3 --+ 12 and 3 --+ 4, and asked the pupils to find a suitable division operation to perform these changes. This had considerable shock value. They had then to write down rules governing the choice of numbers, and also to write problems to fit each calculation. The discussion was focused on these rules. Games Two games (with variations) were also devised, both involving choice of numbers with multiplication or division to achieve a required answer. One was a table football game with a calculator in which two players in tum had to multiply the existing number on the calculator by a suitable chosen number to give a result within the range (say 30-40) belonging to the player she wanted to pass to. In Shell Shocker, spaces were 'bagged' by choosing a pair from a given set of numbers which when multiplied or divided, gave the number of the space desired. In this game, players take turns to try and place counters on the board. The winning player (or team) must have a line of counters connecting two opposite edges. These games involve the pupils in a great many choices concerning the size of answer, and thus to become successful one has to become aware of the correct generalisations about number size and the operations. But, interestingly, we found (on the one occasion when we tested this) that only modest improvements on a short test of the numerical misconceptions were achieved even after playing the game quite intensively for about half a lesson. In the following lesson
- 159. 147 there was a conflict -discussion in which the teacher encouraged the pupils to express the number generalisations governing their choice quite explicitly, e.g. 'to make a number smaller you can multiply by a number less than 1'. This proved difficult; the students could quote numerical cases e.g 3 + 6 is 1/2, but found it very hard to express general statements. Following this discussion a much greater improvement on the test was noted. Shell Shocker A calculator game for two players (or teams). Each player (or team) will need a set of coJters. Take it in turns to: i) Choose any two numbers from here ii) Multiply or divide one number by the other with the calculator iii) If the answer js on the board cover it with your counter One player (or team) must start at a white edge and the other player (or team) at a black edge. To win, there must be a connected line between edges of the same colour. Football game with a calculator
- 160. 148 Making up Questions (1) The Bag of Flour task is illustrated below. A variety of information is provided but the questions asked are the reverse of the usual ones - pupils are asked to supply the questions to which the given calculations would provide the answers. ｾｾｾｾ＠ ｾ＠ These are generally harder than the usual questions. In the example shown, most of the answers are correct, but some are incomplete (iv, vii) and two are reversed (vi and additional question). In the lesson from which this example is taken, a great deal of conflict was aroused by 42 + 1.5; suggestions included the cost of a bag, a gram and 1.5 kg, as well as the correct lkg. In this case, the question was largely resolved in the discussion by considering what the answer would be if the flour weighed 2 kg instead of 1.5 kg. Often, changing the numbers in a problem is regarded by many students as potentially changing the operation - they do not see the operation as embedded in the structure of the problem. In this case, it may have been the way in which the numbers were attached by labels in the diagram which helped them to envisage a number change without loss of operational structure. It is also worth noting that the research on these problems has shown that a number such as 1.5 makes for greater difficulties in recognising the operation than does for example, 3.5 which is seen as a number much like 3; one reason may be that rounding to 1 gives no clear help in conceptualising the operation.
- 161. 149 Making up questions (2) Students early attempts to make up questions showed this was surprisingly difficult. Frequently the questions were unanswerable. Sometimes they simply gave data but asked no question; in other cases they gave insufficient information. Two examples were: Mrs Vel6S ｢ｾ＠ 0.11. !WcE1liY,S ｾ＠ f& .UEIAl ｾｈＮ｣Ｑ＠ dJ 1t ｾＩＮ＠ A mCln lxlk ｨｾｳ＠ qoq Or 0. v,tlk, ｾ＠ rdher 0. doq 1,01. ｾ＠ ti'dn ｾ＠ a ｾｴ＠ tow MJt htW d'd'k ｾｾ＠ -h cp DJ ｾｉＬｾｅ＠ ｨｾｍ･ｴｴ･ｳ＠ cr ｾｉｏ＠ ｾｉｯｾ･［＠ ｾ＠ hmr. O·'3)<.10 ＺＺＺＺＳｨｃａｬｲｾ＠ In later attempts students gave more coherent questions, and these showed up some important misconceptions. The following examples show a common failure to recognise correctly the numerator and denominator roles of the two quantities in a rate. This is particularly so when expressed, for example, as miles in each minute. In the task Double talk, quantities had to be chosen from a given set and put together to form a correct question and its answer. In this case the numbers are chosen so that a real choice has to be made to get the correct operation and the quantities in the correct order, since the numbers will make wrong problems as well as the right ones. Some of the incorrect problems quoted above are derived from this exercise. In another form of making up questions Picture the problem, students are given a set of related quantities, but just one numerical value and were asked to write two possible questions, one containing easy numbers, and one with hard numbers. This gave the students the challenge of recognising what were 'hard numbers' in these problems, as well as having to operate with them. These tasks offer a variety of different demands and constraints; each has some element of freedom for creativity, and also some means of ensuring that the more difficult number and quantity combinations are faced as well as the easier ones. And each, I hope, has some degree of attractiveness in presentation. All are intended for use to expose conflicts by identifying incorrect questions and discussing them, generally in pairs or small groups initially, then with the whole class.
- 162. 150 Marking Homework These exercises, shown below, lend themselves in a somewhat different way to identification and discussion of misconceptions. P Blackwell is a fictitious pupil who has all the common numerical misconceptions; so the second (b) question of each pair is wrong. As with the other tasks, the important learning is likely to come not simply from solving the problem but in the discussion which can be developed from the conflict between different suggestions for the solution. In this case, formulation of the rearranged calculation needed to yield the unknown can give rise to discussion of relations between multiplication and division. 3R The aim was that students would identify the errors, explain what the right approach was, and also suggest how the mistake came to be made - that is, explain how the misconception occurs. This, we felt, would provide the best safeguard for the students against the misconception. In the event, they did not in general reach this third stage. There were some different kinds of explanations of how the errors were recognised; some referring to the size of answer e.g 80 + 50 would be nearly 2 and you need something less than I, others appealing to the quantity relation. You should divide the distance by the time, because you need to find the answer in kilometres per hour. In the material described here the main differences between the earlier and later teaching materials lay in the shift of emphasis from worksheet to conflict-discussion. In both cases there was discussion of the same misconceptions but in the earlier cases it was 'expository discussion' and was followed by the worksheets (self checking where possible), which were envisaged as providing the main learning experiences. In the later series we saw the 'conflict- discussion' as the main learning experience, and the written work as introductory, giving the
- 163. 151 opportunity for opening up the situation and allowing some mistakes to be made which led to the conflicts and hence the discussion. Following the discussion we gave some written work to 'consolidate' the understanding gained. This consisted of similar problems, but with feedback enabling immediate correction to be made if necessary. Many people feel that class discussion in mathematics is difficult. This may be partly because we try to make it too convergent, aiming at appreciation not only of a single correct result but also the single correct line of reasoning towards it. But, as can be seen from any discussion based on a strongly felt conflict, many factors and many connections contribute to pupils' convictions. These all need to be brought out and aired. Discussion may sometimes appear to be repetitive; normal discussions on any subject do, because it takes more than one cogent argument to shift an established view. Perhaps the most striking observation from all this work is that back-sliding is the norm. Even after clearly effective lessons with learning visibly taking place, in the next lesson most of the class could slip again into the original error. True, the second recovery was quicker than the first. The method of conflict-discussion provides a more effective way of dealing with this widely recognised phenomenon than simply reteaching; this is shown clearly by the experimental results quoted below. The same key conceptual points do need to be the focus of discussion repeatedly; but they need to appear in different contexts and modes of presentation. The group discussion of division, from which extracts were quoted above, was one of the most successful lessons in these sets. Discussion in groups first allows more people to be expressing their own views and insights (and misconceptions) in a less threatening atmosphere than that of speaking to the whole class. Following group discussions by a whole class session then helps to ensure that if a whole group has accepted an erroneous conclusion, it can be exposed and countered. But it is important to establish with the class that the aim of their work together is to discuss situations chosen so as to help people to get wrong conceptions dealt with; and that reluctance to voice a possible wrong view is a hindrance to learning. We need to develop the awareness that a discussion of meanings is in itself a valid type of mathematics lesson - just as a discussion of insights into a play or a book is recognised as a valid English lesson. Experimental Results The material discussed above was used in a substantial experiment in which the method of conflict and discussion was compared with a more usual method of problem solving activity with teacher help. Five teachers taught classes using the experimental method, and two of them taught other classes by the control method. The results followed a pattern which was repeated in a number of similar experiments - somewhat greater gains with the experimental method between pre and post tests, and much greater retention between post and delayed post test, than with the more usual methods.
- 164. 152 Mean Percentage Scores of the two Groups The two graphs, Figure I and Figure 2, show the scores of each pupil on each of the three tests, and so give a fuller picture of the result. It is clear that the scores of the conflict group might well have been even higher, if there had been more headroom in the test. Similar results were obtained in the other experiments. Test Mark Pretest Posttest Delayed Posttest Results for each pupil in the booklets group Figure 1 Test Mark .. Pretest Posttest Delayed Posttest Results for each pupil in the conflicts group Figure 2
- 165. 153 Table 1: Experimental VerSUS control group results Gain pre-post Experimental classes Control classes +20 +10 Gain post-delayed -1 -5 The general pattern of result, in which the conflict group gain somewhat more initially and lose much less over time, was repeated in three similar experiments. These were on Reflections, on Fractions and on Decimals. In the first two experiments, the comparison was with teaching based on a popular individualised booklet scheme. The 'conflict' material was devised by the teachers concerned, to cover the same ground as the corresponding booklets. The work on reflections was conducted with two parallel first year secondary mixed ability classes in the same school, though in this case the classes did not have the same teacher. The 'conflict' teaching focused, as before, on observed misconceptions (such as that horizontal or vertical objects always have horizontal or vertical images, and that any line which divides a figure into equal parts is a line of symmetry). But it also aimed to establish, explicitly, through the conflict-discussions, the correct principles for relating object and image (straight across and the same distance, with appropriate interpretation). But whereas the booklets contained large numbers of fairly easy questions, the conflict teaching was begun by giving one or two much harder questions of the same general type, to be argued out in small groups; further questions were then made up by the groups, to give to other groups, finally there was a discussion among the whole class. The two classes were matched on a general mathematics test (NFER II), but on the Reflections pre-test, they were not particularly close. Table 2: The mean scores of the groups Conflict Booklets Pre 48 32 Post 79 70 Delayed 82 54
- 166. 154 Conclusions The overwhelming conclusion from these experiments is briefly, that for effective learning and retention, the key concepts and relations need to be exposed and intensively discussed. Focusing on likely common points of misconception, provoking and exposing and discussing the error - these lead to effective learning. Forewarning of the error is less effective; and working through graded problems without this intensive focussing can result in apparently successful experiences which, in the medium or longer term, leave little trace. References 1. Bell, A. W.: Teaching for the test Times Educational Supplement, 27th October 1979 2. Bell. A. W.• Costello. J., & Kuchemann, D.: A review of research in mathematical education - Pan A : research on learning and teaching. Windsor: NFER-Nelson 1983 3. Brownell, W. A. and Chazal. C. B.: The effects of premature drill in third-grade arithmetic, Journal of Educational Research, 29,17-28 (1935) 4. Brownell, W. A., & Moser, H. E.: Meaningful versus mechanical learning: A study in grade II subtraction. Duke University Research Stud. Ed (8), Durham, NC: Duke University Press 1949 5. Freudenthal. H.: How to teach mathematics so as to be useful. Educational Studies in Mathematics. 1 (1968) 6. Peel. E. A.: psychological and educational research bearing on school mathematics. In: Teaching School Mathematics (W. Servais & T. Varga, eds.). Harmondsworth: Penguin Education for UNESCO 1971 7. Steffe, L. P. & Parr. R. B.: The development of the concepts of ratio and fraction in the fourth. fifth and sixth years of the elementary school. Madison: University of Wisconsin 1968
- 167. Pre-Algebraic Problem Solving Ferdinando Arzarello Dipartimento di Matematica dell'Universita di Torino, Via Carlo Alberto 10, 10123 TORINO, Italia Abstract: Typical solution processes of (pre-)algebraic problems live dialectically between two opposite polarities: procedural and relational. The fonner is a-symmetric; is ruled by "the logic of when"; is close to the meaning of symbols; its main epistemological style is arithmetic. The latter is symmetric; is ruled by "the logic of iff'; is syntactic, insofar concrete meaning have evaporated; its epistemological style is anti-arithmetic. But procedural thinking allows pupils to do concrete experiments and get feedbacks from the problematic-situation; and this, in the long run, is useful in order to jump to relational polarity, where more formal algebraic manipulations can be done. Keywords: pre-algebra problem solving, procedural polarity, relational polarity, arithmetic, algebra, epistemological obstacles, validation, a-validation, algebra teaching Introduction This paper is devoted to discuss last two years' researches of the author in pre-algebraic problem solving; a precise definition of what pre-algebraic means will be given later. For the moment. good examples of pre-algebraic problems are those in Appendix 2 (except problem 0), where pupils are requested to discover some rule from numerical data, which they have elaborated themselves (see also [7,14]). Solution's processes of pre-algebraic problems in pupils from 10 to 16 years old (and over), presuppose and include all the aspects of the solution's processes of arithmetic problems, such as have been described in Arzarello [I, 2]. Particularly, the notion of conceptual model, is basic to understand and describe pupils' perfonnances. (A brief sketch of this notion is given in Appendix 1). But even if the genns of pre-algebraic thinking are already present in the processing of many arithmetic problems (see [14]), things are obviously more complicated in pre-algebra than in simpler standard arithmetic problems. Two main novelties enter, namely: a) Arithmetic is an epistemological obstacle for constructing a pre-algebraic knowledge (for the notion of obstacle, see [3]): conceptual models, which pupils have elaborated in previous, purely arithmetic activities, become too rigid in the new context and block those typical ways of thinking, which have an algebraic flavour, and allow them to attack the problem properly. To
- 168. 156 be more precise, in pre-algebraic problem solving, pupils are requested to use more than one model at once, and also to use them with an epistemological style, which is the exact opposite of the arithmetic one. So they may lose the control and can no longer integrate their models in a global strategy. For example, in problem 2, after the "local" law has been found relatively easily (namely, something like Pn+1 = Pn + n+1), even those pupils who already know the relation for the sum of first n integers, do not see it in the foregoing formula: numerical data which they have elaborated (and usually arranged in a table) and the formula are not put together. The very arithmetic epistemology inhibits students from looking at numbers and variables in the right manner, that is Arithmetics inhibits algebraic thinking. b) Integration of this obstacle in a new knowledge, when and if it happens, modifies the main features of validation (for a discussion of this notion, see [4, 6, 11]): a more "abstract", less "tangible" and new form of validation appears, which will be called a-validation or weak validation. This change is a very delicate point: it represents an obstacle, which involves mainly metacognition, and may inhibit pupils from solving pre-algebraic problems. For example, in problem 3b, whose algebraic content is mixed with geometry (i.e., number geometric configurations), 15-16 years old pupils find very difficult to manipulate the formula they have got for Tn [namely n(n+l)/2] to show that 8Tn +1 is a square (a formal job which they find easy, if explicitly requested to do it as a manipulation of formulas): some do not do anything at all; others use a great ingenuity to prove the claim by means of non-algebraic methods (Le., using geometrical models: see Figure 1). • II II II • • • • • II II • • II • • • II • II II • • • + • • • II II • II • • • II • • II II • • • • • II II II • Figure 1
- 169. 157 The aim of the paper is to give empirical evidence to points a) and b), illustrating empirical research made with 10 years old pupils to university students, while solving (pre-)algebraic problems. Obstacles It is very well known that many pupils, from middle school up to the university, solve algebraic problems more in a syncopated style than in a symbolic one (see 7, 8]). 16 years old pupils do not yet use the algebraic code spontaneously while solving simple algebraic questions (e.g., see the very well known problem 0 in Appendix 2), even if this may cause them troubles and long detours (I have verified awful not-algebraic detours also with my undergraduate mathematics students, while solving problems 3 and 4). Their preferred conceptual models are still arithmetic ones and this may cause conflicts with the algebraic way of thinking. Moreover, the only way, which seems to allow them to master semantically the situation is mainly when they can represent their model using a syncopated language, which allows them to manage things avoiding conflicts between the situation and the model they are using. The point is that algebra is understood only in abstract, as a formal and general method, but it is not concretely used as a method of justification and generalization in specific situations; namely, generalization as an effective and operative method is used with great difficulty by pupils while solving algebraic problems. Even if they seem to manage formalism, they do not use it spontaneously in concrete problematic situations and seem to live more comfortably with its substitutes, namely those arithmetic conceptual models, where the meaning of involved things is expressed more directly by the syncopated language of arithmetic: see the sketchy discussion on problem 2. For another example, see problem 4. Even if it may seem very difficult at a first glance, the problem can be managed relatively easily by 15-16 years old pupils up to a certain level. In fact pupils can grasp the rule for So describing it in a syncopated arithmetic way; so they can postpone algebraic reasoning till the end (when most of them get lost), which allows them to control semantically the situation for a long time. On the opposite, their teachers have attacked the problem immediately with algebraic means and some have got serious troubles; for most of them only a detour to arithmetic syncopated language and to a procedural style (see later, for this notion) has settled the question. Students generally learn at school how to solve specific problems with algebra (e.g., using unknowns), but meet major difficulties when required to use it as a symbolism to express general solutions, to discover, generalize and prove laws behind numerical relations (as in Appendix 2 problems). The main difficulty consists in integrating formal (algebraic) algorithms with arithmetic models. Also here one is faced with the phenomenon of opposition between model and situation, so that to overcome difficulties, one must adapt the model to the different
- 170. 158 situation ｾ､＠ get a new model, possibly unstable (see the discussion in Appendix 1). But this phenomenon has a different conceptual character. The opposition is not between the problem and the model, which is not adequate; the point now is that conceptual arithmetic models are adequate to attack the situation, but only from an arithmetic point of view: the opposition now is between the very way in which one has the habit of using the model, and the necessity of looking at it in a different, more abstract fashion. Conceptual models must be transformed, but a partial modification is not enough: they must be inserted into a more general framework. We have seen that in problem 2 many students do not see things, even if they are under their eyes, and this happens in all problems we have tested. The difficulty is to look at arithmetic things with a new (algebraic) eye; so to say, pupils must learn to speak after a new code. The main novelties of this operation have been pointed out by C. Laborde in her thesis [9J. More precisely, they represent the main functions of the algebraic code, namely individualization and linkage. The transformation of conceptual models required to face (pre)algebraic problems are mainly these. But this kind of operation costs very much to pupils, who like better to use more stable models. even if inadequate (in this sense, the solutions of problem 3 are typical), namely to substitute such algebraic functions with extralinguistic properties. The language of pupils faced with problems like those in Appendix 2 is plenty of linkages to different aspects of the real space-time situation in which they act, while producing their own solution of the problem. In their protocols it is easy to find a melting pot, whose ingredients are both mathematical and extrarnathematical, extralinguistic, and so on. Typically, mathematical objects are referred to by means of subject's actions (e.g., processes of calculus made by the subject him/herself, or by somebody else); algebraic laws are put into the flowing stream of time. All this makes it easier for pupils to remember the meaning of formal things they are speaking of, and to control semantically the situation. This make it possible for them to use mainly stable, arithmetic conceptual models, even if their use has only a local character (i.e., they allow the construction of recurrence laws, not explicit global formulas) and costs a lot in term of memory; this style decreases very slowly in the course of years: I have found a relatively low difference between 10 years old children, who have solved problem 1 and my undergraduate mathematics students, when engaged in solving problems 2,3, and 4. Pupils (but also mathematicians' speeches) express the meaning of mathematical objects, relating them in some way to: subject's actions, the very processes of their constructions and generations, every other extrarnathematical information about them. This is a major root of the obstacles to symbolization and to purely syntactic manipulation of symbols (particularly when condensed in literal blocks: see [12]); in fact, this causes an evaporation of extra-mathematical data and, consequently, a possible dramatic loss of meaning. Evaporation is one of the main features needed in order to develop (pre-)algebraic models, from arithmetic ones. Without evaporation, no one can get the general (global) formulas for solving problems 2, 3, 4; but this may cause big conflicts with arithmetic models, which have allowed pupils to construct and
- 171. 159 grasp local formulas for the same problem. For example, in problem 2 (discussed above), to see the formula which expresses the sum of fIrst n integers in the local law of generation, pupils must forget the meaning of things and to look in another way to the very formula, they have written. The Double Polarity Every solution of a pre-algebraic problem, like those in Appendix 2 lives dialectically in a double polarity (see [13], for a theoretical framework of this kind of ideas). From one side, there is the subject, who solves the problem, with his actions in the flowing of time: the algebraic code is interpreted essentially in a procedural manner.. So arithmetic models can be easily adapted and integrated to this way of thinking. From the other side, the algebraic code must be interpreted in an absolute way, independently from the actions of anybody: it is a contemplation of relations and laws sub specie aeternitatis; neither procedures nor products of actions are involved: in fact only the abstract-relational aspect remains and its privileged code is symbolic. For example, 10 years old pupils who attack problem I, fIrst mimic concretely messenger's and prince's trips, and score in some way the exact time when they meet each other; after the third meeting, concrete manipulations become too diffIcult, so they try to foresee "the rule" and check it (procedural polarity, using both additive and multiplicative arithmetic models). When requested to justify the general rule, they have found (essentially, some formula which involves powers of 3, but expressed in multiplicative language), things are viewed from a relational point of view, and their reasoning can be summed up as follows (I use letters, like d, S, etc. for conciseness' sake; children like better other detours, e.g. the use of concrete numbers as variables): "suppose the messenger and the prince separate after d days, say in point S; it takes d days more to the messenger for reaching the castle and coming back to S; in the meanwhile the prince arrives at a point I; in the end, exactly in d days the messenger reaches him at a point E, symmetric of S with respect to I". The transition from procedural to relational polarity is even more marked for strategies of solutions in problems 2,3,4 (protocols from 14 to 16 years old students show a big jump: only the best reach relational polarity without any help). The main features of the double polarity with respect to the birth and development of algebraic thinking can be sketched as follows. Procedural polarity is a-symmetric, has a privileged direction, is controlled by means of tense adverbs and prepositions; its logic is the "logic of when". Relational polarity is ruled by logical and equivalence laws, which are typically symmetric: its logic is the "logic of if and only if'. The former allows a strict semantic control: extra-linguistic facts link steadily the speech to subject's actions in the flowing stream of time. The latter, on the contrary, is typically syntactic:
- 172. 160 concrete meaning has evaporated and only symbolic objects have been preserved. It has its own semantic, but it is a very formal one, as depends upon symbolic code and no longer on previous extra-linguistic facts. Procedural polarity is a sort of fictitious "real world", where one can do experiments. Its reality depends on the situation (namely, the problem, the pupils, the negotiation in the class, etc.), but it gives real feedbacks to the pupils who have built it, and allows them to activate their own conceptual models. Compared with relational polarity, procedural dimension in most of examined cases is more suitable to avoid stumbling-blocks; in other words, it allows a genuine validation (see later) and facilitates all those processes, typical of problem solving, by which data can be selected, processed according to one's conceptual models, integrated into new pieces of knowledge, etc. To be more precise, in problems 2 and 4, pupils first work in procedural polarity and find the recurrence law, which expresses locally the solutions (e.g., Pn+1 =Pn + n+l in problem 2; the law is expressed in an arithmetic, syncopated and typically procedural way in problem 4); afterwards, someone manipulates it more formally, almost forgetting the original meaning of the symbols (evaporation), in order to eliminate the parameter n (i.e. time) from the formula and to get the global solution. Here one meets the major obstacles: in fact the transition from procedural to relational style is really on the spot and it is very difficult to analyze. In any case, it is very rare (also at university level) that students do not start at the procedural level; it seems that the abundance of data (for example, to check many concrete examples) is a condition which aids subjects for a sudden transition to a relational style, after that a lot of experiments with numbers have been done, collected and compared. In the procedural polarity, the dominating epistemological style is arithmetic. Calculations are performed as soon as possible; every (syntactical) term is developed at once, until it is reduced to an irreducible one (number written in a canonical form). The epistemological model compels to reduce formal complexity of terms, without caring of their numerical complexity, namely the number of digits required to write them, e.g. in base 10. On the contrary, the epistemological style of algebra sometimes requires increasing the formal complexity of a term, at least locally, possibly preventing the growth of its numerical complexity: e.g., to solve problem 0 one must extract the identity (n+1)(n-l) =n2-1 from numerical examples, like 7x9 = 8x8 - 1, i.e., one must not only stop calculations but also increase the formal complexity of terms 7, 9, writing them as (8-1), (8+1). That is, one must rule things against arithmetic style. Analogous observations are true for the manipulations needed to pass from (recurrence) local formulas to (explicit) global ones in problems 2, 3, 4. The most difficult point is that such manipulations become active if (and generally only if: see above observation on school teachers who solved problem 4) are approached from procedural polarity (with a rich numerical base), but can be done effectively only destroying
- 173. 161 extra-mathematical tracks, that is eliminating this very polarity (evaporation), which is the semantic base of syncopated algebra. Sometimes, a main product of this process is what I call condensation, a typical phenomenon of algebraic thinking. It means that using the symbolic code, one can write concisely and expressively the amount of information of a term, whose complexity cannot be easily ruled with the syncopated language of arithmetic: typical products of condensation are global general formulas. Condensation is not the result of a process as much, but rather a general attitude with respect to the very process (what I call a procedure). Condensation marks deeply the passing from a procedural moment to a more abstract and relational one; it appears at once in the strategies of solutions of pupils and, as all processes which happen on the spot, is very difficult to analyze. For example, in all examined cases, most pupils who worked successfully at our problems, could solve them passing from procedural to relational polarity, with a sudden change of their style. However, even when explicitly asked, they were not able to explain what had happened. Only in problem 3 (and in similar ones), some pupils who had got the "formula" arguing in not algebraic terms (for example, using geometric patterns like those ofFigure 1) were able to explain it with a lot of details. Some Final Remarks To conclude, some general remarks and some sentences which yet need to be supported by empirical evidence. The discovery-construction of an algebraic rule is not a trivial process of generalization from particular to general, but it is stirred by the strained connections between the two polarities. Typically, the dialectic between the two polarities marks the birth of algebraic work. Relations between the aims of algebra to answer a question and the very algebraic work have been identified (see [11]) as the roots of validation (of the product) and control (of the procedure). In fact, aims live in a procedural dimension, while algebraic work develops in a relational one: control and validation sometimes involve both, sometimes there is a form of weak validation, where only the second polarity seems to be involved. In this case a real validation is very problematic; one can get it only at a very abstract level: I suggest for this sort of weak validation the name of a-validation. For example, this type of validation has been observed, even if in an unstable form, in all pupils who have solved algebraically problems 2, 3: in all these cases a-validation has been accompanied by condensation. At this point, it is possible to give the promised definition ofpre-algebra: it is every activity which involves both polarities: the first in an effective way, the second in a more problematic way (and it is so not only because arguments are so, but also because of the "didactic contract"). In other words, in pre-algebra the procedural aspect is still prevailing, but it is not a quiet way of life: the tension with relational aspects is smouldering under the ashes.
- 174. 162 Last, but not least, some comments from the point of view of mathematics teaching. Previous observations on algebraic thinking make sense as far as algebra is not taught as a set ofpurely mechanical and formal manipulative rules, but is used as a tool to solve problems, to justify pieces of reasoning, etc. The given problem must be an intellectual challenge for the pupils, which motivates and sustains all their work; consequently, it may contain the roots for transforming knowledge, provided one keeps an eye on such long-term aspects of teaching mathematics as the habit of contrasting and discussing in the classroom (formal mathematics can grow only on the base of a mathematics of spoken and written words). A typical example of this teaching style is the devolution of problematic situations to pupils (see [11]). In devolution (of a stimulating problem) are the roots of the continuous transformations of the problem - necessary to solve it - as well as of the dynamic process from problem posing to problem solving and vice-versa; devolution marks also the building of fictitious worlds, where pupils can do experiments and have feedbacks, which allow them a fruitful interplay between the problem and their own conceptual models. As well as the mastery of an algorithm may contract reasoning, so the use of a syncopated language makes it possible to elaborate and integrate one's knowledge into new conceptual models, where new algorithms possibly have their own roots. The natural consequences of this point for mathematics teaching are that, in order to base a good construction of algebraic knowledge, it is better to postpone formalization, to anticipate pre-algebraic problems, and to stimulate in pupils the developing of numerical experiments. Appendix 1 - Arithmetic Conceptual Models A summary of the notion of conceptual model is sketched, in order to make clear some references given in the paper. Following Lesh [10], a conceptual model is conceived as a system which integrates the organized mathematical knowledge (both concepts and processes) with the activity of problem solving in concrete problematic situations. Conceptual models are used by children from the very beginning in their activities of problem solving. Here is an example. To solve the following additive problem: (A) "Yesterday at noon there were 28 degrees (Celsius); today there are 4 more. How many degrees?", many children use a counting on model (see [5]). That is, they count 29,30,31,32 (perhaps using 1,2, 3,4 fingers while counting). Sometimes the problem can be solved directly by very simple, immediate and interiorized conceptual models (e.g., problem A with counting on). In such cases the models are very close
- 175. 163 to the intuitive and direct meaning of the words in the problem (in the example, 4 more means +4). But sometimes this is not the case and the model at disposal of the pupil may not be adequate to represent and solve the new problem; so it must be transformed and translated into a fresh one, possibly at a more fonnallevel. Examples: (B) "Today at noon there are 32 degrees; yesterday there were 28 degrees. How much more today than yesterday7"; (C) "Today at noon there are 32 degrees, 4 more than yesterday. How many degrees yesterday at noon7"; (D) "Six months ago there were 32 degrees (Fahrenheit). Today there are 54 degrees more. How many degrees7". In such cases the model counting on does not work directly. It must be transformed and adapted to the new situation (cases B, C) or it must be translated into a more formal and sophisticated model (case D). The major point now is that the text and the context of the problem may help or oppose such transformations, which, consequently may be performed in conformity, neutrality, or opposition with respect to them. For example, using the classification discussed in Carpenter [5], typical change or compare problems like a + b =7 can be solved using basic models, which are in conformity with the (standard) text of the problem. Confonnity means that the problematic situation, the text and the conceptual model are isomorphic copies ofeach other (so these are easy problems). In other cases (for example, in change or compare problems like a ±7 = c), basic models must be deeply reorganized in order to solve the problem: the produced model is not any longer a copy either of the problematic situation or of the text. It is already a reflected representation of the problematic situation, drawn out from the text, which from its own appears opaque. I call this situation of neutrality. Of course, models got in such a situation are less stable than those got in a situation of confonnity. Things become more difficult with change or compare problems like 7 ±b =c. In such cases pupils do not have at their disposal other concrete models, which can fit to the new problems. So they are forced to adjust their models to them, but unfortunately the text and the context of such problems make things difficult. In fact, they must make their changes in a situation of opposition with respect to the text of the problem. This makes such models very unstable and explains the difficulty of these problems: first, they are got as the result of a long chain of conceptual transformations; second, they are not good models, because of the opposition with the text. To manipulate properly such models, pupils must work at a more formal level than in previous cases, where conformity or neutrality makes it possible to mimic reality in a more concrete fashion. Conceptual models integrate concepts and processes in the organized network of mathematical knowledge. For each field of problems (in the sense of Vergnaud), the relative
- 176. 164 network is generated starting from some basic models, adapting them to the context and text of specific problematic situations, by means of translations and transformations. A conceptual model can be mastered by a pupil with more or less rigidity or instability: the former, if (s)he has low flexibility in reorganizing her/his models to new situations; the latter, when the various models generated by basic ones are poorly connected each other. It is clear that conceptual models may be studied from two points of view, at least: a) in connection with pupils' organized mathematical knowledge; b) with respect to the plethora of examples in a given field of problems (varying also the text and the context). The former supplies the so called semantic structure of problems and is a typical conceptual analysis, concerned with the meaning of formalized language in mathematics. The latter, on the opposite, focuses on processes and natural language and tries to pick out: (i) basic conceptual models used by pupils; (ii) typical transformations and translations which reorganize basic models in order to adapt them to concrete problematic situations. Conceptual models relative to a field of problems are properly described only by crossing both kinds of analysis. Appendix 2 - Problems Summary of problems quoted in the paper: in brackets information on the age of pupils, to whom the problem has been posed. For space's reasons, the form in which problems are quoted is not the same as that given to pupils. o. Take three consecutive numbers, calculate the square of the middle one, subtract from it the product of the other two; what is the result? now change numbers, and try again.... Explain if and why the result is always the same. [13 -16 years old]. 1. A prince decided to make a trip along his land and started with his followers. In one day they travelled 50 km. Next morning, the prince sent back a messenger to his castle, while he continued his trip. The prince went on travelling 50 km every day, while the messenger rode 100 km each day. How long a time before the messenger reached his prince? And if the history goes on, with the messenger who rides back and forth from the prince to the castle 100 km a day, while the prince goes on 50 km every day, how long does it take to meet the second, the third, the nth time? [10-12 years old]. 2. With a single cut, a big pizza can be divided into 2 parts; with 2 cuts (suppose to do straight cuts), a pizza can be divided at most into 4 parts; with 3 cuts one gets
- 177. 165 at most 7 parts, etc. Which is the maximum number Pn of parts one can get with n cuts? [14 years old and over]. 3a. Square numbers ｾ＠ are: n 1 On: ••••• •••• ••••• ••• •••• ••••• •• ••• •••• • •••• • •• ••• •••• ••••• Figure 2 2 3 4 5 4 9 16 ..... 6 7 25 36 Qn + 1 = Qn + 2n + 1 (recurrence formula) Qn =n2 (explicit formula) 3b. Triangular numbers Tn are: • • •• • •• ••• • •• ••• • ••• • •• ••• •••• • •••• Figure 3 n 1 2 3 4 5 6 7 T . n . 1 3 6 10 15 21 Find recurrence and explicit formulas for Tn . 49 ... 28 ... Show that 8 times a triangular number plus one equals a square number. 3c. Isosceles numbers In are: •••••• •• •• •••• • • • • • • • • • • Figure 4
- 178. 166 n 1 2 3 4 5 6 7 In: 3 7 13 21 31 43 ... Now solve: "The set Q of square numbers and I of isosceles numbers are not disjunct: the number 1 belongs to both. Find other numbers, which are both square and isosceles. How many are they? Justify your answers". [16 years old and over] 4. There are n people at a round table, and we eliminate every second remaining person until only one remains. For example, for n=lO, the elimination order (starting to count from n.1) is: 2, 4, 6, 8, 10, 3, 7, 1, 9 and 5 survives. Determine the survivor's number, Sn . [14 years and over]. References 1. Arzarello, F.: Strategies and hierarchies in verbal problems. In: International Congress on Mathematical Education, Short communications, p. 15. Budapest 1988 2. Arzarello, F.: The role of conceptual models in the activity of problem solving. In: Actes de PME XIII, pp. 93-100. Paris 1989 3. Brousseau, G.: Les obstacles 6pistemologiques et les problemes en math6matiques. Recherches en Didactique des MatMmatiques 4(2),165-198 (1983) 4. Brousseau, G.: Fondements et m6thodes de la didactique des matMmatiques. Recherches en Didactique des Math6matiques 7(2), 33-115 (1986) 5. Carpenter, T. P.: Learning to add and subtract. In: Teaching and learning mathematical problem solving (E. A. Silver, ed.). Hillsdale, NJ: Lawrence Erlbaum Associates, 1985 6. Chevallard, Y.: La transposition didactique, La Pensee Sauvage: Grenoble, 1985 7. Harper, E.: Ghosts of Diophantus. Educational Studies in Mathematics 18,75-90 (1987) 8. Kieran, C.: A perspective on algebraic thinking. In: Actes de la l3erne Conference Internationale PME, Vol. 2, pp. 163-171. Paris 1989 9. Laborde, C.: Langue naturelle et &riture symbolique, These d'Etat, Grenoble, 1982 10. Lesh, R: Applied mathematical problem solving. Educational Studies in Mathematics 12,235-264 (1981) 11. Margolinas, C.: Le point de vue de la validation: Essai de synthese et d'analyse en didactique des math6matiques, These, Universite J. Fourier, Grenoble 1989 12. Norman, F. A.: Students' unitizing of variable complexes in algebraic and graphical contexts. In: Proceedings of the VIII Annual Meeting of PME-NA (G.Lappan & R Even, eds.), pp. 102-107, 1986 13. Sfard, A.: On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics 22,1-36 (1991) 14. Vergnaud, G.: L'obstacle des nombres n6gatifs et I'introduction 11 I'algebre, Construction des Savoirs: obstacles et conflits, ARC, Ottawa, 1989
- 179. Can We Solve the Problem Solving Problem Without Posing the Problem Posing Problem? Judah L. Schwartz Massachusetts Institute ofTechnology & Harvard Graduate School ofEducation. Cambridge. MA 021138. USA Abstract: The first section of this paper deals with the inter-related nature of problem solving and problem posing. Although some time and attention is devoted in schools to problem solving activities in mathematical domains. problem posing as an intellectual activity is almost totally neglected. This is particularly unfortunate because the intellectual progress of mankind in mathematical and scientific domains depends on our being able to make and explore conjectures. i.e. problems that we pose for ourselves. We are thus confronted with a two-fold problem. First how might we change the way we educate people so as to help them make and explore conjectures.. Second. not all conjectures are interesting or productive. How can we help people to develop and exercise taste and judgement with respect to the conjectures that they make and explore? How does one decide what problems are worth working on? Information technologies offer the possibility of constructing "intellectual mirror" software environments that can scaffold the posing of powerful problems. The properties of such environments will be described. Finally some of the implications and consequences of the use of such environments in classrooms will be discussed. Keywords: problem posing. conjectures, intelectual mirror software, complex thinking On the Inter-related Nature of Problem Solving and Problem Posing The dominant paradigm in most schools around the world is, understandably, the transmission of the culture of the society to a next generation of youngsters. By and large this means that a considerable fraction of time in school is devoted to the teaching and learning of factual information or what is sometimes called "declarative knowledge". If all schools had to do was to help youngsters to learn what is already known, it could be argued that the process could be accelerated and its efficiency improved by the use of Computer Assisted Instruction (CAl) technology. Certainly in those areas of society in which we feel it is important to train people to know the already known, such as training soldiers to clean rifles and automobile sales people to sell new and used model cars, this technology has more than amply proved its worth.
- 180. 168 However, we have broader ambitions for schools in democratic societies. While it is still the case that we expect schools to be places for the transmission of the culture and for teaching the already known, it is also the case that we expect schools to teach our youngsters to think critically about their societies and the world around them. In the rhetoric of the education community, we would like them to become good "problem-solvers". In the United States in recent years there has been a growing amount of attention paid to the teaching and learning of problem-solving. Most of the efforts of researchers and teachers in this area fall clearly into one of two, largely non-overlapping camps. The ftrst camp consists of people who want to teach general problem-solving strategies that can be used broadly across intellectual domains. The second camp is largely ftlled with those who think that learning to solve problems is best done in the context of particular domains. I belong to the second camp. Aside from certain simple meta-strategies I do not think that there are problem solving heuristics that are realistically useful across domains. Getting from Boston in the United States to Oporto in Portugal, conjugating irregular French verbs, solving a differential equation and ftnding a decent restaurant in Moscow are all problems. It is hard for me to believe that there are any powerful commonalities among the strategies that a reasonable person would pursue in attempting to solve this diversity of problems. For those of my persuasion who are interested in the teaching and learning of science and mathematics, the pedagogic problem becomes one of ftnding ways of getting students to appreciate the power of some of the heuristic strategies that have proven to be valuable in science and mathematics. Here I refer to such strategies as shifting one's frame of reference, or invoking symmetry and invariance, or approaching a complex situation through a successive series of approximations, each modeling the problem situation a bit better than the previous one. Perhaps the most seminal of these problem solving strategies is one that Polya captures with his dictum, "ftnd a similar but easier problem and solve it! ". I ftnd it interesting that this most powerful of strategies for solving problems asks people to pose a problem as a step on the road to solving a problem. Is there something to be learned from this about the way we build curricula? Can we make the posing of problems an important part of the way we teach and learn mathematics and science? At the moment, it is certainly the case that our curricula in these subjects do not attach great importance to problem-posing. The Role of Conjecture in Intellectual Progress Students have odd notions about the nature of mathematics and science as school subjects and as intellectual disciplines. As subjects in school that they are asked to learn, many students, and teachers, believe that their tasks are, respectively, to learn and to teach the science and
- 181. 169 mathematics already made by other people. It is not hard to see the probable reason for this attitude. The problems that students solve and that teachers grade are already there printed neatly in the text. Moreover, the problems are carefully fashioned to have relatively clean solutions using methods that have just been explained in the preceding section. With this image of science and mathematics as school subjects, it is not surprising that the attitude that carries over to science and mathematics as intellectual disciplines is one of a body of knowledge with clean, right and wrong answers and with little, if any, uncertainty. Moreover, the disciplines are believed to be complete, with no further intellectual development required or possible except at the "frontiers of science". Needless to say, these frontiers are believed to be necessarily abstruse and complex and certainly beyond the comprehension of normal mortals. But the essence of good science and mathematics does not lie only at the cutting edge. It is entirely possible to do first rate science and mathematics in those parts of the subject that are believed to be sufficiently well understood to be taught in schools. Changing Attitudes and Expectations Clearly this is not the case with the teaching and learning of science and mathematics in our schools now. What must change so that we might move our schools in this direction? I believe that to do this we must expand dramatically the time and attention we devote to the posing of problems. By posing of problems, I mean both the formal posing of new problems by teachers as well as, and perhaps more importantly, the development and exploration of conjectures on the part of the students. A major aspect of the problem confronting us is the need to change the attitudes of people in and out of schools about the importance of problem posing and the making and exploring of conjectures. Doing this amounts to a strong redirection of the culture of schools and the expectations of various publics about what schools can reasonably be expected to do. This redirection will not occur easily or readily. It is unlikely that preaching and forecasting of educational doom will help, despite the fact that current practice fails dismally to educate more than the merest sliver of our youngsters to anything like scientific and mathematical competence. Lest it not be clear, my sense of competence includes the development of students' ability and willingness to make and explore conjectures. It seems to me that the only viable strategy is to adopt the advice of the late Jerrold R. Zacharias. He suggested that the way to bring about viable education reform was to "wheel in a Trojan mouse". By this he meant, to introduce a change in materials and practice that is perceived to be contiguous with and consistent with what is current in the schools, but which contains the germ of long-term, deep and systemic change. Considering the complexity of the educational system this is a major problem.
- 182. 170 Somehow, we must act in such a way that attitudes of students and teachers and the public all begin to change. Further, we must do this in a way that does not manifestly threaten what students and teachers and the public think ought to happen in schools. Thus, simple minded cries for new curricular content will not do. Curricular content is probably the most salient aspect of what goes on in schools, and any call to change the manifest content is bound to arouse suspicion and distrust. This is not to say that the curriculum should not change, but rather it is a word of caution about pinning extensive hopes on the effectiveness of a strategy that is based on changing curricular content. By the same token, it will not do to simply call for a longer school day, or a longer school year or more homework. While these changes in themselves are no doubt desirable, it is not clear that they will have the kind of qualitative effect we seek to have on the teaching and leaming of science and mathematics. These subjects are not taught adequately anywhere in the world. The most likely outcome of American youngsters spending as much time in school and doing as much homework as Japanese youngsters is that American students will do as well as Japanese youngsters in solving the problems that are used on the cross national tests of scientific and mathematical competence. As desirable as that may be, that is not the kind of change that is needed in the US. The kind of change that is needed in the US is precisely the kind ofchange that is needed in Japan and indeed, everywhere. We need to begin to educate youngsters to conjecture and to pose problems and to question evidence and authority. We need to have succeeding generations ask naturally and spontaneously about everything they see in the world around them, "What is this a case ofl". Where can we turn, if not to completely new curricular content and not to dramatic expansions of time in school? I would like to suggest that there is a kind of answer provided by technology that may help in just the ways we have outlined. I hasten to add that I am not referring to traditional CAl uses of microcomputers but rather to a use which is conceived of within and consistent with an entirely different pedagogical framework, i.e. guided inquiry. It is to this form of computer use in education that I now turn my attention to. A New Role for Information Technology Information technologies offer the possibility of constructing "intellectual mirror" software environments [1] that can scaffold the posing of powerful problems. In this section I will describe some of properties of such environments, as well as discuss some of the implications and consequences of the use of such environments in classrooms. What is "intellectual mirror" software and why do I believe that software of this sort offers a reasonable promise of helping to move us in the direction discussed earlier in this paper?
- 183. 171 Renecting the Consequences of Users' Actions The first, and perhaps most important property of "intellectual mirror" software environments is based on the fact that humans are not very good at making intricate and concatenated chains of inferences. To the extent that we wish them to be able to do so, it is important that we provide them with intellectual prosthetic devices that will help them. In science and mathematics it is often the case that there is a substantial logical distance between the starting points offered by nature and our conjectures about nature and the detailed implications of our models. It is precisely in this arena that appropriately crafted software environments can aid dramatically in extending our ability to explore our formal models. Briefly stated, therefore, intellectual mirrors are software environments that allow users to explore the entailments and logical consequences of the formal descriptions and models that they make of natural phenomena or of rule-governed systems. These software environments have no built-in pedagogic agendas. They do not ask the user questions, nor do they evaluate the "quality" of the user's efforts. A Short Discourse on the Nature of Primitives More needs to be said about "intellectual mirrors" because what has been described so far is a reasonable description of almost all programming languages. As wonderful as I think programming languages are, I do not think of them as practical "intellectual mirrors" because of their generality. Most programming languages can be used to in truly wondrous varieties of ways. This flexibility and diversity is a direct consequence of being able to build explicitly increasingly complex procedures from very simple primitives. One way to think about an "intellectual mirror" environment is as a special purpose programming language with a severely limited domain of application. Because of this limited domain, it is possible to imagine a different sort of primitive than the kind of primitive one normally finds in programming languages. Let us consider this difference. To make the discussion concrete, let us digress for a few paragraphs to describe one such environment The Geometric Supposer is an "intellectual mirror" environment meant to encourage and invite the making and exploring of conjectures in Euclidean geometry. Users' may make elaborate Euclidean constructions on a geometric shape of their choice, e.g. isosceles triangle, a pair of tangent circles or a user constructed quadrilateral whose diagonals form particular pairs of vertical angles and whose lengths are in a specified ratio. The "construction tool-kit" consists of such primitives as the drawing of line segments, perpendiculars, parallels, angles bisectors, inscribed and circumscribed circles etc.
- 184. 172 Measurement tools are available for the user to inspect the properties of the construction. It will often be the case that seemingly "interesting" properties of the construction will be suggested by such measurements. In order to inspect the generality of such "interesting" properties, the environment allows users to repeat the construction and measurements on arbitrarily many instances. Although I shall continue describing the Geometric Supposer later, the description to this point is sufficient to allow us to return to the discussion of logical and psychological and/or pedagogical primitives. To be workable at all, a programming language must have a manageable number of different primitives. Therefore, there may not be too many of them. Because of generality of the language, however, it is necessary for these primitives to be truly primitive symbol manipulating operations. One can think of these primitive as the logical primitive objects and operations of the symbol system. The usual way this problem is addressed in a computer-literate culture is through the building of "macros" and procedures that can be used to extend the language. Thus one can fashion complex "primitives" out of simpler ones for special purposes, limited only by the constraints oftalent and imagination. In an "intellectual mirror" environment, the primitives are of a different sort. They can be thought of as the psychological or pedagogical primitive objects and operations of the domain. Let us consider how this distinction applies in the case of the Geometric Supposer. The logical primitive objects ofEuclidean geometry are the point and the line and the logical primitive operations of the subject are those operations that can be performed with idealized straight edges and compasses. In the Supposer the primitive objects are clearly more complex than the logical primitive objects of the subject. Similarly, the primitive operations are more complex than the logical primitive operations of the subject. I believe that the objects and operations of the Supposer are better pedagogical primitives in geometry for most people than are the point, the line, and the straight edge and compass operations. It is, of course true, that the Supposer objects and operations can be understood in terms of the logical objects and operations of the subject. We have the experience ofcenturies of geometry teaching and learning to support this assertion. A Pedagogy of Intellectual Depth in Bounded Domains What I am suggesting is that we begin the subject with these pedagogically primitive objects and operations and work in two directions to develop the subject. One direction is the classical one of establishing how each of these primitives can be thought of as concatenations of the logically simpler objects and operations of the subject. The other direction is rather different.
- 185. 173 Because the primitives of the Supposer environment are moderately complex and because the environment makes it easy for users to concatenate these primitives richly while still being able to inspect easily the consequences of such concatenation, it is an environment that entices users into "supposing", i.e. making conjectures and exploring them quickly and in some depth. The reader is entitled to ask why I recommend starting "in the middle" and working in two directions when it is possible to start "at the beginning" with the logical primitives and build "macros" that can bring one to the starting point I am recommending. I hold the position I do for two reasons. The first is that the explicit, but reasonably abstract, notion of procedure has still not permeated the culture deeply enough for us to be able to build on it. Evidence to support this observation comes from as widely diverse sources as the difficulties youngsters have in learning to program computers to the almost universal avoidance by adults in the business world who use spreadsheets and word processors of the facility to write "macros". The second reason has to do with the nature of what motivates many, if not most, students. It seems that only a very small number of them share the a priori enthusiasm that many scientists and mathematicians have for the reductionist aesthetic that celebrates the building of intellectual edifices starting from the sparsest of ingredients. A large fraction of our students, and the larger population that they eventually join, have a more pragmatic set of criteria for intellectual worth. To be sure this often means, can the knowledge of a particular piece of subject matter be turned to advantage in the outside world in which one lives. But people are engaged by more than just the pragmatic. They are often engaged by interesting complexity, particularly if it is complexity of their own making. We can think of "Intellectual mirror" environments as offering people the opportunity to fashion and explore complex situations in domains that our culture has come to regard as important. It is evident that there are substantial implications of the use of such environments for enhancing the role of problem-posing in our schools. Capturing Particularity and Inferring Generality I wish now to describe a second feature of "intellectual mirror" environments and examine its consequences. Before doing so, I shall return, for the sake of concreteness, to the description of the Geometric Supposer. There is a focal problem in the teaching of geometry. The human cognitive apparatus seems to require diagrams and images in order to aid thinking about spatial matters. The diagrams we construct, however, are of necessity, specific. For example, although we can construct a regular 8-gon and a regular 12-gon, we cannot construct a regular N-gon. We posses no visual notational scheme for shapes that approaches the generality of our notation schemes for algebraic constructs.
- 186. 174 On the other hand, the mathematics we seek to construct within the framework of geometry deals not with the properties of individual shapes but rather with properties of classes of objects. It is relatively common for students, and even teachers, to be deceived by the artifactually particular properties of particular diagrams and to reach inappropriate generalizations that are rooted in the particularity of the diagrams we use. By allowing the user to make constructions on particular shapes, and then enabling the construction as a procedure to be separated from the original shape and to be repeated on another exemplar, the Supposer scaffolds the transition from the particularity of the diagram to the generality of the mathematics. This property of the Supposer environment enables and encourages the making and exploring of conjectures. It is easy for a user to explore whether a property discovered to be true in a particular case is true in other cases. If there are other cases for which the discovered properties is true, then the characterization of the nature of the range of cases for which the property obtains becomes an apparent and important problem. A central goal of education is to get our students to internalize the need and desire to continually ask of everything around them, "What is this a case ofl" This continuing fugue between particularity and generality permeates every intellectual discipline and activity. The student of literature must come to understand that the greatness of the nineteenth century Russian novelists lies in their ability to explore the generality of the human condition through the particularity of three very different brothers or a fatuous landowner. The student of biology must come to understand that the genetics of drosophila can shed light on the ways in which all organisms transmit their characteristics to succeeding generations. It hardly needs to be pointed out that the making and exploring of conjectures is, in its very essence, a way of asking the question "What is this a case of?" On the Need for Abstraction There is yet another property of "intellectual mirror" environments that needs to be examined. Here I refer to the need for the program to "know" with precision and accuracy what the user is talking about. Once again, it is useful to draw on a specific example from the Geometric Supposer. Many people who see the Supposer for the first time are somewhat surprised that it is not possible to enter a triangle, for example, by clicking' the mouse on three non- colinear points on the screen. Instead, the user is required to obtain a triangle on which to make a construction in one of the two ways. The first way allows the user to choose a randomly generated triangle from within a defined class, such as obtuse isosceles or scalene right. The second way allows the user to specify a triangle by a
- 187. 175 triplet of measures such as SIDE SIDE SIDE or ANGLE SIDE ANGLE. In either case the program "knows" what kind of triangle the user intends to work with. Suppose, on the other hand, that the user had to input a triangle by clicking on three points on the screen. Suppose further, that in an attempt to enter an equilateral triangle, the three points that the user clicked determined a triangle with angles of 59, 60 and 61 degrees. What should the program conclude? The power of an "intellectual mirror" lies in its neutrality with respect to the actions of the user. It must simply be a mirror that reflects, with as little distortion as possible, the consequences of his or her actions without attempting in any way to make inferences about the users' intentions. Imprecision in specifying the "object" of mutual concern to the user and the program is inimical to the achievement of this goal. I would like to argue that this need for an abstract description language that allows the user to communicate with the software environment is an asset rather than a disadvantage in terms of our long-term educational objectives. Despite all of the rhetoric surrounding the need for more "hands-on" experience in the education of our youngsters, we should not lose sight of the fact that "hands-on" experience is merely a stepping stone to a more important goal, namely "minds- on" experience. In a deep sense, getting youngsters to abstract generality from the particularity of their own experiences is among our most important ultimate aims as educators. Some Concluding Remarks In writing this paper, I have made a variety of assertions and analyses about the desired role of problem-posing and conjecture making in mathematical and scientific education. These assertions and analyses have been based on the experiences that my colleagues and I have had, both directly and indirectly, with large numbers of geometry students and teachers using the Geometric Supposer. There have been many reports [2] of the effects of this mix of new tools and habits of mind with traditional curriculum content on teachers and students. The settings from which these data are drawn vary widely as do the students and the teachers within those settings. The outcomes can be categorized in many ways. There are performance measures using traditional assessment instruments. Using these measures there are small differences in performance almost always in favor of the students who have become accustomed to making and exploring conjectures. More important, however, and probably more illuminating are different outcome indicators. These students come to school early in order to work on "their" mathematics on their own time. They discuss and argue mathematics with one another, and they end up taking further mathematics courses that they would otherwise not have taken.
- 188. 176 In addition, we have had a smaller and newer but equally encouraging collection of experiences with similar environments designed to encourage the making and exploring of conjectures in algebra at the secondary school level. I do not claim that expanding our conception of science and mathematics instruction so that it gives problem posing a central role will, by itself, repair the ills of our educational systems in these domains. But the insufficiency of problem posing as a prescription does not in any way detract from its necessity. References 1. Schwartz, J.L.: Intellectual Mirrors - A Step in the Direction of Making Schools into Knowledge Making Places, Harvard Educational Review, 25(1), 51-61 (1989) 2. Schwartz, J.L., Yerushalmy, M., & Wilson, B. (eds.): What Is This A Case 01'1 A Geometric Supposer Reader, Hillsdale NJ: Lawrence Erlbaum, forthcoming
- 189. Problem Solving in Geometry: From Microworlds to Intelligent Computer Environments Colette Labordel , Jean-Marie Laborde2 1Department ofMathematics and Statistics, Concordia University, Montreal H4B lR6, Canada 2Laboratoire LSD2, IMAG, Universite Joseph Fourier-CNRS, BP 53 X, Grenoble, France Abstract: The role of the computer environment on the problem solving processes is investigated in two kinds of situations in geometry: (a) situations in which the computer giving "objective" feedback is used as a tool and (b) situations in which the computer provides an aid based on an evaluation of the performance of the student (guided activity). In the first kind of situations, we analyze to what extent the constraints and feedback of a computer environment may affect the solving processes and the kind of solution elaborated by the student. After presenting the general principles underlying what an intelligent help provided by a computer environment can be, an example is proposed in the case of a specific geometrical task for which a prototype "Hypercarre" has been designed. Keywords: computer environment, learning environment, guided discovery, feedback, problem situations, geometry, geometrical figure, inductive and deductive reasoning, model of student's knowledge I - Problem Situations: Assumptions and Starting Points The role of problem solving in mathematics learning has been very often emphasized. Starting from this claim, the purpose of this paper is to identify the features of problem situations in computer based environments which may affect the solving processes and consequently favour the construction by the learner of new solving tools since this construction can be viewed as the potential source ofknowledge acquisition. It is generally assumed that solving processes occurring in problem situation depend on interactions between three main elements: the student as a cognitive subject, the mathematical problem, and the context. The fact that the computer is part of the context may lead to strong changes into these interactions and therefore affect two kinds of processes: - The "devolution" of the problem [4], i.e. to what extent he/she is really in charge of finding a solution with his/her own intellectual means and all his/her knowledge - The solving processes. As Pea [13] has stated, computers play not only the role of conceptual amplifiers but also of conceptual reorganizers as providing new
- 190. 178 facilities and therefore new solving tools. The same task which can be performed in a paper and pencil environment with certain solving strategies may require in computer environment completely different strategies. In the design of teaching-learning sequences, teachers can play on these changes to improve the learning processes. For this purpose, a better knowledge of critical aspects of the role of computer environments is required. We address this question in the specific case of the teaching of geometry. Geometry provides an appropriate field of investigation because of the important role played by external representations (usually called figures) and the new ways of using these representations which are made possible by software specifically designed for geometry. We will consider two types of problem situations in computer based environments: - Situations in which the teacher does not intervene, i.e. neither suggests ideas, nor corrects mistakes or erroneous procedures: these situations are called in French "adidactical situations" [4], the student is involved alone in the solving process and is alone in charge of finding a solution; - Situations in which the teacher or the computer as playing the role of the teacher provides the student with feedback taking into account an hypothetical state of the knowledge of the student which is inferred from the actions of the student at the computer and from his/her solution. Some criteria will be considered in this paper to analyze both kinds of situations. They were chosen because of their potential impact on the solving processes developed by the students: - To what extent is the problem open-ended? We focus our attention on problems which students can tackle with their prior knowledge but cannot completely solve without elaborating new solving tools. Routine problems, or problems too hard for the students, preventing them from any possible search for a solution are not considered here. The type of the question may affect the open nature of the problem. Questions providing the answer may lead students to use any means to come to the answer and in a sense may prevent them from really entering the problem: students are partially freed of the responsibility of the answer. A kind of question in the Varignon problem (see § III.I) like "show that MNPQ is a parallelogram whatever the quadrilateral ABeD is" does not induce the same solving strategies as a more open ended question calling for conjectures like "how to choose D such that MNPQ is a parallelogram ?" . - The tools which are available or the actions or operations which the student may perform directly (primitive actions); the solving strategies heavily depend on these features of the context of the problem. - Feedback provided by the situation enabling the student to become aware of the possible incorrect nature of his/her answer; this kind of feedback is a factor of evolution of the solving processes and consequently of the possible impact of the problem situation on the learning of mathematics by the student.
- 191. 179 II - The Specific Case of Geometry 11.1 - The Nature of the Production of the Student Geometrical tasks call for different kinds of productions. Students may have a better control of certain productions than of other ones. In particular they may be more easily aware of the incorrectness of their production in certain cases. Producing a proof and producing a diagram strongly differ from this point of view. Perception allows the student to infer information about the diagram he/she has constructed and to control it, whereas it is very difficult for a student to make a judgement about a proof he/she has elaborated1. Construction problems or optimization problems (for example, fmding the shortest distance between two varying interrelated points or the largest area of a varying shape) are such problems where a perceptual control is made possible. 11.2 - The Role of Perception in Solving a Geometrical Problem Perception is one of the possible tools which can be used to solve geometrical problems. But the power of geometrical knowledge lies in that it allows to solve problems which cannot be solved only by a perceptual activity. Therefore tasks in which perception provides the solution differ from a learning point of view from tasks which cannot be entirely solved by perception. The latter ones can be used to foster the learning of geometrical knowledge and to be designed so that perception may constitute a feedback giving information to the student about the path to a solution, or the correctness of his/her solution In these situations the learning environment is so organized that perception is not the only instrument of solution but is also an instrument of validation. 11.3 - The Ambiguous Status of Diagrams: Drawing and Figure Figures in geometry playa complex role which cannot be reduced to illustrating geometrical knowledge. They have both a function of a "signifier" and a function of a "signified". Parzysz [12] has introduced the distinction between "dessin" (drawing) and "figure" (figure). As material representations (drawing) figures give rise to visual impressions while pointing out on theoretical concepts. The theoretical and perceptual aspects may interfere, they may fit in each other or they may be conflicting, as shown by Duval [5]: a drawing can give rise to a visual 1This does not mean that students cannot control the proofs they elaborate in using diagrams and changing conditions of the problem. However it is not a spontaneous behaviour and has to be learnt.
- 192. 180 impression leading to an analysis of the figure which is not adequate for the mathematical problem, for example leading to break: up the figure into constituents which are not the right ones to be considered for solving the problem. In the Varignon problem (see § flI.1), the students are generally attracted by the sides of the initial quadrilateral and not by its diagonals because these latter are not drawn on the figure, but the solution is based on the consideration of these diagonals. Another element of the complexity of the notion of figure derives from its intended and implicit generality: a figure does not refer to one drawing but to an infinity of drawings. What is invariant in this class of objects are the relations between the objects. Furthermore only some geometrical relations are relevant for the problem, and sometimes they are less visible on the drawing than other ones. For example, the size of angles and sides of a triangle does not playa role on the property ofintersection of the right bisectors of the sides of the triangle. This is a real problem for students, when they move from the geometry of observation (geometry of drawing) to the geometry of proof (geometry of figure) as it has been observed by Schoenfeld [14]. The obstacles caused by perceptual aspects of drawing have been for a long time brought out (cf. for example Zykova [14] or Fisher [18]). Recently they have been summarized by Yerushalmy and Chazan [17] in three categories: - The particularity of a drawing may lead students to include irrelevant characteristics in the drawing; - Standard drawings cause difficulties in interpreting non-standard drawings: students were much better at recognizing right angles in an upright position, than when the right angle was "at the top"; - The inability for students, to see a drawing in different ways to attend selectively and sequentially to parts and whole, as Mesquita [11] could observe it in her experiments with students of middle school. Visual possibilities of computers have been used to design softwares materializing the multiplicity of drawings attached to the same figure, and/or the notion of variability of a drawing, briefly in materializing the theoretical notion of figure (which is sometimes called "class of figures" as in the French curricula) as for example Geometric Supposer (presented in Yerushalmy & Chazan [17] and in Schwarz [15]), Geocon [2], Cabri-geometre (presented in Laborde & StriiBer, [9]): drawings are no longer stereotypes, critical cases can be visualized, a geometrical situation can be considered from several points of view. In this kind of software, a necessary condition for a construction to be correct is that it produces drawings preserving the expected properties (for every position of the drawing in Geometric Supposer, when one element ofthe drawing is dragged in Cabri-geometre).
- 193. 181 The geometry of this kind of software seems to better materialize incidence geometry than paper and pencil geometry in so far as correctness of a drawing depends only of incidence properties and not of elements such as the size of the drawing. III - Problem Situations in a Computer Environment Without Teacher Intervention 111.1 - Influence of the Features of the Software on Elaborating the Solution We consider here mainly the heuristic phase of the solving process. Two criteria have been defined: (a) available primitives in the program, and (b) feedback provided by the program. They will be discussed below. The actions made possible by the software guide the student in his approach to the problem since even if the student does not carry out a trial and error approach trying each possible action, he/she is constrained by the ways of action offered by the software. It will be illustrated here by the example of the Varignon problem already used by Artigue [1] and Gras [7] in teaching experiments based on computer environments. The Varignon problem. The problem is the following: ABCD is a quadrilateral. M, N, P, and Qare midpoints of its sides (see Figure 1). 1 - How to choose D such that MNPQ is a parallelogram ? 2 - How to choose D such that MNPQ is a rectangle ? 3 - How to choose D such that MNPQ is a rhombus? 4 - How to choose D such that MNPQ is a square? It is clear that the use of software enables the student to produce several drawings and that he will be easily convinced of the fact that MNPQ is a parallelogram, more easily than in a paper and pencil environment with only one drawing. In the teaching experiment in a paper and pencil environment reported by Gras [7], the teacher was led to ask the students to draw several drawings in various positions replacing in some sense the facility offered by the computer. Using this problem with both programs Cabri-geometre and Geometric Supposer in a teacher training at Concordia University (January 91-ApriI91), we could also observe that the evidence given by the figure was clear to the students with both programs. But after using Geometric Supposer some of the students concluded that MNPQ could only be a parallelogram if ABCD was a parallelogram, or a kite or a trapezium. It came from the fact that a quadrilateral is drawn randomly by Geometric Supposer after the user selects one of the possible categories
- 194. 182 A M D ｾ＠ __________ｾｾ＠ __________--4C p Figure 1 among the following: parallelogram, trapezoid, kite, quads/circles (i.e. quadrilateral containing an inscribed circle or circumscribed by a circle), your own. The "your own" possibility actually is seldom used by students, firstly because it is the last one in the list (!) and secondly because it requires the user to give measures of sides, diagonals, or angles to let draw the quadrilateral. The problem being proposed only in a pure geometrical setting without indications of measures, an implicit contract prevents students2 from using measures: a change of setting is often perceived by students as a break ofcontract (We refer here to the notion of"didactical contract" as introduced by Brousseau, [4], i.e. to the implicit rules underlying the mutual behaviours of students and teacher). The only students who used the "repeat facility by deforming the present shape" offered by Geometric Supposer could be aware of the invariance of the property even if the initial particular shape was not preserved. In this teacher training the students had little time to become familiar with the program and few of them managed to use it extensively. It is a sign of the importance of the introduction phase to the use of a program; the mere fact to give to a student a program does not imply a spontaneous use of all possibilities by him/her (this can vary from one package to another). It is also interesting to contrast the use of different programs for the second question with this one for the third question. Artigue [1] reports that after the first question students using BucHde (a programming language made of LOGO commands and LOGO procedures) performed such a big effort in writing a correct procedure producing the drawing that they considered the problem as achieved after the first question and had to be strongly pushed by the teacher for looking for particular configurations of MNPQ. They did not succeed in solving the questions with BucHde: they often obtained particular parallelograms MNPQ but not 2 Students are able to decide alone a change of setting if they are taught to practice this but it is not a spontaneous behaviour; a teaching about moving from one setting to another one leads to a modification of the didactical contract
- 195. 183 deliberately, and they were unable to find the corresponding conditions on ABCD. The indirect way of choosing D (either by coordinates in a procedure or in a discrete way by placing D on the screen with an optic pen for each drawing) made very tedious the search for conditions only by means ofthe program. That is probably the reason why the teacher finally decided to achieve the work on these questions in a paper and pencil environment. With Geometric Supposer the teacher students were lead to the same kind of partially correct answers as they were for the first question. They concluded that if ABCD is a rhombus, MNPQ is a rectangle and if ABCD is a rectangle, MNPQ is a rhombus. This answer does not cover all possible solutions which are given by the following properties: if and only if the diagonals of ABCD are perpendicular, MNPQ is a rectangle, if and only if the diagonals of ABCD are equal, MNPQ is a rhombus. In this case the "repeat" facility by deforming the present shape and by moving step by step the vertex D was without help for those who tried to use it because it destroyed the configuration ABCD giving the expected MNPQ and because it was too difficult to move in a discrete way the vertex D in preserving the relation between the diagonals. As in the case of Euclide the facilities of the program did not lead to an economical search. With Cabri it is easy to obtain the particular configurations for MNPQ by dragging the vertex D, the continuous dragging enables the student to deliberately and rapidly produce the intended drawing. It does not imply that students automatically find out the necessary and sufficient geometrical conditions but Cabri offers the possibility to make observations on the realized configuration. The role of the teacher should also be important in this case: it consists of teaching the students how to extract information when using the dragging mode in order to be able to solve the geometrical problem. This point is discussed in § III.2. Concerning the feedback given by the programs, it is clear that the perceptual feedback worked very well for the first question since every student was convinced to obtain a parallelogram. Artigue [1] notes that perceptual feedback was often misleading for questions 2 and 3: students must devise some tests to be sure that they obtained a particular parallelogram MNPQ and very often the tests disqualified the expected property. In the same way the Supposer and Cabri offer the possibility to check the validity of visual impression by the measuring facility: measuring of lengths and angles. It is interesting to note the interplay between the different kinds of feedback given by these programs and to observe that students were not satisfied by the only visual aspect of the drawing. We think that it is possible to extend this interplay to the interaction between inductive and deductive reasoning in the case of software offering a continuous dragging of the drawing.
- 196. 184 111.2 Interaction between Inductive and Deductive Reasoning One of the dangers of software which has often been mentioned is of inductive kind: the. student would naturally conceive theoretical objects or relations from the only perception. Mathematics teaching could very easily avoid the deductivist danger but meet the inductivist danger peculiar to experimental sciences as Lakatos wrote [10, p. 74]: "On the other hand those who, because of the usual deductive presentation of mathematics, come to believe that the path of discovery is from axioms and/or definitions to proofs and theorems, may completely forget about the possibility and importance of naive guessing. In fact in mathematical heuristic it is deductivism which is the greater danger, while in scientific heuristic it is inductivism". Lakatos explains that in deductivism as well as in inductivism, the notion of counter example is not taken into account. It seems that the danger of a pure inductivism enhanced by visual possibilities of computers may be avoided if students can be taught how to look for visual counter examples, and make inferences about the reasons giving rise to counter examples that is about the initial conditions of the figure which are not satisfied by the counter example. That was the idea underlying the facility "by deforming present shape" offered by Geometric Supposer but it cannot be used in practice because of the lack of direct manipUlation: it is impossible to handle in a precise way the use of arrows That is also one of the ideas underlying the dragging mode of Cabri which can be really achieved through the direct manipulation. Feedback offered by the dragging mode of Cabri-geometre is of double nature: (a) giving evidence of the incorrectness of the solution, and (b) providing information on relations between various elements of the figure. - If a drawing is not constructed by means of geometrical relations but is only visually correct, its properties are not preserved by the dragging mode which keeps only geometrical relations used for constructing the figure or which can be deduced from them. The dragging mode is a potential means of producing counter-examples to a construction and thus provides the students with a kind of control on their productions. - It is also possible to extract information from the changes of a drawing through the dragging mode. Let us come back to the Varignon problem. The question is to identify the quadrilaterals ABCD leading to a rectangle MNPQ. It is very easy to visually produce a rectangle MNPQ while dragging the vertex D and observing the variations of the measure of one angle of MNPQ. This measure is continuously indicated on the drawing when using the facilities "mark an angle" and "measure". Observing the produced rectangle very often does not give any idea of the reasons why it is a rectangle. But when dragging D in controlling its trajectory the user can infer the dependence relations between D and MNPQ. If D is dragged so that line BD is invariant, MNPQ remains a rectangle. If D is moved aside the initial line BD, MNPQ is no longer a rectangle. The solver must then look for the fixed or invariant elements of the drawing: the direction BD and the
- 197. 185 B c o p Figure 2 points A, Band C. This first step is of perceptual nature. A second step starting from this visual information involves the search of invariant relations between MNPQ and ABCD, and the inference of a link between MNPQ and the diagonals of ABCD since AC is fixed and BD remains globally invariant when D is dragged in keeping MNPQ as a rectangle. The solver is then lead to draw the diagonals of ABCD. The third step is of deductive kind and consists in analyzing the figure in a traditional way: what are the relations between AC, BD and the sides of MNPQ? Such a process combines the two aspects of a proof distinguished by Hanna [8], since the proof "that proves" is the third step coming at the end of "an approach looking for insight into the connections between ideas", which according to Hanna is a characteristic aspect of "a proof that explains". In a paper and pencil environment, as mentioned by Artigue and Gras, as an effect of the didactical contract, the students generally do not draw the diagonals of ABCD if they are not asked to do it. But the visual impression does not lead to the deductive solution as long as the diagonals are not drawn. In the described use of Cabri we would like to stress that the changes in the solving process brought by the dynamic possibilities of Cabri come from an active and reasoning visualization, from what we call an interactive process between inductive and deductive reasoning. A passive visualization even of dynamic process is of little help. It is when analyzing the changes of the drawing under the dragging mode, and not only in seeing them, that the solver may find out some geometrical relations between elements of the figure. This way of solving the problem is strongly related to the fact that the solver does not work on the drawing but really on the figure;
- 198. 186 in this approach the figure is considered as a set of relations between variables and the dragging mode is a powerful means of externalizing these relations. It is interesting to note that a specific program for this problem with the same kind of dragging mode has been designed by Giorgiutti and experimented with students of grade 9 and younger students [7]. The relation of orthogonality between the diagonals of ABCD through the use of the dragging mode was perceived by almost all students but Gras notes that in the third question the isometry of the diagonals is not so easily found out (probably because the corresponding trajectory ofD is more complex: it is a circle). Gras also notes that students must be prompted to develop a complete deductive proof in a final step. The same interactive process between deductive and inductive approach has be organized on the following problem by in a teaching of geometry including the use of Cabri in a 8 grade class (the teacher was Capponi and the school is located in the surroundings of Grenoble): the right triangle. Let ABC be a right triangle with A vertex of the right angle. D is a point on BC. Let be I and J the feet of the perpendicular lines drawn from D to AB and AC. How to choose D to minimize the length of IT? Such an active and reasoning way of elaborating a solution is of course not spontaneous and must be learnt by the students. It must be part of a teaching of geometry involving the use of computers. It would be illusory to believe that such a teaching is easy but the resulting learning is certainly very powerful in tenns of conceptual knowledge in geometry. This leads us to focus on time aspects implied by the use of computers. 111.3 - Long Term Learning and Teaching The computer by its capabilities is a powerful tool of solving problems but because making possible complex actions introduces a new complexity. An elaborate interpretation of the dragging mode and of its effects is certainly constructed by the students through a long process made of interactions with various problem situations on the computer. According to the notion of "experience field" proposed by Boero [3] we think that before being an experience field for the student, a software is a field of experimentation as part of the environment in which problem situations given to the students take place. In interaction with the use of the software for solving problems, the student constructs a subjective experience ("internal component of the experience field" according to the tenninology of Boero) of the software which becomes richer. This dialectic evolution of the software as both experimentation field and experience field is a long tenn evolution in which the teacher plays a decisive role through the choice of problem situations.
- 199. 187 IV - From HyperCarre to a General ITS Architecture IV.1 • Short Description of HyperCarre Starting in March 89 a group of researchers including both authors as well as Bernard Capponi, Rudolf Strii6er and Vassilios Dagdilelis started an experiment on the following ideas: to link two existing tools, Cabri-geometre and HyperCard to achieve a system able to guide a limited construction task in geometry. The group consists of researchers in mathematic education, teachers, and computer scientists. Its aims were: • to structure the knowledge in terms of tasks (problems to be solved) - in a reference to a constructivist hypothesis; • to emphasize exploring type activity - in reference to a microworld environment; • to provide the student with specific help. The task proposed to the student consisted of constructing a square from a given line-segment as intended side in the microworld of Cabri-geometre, i. e. to realize on the screen using the tools of Cabri the shape of a square implicitly retaining its characteristic properties when dragging one of the endpoints of the initial line-segment. This task was selected as satisfying some important characteristics • relatively "simple" task, despite the existence of many different possible solving strategies (for the grade 5-6); • unambiguous visual feedback; • existence of so-called "related task" (see below); • evidence of some misconceptions. For a presentation of the initial research and of a report on the classrooms experimentation of the realized prototype refer to [9]. IV.2 • A Proposal Considering a problem to solve like in the square task it appears that solving the problem involves knowledege at different levels [16] • the perceptive familiarity about the geometrical objects involved in the task (in our example the squared shape), • the ability to make explicit characteristics of objects involved in the task (here the definition of a square in terms of right angles and equal length sides) (theoretical knowledge), • the ability of effective performing of actions in a sequence which results in an object with the expected properties (procedural knowledge).
- 200. 188 Of course these levels are interrelated; for instance starting an activity always requires perceptive familiarity and procedural ability. For a given task, we propose to design a student model as the list of Boolean entities representing the state of the student with respect to the preceding levels. In the case of the square-task we consider - the familiarity with the form of a square, - the awareness of right angles in a square, - the awareness of the equal lengthy sides, - the ability to produce right angles, - the ability to produce equal length line-segments, - some implicit knowledge about the status of intersection points. We attach then a to a task a structure base on the following pattern: mastered value is YES or NO theoretical knowledge { conditional knowledge { } set of related tasks { In order to illustrate the way the preceding structure for a task could be designed, we give below what it can be in the case of the square task: mastered value is YES or NO theoretical knowledge { familiarity with the shape of a square, knowing about right angles in a square knowing about equal length sides in a square conditional knowledge { knowing the use of perpendicular lines knowing how to realize equal lengthy sides knowing how to manage intersection points of objets
- 201. ) set of related tasks ( 189 use of the perpendicular line item from the menu task of carrying a given line-segment to realize an other segment with same length look at an explicit demonstration of the solution The set of fields corresponding to theoretical knowledge constitutes the declarative knowledge part and is supposed to be a set of necessary conditions. whose union is sufficient to assert that the student succeeded in the task. The set of fields corresponding to procedural knowledge includes both correct or incorrect procedures attached to misconceptions. The set of related task constitutes a set of tasks supposed to be mastered in order to performed the current task in a correct way. This could be the same task but in a different situation: consider the carry out of a length from one line-segment on a perpendicular line and the same task when the angle of line with the line-segment is rather small. This could a part of the current but in another context: consider the extraction from the whole task of constructing a square. the task or carrying a line-segment. A run of the tutor consists of four steps: 1) manipulation and displaying of the goal to achieve. 2) providing the problem. i.e. a formulation of the task. 3) course of activity with specific tools inside the microworld. 4) evaluation of the figure produced by the student by the system3• i.e. the system fulfills values in the different fields related to theoretical and procedural knowledge. At the end of the run different situations can occur - All the necessary conditions are fulfilled and the system put the value YES in the field "mastered" and the control passes to the calling task as follows; if all the related tasks have their "mastered" flag to YES. then the flag of the calling task is also marked as YES. - If some fields from the declarative knowledge do not have a positive value. then according to a pre-established mapping'! from all possible cases in the set of the related task. the system takes out one of task that has not yet a value YES in its "mastered" field, and the process continues with this "replacement" task. 3 Actually the evaluation starts just on request on the student, but this point is not essential and in some cases it would be preferable to have the tutor evaluating the solution of the student on the fly and providing the student with some message in the case its activity is too deviant. 4 This mapping fullfils the requirement that in case all the related task have their flags marked to YES, then the same task is proposed again. If the student becomes stuck again the degenerated task is proposed which evaluation gives by definition the value YES to its flag "mastered".
- 202. 190 A special case of interest is the case of a degenerated replacement task where the student is just presented a message in a fonn of a more or less complete hint. This is used when all the related tasks has been visited and obtained the mark "mastered" but the student is still unable to perfonn the task itself, in that case the student is provided with a complete demonstration of who to proceede and may be, just after, asked to redo what has just been done. This special feature ensures that the whole system continues to evolve and cannot fall in a dead end and This situation to appear for many students could be connected with the so-called epistemological obstacle and be a revelation for them. If it is the case there would be no wonder that the system in some sense fails here, giving room for the parallel intervention of a human tutor. This leads to an organization of tasks as a net ofinterrelated tasks (figure 3). dummy task Figure 3
- 203. 191 The whole process is initiated by considering a special task at the top of the net, with related tasks. The evaluation of this task is empty, and its "mastered" flag tum to YES just in case all the related tasks have the corresponding flag to YES. Starting from the top and going downwards the graph the student reaches a task at a certain level which represents really a problem to solve for him. We could try to list some of the features that give to the presented architecture some advantage in comparison to other systems. • First of all the modularity of the system allows a progressive realization of the tutor covering a given field, as it is always possible to design for a while the set of the task related to a given task just as demonstrative trivial task (the student is just faced with messages showing how to solve the task) • The system presents a character of self-adaptability, in the sense that when the student starts with the task at the top of the graph, the system by successive task replacements and going downwards the graph, will provide him with a task that is really a problem to solve for him. References 1. Artigue, M., BelIoc, I., & Touaty, S.: Une recherche men6e dans Ie cadre du projet Euclide, IREM de l'Universite Paris VII 1987 2. Ban, W., & Holland, G.: Intelligent tutoring systems for training in geometrical proof and construction problems. In: Learning and instruction, European research in an international context (H. Mandel, E. De Corte, N. Benett, & H. F. Friedrich, eds.). Oxford: Pergamon 1989 3. Boero, P.: The crucial role of semantic fields in the development of problem solving skills in the school environment (In this volume) 1991 4. Brousseau, G.: Fondements et methodes de la didactique des mathematiques. Recherches en Didactique des Mathematiques, 7(2), 33-115 (1986) 5. Duval, R.: Pour une approche cognitive des problemes de geometrie en termes de congruence. In: Annales de didactique et de sciences cognitives, Universite Louis Pasteur et !REM de Strasbourg, Universite Louis Pasteur, Vol. 1, pp. 57-74 (1988) 6. Fisher, N.: Visual influences of figure orientation on concept formation in geometry, in Recent research concerning the development of spatial and geometric concepts, ERIC, pp. 307-21 (1978) 7. Gras, R.: Une situation de construction avec assistance logicielle. Recherches en Didactique des Mathematiques, 8(3), 195-230 (1987) 8. Hanna, G.: Some pedagogical aspects of proof. Interchange, 21(1) (The Ontario Institute for Studies in Education) 1990 9. Laborde, I.-M., & StrliBer, R.: "Cabri-geometre": A microworld of geometry for guided discovery learning. Zentra1blatt fiir Didaktik der Mathematik, 90(5), 171-177 (1990) 10. Lakatos, I.: Proofs and refutations, the Logic of Mathematical Discovery. Cambridge University Press 1976 11. Mesquita, A.: L'intluence des aspects figuratifs dans l'argumentation des eleves en geometrie: elements pour une typologie, these de l'Universite Louis Pasteur, Strasbourg, 1989 12. Parzysz, B.: Knowing vs seeing, Problems of the plane representation of space geometry figures. Educational Studies in Mathematics, 19(1), 79-92 (1988)
- 204. 192 13. Pea. R. D.: Cognitive technologies for Mathematics education. In: Cognitive science and Mathematics education (A. Schoenfeld. ed.). pp.69-122. New York: Lawrence Erlbaum Associates 1987 14. Schoenfeld. A.: Mathematical problem solving. Orlando: Academic Press 1985 15. Schwartz. J.: Can we solve the problem solving problem without posing the problem posing problem? (In this volume) 1992 16. Tuyet, A.: Un ､ｾｶ･ｬｯｰｰ･ｭ･ｮｴ＠ A partir ､ﾷｈｹｰ･ｲ｣｡ｲｲｾＮ＠ ｍｾｭｯｩｲ･＠ de DEA (Didactique des disciplines scientifiques). Grenoble. June 1991 17. Yerushalmy. M. & Chazan. D.• Overcoming visual obstacles with the aid of the Supposer. Educational Studies in Mathematics. 21(3). 1990 18. Zykova. The psychology of sixth grade pupils' mastery of geometric concepts. Soviet studies in the psychology of learning and teaching mathematics. School mathematics Study Group Stanford University and Survey of Recent East European Mathematical Literature. University of Chicago. Vol. I. pp. 149-88 (1969)
- 205. Task Variables in Statistical Problem Solving Using Computers! J. Dfaz Godino, M.C. Batanero Bernabeu, A. Estepa Castro Departamento Didactica de 1a Matematica, Universidad de Granada, Campus de Cartuja, 18071 Granada, Spain Abstract: In this work an analysis of some task variables of statistical problems which can be proposed to the students to be solved on the computer, are presented. The objective of this didactical - mathematical analysis is to provide criteria of selection of the said problems, directed to guiding the student's leaming towards the adequate meanings of the statistical notions and to the development of their ability to solve problems. Keywords: teaching statistics, data analysis, statistical laboratory, computers in mathematics teaching, task variables Computers and Learning of Statistics The use of computers in mathematics teaching is a phenomena which is more and more stimulated by the teachers and researchers in Mathematics Education. The impact which it will have in the design of the curriculae of this decade is shown in documents like the NCTM Standards [5]. Nevertheless, this use creates problems in the field of research of the Didactics of Mathematics which can be summarized in the following questions: (1) How does the availability of computers affect the contents which should be taught? (2) What new problems, based on the use of these tools should be proposed to the students? (3) What didactic consequences are produced when this change is carried out? In particular, these questions are especially important in the field of Applied Statistics, due to the growing demand for formation on the part of pupils of different specialities, whose interest in the subject is essentially instrumental. The fact that a package of programs has been used in the Statistics teaching at those levels, can produce changes which are necessary to foresee and assess about the mathematical contents to be taught: certain rules and properties (as for example, the algorithms of abbreviated calculus or the use of the distribution tables) loose their validity. Other topics arise as a consequence of the availability of software, such as the 1This report fonns pan of the Project PS88-0104, granted by Direcci6n General de Investigaci6n Cientffica (MEC), Madrid
- 206. 194 exploratory data analysis. The way in which certain topics can be approached can also change: for example, the possibility of carrying out a multiple regression iteratively including or excluding cases during the process of analysis. On the other hand, the same statistical packages are converted into objects to be studied. Likewise, it is necessary to study the sequencing of the traditional type of teaching and laboratory activities with a computer, in such a way that the main aim of the statistical learning is not substituted by that of the learning of the calculus packages, thus producing a phenomena of "glissement metadidactique" [I]. The computers and the whole range of resources which this instrument provides, offers the pupil new uses of the statistical concepts and procedures, and so, new meanings for the same. However, within the potentially unlimited set of uses, we should ask ourselves: which ones should we select? With what criteria should we choose some problems or others for the different groups ofpupils? The didactical-mathematical analysis of the specific tasks variables to be developed in the computer environment are shown as a prior and essential step to be able to construct significant didactical situations. Role of Problem Solving in Teaching Statistics with Computers Stanic and Kilpatrick [6] indicate that the role of problem solving in the mathematical curriculum has been characterized within three modalities: as a context to reach other didactic goals, as a skill to be learnt and as an art of discovery and invention. Within the first modality the goal aimed at could be: the justification of the contents to be taught, to stimulate the motivation of the pupil, to provide a vehicle for the transmission of knowledge and the reinforcement of the learning by practice and this can even be used as a recreative activity. Each one of these roles can be affected, in the case of the problems referring to statistical concepts and procedures, by the use of computers. If the pupil has a computer available then he is released from the task of calculating the different statistics and from carrying out the graphic representations of the distributions. Moreover, it is possible, in the didactic time which is usually available, to solve a greater number of problems given the great processing speed of the computers. As a result of this, it is possible to wonder what the type of practical activity which can be proposed to the pupils is, and what roles could this method of problem solving play in the learning. We can classify these activities to be carried out with the computer in an overall way, in the three modalities which are described as follows: (1) Discovery, using experimentation and simulation. of some of the mathematical properties of the distributions or their statistics. as would be for example. the convergence of the frequency polygon to the density function by increasing the number of cases and subdividing the intervals.
- 207. 195 When a certain mathematical property is taught, the role which in the deduction of the same has had the intuition and the observation of this property in particular cases, is very often forgotten. In the field of probability and statistics, the computer is a very powerful instrument to be able to provide an intuitive approximation, prior to the deductive proof of many properties. This type of activity would reinforce the role of problem solving as a vehicle for the learning. (2) Invention of questions from a set of data by the student himself. One of the main difficulties which is put to the statistician in the analysis of the data provided by a user, is that often the person who has collected the observations, does not know what can be obtained from the same. This implies that in many of the cases the fact that a greater or lesser number of variables and observations than were necessary for the analysis have been collected. So, it is in our interest to propose to the pupils that they ask questions relative to a given file to increase their capacity of significant and solvable interrogation on a data set. This capacity may be developed using a special type of problems and it is made easier with the availability of computers. (3) Solving data analysis problems. The capacity to analyze data can not only be considered as a skill which is necessary in many professions, and which to a great extent justifies the teaching of the statistical concepts. It may also be seen as an art, since there are many levels in this capacity. Moreover, the work with files of real data is a source of motivation for the pupil. Finally, an adequate sequencing of this type of problem may be an effective means of learning or reinforcement of concepts. Task Variables in the Data Analysis Problems From the classification of variables of the problems given by Kilpatrick [4], in Table 1 we show some task variables, corresponding to the data analysis problems. In the following sections we describe these variables in detail and we propose specific examples of different types of problems which can be considered from this classification. Data Files: Way in which they are Obtained and their Structure As Jullien and Nin [3] indicate, to choose the set of data on which teaching should be based, the domain of the reality chosen should be relatively familiar to the pupil and sufficiently rich to make questions of didactical interest arise. From these applications, a first group of task variables which are going to condition the types of problems which can be proposed to the students, arise. These variables are the following:
- 208. 196 Table I: Task variables in the data analysis problems A: Way of obtaining the data: Obtained by the pupils Proposed by the teacher B: Field of application (Biological, Physical, Educational...) C: Number of statistical units D: Number of statistical variables: in the problem in the data set E: Characteristics of the statistical variables E1: Type (Qualitative, Discrete, Continuous) E2: Distribution characteristics Central position values Variability Form Skewness Outliers E3: Values: Number of different values Number of digits F: Mathematical content: Implicit Explicit G: Number of samples and their relationship: H: Opening of the problem the whole file a sample several samples related independent A: Way of obtaining the data files. One first distinction which we can make is the way in which these files are obtained. In the first place, the data can be collected by the pupils themselves, using a survey made on their companions or another similar project. In Table 2 the variables included in a survey of this type which will serve to illustrate the different types of problems which we are going to discuss, is shown. Likewise, a classification of the statistical variables of this file are presented according to the characteristics of the same which we will discuss in the following section. Due to the time which is required to collect data (a process which without doubt is also of great interest), it will not be possible to show the students the whole range of statistical
- 209. 197 Table 2: Example of the data files collected by the pupils (SURVEy) Name of the variables Sex Smoker/non smoker Weight Height Pulse at rest Pulse after30 press ups Pts. he carries in his pocket Characteristics Dichotomous Dichotomous Continuous, bimodal, without outliers Continuous, bimodal, without outliers Discrete, symmetric, without outliers, requires grouping Discrete, symmetric, without outliers, requires grouping Discrete, asymmetric, with outliers, requires grouping applications, if we limit ourselves to using this type of file. For this reason, a second type of data set should be provided by the teacher from the real applications in areas of interest for the pupil. However, the previous manipulation of these data files will be necessary, in the sense of reducing the number of statistical variables or number of cases, in such a way that an application which can be reasonably understood by the pupil can be obtained. The selection of those variables or values of the same which could be more significant for the introduction of the statistical concepts and procedures to the pupil will also be required. B: Field of application. These different types of data sets and the problematic situations of analysis which have lead to obtaining the same, give rise to a second variable: the field of application (educational, biological, geographical, etc.) which serve as a specific context for the problems. This variable can carry out a relevant role to capture the interest of the pupils. C: Total number of statistical units. This could also affect the complexity of the situation of analysis and of the problems which are considered about the same. In the case of a small number of cases it is possible that, by visual inspection, the student obtains a first idea of the characteristics of the statistical variables, and from this he can deduce, a priori, the most adequate type of analysis for a question considered. This is not possible with a high number of records, so in this case the probability that a student should modify his initial strategy of analysis grows, when the results show him certain unexpected characteristics of the variables. D: Number of statistical variables. Also of consideration is the number of statistical variables included in the problem as well as in the file. Both of them affect the complexity of the analysis situation, since the number of comparisons of pairs of variables (association studies) or of selection of parts of the data files (study of conditional distributions) grows with the square of the number.
- 210. 198 Characteristics of the Statistical Variables A second group of variables is given by the statistical variables themselves presented in the statement of the problem and also by those which have been included in the file and which could eventually be employed for the solution of the same. We have named this group Characteristics of the statistical variables (E) and we have subdivided it into three aspects: EI: Type of statistical variable. Variables may be qualitative (and within this dichotomous or not), discrete and continuous. E2: Characteristics of the frequency distribution, in particular are of great importance: E21: The central positions values. It is interesting to distinguish whether it deals with data coming from only one population, or whether the distribution is a mixture of two (eventually more than two) populations. In this case a feature of data which stands out is the bimodality, as occurs in the HEIGHT AND WEIGHT variables of the example shown in Figure 2, since these characteristics are well differentiated between the males and the females. E22: The greater or smaller dispersion, and whether this dispersion is or is not a function of another variable. E23: The form of the distribution (symmetric or not), since many of the statistical procedures based on the normal distribution couldn't be applied in cases of very strong skewness or else they would need a previous transformation of the variable. E24: Possessing outliers when the distribution is asymmetrical makes the graphic representation difficult and leads to having to employ preferably the order statistics in the analysis. The number of outliers is also conditioning the procedure to be used, since if there are one or two cases, they could be suppressed, but this distortions the results more and more when increasing the number of outliers. The magnitude of the differences between those atypical values and the average values is also a condition of the analysis. The type of variable as well as the characteristics of its distribution will affect the statistical procedure (graphics, measures of central tendency and dispersion, measures of association, etc.) which the pupil can choose to solve the problem. The existence of all the variants of these characteristics supposes a great diversification in the type of problems which we can propose to the students, especially when a joint study of two variables is carried out. For example, we consider the study of the bivariate regression. In traditional teaching the calculation of the regression line and the correlation coefficient by hand or with a calculator, are usually effective. Due to the time necessary for the training of the pupils in the skill of calculus, only a few examples which are chosen adequately in order to present a good correlation with few data, are solved. But, in practice, this form of procedure is not usually adequate. What happens if one or the two variables are discrete? As we know, the predictions should be carried out by interpolation
- 211. 199 for interior values to the observed range of the independent variable. In the case of one, or a few very high atypical values, the case of a strong value of the correlation coefficient may be produced which, however does not correspond to the existence of a real association between the variables. These values have to be eliminated to carry out the analysis with the remainder. Another case is that of the mixture ofpopulations, especially continuous and symmetric, in which there are really two different regression lines for the two populations, as occurred for the heights and weights of the males and females in the survey carried out by the students. The implicit mathematical model would be the covariance analysis. It is possible that one of the variables possesses a strongly asymmetric distribution, but without outliers; that could correspond to the case of nonfulfillment of the homocestadicity hypothesis. This example is presented by studying the regression of the cellular surface and the nucleus of the neurons with data of an application of the field of biology, due to the fact that with a similar size of the nucleus, there are different types of neurons which for the larger or smaller quantity of branches possess a larger or smaller cellular surface. In this case, we cannot separate the populations, but we can observe a proportional variability to the size of the nucleus, thus indicating that the precision of the prediction for the dependent variable depends on the value of the independent one. Finally it is possible that the linear fitting is not good, but a linear approximation may be used in part of the variation range of the independent variable, or that a piecewise linear approximation may be used or else an adequate linear fitting could be obtained previously transforming the data. By releasing the pupils from the calculations, we can dedicate more time to the presentation of various cases and to their discussion. The graphic part enables us to easily appreciate aspects which would otherwise go by unnoticed. This may be one of the greatest contributions of problem solving with the computer to the teaching of statistical concepts. E3: Values of the variables. In the case of variables where the grouping in class intervals is necessary we must consider the following factors: the total number of digits, the existence of the decimal part and the number of decimal digits or not. The total number of different values is going to influence in taking the decisions like the choice of the extremes and the amplitude of the intervals or the number of digits in the steam and leaf graph, which will affect the precision of the results. Other Task Variables in Data Analysis Problems As well as the previous groups of variables which arise when we consider the data file and statistical variables included in the same, we can also identify the following task variables in this type of problem:
- 212. 200 F: Mathematical content. One fundamental point is the particular content that may be used in the solutions of these problems, that is, the mathematical concepts and procedures "put in play". Within this category we should study the whole traditional range of variables of mathematical content and match the way in which these variables are affected by the availability of computers. One possible discussion is whether the mathematical content appears explicit in the verbal statement of the problem, and, in the case of being implicit, whether the concepts have been put forward previously in class or whether there is some type of anticipation or modulation of the learning. We must emphasize that in some activities of data analysis it is possible to anticipate in an implicit way concepts which will be treated later on a formal level, thus serving as examples and previous motivation of the said concepts. As well as the ideas of association and sampling, of which we have already spoken, ideas such as that of marginal and conditional distribution and their statistics, mode and order statistics can be used implicitly before having studied it. In particular, the modern approach of exploratory data analysis permits an intuitive introduction to the contrast of hypothesis by enabling the consideration of problems in which the pupil, by using the empirical evidence of the data, can confirm or reject theories. Nevertheless, the aspect of the possible extension of these conclusions may be more delicate. We can for example, propose the following question: PROBLEM I: Some parents usually give more money to their male children than to the girls for their weekly expenses. From the data in the sample, could you contribute some evidence in favor or against this affirmation in the case of these students? In the example proposed, we find ourselves with the fact that not all male children receive more money than their female companions. We need to compare the average values. Moreover, other possible factors could introduce a false association, for example the number of brothers and sisters. On the other hand, what is the magnitude of the difference from which we decide to take this into account? Should this magnitude be the same for a big sample as for a small one? G: Number of samples and the relation between them. One of the complexity levels of the data analysis problems is conditioned by the fact that the study is carried out in the whole file, in only one sample of the same, in two or more independent samples or in two or more related samples. In this way we obtain the following categories. GI: Analysis of only one variable (or joint analysis of several variables) with the totality of the data in the file, as in the following problem: PROBLEM 2: Graphically represents the weight of the pupils in terms of their height, in the me SURVEY. What type ofrelation exists between the variables?
- 213. 201 The conceptual and procedural knowledge "put in play" in this problem are the following: - Interpretation of a verbal statement and the translation of the same, in such a way that it can be answered using a statistical calculation or graph. - Capacity to choose a program which provides the said results. - Capacity to operate this program, responding with adequate data to the different questions of the same, if it deals with an interactive package. In the example, as well as knowing how to provide the relative data to the file and the variables to analyze, it is necessary to identify which of the variables are the dependent or independent variable. - An adequate interpretation of the different results of each program, which are usually varied. So, the minimum, maximum, mean and variance of the variables, the equation of the regression line, the correlation coefficient, etc. are calculated with the regression program as well as the graphic representation of the scatter plot for the example mentioned. G2: Analysis of only one variable (or joint analysis of several variables) in a sample obtained from the data file, which is specified by certain values of other variables, as in the following example: PROBLEM 3: If we only consider the data of the girls in the file SURVEY, what would the average weight of a student be if her height is 1.70m? The selection of cases for the analysis within a set of data given is one of the usual operations in the initial analysis of real cases. As well as the activities we have enumerated for the previous case, the pupil needs to identify the variable and the range of variation of the same which should serve as criteria to select the subgroup of units of the file given. He should also acquire the ability to operate the selection option using the software available. These activities also serve to show the idea of variability of the sampling process. G3: Comparison of one variable (or of the joint study of several variables) in independent samples obtained from the file, as the problem which we are now going to present: PROBLEM 4: What is the proportion of smokers in the class? Is it the same between males and females? In this section we include those questions where the student is asked to compare one determined graph or parameter of a certain distribution in two or more independent samples. These samples can be obtained from the data file, selecting records according to the values of another variable included in the same. This category of exercises is produced by the reiterated application of the previous one, so, the concepts and skills which we have described for this case come into play. But, moreover, an activity of comparison of the said distributions should be carried out and this would imply a first idea of the association between two variables. In fact, from the study of the contingency tables, this type of problems can also be solved by applying this program only once and so from this moment, it will be included in the first section.
- 214. 202 G4: Comparison of related samples, that is to say, the comparison of the distributions or statistics of different variables in the whole file: PROBLEM 5: Which of the variables "number of pulsations at rest" and "number of pulsations after 30 press ups" has more dispersion? What is the average difference of pulsation between before and after doing the press ups? This type of problem implies the repetition of one same process of calculation with different variables in the complete file, and so there is not a selection process. This corresponds to the study of the differences between related samples, which is also equivalent to that of the association between one variable included in the set of data and another implicit qualitative variable. H: Opening of the problem: In the previous sections we have identified several different categories of variables; likewise, from the examples shown, we can also deduce a great range of variability in the opening of the problems: in general, there can be several methods of solution and even several possible correct solutions for the problems considered. In particular, one interesting type of problem is to propose to the pupils to choose the most adequate among several possible solutions to a given problem. For example, to decide what is the most representative measure ofcentral value of a certain characteristic, or as in the following case, in what the solution is in some extent subjective: PROBLEM 6: Build several frequency histograms for the PTS in pocket variable (SURVEY file). In your opinion, what is the most adequate amplitude of interval? Reason your answer. Final Notes In this paper some of the specific task variables of the data analysis problems have been described, showing the diversity of problems which give rise to the classification presented. This variety will be even greater when the study which we have initiated here is completed with that of other task variables, non specific in the field of statistics, described in the research about problem solving. Moreover, it is necessary to take into account the relevant role for the pupil's learning of the variables linked to the role of the teacher and to the situations in which the problems are solved. As Dfaz Godino et al. [2] show, it is not enough for the student to face the solution of the realistic problems of data analysis, provided with powerful resources of calculation, to be able to acquire a better understanding of the conceptual mathematical objects. In this research we point out the limited comprehension of the notion of statistical association achieved by the pupils after a teaching process based on the intensive use of package of data analysis programs.
- 215. 203 We believe that the demand for statistics education, linked to the availability of computers in the teaching centers, is going to promote the interest towards this field of research in the years to come. The systematic studies of the specific variables of the data analysis problems will enable us to design didactic situations which make the acquisition of statistical concepts and procedures and the development of the pupils capacity in problem solving, easier. References 1. Brousseau, G.: Fondements et ｭｾｴｨｯ､･ｳ＠ de la didactique des mathematiques. Recherches en Didactique des ｍ｡ｴｨｾ｡ｴｩｱｵ･ｳＬ＠ 7(2),33-115 (1986) 2. Diaz Godino, J., Batanero Bemabeu, M. C. & Estepa Castro, A.: Estrategias y argumentos en el estudio descriptivo dela asociaci6n usando microordenadores. In: Proceedings of the XIV PME (G. Booker, P. Cobb & T. Mendicuti, eds.), Vol. III, pp. 157-164, Mexico 1990 3. Jullien, M. & Nin, G.: L'E.D.A. au secours de L'O.G.D. ou quelques remarques concernant l'enseignement de la statistique dans les ｣ｯｬｬｾｧ･ｳＮ＠ Petit x, No. 19,29-41 (1989) 4. Kilpatrick, J.: Variables and methodologies in research on problem solving. In: Mathematical problem solving: Papers from a research workshop (L. L. Hatfield & D. A. Bradbard, eds.). Columbus. Ohio. ERIC/SMEAC 1978 5. NCTM: Curriculum and evaluation standards for school mathematics. Reston. VA: NCTM 1989 6. Stanic, G.M.A., & Kilpatrick, J.: Historical perspectives on problem solving in the mathematics curriculum. In: The teaching and assessing of mathematical problem solving (R.I. Charles & E.A. Silver. eds.), pp. 1-21. Reston, VA: NCTM 1989
- 216. The Computer as a Problem-Solving Tool; It Gets a Job Done, but Is It Always Appropriate? Joel Hillel Department of Mathematics and Statistics, Concordia University, Montreal H4B lR6, Canada Abstract: Several problem-solving situations involving BASIC, Logo and Maple are presented. Because in each, computer feedback is quick and essentially 'cost-free', a solution strategy of computing, feedback and patching-up becomes the dominant heuristic wheter it is effective or not. This raises questions as to the way computers influence the nature of the problem, heuristics. monitoring and assessment, and beliefs. These questions are discussed in relation to the specific situations presented in the paper. Keywords: problem solving, computers in mathematics education, computer feedback, strategy, students' beliefs Prologue This paper is presented as four one-act plays, each act describing a different problem-solving activity which involves the use of computers. Though the scenarios are quite different from each other, in terms of the type of problems, the age of students, and the use made of the computer, I will weave a thread that connects them together. In each case I will raise the questions as to the role of the computer in the solution process, the suitablity of the problems for a computer environment, and how the same or equivalent problems might have been solved without the availability ofcomputers. The central notion of the computer as an investigative and problem-solving tool in mathematics hinges on its ability to provide a variety of visual, numerical and analytical feedback. Judah Schwartz, for instance, has emphasized this point on numerous occasions when discussing geometry problem-solving activities with software such as 'The Geometric Supposer' or 'Cabri-geometre'. He particularly refers to the fact that the computer affords users quick and essentially unlimited feedback at 'no cost'. Thus problem-solving work with a computer gives a solver the opportunity to quickly test ideas, to observe invariants, to do some auxillary calculations and, generally, to be bolder about making generalizations.
- 217. 206 However, when we speak of problem-solving, we should keep in mind that there is often a mismatch between our aims, as mathematics teachers, in giving students problems, and their own view of the nature of problem-solving activities. For students, the goal of problem- solving work is often focused on the ends and not the means. Their agenda is governed by practical rather than theoretical considerations and, as N. Balacheffreminds us, their concern is "to be efficient not rigorous; it is to produce a solution, not to produce knowledge" [2]. It is this issue, in particular, that will animate my evaluation of the role of the computer in problem- solving. I will look at the other side of the 'feedback-at-no-cost' coin and raise the question whether it, at times, leads to unintended and undesirable effects on the solvers. In particular, I will give examples from our research that point an accusing finger at solvers' reliance on computer feedback as the cause of an undue emphasis on 'producing solution' rather than 'producing knowledge'. ACT I: The Art of Avoiding Concepts and Techniques by Developing Brute Force Calculations Which Enable One to Travel More Lightly In preparing for a recent course for teachers on problem solving, I have had a second look at some papers and reports on research conducted in the late 70's. One of those was a final technical report of M.G. Kantowski [5] on two exploratory studies on the use of heuristics in problem solving. One of the reports is among the earliest studies to examine the effects of the use of computers on heuristic processes, in this case in the solutions of number theory problems. The report contains summary information as to the kind of use made of the computer (writing a complete program, a partial program, or using it strictly as a calculator) across the 10 students who participated in the study and across the 15 problems. Kantowski reports that when students were asked after solving each problem whether they would have solved it differently without the computer, "in high percentage of cases the students said they would not even have attempted to solve the problems if the microcomputer had not been available" (My underline). For example, 80% of the students said they would not have tried the following problem: "Find the five digit decimal integer ABCCA whose Cth power is the fifteen digit integer CCCCCDEBFEGFGFA". Indeed the percentage of students who used the computer as a calculator in solving this problem was also given as 80%.
- 218. 207 My interpretation of these results now tend to be quite different now than when I read the study for the fll'St time, nearly ten years ago. On initial reading, the students' claim that they would not have even attempted the problem without the availability of calculating power, led me to conclude that the computer was indeed the critical catalyst to launching the solution process. However, because of our own observations of problem-solving activities in computer environments that were conducted in the interim (some of which will be discussed in this paper), I now view the role of the computer in solving the ABCCA problem in a different light. For one thing, I am struck by the fact that the students who participated in the Kantowski study were not your average high-school students. They came from grades nine through twelve, have already had previous instruction in problem-solving techniques, were members of their school mathematics club as well as participants in mathematics competitions. It seems to me that they were the kind of students who were perfectly capable of having a good go at this problem without any use of calculating devices and the fact that 8 out of the 10 students stated that they would not have even attempted the problem suggests that their perception of the demands of the task and their subsequent solution strategies were shaped by the presence of the computer. This then raises several questions: Was their problem-solving behaviour consonant with the initial aim of giving such a problem? Was the presence of the computer actually detrimental to the use of heuristic reasoning? The ABCCA problem is what is generally referred to as "non-routine" and the purpose behind giving such problems is, at least implicitly, to challenge students and foster sound problem-solving strategies. It is a tailor-made problem for the pencil-and-paper world since it is sufficiently complex as to make its assault by brute-force hand calculations a very unattractive option. Unlike a "similar" problem such as "AB raised to the power A is CAB" which is amenable to a computational trial and error strategy, the intent of the ABCCA problem is precisely to entice students to be clever and to tap their mathematical resources, particularly their knowledge of base 10 notation. With such knowledge, it would be fairly straightforward to find that C =3 and A = 6. Finding the last missing digit, B, requires a little more work. The value B = 9 can be obtained by looking at the magnitute of the two-digit number 6B raised to the third power, which is bounded above by the number 33333D. Of course, there would be nothing wrong in using a calculator at this stage since the method ofsolving the problem would have essentially been cracked. It is an approach that one would have reasonably expected of students such as those who participated in the above study. Yet, given access to computers, these students felt that the computer was essential to solving the problem. While I have no details as to what calculations the students have made when solving the ABCCA problem, my guess is that they initially tried to get a hold of the value of C (which is
- 219. 208 the exponent in the problem) and that it did not take them many computational trials to settle on C = 3. The values of the other digits were probably obtained by either via computations or by extracting some implicit data from the problem ( such as "the end digit of A3 is A"). We can say that, from their perspective, the students used solution strategies which were optimal in terms of using the computer for finding a solution, i.e. they met the condition ofefficiency over that of rigour as stated in the above citation of Balacheff. But the availability of the computer has changed the character of the problem. It certainly made it into an easier-to-solve problem and we need to ask ourselves whether too much of the essence of the problem has been given away. Since the availability of a computer naturally led to strategies involving computational trial-and-error, the issue to consider here is about the status of empirically derived results such as C = 3 for the solvers. If it was simply accepted by the solvers as a fact then we may ponder if it is any different than giving C=3 as part of the problem-statement. On the other hand, such a result may have triggered a post factum awareness that C=3 is derivable directly from the givens by thinking of ABCCA as a number between 104 and 105. We could then say that the computationally-based solution has led to strategic knowledge about how to handle such problems, so the use of the computer has eventually resulted in 'producing knowledge' rather than simply producing results. We would then have the expectation that students would be weaned of their reliance on calculators if several problems of the same type were given over time. The situation here is quite different than, say, using software for geometry problem- solving where the computer provides empirical evidence of relationships which then have to be proved deductively. In the case of the ABCCA problem, the computer can provide most of the information about the digits and a tool for verifying that the answer is correct. Its use encourages a certain computational strategy which is effective for coming up with a correct answer but may fall short of providing insights about the nature of the problem. It is this aspect that seems to be addressed by the mathematician Michael Attiyah who commented on the limitation of the use of computers [1] by speaking of mathematics as "the art of avoiding brute- force calculation by developing concepts and techniques which enable one to travel more lightly". For Attiyah, no doubt, 'travel' refers to a life-long journey through mathematics. But for our students, 'travel' may be a short journey through a single problem. As the title of ACf I suggests, the availability of the computer gives legitimacy to their feeling that the exact contrary of Attiyah's statement is true.
- 220. 209 Intermission Since 'problem-solving' is a general banner which subsumes many different types of activities, I would like to delineate several general categories of problem-solving activities which are part of the scenarios discussed in this paper. First, let me point to the obvious. In some problem- solving activities, the ends rather than the means are important. We may be in need of some results and we would be willing to beg and steal in order to get them. In such cases, it matters little if the results are obtained via elegant methods or by a brute-force exhaustive search. This is the practical aspect of problem solving and one which is not necessarily intrinsic to the problem but rather depends on the context in which the problem comes up. It is not only evident in the domain of so called 'real life' or applied problems. Finding some normal subgroups of a particularly nasty permutation group may be as practical a problem to a 'pure' mathematician as computing some trajectories of ballistics for an 'applied' one. But the purpose behind most problem-solving activities within the school setting tends to have a slightly different focus. As teachers of mathematics we give our students problems that fall into the following, rather broad, categories: a. Problems to reinforce some newly learnt techniques and concepts; b. Problems to challenge students - cultivate interest, curiosity, excitement as well as develop problem-solving skills; c. Problems to develop some new mathematical knowledge (about objects, relationships, properties); One difficulty with problem-solving in instructional settings is to convince students to share the same goals as we have (when we make our goals explicit) and the use of computers in problem-solving may (unintentionally) make the job of convincing even harder. For example, the ABCCA problem falls squarely into category b., but as suggested in the first act above, the computer may have deflected the instructional aims by rendering the problem less challenging and by fostering strategies which are appropriate only in computational environments. The next two acts involve the use of computers in the solution of problems which fall into category c. I will start by extracting several episodes from the Logo studies which I conducted with Carolyn Kieran and Jean-Luc Gurtner [3]. ACT II: Successful Computationally-Derived Solutions as Obstacles to Generalization Here I consider some very common Logo tasks, namely those of reproducing figures with n- fold rotational symmetry. The implicit aim of such tasks is for students to arrive at the
- 221. 210 relationship governing n, 360, and the angle of rotation, hence the tasks fall into category c. described above. In the case I will be discussing, the solvers are 12-year olds who, through their previous 12 hours of work with Logo, have some notion of the relation of 360 to a complete rotation, though this knowledge manifested itself as a 'theorem-in-action' [6] rather than as an explicitly-stated knowledge. Before examining the nature of the solutions of these specific tasks, there are comments to be made about the effects of the particular Logo context on problem solving. It has been our experience that work with Logo fosters in pupils a particular set of beliefs that guide their problem-solving behaviour. In particular, the metaphor of 'drawing with the turtle', which is successful in initiating children into this computer environment, often leads to a too strong association of the activity with that of hand drawing. Hence, children tend to conceive of goals in 'more-or-Iess' terms so they will consider as proper solutions their productions of figures, even if these only approximate the task figures. The situation is also confounded by their awareness that figures appearing on the screen or on printouts are often distorted and so a high- fidelity replication is not really 'part of the game'. There is another level of complication with children's work, not specific to Logo, but which has to do with the very presence of geometric figures. It is well-documented that students have difficulties discerning the essential characteristics of a given figure. For example, in the task illustrated in Figure 1 below, our intention is to give a prototype of a figure with 3- fold symmetry (the children already possess a Logo-procedure to construct an 'arm'). Implicitly, some of the visible attributes of the figure such as its size, orientation and its particular location are not relevant for the task of writing Logo instructions to reproduce the figure on the screen. The important feature, from our perspective, is the invariance of the angle of rotation between the adjacent arms of the figure. But to the solvers, the salient features of the figure might be quite different. From our experience, the most striking is the fact that one of the arms is in a vertical position. But children might equally well focus on the three arms emanating from the same point, or possibly, on the vertical symmetry of the figure. Thus, in this geometric-computer context, not only is the criterion for having reproduced a figure guided by the belief that the screen output need only be approximate, but the figure to be reproduced is often misinterpreted. In time, we became aware of the need to be explicit, both about the given and the goal governing a task (e.g. "all the turns are the same in this figure"), so our 'experimental contract' was more transparent. Our insistence that the solvers' solutions had to meet all the stated conditions of a problem meant that we were increasing the level of demands on the solvers. We hoped that the that this would result in an increasing awareness of embedded relationships in the task that could be exploited in the solution.
- 222. 211 Figure 1 The children's solutions of such 'well-structured' tasks such as those of rotational symmetry were marked by rather consistent solution strategies. For example, among the rotation tasks, the fIrst one involved six-fold symmetry. A typical solving behaviour can be nicely exemplifIed by that of Ben, who began by constructing the vertical arm and continued with a repeated 45-tum. Once he noticed that the number of atms on the screen fIgure was getting too large, he changed his solution to one involving the sequence of 45, 90, 45, 45, 90 turns (the sixth tum being unnecessary for completing the fIgure) which, because of the vertical symmetry of the resulting fIgure, he considered as a solution. When it was pointed out to him that the turns were not all equal, he tried repeated turns of 50 then 65 then 55 and finally settled on 60. At this point, we might say that the status of his understanding of the nature of the problem is not very different than the empirical observation that C =3 in the ABCCA problem. The number simply works. So far, the above problem-solving behaviour could be termed as a reasonable and efficient use of the computer, as feedback has led to adjustments till the correct rotation was found. However, in contrast to the situation described in ACT I, in this study we intervened in a way that we had hoped would direct attention towards some explicit relationships (so Ben, for example, was prompted till he ended with the REPEAT 6[Arm Tum 60] construct). But more importantly, we also looked for evidence of 'producing knowledge', at least in some form of inductive reasoning, as we observed Ben's and the other children's behaviour over five other tasks involving rotational symmetry (and over 4 weeks) . Despite the similarities in the task, Ben never attempted to wean himself of the reliance on the computer in order to fInd the appropriate angle of rotation. In fact, he approached each of the subsequent tasks 'from scratch' using the same strategy involving the cycle guess ｾ＠ feedback ｾ＠ patch-up. This behaviour was rather typical of many other children whom we have observed over the years. What was perhaps more striking in Ben's case was that even though he was a boy-scout and had the contexualised knowledge that "360 makes a full circle" from using a compass, he never brought this knowledge to bear in his work on the rotation tasks. The availability and the 'no cost' of
- 223. 212 feedback completely dominated his solution strategy and, like so many of the other children, he did not search for a possible connection between 360, the number of arms and the rotation angle. The ABCCA problem is basically a pencil-and-paper task for which the computer was parachuted into the problem setting. We have some reasonable grounds for comparing students' solving behaviour with and without the computer, if not exactly on the given problem then on closely related problems. On the other hand, the rotational symmetry tasks are rather specific to the use of computers and are meant to build upon pupils' previous Logo experience. In this case, it is a little harder to imagine an appropriate equivalent task in a paper-and-pencil mode, in order to compare solving approaches. Since children's work on the rotation tasks was almost exclusively done in direct-mode, we might conclude that the computer essentially acted as a ruler (drawing the arms) and a protractor (making a tum) and this suggests one possible basis for a comparison. Pupils of equivalent mathematical or Logo background might be observed as they attempt to reproduce the same sequence of figures with rotational symmetry, using a ruler and protractor. Such a study (which I don't believe has been tried) would enable us to better ascertain whether the computer was an obstacle to generalization about the relation of the angle ofrotation to the number of arms. ACT III: Unsuccessful Solution Strategies Using a Computer which, nevertheless, Are Hard to Abandon In the previous act, I have suggested that a 'successful' solution strategy based on computer feedback followed by adjustments became a hinderance to a process of generalization across the rotation tasks since, being 'at no cost' to the solver, the same strategy could be evoked anew for each task. The rotation tasks involved essentially a single condition (equal turns) and involved one parameter (the number of arms). In my next example, which is also Logo-based, the problems involved two or more conditions so the strategy used by solvers on the rotation tasks was not likely to be successful. The children were given a sequence of tasks of reproducing geometric figures built out of rectangles, after they had in their possesion a two-variable procedure RECT (constructed with the turtle starting and ending at a vertex, and parametrized in the obvious way). All the figures in this sequence of 'centering tasks' had the stated condition that one rectangle was centered relative to another, as well as some other explicit constraints (both on relationship among the subfigures and on the order in which they were to be constructed). The simplest of these tasks involved constructing the figure Tee, shown in Figure 2, which comprised of two congruent rectangles.
- 224. 213 Figure 2 The full analysis of these tasks is found in [3]. Let me point out that despite the apparent simplicity of the figure, this is not a trivial task. First of all, the computer environment was constrained so the allowable moves consisted only of drawing rectangles and of displacing the turtle laterally (in particular, no line segments could be drawn). A solution which met both conditions (congruency and centering) required the coordination of the two magnitudes of the rectangle. In particular, regardless whether the stem or the bar rectangle was constructed first, the turtle had to be displaced by a distance (see boldfaced segment in Figure 3) which equals half the difference of the base and height of the rectangle, in order to 'interface' correctly with the next rectangle. Clearly, coming up with the relationship Ibase-heightl!2 required anticipating where the second-constructed rectangle will appear in relation to the first. Figure 3 After being given a printout of the figure, the children were asked to write a complete solution (as a procedure) prior to working on the computer. This meant that their initial use of the computer was strictly for verifying their first attempt at a solution. However, once started on the computer, they generally resorted to direct-mode work, relying exclusively on visual feedback and generating a cascade of patch-up strategies. Because these centering tasks involved multiple conditions the patch-up strategies, in turn, often moved the attempted solution further and further away from the goal. Unlike the rotation tasks, the inefficiency of these strategies for the centering tasks might have triggered a more analytical approach and an effort to draw out the relevant relationships in the figure, though we saw very little evidence of such switch of strategy.
- 225. 214 Once again, it is useful to compare the above solving behaviour with those of students in a somewhat analagous non-computer problem situation. If we look at Figure 2 above, it satisfies two explicitly stated conditions - the rectangles are congruent and they are centered with respect to each other. With this task as well as the other centering tasks, our solvers often focused on one condition which failed to hold and in the process of trying to fix it, they would 'de- structuralize' other conditions that were already met. For example, in the Tee task, if a solution had two congruent rectangles but a stem which was not centered relative to the bar, the stem was reconstructed with a narrower base which made it look more centered but violated the congruency condition (see Figure 4). Figure 4 This behaviour seemed to us quite reminiscent of students of the same age working in a non- computer setting on problems such as: "In a parking lot there are 40 vehicles. Some are cars and some are motorcycles. Altogether there are 100 wheels. How many vehicles of each kind are there in the parking lot" Our analysis of the solution process for the above problem [4] has also pointed to a constant shift of attention from one condition (100 wheels) to the other (40 vehicles) and to a difficulty in coordinating both conditions. However, when we compared the solving behaviour in these two very different settings, we were mainly struck by one thing. In the computer environment there was almost a total absence of 'blockages'. Yet, blockages are extremely common phenomenon of paper-and-pencil solutions of non-routine problems. Our computer solvers were never stuck; every visual feedback immediately suggested some patch-up action. The quickness of the feedback seems to have prompted an equally quick respond. Our solvers rarely gave up; they just kept on going until they felt either that they had solved the problem or that an approximate solution was 'good enough'.
- 226. 215 The fact that the solving of problems in a computer setting is 'action oriented' is a two- edged sword. Blockages can either make or break a solution to a problem. Solvers who are blocked may remain stuck and simply give up. But they may be, equally well, triggered to shift focus or strategy. As I have mentioned above, the attempts to solve the centering tasks rarely led anyone to abandon a problem, at least not before a substantial expenditure of time. Our analysis show that almost every solution attempt was a classical wild goose chase which tended to go on without interruption from the moment the work on the computer started. The total lack of any monitoring of progress, especially for tasks with multiple conditions, led to progressive solutions which were meeting fewer and fewer of the required conditions (we have spoken of the 'destructuralization' of a task in this context). We attributed this behaviour to the almost compelling nature of the interaction between the solver and the computer which seemed, at time, to take on a life of its own leaving the task on hand by the wayside. ACT IV: The Transposition of a Verification Tool into a Solving Tool In this scenario, I want to look at a very different use of computers. We have just completed a phase of a three- year long study involving the use of a Computer Algebra System (Maple) in the teaching of a basic course on functions1. This study evolved within the context of a university 'make up' course with a clientele consisiting mostly of students whose formal education was interrupted for several years and who needed basic mathematics as a prerequisite for their chosen disciplines. While this is not the place to give the rationale for the use of such powerful mathematical systems I would like to say that, among other things, we saw it as an opportunity to give the students a very different mathematical learning experience. Their time in the computer lab, which was half the total course time, was structured as a set of activities (given in terms of worksheets) which involved some directed and undirected exploration work with Maple for each particular function topic. These activities were then followed by posing some problems that were to be answered by the students without the use of the computer and then to be verified by accessing Maple once more. These problems were meant to consolidate notions that were being explored in the computer lab or discussed in class. I will mention two specific though typical examples. The study of the graphical representation of trigonometric functions first involved exploring different Maple plots as one 1 The Maple research is supported by Social Sciences and Humanities Research Council of Canada, Grant #410-89-1174 and by the Ministry of Education ofQuebec, FCAR Grant # 90-ER-0245.
- 227. 216 varied the graphical window or the period and the amplitude of the function. Then we posed pencil-and-paper problems such as the following: "Sketch below what you think the graph of sin(x) would look like in the interval 5ht..S31t. Check your answer by plotting with Maple" Similarly, with work on graphical representation of linear functions (which, by the way, was undertaken only at the very end of the course since there was nothing particularly 'easier' about them) we followed the initial exploratory work by problems such as: " Look at the plot of y =2x+1 in the window x=-S..S, y=-S..S. Predict the plots of y = 4x+l, Y= (l/4)x+l and y = -4x+l in relation to the initial plot. Check your predictions by Maple plotting these functions in the same window as the initial function" These are clearly the kind of problems that fall into category a. since the intention behind such problems was to consolidate the learning of a some mathematical concept. The intended role of the computer for these problems, which were posed at the end of each cycle of activities, was to give verification to the students of their own conceptualizations. However, when we started to pay closer attention to the way students worked, it became apparent that some of them were 'playing the game' very differently. As they grew more adept in using Maple, they completely bypassed the part of the activity in which we wanted them to reflect on their learning and they went directly to the computer for answers. With the computers in front just waiting to be prompted, the temptation was too great. We might say that this kind of behaviour is neither unexpected nor unfamiliar. It is the equivalent to that of students going directly to the back of the book and looking for answers (at least, to the odd-numbered questions!) when they are assigned textbook problems. But one of the reasons for our integration of a computer component into the course was precisely to bring about a change in attitudes and habits. Implicit in our thinking was the idea that the move away from a passive style of learning to one where there is an active interaction with a mathematically sophisticated environment will increase the level of intellectual engagement as well. Now, even casting a cursory glance at students working at a computer lab one is bound to be struck by the high level of activity. Students are personally engaged, be it in entering data, fixing syntax, manipulating different aspects of the graphs of functions or going through the worksheets. They often seem riveted to the computer screen to the point that, when the instructor wants to bring up an idea for discussion, hardly anyone pays any attention. But the question we must ask is whether this apparent intensity is not at the same time linked to a certain intellectual
- 228. 217 passivity. I do not mean to say that no learning takes place in this environment. Rather, I am wondering if for these students, just like for the Logo children I have described before, the constant and rapid level of interaction with the machine comes at a price. It seems to create an atmosphere which is more conducive to produce answers than to reflect on one's actions. How much of this attitude is context-bound to the computer lab and how much is carried to other contexts is still to be determined. Epilogue Each of the four one-act plays looks at a particular problem-solving situation and the effects of the computer on the problem, on the solver, and on the solution process. What connects these different situations is the notion that, in each, the solvers' behaviour was 'natural' to the setting. The presence of the computer evokes a particular strategic approach to the assault on problems which is based on computations (in a general sense, including numerical, symbolic, graphical and geometrical computations), and responses to feedback. The issues I have raised about the extensive use of such a strategy touch on the following: - Do certain problems lose their raison-d'etre as problems once a computer is used in their solution? What is learnt (about the problem and about heuristics) when a computational approach solves the problem? (ACT I) - Does a successful and 'cost-free' computational strategy put blinders on embedded relationships in a problem and on the possibility of generalization? (ACT I & II) - Is the reliance on a computational strategy so strong that, even when it is not profitable, it is hard to abandon and to look for alternative strategies? (ACT III) - When computers are on-hand and are known to possess answers to problems, is it realistic to expect students to solve the same problems on their own? (ACT IV) There are many problem-solving situations, some reported in this volume, in which the computer serves as an exciting problem-solving and investigation tool. I have deliberately raised issues relating to possible pitfalls and disadvantages with the purpose of putting us on guard against overstatements. The question of assessing the role of the computer in problem-solving is inextricably tied to the choice of tasks, individual solving styles and to educational goals, both of the poser and the solver. This brings us back to the issue raised by Balacheff about goals of 'producing knowledge' versus those of 'producing results'. For many students who solve problems with the aid of a computer, the precarious balance which we try to maintain between these goals gets tilted in favour of calculations and results.
- 229. 218 References 1. Attiyah, M.: Mathematics and the computer revolution. In Howson, A. G. & Kahane, J. P. (Eds.), The iriflUl!nce ofcomputers and informalics on mathematics and its teaching. ICME Study Series, 43-51, 1986 2. Balacheff, N.: Cognitive versus situational analysis of problem-solving behaviors. For the learning of Mathematics 6(3) (1986),10-12. 3. Hillel, J., Kieran, C., & Gurtner, J.-L.: Solving structured geometric tasks on the computer: The role of feedback in generating strategies. Educational Studies in Mathematics, 20, 1-39, 1989. 4. Hillel, J., & Wheeler, D.: Problem solving protocols: a task-oriented method of analysis. Mathematics Department, Concordia University, Montreal, Canada 1982. 5. Kantowski, M. G.: The Use ofHeuristics in Problem Solving; An Exploratory Study, Final technical report, NSF Project SED 77, 10543 6. Vergnaud, G.: Pourquoi une perspective epistemologiqUl! est-elle necessaire pour la recherche sur I'enseignement des mathimaliqUl!s? Proceedings, 5th annual meeting, PME-NA, Montreal 1983.
- 230. Insights into Pupils' and Teachers' Activities in Pupil- Controlled Problem-Solving Situations: A Longitudinally Developing Use for Programming by All in a Primary School Chronis Kynigos Department ofInfonnatics, University of Athens, 19 Kleomenous St, 10675 Athens, Greece Abstract: This chapter discusses a long tenn attempt to inject a pupil-centred problem solving pedagogy in a primary school in Greece. The project involved the use of Logo programming, and aimed at cooperative work, active thinking and the self-initiated solving of problems. All pupils and teachers took part. The projects' elements: teacher education, curriculum development and research into pupils' and teachers' strategies, are described. Insights into the process of structuring openended problem-solving, the process and content of children's learning and the developing of teachers' agendas for intervention are offered. Keywords: Logo, open-ended problem-solving, active thinking, cooperative problem- solving, children's learning, teacher strategies, teacher education, curriculum, programming, investigations Introduction The historical coincidence of the emergence of infonnation technology and of the developments in our knowledge concerning the nature of learning, is giving rise to a rather fundamental reflection on how and what we want students to learn in schools. On the one hand infonnation is becoming increasingly varied and easily accessible, thus reducing the hitherto prevailing importance of transmitting infonnation as the school's key function. On the other, we are realizing that learning happens through a continuous re-organisation of personal experience rather that the accumulation of infonnation. Education in the near future may therefore be more meaningful when it aims to enable individuals to critically select and use infonnation. to actively pose and solve problems and fonn aims for themselves, to practice and develop their creativity both in the Arts and in the Sciences. In one way or another, as educationalists we are thus studying methods by which to strive towards a society of "learners" rather than one of "knowers". Technology is providing us with useful tools with which we can learn what it is to become better learners and subsequently how to achieve it. For instance, we now have some longitudinal research examples of interactive, dynamic classroom environments using
- 231. 220 microcomputers as tools for the students themselves, where the students engage in meaningful problem solving activity [7, 13]. As we now know, one of the most well researched computer environments in education, Logo, can become the means to generate learning situations where children use ideas first and then discriminate and generalize them [3, 5]. In this way, the learning of content originates from, and is closely related to, experiences which are personally meaningful to the children [8]. The computer environment is then used as "scaffolding" [4] to achieve goals even when ideas and concepts are understood only partially and locally. More recently, Logo based microworlds have been developed, where children may use specific primitives as a support in focusing on specific content areas. These primitives are either very fundamental, so that children may only build with them [11, 17], or more complex, so that there is also ground for discriminating the ideas embedded within the primitives themselves [2, 6, 12]. Moreover, although not fully acknowledged in early Logo research, the teacher is an indispensable factor in this process; we now know that children might employ specific concepts when working with Logo, but they are not often aware of the power of these, or even that they are using them at all [1, 9, 10]. A teacher who understands which concepts children are learning, may encourage them, for instance, to reflect on, generalize and use these concepts in different contexts [3, 5]. But how can this come about in a classroom where the teacher has been transmitting knowledge since the beginning of his/her career and the children have been consequently conditioned to passively sit and listen and then try to memorize at home? What is most likely to happen when this teacher is given the technology to use Logo in the classroom and is "encouraged" to do so, given a user friendly manual type book? Computer environments like Logo, often fall victims to their power and flexibility as learning tools since they may easily be used even in a classroom where knowledge is transmitted by the teacher and the computer application becomes just another chunk of information the children have to memorize. It is often the case that the issue of the way Logo might be used does not even arise; the computer is just locked up in a cupboard, or the teacher will pass the "responsibility" to an enthusiast colleague, who in tum will teach enthusiast children - those who most likely would use computers anyway. In many cases where the schools can afford it, a computer "expert" is hired to teach Logo (!), most of the time having little or no experience with educating children or teachers. In any case, there is a tendency for the use of computers to become a privilege of the few. Large scale implementations of the above research ideas about how to exploit Logo as a problem-solving tool for all children in normal classrooms, within the day-to-day function of a prescriptive educational system, consequently present serious problems that ideal research situations do not have to face. If environments like the above are to be created, educational principles have to be questioned, teachers have to be trained, often in contradiction to what they have learned so far. Furthermore results cannot be measured in the same ways as before since
- 232. 221 conventional measuring instruments and expectations of the children's performance would be irrelevant [15]. Providing children with the opportunity to take control of their learning and to solve personally meaningful problems in classrooms thus has to be a process of negotiation between innovation and relevance. For instance, there is no recipe for the "right" way to use Logo in the classroom or the "right" way to train teachers and changes in educational practice will certainly not come about as a result of short-term training [14]. Allowing children to learn by doing and teachers to reflect on their children's learning and perhaps to re-formulate their practice, may currently be the most important educational potential for which such open-ended computer environments may be used. Within this framework, although what is actually taught may vary widely, it is always of educational value. This paper discusses an attempt at a long-term injection of a pupil-centred problem-solving pedagogy into the culture of a primary school, involving all the school's teachers and children, within an educational system predominantly based on the prescriptive transmission of knowledge from teacher to children. While claims to any outstanding success are not intended, the continual negotiation of the above ideas and arguments within the entirety of the school's society (all non-enthusiasts included) may be worth sharing. The project is taking place in a private primary school in Greece and is now at the end of the final year of its planned six-year duration. Inevitably, the school itself being one of the privileged private schools in the country does not facilitate straightforward generalizability of the project's experience to primary schools within the state system. However, the project is meant to generate potential for a pedagogically-aware use of technology through Logo within the Greek educational system. Thus, the early availability of technology and specialized teacher education in one school may be exploited in order to provide a bank of ideas for wider application in the near future. An Outline of the Program's Main Features There are two main factors which enabled the program to take place: the acquisition of the technology (details are given below) and the availability of one hour per week for each class to work with the computers for all classes from the 3rd to the 6th year inclusive. However, there was no possibility to find extra hours for teacher training apart from 10/20 hours before the beginning of each school year and on occasions within the school program where the teachers of one grade had a free period (e.g. when the children did their P.E. or their English). Furthermore, the teachers were not expected by the school's direction to receive training for and to undertake an additional topic to teach as an add-on to their already heavy schedule. It was therefore essential to begin using the computers as a tool for educational objectives which were already present. This was already a lot to take on, since none of the
- 233. 222 teachers had had any experience with computers. There were issues of technical know-how to be resolved, but more importantly, the teachers had to learn to use Logo in order to generate learning situations amongst their pupils according to Logo's philosophy. Learning more about the content of Logo programming and Logo mathematics would come gradually for the teachers, possibly shaping their intervention strategies vis a vis the children. All 24 teachers of the school are taking part (16 of them each year) together with all the 500 or so children from the third to the sixth year of primary inclusive (age range 8 to 12). The main aspects of the project maintained throughout its duration are: a) teacher training; b) curriculum development; c) informal evaluative research on the development of the teachers' strategies and the children's learning. After a pilot first year, implementation, evaluation and development were concurrently maintained throughout the project. The main educational features of the project are analyzed in more detail below. Educational Objectives The experience of a classroom of 20 children learning Logo during the pilot year highlighted their dependency on the teacher for learning, their reluctance to explore an idea voluntarily or solve a problem and their overwhelming inexperience at working within a process of constructive communication amongst themselves [12]. The results from the pilot year, the available conditions for implementing the program (one teaching period a week for each class, very little time available for teacher training) and the aim to exploit Logo's educational strengths were consequently the main factors in formulating the educational objectives with which the program began. These were to generate classroom environments encouraging: a) active thinking (e.g. to solve own problems); b) initiative (in thinking, creativity and decisions); c) cooperation (cognitive, affective, social). These objectives were not mediated to the teachers in a prescriptive manner. The way in which they are phrased above is the result of them having been reformulated through discussion with the teachers who subsequently perceived them as an educationally meaningful reason for using Logo. It is evident from the objectives themselves that the initial focus of the program was on setting up and establishing a child-centred, investigational classroom atmosphere, in comparison to what was common practice in the day to day school curriculum. A gradual injection of content within this process, in the form of Logo programming ideas such as the use of iteration, procedures, subprocedures and variables, began later; implicitly at first, and then developing into an organized and explicit "curriculum" (see below). However, content is not only perceived as a curriculum of specific Logo ideas, slowly integrated within the program's working structure. The teachers are also using their developing knowledge on Logo
- 234. 223 programming in order to intervene meaningfully in the children's working process. The content areas predominantly influencing their interventions are: a) mathematics; b) programming; c) written expression. The way in which the teachers encourage the children to use and reflect on the mathematical and programming ideas which are embedded in the Logo programming environment and to coherently write about these and about their cooperation are analyzed below. Classroom Setup The setting is as follows. There is one computer room with ten Apple lIe's each linked to one of the three available printers (Imagewriter II's). The version of the Logo we are using is L.C.S.I.'s Apple Logo II. One computer period a week is allocated to each class of 30 children, during which they work in freely formed but permanent groups of three with their own teacher. The teacher's role is to encourage the children to cooperate within their groups and to develop control over their own learning. The teacher therefore avoids adopting the role of the transmitter of factual knowledge and instead supports the children in their exploration of ideas. Each group of three children uses one computer, disk (where applicable), and writing book. There is free collaboration and the groups are responsible for presenting results. A Structure for Working Encouraging a non-directive approach to the children's work with Logo did not in effect imply the absence of some working structure and some tangible result of the children's efforts. On the contrary, it was considered necessary to layout clear cut "rules" for a working method that was, after all, relatively new to everyone. However, the aim of the working structure was to facilitate the encouragement of the program's process-related educational objectives. The main component around which the program is organized has been given the name of an investigation. An investigation consists typically of a project carried out by a group of three children and a subsequent written presentation of the project. The duration of the children's work with the machine is four teaching periods for each investigation. They are subsequently given time to prepare their presentation which is commented on by their teacher in writing. Finally, one teaching period is spent in the classroom, where groups of children orally present their work to the rest of the class, followed by a discussion where the audience poses "critical" questions on the presentation. An investigation starts and finishes at the same time for all children in a classroom.
- 235. 224 Importance is given to the children's presentations of their investigations. Each presentation essentially has three parts; a) a written essay on problems met during the project, points of interest and issues referring to the group's cooperation; b) a written record (or a printout) of the commands and procedures written during the project; c) a graphics dump of the constructed shape(s), design(s) or figure(s). In effect, therefore, each investigation is completed in six weeks. The children's presentations are useful in four different ways. Firstly, it is an integral part of the children's Logo work, since they use three different symbolic methods (writing, Logo commands, graphics) to express a meaningful reality, also an important aspect of working in a turtle graphics environment [13, 16, 18]. Secondly, as a tangible result of the children's work, which facilitates reflection, social mediation in the classroom and the opportunity for the teacher to provide further feedback to the children. Thirdly, the presentations are an important part of the collected data for evaluative research, since they provide a picture of both the children's work and the teachers' strategies through their written comments. Finally, the presentations are a means by which the program's objectives and the children's work is mediated to interested parties outside the school's everyday life, i.e. the parents, the board of directors, other schools and educators, educational conferences. Content As the project progressed, it became common experience amongst the teachers that what the children learned in the Logo classroom was to a large extent not strictly intended or expected by the teachers themselves. However, an intended content was developed gradually by the writer in negotiation with the teachers, as a function of what was actually going on in the classroom and of the teachers' and children's growing expertise with Logo programming and technical know-how. The content focused on the features of the Logo programming language in the role of thinking tools, in the sense that the children would be provided with these tools and encouraged to use mathematical and programming ideas during their investigations and structure their written essays in their presentations. The content for each year (age group) is outlined below. The actual Logo commands and programming methods are being introduced to the children in between investigations. During investigational work, the teachers only encourage a group ofchildren to use a particular idea (e.g. procedure) if and when they feel that the group will see the need for it. a) year 3 (8-9 years old) - The children play turtle in the playground and have preliminary experience with the computers (but not within the structure of "investigations"). - They complete two investigations in total. The focus is on the working method, the classroom dynamics, the writing of presentations and the detailed explanation of what is meant by an "investigation". The Logo content is mainly direct-driving.
- 236. 225 b) year 4 (9-to years old) - The children are asked to reflect on and to structure the commands section of their presentation in two ways; firstly, to "tidy-up" their commands by discarding those which are unnecessary (e.g. RT to RT 60 = RT 70); secondly to split their commands into sections, whereby each section has a title related to what the turtle does by its commands and shows a drawn picture where the beginning and the ending state of the turtle is shown. They are introduced to the REPEAT command. Two investigations are written. - They are introduced to the writing of procedures, editing, saving and loading on disk. Two or three investigations are written. c) year 5 (to-II years old) - The children are encouraged to process procedures (edit and debug) as pan of their method for working. - They are asked to reflect on how to present and to structure their procedures in their presentations. - They are introduced to the notion of subprocedure and encouraged to include subprocedures in the structuring of their work. - The role of interfacing procedures is made explicit. - Four to five investigations are completed during this year. d) year 6 (11-12 years old): - Structured programming is made more explicit. - Commands are printed out and backup disks are used. - The notion of variable is introduced. - The calling ofvariables amongst subprocedures is introduced. - Four to five investigations are completed in year 6. As can bee seen from the above, the content of the "curriculum" consists of specific powerful programming tools, Le. iteration, procedure and subprocedure, editing and debugging, and variable, and of increasingly powerful ways in which to use them. During the last two years, most teachers actually request that the children use some of these tools, especially procedures in the fifth year and subprocedures and variable in the sixth. However, there is always plenty of scope for the children's own decisions on the way they will use the tools and of course on what and how they will explore or construct. Examples of the way the "curriculum" is mediated and implemented are shown below. Teacher Training As mentioned above, the program's aim was to integrate the use of Logo in the school's everyday life to the extent made possible by the existing conditions and the school's curriculum. Logo was not to be an aim in itself, an additional curriculum topic or a new technological gimmick, soon to be outmoded. Using Logo was not to be given the role of something special, but just that of a useful classroom tool. Teacher training was therefore given
- 237. 226 a low profile, and was restricted to school working hours. Under this framework, all the teachers agreed to give it a try--even those who were far from enthusiastic about "computers". From the outset, the main concern regarding the training of the teachers was to encourage them to take control of the way in which they·would implement the above educational objectives. Thus, the reason for focusing on process, was for the teachers to use Logo as a tool with which to teach something that they considered familiar, relevant and useful in relation to their experience so far: they felt that the children had very little chance to engage in active thinking, take initiatives and cooperate in their normal classrooms and that these experiences would be of great value to them. Once a culture of children building control over their learning could be established, with the teachers feeling that they were using this new tool meaningfully, then it would be time to progressively throw more emphasis on exploring the mathematical and programming ideas embedded in the Logo language. In effect, focus on content was made explicit during the final two years of the program. Research and Evaluation Objectives One of the objectives of the program is to study the longitudinal social and learning characteristics built within the dynamics of the Logo classroom. A year-to-year record of the program's progress is kept and analyzed, on the one hand with the aim to evaluate and reformulate the program itself and on the other with the aim to collect a corpus of data from which more formal research questions will be developed. The research problem has therefore been given two dimensions: 1. To study the extent and the way in which the program's educational objectives are implemented and reformulated year by year. 2. To formulate research questions concerning the development of the teachers' strategies, the children's learning processes, what meanings they bring to their working environment and the social dynamics at the level of the classroom as a whole and at the level of one group of children. Data Collection and Evaluating Method As mentioned above, using the computers was a normal everyday function of the school and teachers and children have not been asked to spend time on top of their working schedule. Within this framework (apart from the pilot year), research has been informal, data collection has not required any extra activities on the part of teachers or children and the hitherto general research objective has been to evaluate the program's progress in the view ofreformulating it. A detailed account of the collection of data and some summative results from the first three years of the project can be found in [II]. In brief, the following data has been collected.
- 238. 227 The Pilot Year The pilot year involved a rather detailed collection of data, i.e. classroom observation notes, dribble ftIes and verbalization transcripts of children's attempts to solve structured tasks, questionnaires concerning their attitudes towards computers and mathematics, children's written accounts of their experiences with Logo. The Main Program The data is collected and analyzed in one year cycles. Towards the end of the second term (March or April), the researcher carries out some pilot classroom observations and then visits each class for one teaching period and takes notes on both the teacher's and the children's activities. Concerning the teacher, the observation focuses on his/her attitude towards the children's learning, the type of interventions made (factual, process-oriented, social management, content related), the frequency with which he/she intervenes, whether the intervention was requested by the children or not, the time spent with one group of children, his/her confidence with the technology. Concerning the children, observation focuses on general issues such as the extent to which the children seem involved in their projects, whether all members of a group are actively participating. There is also interest in specific issues, such as the awareness of specific children about the program's structure, e.g. how they explain what they are doing at the time in relation to the whole of the investigation, or whether they are aware of the phase they are currently at, for instance "second period, third investigation". The children's presentations of their investigations are also important data for the evaluation of the program. In one school year, four investigations are carried out by each class, on average (except the third graders, where the children carry out two investigations). Consequently, around 160 presentations are collected every school year from year-groups 4, 5 and 6, and 80 from year-group 3. This means that 560 investigations are produced each year, almost all of which are collected and analyzed in the following summer holidays, influencing and formulating the program for the next year. The analysis of each investigation concentrates on both the children's work and on the teacher's comments. The comments written by each teacher throughout the year are then analyzed in relation to the observation data on hislher classroom activities and a composite "profile" of his/her yearly progress is made and used in the teacher training seminars before the beginning of each school year. The children's work is analyzed with respect to process-related issues, awareness of the program's structure and content-related issues. Examples of presentation analyses are shown below.
- 239. 228 Results Learning Process and Content: an Analysis of Children's Work and Teachers' Written Feedback Three characteristic features of the program in its current state of development are analyzed in some detail. Firstly, the strategies a teacher has developed in order to encourage learning within the process-oriented goals mentioned above. Secondly, the process by which children learn as a consequence of their own actions. Thirdly, the programming content many children have started to master towards the end of primary. These issues are discussed through an analysis of three corresponding examples, each taken from the presentation of the investigation of a group of children. To provide a picture of the content of the children's work in different ages, the three examples are from years 4, 5 and 6 respectively. A Teacher's Input to the Learning Process Example I shows the presentation of the fourth investigation in the fourth year, i.e. the children are aged 9 - 10. The teacher's contribution to the process-oriented educational targets of the program (active thinking, initiative and cooperation) can be split into three categories; the requirements, or rules of the game, that are imposed as a pre-requisite for the investigation and for its presentation; her method of intervening during the investigation in the classroom; her written comments on the presentation in order to encourage the children to reflect on the above issues. Example I Thursday 22 March 1990 4th Investigation 22/2/90 - 22/3/90 At the beginning we wanted to make an egg but in the end we made a nAl:XA (Easter in Greek, C.K.). Our cooperation was very very good (teacher'S comment here: "how did you manage that"). We finished the drawing in the second session but our teacher told us to do more. So, we made a signpost that said nALXA. We began the drawing in the middle left and finished in the middle at the bottom. We first made the n, then the A, the l: after that, the X and then the A. (teacher's comment: "what about the signpost?).
- 240. 229 Invl.lbl. n Invl.lbl. A Invl.lbl. I PU FO 30 PU PO PU PO RT 270 LT 90 LT 135 FO 30 BK 20 BK 20 FO 80 RT 180 RT 90 RT 270 RT 315 LT 270 FO 20 FO 30 FO 10 FO 15 PO RT 90 BK 60 LT 90 LT 90 FO 30 FO 20 FO 40 FO 10 RT 135 FO 20 Invl.lbl.lln. of X Invl.lbl. lin. of X Invl.lbl. A PU PO PU PO PU PO FO 10 FO 25 LT 135 FO 25 BK 30 FO 25 RT 45 FO 20 RT 225 BK 25 FO 20 RT 180 FO 25 RT 270 BK 20 LT 45 RT 135 FO 25 FO 25 BK 13 BK 25 RT 135 FO 15 Invl.lbl. .qu.r. Invl.lbl. colour PU PO PU PO RT270 F0200 BK30 FILL F019 RT90 F050 RT90 F050 I F010 RT90 ｾｌｘＺａ＠ I RT90 F0200 BK40 RT90 F050 RT90 FO 150 I RT 180 F050 RT270 F050 Figure 1 (teacher's comment: "In all these commands I don't see the REPEAT command anywhere. Why?". Her general comments at the end: "Your presentation was quite good, but short. Since the drawing was symmetrical the REPEAT command could have helped you.) As can be seen from the children's essay, the teacher asked the class to make something that was related to the Easter Holiday. In doing so, she gave the children a framework for working, but also the chance to make their own decisions and solve their own problems. With respect to their presentation, the requirement for the essay was that they would write about their cooperation and about what they constructed, how they did it and what interesting or difficult problems they met on the way. The requirement for the commands was that, apart from being clearly presented, the children would group them into sections according to their own decisions and they would give each section a title. This process is used as a preparatory arena for the children to later relate the writing of procedures to the structuring of their work. This teacher, for instance, encouraged the children titles to interfacing sections. The fact that they gave the
- 241. 230 same name to all of these ("Invisible") could serve as a root experience for generalizing the notion of interfacing procedures, something that does not come about automatically with children [13]. Furthennore, next to the commands of each section, there would be a drawing or a printout referring to the commands-a pre-amble to the injection of modularity to their programming (in example 3, involving 6th year children, structure and modularity are much more apparent in the children's work). Finally, the screen dump would show their final product. The teacher's strategy during the investigation was to allow the children to decide what they would do and not to intervene in a prescriptive manner, or to readily provide requested or unrequested factual information, as long as the team seemed to be engaged in the project. However, as the children write, they finished the letters of the Greek word for "Easter" earlier than the end of the investigation time. The teacher intervened at that point to encourage the children to use their creativity and amongst them negotiate expanding their investigation by adding something relevant to their construction. The teacher's comments on the essay aim to encourage the children to impose some structure on the topics they refer to, a problem that is very common in the children's writing in the school. Not surprisingly, the children have great difficulty in structuring the topics they write about - most often they either go into extreme detail and stop when they get tired, or they write vague generalities like the first phrase in the example. In the commands section, it is evident that there is a lack of discrimination between giving a written title to each section and showing the graphics it produces, perhaps due to the coincidental matching of the two in this case. The teacher has consequently asked for the position of the turtle to encourage the children to think about the structure of their work not only on their commands but also on the graphical outputs of these commands. Furthermore, she has given them food for thought on using the REPEAT command, i.e. noticing whether there are similar patterns in their shape. In general, the teachers' strategies concerning the program's objectives have been developing mainly as a result of their own common sense, teaching experience and familiarity with their pupils' specific needs. The seminars and training they receive function as an opportunity for reflection and for a gradual advancement of their own experience with the technical know-how and the programming and mathematical content. Finally, the developing "curriculum" is discussed, and reformed at the beginning of each year. The Process of Children's Learning Example 2 is used to discuss the dialectic between the children's working process and their learning of content, in this case using procedures to program the computer. The example shows the presentation of a first investigation in the 5th year, the children aged 10 - 11. The prerequisites here include the use of procedures, but the extent and the way in which they are used and ofcourse the topic for investigation, is up to the children.
- 242. 231 This was the children's third investigation with procedures (the previous ones were carried out when the children were still in the fourth year). In the earlier investigations, the children used procedures only as a means to store information and economize in time and effort; for instance, the contents of their procedures would be long columns of direct-drive style commands, only one procedure written within a teaching period and used the following week as a means to start off where they had left. So, during the investigation in the present example, the children had progressed to using procedures to sectionalize and begin to structure their work, even though they still considered procedures as product. Research has shown us elsewhere [7] that children spend time using procedures as product before gradually shifting their focus to perceiving procedures as process. The presentation in the example, reveals how the children began with a concrete aim, incorporated the writing of procedures in the process of constructing a small part of their project at a time and discovered after the third session that the figure was too large to fit the screen. This is the point where their coherent use of programming broke down. Far from realizing that all that was needed was, for instance, a parameter change in one of the written procedures, the children started to construct their figure all over again in a new procedure (so that the prerequisite of using procedures would be fulfilled). What is more, due to time pressure, they apparently thought it would be quicker to incorporate all the commands in one procedure, in effect falling back to direct driving within a procedure. A study of the geometrical ideas they used in their programs indicates that this "regression" was not due to the difficulty they had with the figures' mathematics; their final procedure (BILLY) indicates that they had worked out a limited but functional method to change the size of circles by changing the turning quantity of the turtle (children's localized and limited generalizations of mathematical ideas is discussed in the following section). So the children's falling back to a more naive use of procedures seems to have been a step in the process of learning the power of a programming tool by means of concrete consequences arising from the way in which the tool is used. Example 2 a) responsible/or the disk: D. M. b) responsible/or the note book:F. K. c) responsible/or the presentation book:M. X. d) responsible/or the writing book o/important notes: M. K. Investigation 1 Our team is made up by F.K., M. K., D. M. and M.T. During the first day of our investigation each one of us would say some idea and in the end we decided to make a bomb. In the first two sessions we had problems about how to make the circle since we had forgotten from last year. When we made the circle, we made the fuse. We saw that the screen was not big enough for us and half of the fuse came out underneath. So we made the circle smaller. After many difficulties and problems
- 243. 232 which we mention above we managed to make our drawing, the original one. In the end of the fourth session we printed it The first day we went to the computers we had many ideas in our minds about what drawing we would make but in the end we decided to make a bomb. The remaining time we tried to make the circle but we didn't manage it. The second day, after we went to the computers we started to draw the circle. That day we had problems with the procedures but with the help of our cooperation we solved that problem of ours. The third day we decided to make the fuse. After we made it we saw that half the fuse came out underneath. But the bell rang and we didn't have time to fix it. The fourth day we corrected the circle and we made it smaller. In this way, our fuse came out nice. lOCH: lO'MQ PU BK50 LT90 FD30 RT90 LT90 FD15 REPEAT 360 [ FD 1 RT 1) REPEAT 90 [ FD 1 RT 1) LT90 RT90 FD55 PD EN> TOBILl.V PU BK50 PD EN> REPEAT 360 rFD 1 LT ?I REPEAT 90 [FD 1 LT 2) RT90 REPEAT 25 (FD 1 RT 2) PU FD5 PD FD5 PU BK5 RT90 PU TOSIX RT40 REPEAT 40 [FD 1 RT 1) RT90 PU FD5 PD FD5 END REPEAT 6 [FD 1 RT 1] LT45 PD FD5 BK5 RT90 LT 180 LT45 PU REPEAT 9 [FD 1 LT 1) RT 135 LT45 PO FD5 PE BK5 LT55 FD5 PD BK5 PU Figure 2 In this case, it is not clear whether the teacher saw the opportunity to encourage the children to reflect on their work and to gain some insight into the notion of a procedure as an entity, i.e. that once written, it always works just as a Logo primitive and can easily be changed or debugged. Instances such as the above arise very frequently, and the teachers are developing a means of interpreting the children's work and intervening at moments when factual information would be meaningful to the children, as in the above case. However, it is not a coincidence that the "curriculum" for the fifth year focuses on procedure "processing", i.e. editing and debugging, as part of the children's working method. After having had a first experience with the syntax and the know-how of writing procedures and using disks to store them and subsequently using procedures in "naive" ways, most children are ready to bring meaning to
- 244. 233 using procedures in a more powerful way. A first step in the fifth year is the structuring of their work by means of procedures and subprocedures. The Content of Children's Learning Example 3 shows the presentation of the second investigation by a group of children in the sixth year. In this investigation, the teacher had given them a prerequisite "initial idea" which was the number 4. The idea was that the children would make an investigation using the figure 4. It was suggested that they use procedures and a variable, having been introduced to the latter earlier. Example 3 Essay In this investigation we had as an initial idea the number four. At the beginning we didn't like it because the shape was difficult and we were not in the mood to work. But we decided to work. Bit by bit we wrote the number four with variable. After that, the remaining shapes were relatively easy. We did have cooperation problems because one student wanted to do everything by himself taking advantage of having more knowledge than we did in computers. On the one hand, their programs reflect a more sophisticated use of structure and modularity than that shown in the fifth year (see example 2). For instance, the programs indicate a rather confident use of one level of subprocedure (using the procedure K as subprocedure for the procedures Kl, K2, K4, K5, K6, K7) and also the use of a second level of subprocedure (the use of K2 in procedure K3). Furthermore, their project involves experiments with the use of the procedure K as a module, rather than using procedures to sequentially sectionalize the building of a concrete real-life design as in example 2. On the other hand, however, their work is not surprisingly characterized by rather local generalizations of programming and mathematical ideas. The children's use of variable, for instance, is limited to the program written for the initial idea; from then on they not only used the program with a fixed input (K 60), but also did not change the value of that input in almost any of their subsequent programs, apart from one more fixed value of 40. In this way, they neither incorporated the use of variable in their programming nor did they at least use variable empirically, by means of different fixed values of their initial procedure. Moreover, by fixing the input to K, the children employed geometrical ideas involved only with the fixed value and therefore did not seem to consider generalized geometrical relationships; e.g. where they have typed FD 15, it is to move the turtle from the bottom part of the figure to the "base" line, a distance of one fourth of the base line as shown in procedure K. Furthermore, their use of the figure's geometry has not surprisingly been rather limited, as e.g. their moving the turtle by 50 or 25, both distances not being related to the figure's dimensions. Their generalizations
- 245. 234 concerning tum have a similar pattern, as indicated by the relations they attached between the number of repeats and the turtle's tum; procedures K2 and K3 involve a total tum of 360 degrees (K being heading-transparent), but procedures K5 and K6 do not seem to relate the two quantities. TOK4 TOK:A FO:A BK :A RT90 FO :A BK:A/2 LT90 FO:A 14 BK:A/2 EN) TOK2 REPEAT 4 [K 40 RT 90] EN) TO PU LT PO K1 90 REPEAT 4 [ K 60] END TOK3 REPEAT 4 [K2 RT 90] END I J. ｾｐｅａｔＵ｛ｋＶＰ＠ FO 15 LTOO FDSO AT 00] TOK6 10KS AEPEAT 8 [K60 FD 15 LT90 FD 25 AT 135) END TOK7 K60 PU F075 RT90 F030 RT90 PO K60 END REPEAT 60 [K 60 FO 15 LT 90 FO 50 RT 135] EN) Figure 3
- 246. 235 Finally, the children's limited use of programming and mathematical ideas in connection with the way they write about their project, indicates their lack of reflection and understanding of the internal structure of the procedures they themselves build, a finding which supports other related studies [2, 9]. In their essay, the children write about the need to "work" in order to write their initial procedure, but refer to the other procedures (which incorporate much more complicated ideas) as "easy", possibly a result of their exploratory ad-hoc interfaces between iterations of the figure 4, and their resistance to reflect on the precise consequences of those interfaces on the graphical output. A likely reason for this may have been that two members of the group seemed to struggle to keep up with the third, as indicated in their essay. The children's reflection on their cooperation, however, is an important means by which they learn to negotiate in the process of their investigational work. Phases of readjustments in their cooperation, as well as phases of effective communal work as a result of discussing cooperation problems (the children's writings in example 2 indicate that they were in such a phase), are an integral part of the program's process-oriented objectives and of course encouraged by the teachers. Discussion "Teaching" the Change of Control Establishing the use of the computers to bring about and encourage pupil-controlled problem- solving activity, by no means came automatically. This, of course, was not due to some specific conceptual difficulty teachers or children had with understanding how they could use the Logo environment to practice an alternative way of working. What took time was the process of making the important issues socially explicit within the classroom. For instance, the teachers gradually developed a meaningful way to communicate to the children why they were not readily giving them factual answers and why they would often throw the responsibility of a situation back to the children themselves; encouraging the children in example 1 to continue with their investigation, but to take responsibility for what they would do next is an example of an explicit negotiation of this issue both on the part of the teacher and on the part of the children, since they clearly wrote about the episode in their presentation. The Working Structure's Receding from Object to Tool The working structure of carrying out "investigations" and then presenting them on paper was not immediately employed to encourage investigational work. Indeed, the main reformulation of the program after the first year, was for the teachers to exemplify the rules of the game to the
- 247. 236 children and to raise their awareness of the structure within which they were working by means of taking more time to reflect, even within the process ofone investigation. For example, in the third and fourth year, the teacher actually states at the beginning of each Logo session, that this is, say, the second period of the fourth investigation and is particularly firm on receiving "complete" presentations, i.e. consisting of all three main parts, essay, commands and graphics. This is why, for instance, the "curriculum" for the third year provides almost double the time for one investigation; investigational work still lasts four teaching periods, but time is given for reflection in between and after completion. After the first two years of the program, the role of the working structure as an end in itself began to recede, gradually giving place to that of being used more as a means to encourage the children's learning of content and the enrichment of their use ofpowerful ideas in an investigational and personally meaningful spirit. The Long-Term Relevance of the Process Oriented Objectives The process-oriented objectives of the program are still in effect. For instance, by the end of primary education, every pupil has had four years' experience of negotiating ideas, working method and the delegation of responsibility, within a group of three. There is a consensus amongst the teachers that this is not only valuable education for the children, but that it isn't even enough. In some cases, teachers have developed ways for the children to work in groups within their normal curriculum (transfer of learning process is not in focus in this paper). In example 2, for instance, the children use the whole of the front page of their presentation to clearly state who had which responsibility. This is most likely an indication of a phase where this particular issue came into question in the working dynamics of the group, possibly as a result of conflict, but in any case exemplifying and raising the awareness of the need to take "responsibilities". No doubt that in the remaining time of their primary education, this group would bring other issues related to cooperation into focus. In the same way, taking initiatives within their work is an experience which is encouraged throughout the four years. The children's decision to carry on and make a signpost for "EASTER" in the first example and the reaction of two members to their peer's non-cooperative attitude in the third illustrate the point, given that these are two instances where both the issue and the age of the children vary. Finally, thinking things out for themselves is an important aspect of the children's work. Even though in some cases teachers put prerequisite demands on the children's work, such as asking for the use of procedure or variable, as in example 3, or giving an initial topic or idea for the children to work with as in example 1, they have always given ample space for individual thinking and original work, as can be seen in both examples. What is more, they developed strategies to encourage the children to do so, thus confronting an initial tendency for them to perceive prerequisites for investigational work as goals in themselves, put by the teachers.
- 248. 237 Interventions in the Learning of Content In a sense, focusing on the learning of content was and still is, a complicated affair. To begin with, the context within which the program took place did not favour intensive a priori training of the teachers in Logo programming and the mathematical and programming content embedded in the language; very little time for teacher training was made available (a far from rare situation in any school) before and during the program; all the teachers were taking part, with varying attitudes to change, to computers and technology in general, and naturally to taking-on additional work; the teachers had little means of being socially reassured, either by tangible examples of similar work, or by official educational authorities; within that particular educational system, there is anyway very little provision for the teacher to take initiatives in deciding on teaching method or content. It was therefore far from reasonable to expect the teachers to change their teaching method and role in the classroom and to simultaneously teach substantial content within the new state of affairs. The second difficulty with focusing on content is related to Logo's strength as an educational tool. Investigations such as the ones presented in examples 2 and 3, where the children use ideas such as turning relations flrst, and then discriminate and generalize them, are common amongst the children's work. However, the problem lies in trying to influence the learning environment, on the one hand in order to discourage an unreflective use of Logo, as in the direct-driving of the turtle in example I, and on the other, in order to help the children to focus on and become aware of the interesting and powerful ideas which they use in amongst their projects, e.g. the use of procedure in example 2. The need to intervene in the children's work regarding both of the above issues is as real in this school as in any environment where children do Logo. The teachers have two means to make interventions in the children's learning within the program's working structure; interventions in the classroom during the learning process; using their presentations to provide feedback. The analysis shows that they have begun to detect opportunities for such content related interventions, and are now in the never- ending process of enhancing their recognition of ideas and situations such as the ones described above and of reflning and enriching the ways in which they intervene. Conclusion In the cultural, educational and practical contexts within which this particular school project took place, focusing on the process of problem-solving was an essential pre-requisite to focusing on content. The dialectic between these two aspects of learning, however, has now become vivid within classroom practice and the children's and teachers' activities are progressively integrating them. What is of equal importance, however, is the issue of spreading an educationally aware use of information technology to the entirety of school populations and not to an enthusiast elite. Future research will focus on the longitudinal socio-psychological characteristics and implications of the environments generated in this particular school.
- 249. 238 References 1. Hillel, J. & Kieran, C.: Schemas used by 12 years old in solving selected turtle geometry tasks. Recherches en Didactique des Mathanatiques, 8(12), 61-103 (1987) 2. Hillel, J.: Procedural thinking by children aged 8-12 using tunIe geometry. In: Proceedings of the Tenth ]nternational Conference for the Psychology of Mathematics Education, pp. 433-438. London 1986 3. Hoyles, C.: Scaling a mountain - A study of the use, discrimination and generalization of some mathematical concepts in a Logo environment. European Journal of Psychology of Education, 1(2), 111-126 (1986) 4. Hoyles, C.& Noss, R.: How does the computer enlarge the scope of do-able. In: Mathematics. Proceedings of the Second Logo and Mathematics Education Conference. University of London Institute of Education 1986 5. Hoyles, C. & Noss, R.: Children working in a structured Logo environment: From doing to understanding. Recherches en Didactiques de MathOOlatiques, 8(12),131-174 (1987) 6. Hoyles, C. & Noss, R.: Seeing what matters: Developing an understanding of the concept of parallelograms through a Logo microworld. In: Proceedings of the Eleventh ]nternational Conference for the Psychology of Mathematics Education, pp. 354-359, Montreal 1987 7. Hoyles, C. & Sutherland, R.: Logo mathematics in the classroom. London: Routledge 1990 8. Lawler, R. W.: Computer experience and cognitive development. A child's learning in a computer culture. New York: Ellis Horwood 1985 9. Leron, U.: Some problems in children's Logo learning'. In: Proceedings of the Seventh ]nternational Conference for the Psychology of Mathematics Education 1983 10. Kynigos, C.: The tunIe metaphor as a tool for children doing geometry. In Learning Logo and Mathematics (C. Hoyles & R. Noss, eds.), Cambridge: MIT Press in press 11. Kynigos, C.: From intrinsic to non-intrinsic geometry: A study of children's understandings in Logo-based microworlds, Unpublished Doctoral Thesis, University of London Institute of Education 1989 12. Kynigos, C.: Can children use the turtle to understand Euclidean ideas in an inductive way? In Proceedings of the Fourth International Conference on Logo and Mathematics Education. Israel 1989 13. Noss, R.: Creating a mathematical environment through programming: A study of young children learning Logo, Doctoral Thesis, published by University of London Institute of Education 1985 14. Noss, R. & Hoyles, C.: Structuring the mathematical environment: The dialectic of process and content. In: Proceedings of the Third Logo and Mathematics Education Conference. London: University of London Institute of Education 1987 15. Papert, S.: Computer criticism versus technocentric thinking. In: Proceedings of the Logo 85 Conference 1985 16. Papert S. et al.: Final Report of the Brookline Logo Project, (part 2), M]T Artificial Intelligence Laboratory 1979 17. Sutherland, R.: A longitudinal study of the development of pupils' algebraic thinking in a Logo environment. Doctoral Thesis published by the University of London Institute of Education 1988 18. Weir, S.: Cultivating minds: A Logo casebook. London: Harper & Row 1987
- 250. Cognitive Processes and Social Interactions in Mathematical Investigations Joao Pedro Ponte, Joao Filipe Matos Departamento de ｅ､ｵｾＬ＠ Faculdade de Ci!ncias, Universidade de Lisboa, Campo Grande, Lisboa, Portugal Abstract: Mathematical investigations may be important educational activities, proving to be useful in the development of mathematical ideas. The principal difficulties may concern students' content knowledge, reasoning processes, and general attitudes and appreciation. This paper refers to a computer based investigation undertaken by three eight-grade students, discussing in special their cognitive processes and social interactions. Keywords: investigations, computers in mathematics education, cognitive processes, strategies, conjectures, social interactions In mathematical investigations students are placed in the role of the mathematicians. Given a rich enough and complex situation, object, phenomenon or mechanism, they try to understand it, to find patterns, relationships, similarities, and differences leading to generalizations. Mathematical investigations range from quite elaborated and complex tasks, that may require a considerable amount of time to carry out, to smaller activities that may arise by consideration of a simple variation on a well-established fact or procedure. Mathematical investigations share common aspects with other kinds of problem solving activities. They involve complex thinking processes and require an high involvement and a creative stand from the student. However, they also involve some distinctive features. While mathematical problems tend to be characterized by well defined givens and goals, investigations are much looser in that respect. The first task of the student is to make them more precise, a common feature that they share with the activity of problem posing. In the process of carrying out a mathematical investigation it is possible to distinguish activities such as define the objective (what are we trying to know?), set up and conduct experiences (what happens in such or such specific instance?), formulate conjectures (what general rules may we propose?), and test conjectures (what may be critical experiences to ascertain the value of this conjecture? Is it possible to make a proof?). The realization of mathematical investigations has become a fairly popular curricular orientation [5, 10]. However, little is known about what happens in the course of mathematical investigations, specially if carried out in school settings (a point also made by [5, p. 94]). What
- 251. 240 kind of profit do students take from them? What difficulties do they find? What constraints do they put on the teachers' role? The computer, used as a tool, has been proposed as a very useful instrument for carrying out mathematical investigations. It encourages the realization of a large number of experiences, allowing the exploration of quite non-trivial situations and issues. It is also of great interest to know what specific features it may bring to this mathematical activity. These are some of the questions that we set ourselves to respond in this study. Pedagogical Context and Research Methodology As in any other educational activity, in carrying out mathematical investigations, it makes a big difference the way things are designed and organised. We need therefore to clarify the general context of the episode, the way the activity was proposed, the role of the teacher (in this case one of the present authors), and the idea that the students made of their own role in this process. The activity analysed in this study was carried out in an extra classroom setting at a school involved in the MINERVA Project (Pole DEFCUL), during the school year of 1988/89. The students were 8th graders who voluntarily enrolled to work with computers, in the school computer centre, in a specified weakly time slot of 2 hours. Their relation with the centre lasted for the whole school year, working in Logo activities and projects. These students had previous contact with Logo the year before, in the mathematics classroom and before this episode they developed programming activities for six months, most of them based in projects of that they designed themselves. One the students, Maria, was a very good achiever in mathematics and in the other subjects. The other two, Nuno and Victor, were medium towards low achievers in most school subjects. One of the present authors was in the school computer centre during this time slot for the whole school year and was well known of the students. The activity discussed in this study was proposed in one session that took place in 18 April of 1989. The activity was based in a recursive Logo procedure with three variables which draws peculiar kind of shapes. It shows in an interesting way the effect of recursion in a geometrical procedure [1, 3, 9]. The students were given a sheet of paper with the procedure, a sample screen output, and a general formulation of the task (see figure 1). Besides, the researcher explained the purpose of the investigation and worked out one or two examples with the students. During the activity the researcher had a strong interaction with the group, specially in the beginning and in two or three moments. In the rest of the time the students worked just by themselves. The activity ended with a general discussion of the results between the students and the researcher.
- 252. ACfIVITY LEM #15 (File: INSPI) INSPIRAL 241 The procedure INSPI allows you to draw a kind of figure that we will call "inspiral". This is the result of INSPI to 0 to. The figure has two "enrollments". Investigate the nature of the figures that you can obtain with the procedure, trying to elaborate a theory about the number of enrollments and the kind of figure you can get. Suggestion: At the beginning it will be better to take the first and the third parameters as constants (for example, 10), and give successive integer values to the second parameter ranging from 0 to 20. TO INSPI :L :A :1 FD:L RT:A INSPI :L :A+:I :1 END Figure 1: Working proposal The involvement of the students varied during the course of the session and, as we will see, was quite different from student to student. Their general attitude was "we are here to try to do whatever the teacher (e.i. the researcher) asks us to do." In this study data was collected by video-taping the students. The tapes were reviewed a number of times by both of the researchers. Successive analyses of the episodes were
- 253. 242 produced, including a scheme describing the main stages of the work of the students within the activity. From there new observations of the tapes were made in order to clarify new aspects and produce the final written account. A Framework to Discuss Mathematical Investigations Several activities can be identified during the course of an investigation. These can be organised within three main phases of work which will be now discussed in detail: (a) formulation of objectives, (b) definition of strategies, (c) reflection on the experiments carried and formulation and testing of conjectures. Formulation of Objectives. An investigative task may suggest the setting of a great multiplicity of objectives. Some may be more general, and others refer to aspects of detail. Some may be more precise and others more vague. The ability to formulate precise research objectives is one of the most essential aspects of the ability to undertake investigations. Significant questions about the setting of the objective of an investigation by the students are: -How is the research objective initially formulated? -Are there turning points in the process of conducting an investigation that can be referred to change in the overall objective? What can be said about them? -Are there more general aspects concerning the way they look at the situation that may change in the process of the investigation? Professional experience, supported in the literature, asserts that students tend to be not very good on formulating research questions to investigate in a spontaneous way. Even when provided with starting points, they may have difficulty in seeing what more general questions may be asked to extend simple cases already explored [2]. This should be hardly surprising in view of the overwhelming tradition of teaching well organised and formalised knowledge, that students are supposed to acquire, and not introducing them to the process of constructing mathematical knowledge themselves. That is, to teach students "answers" paying no attention to the "questions" they are supposed to correspond nor to the way they were constructed. In mathematics teaching the tasks are usually given to the students completely formulated. What are sensible or senseless questions to ask, what are interesting or trivial questions, etc, is something to which no attention is usually given. Setting research objectives is therefore one of the aspects in which students show great difficulty. Definition of strategies. Strategies used in the course of an investigation refer to three aspects. The first concerns the representation of the situation (including the identification of key features and the choice of a suitable notation). The second concerns the key decisions about the sequence of experiences to carry out, indicating a general line of reasoning. The third has to do
- 254. 243 with specific tools that are used to construct and interpret the experiences. Significant questions can be asked about these three aspects: -Is the representation appropriate (in the sense that it describes important aspects of the situation)? -How is the organization of experiences? Are they relevant for the sought objectives? Are they systematic? -Is the "technical knowledge" of the students preventing them of devising and organizing a sensible strategy? Devising appropriate representations and mathematical notations has been widely recognized as an essential element for carrying out mathematical investigations [4, 8, 13]. Not all the representations of a given situation can offer the same insight. Some offer more than others. It is common that students develop more than one kind of representation and fluctuate between them [2]. Investigations are often regarded as good starters for mathematical work. However, it should not be overlooked the fact that "investigational work often rewards mastery of mathematical technique with success, and punishes mathematical inaccuracies heavily" [13, p. 114-115]. Reflecting on the experiences and formulating and testing conjectures. The realization of experiences should lead to a reflection on the situation, gaining insight on it, perhaps revising some aspects of the earlier conceptualization and hopefully to doing some conjecturing. The results of the experiences performed can be used to better understand the situation and draw up conjectures. The conjectures, once formulated, need to be tested. The processes of conjecturing and testing form a cycle that may run for several times. Sometimes the students come out the cycle to modify some aspect of the set up of the experiences. Sometimes the students may feel the need to come even earlier and modify the overall research goal. Testing can also take different forms. It can be test of specific chosen cases, testing of random cases, or attempts to a proof. Besides our interest in these aspects of the process of conducting investigations, we were also specifically concerned with two further issues: (a) The role of the computer in mathematical investigations and (b) Social interactions. The role of the computer. These investigations where proposed to the students assuming that the computer would be used to help performing them. If fact, in this activity, it would be difficult to see the work being carried without the computer. One should therefore ask what are the consequences of using the computer in the working processes of the students. Some of the possible consequences of the computer concerning this kind ofmathematical work are well-known: -It allows a great number of experiences, encouraging strategies where making a good number of experiences is an integral part.
- 255. 244 -It allows feedback in different kinds of representations. -It facilitates the dialogue, since it becomes a new pole of attention. What is done in the computer is not individual property but public. Ifthe students are programming themselves, as often happens in Logo activities, the act of translating an intuition to a program makes it become more obtrusive and therefore more accessible to reflection [II]. However, in this case the program was already made and the students just interfered with it for changing some of its minor features. One should note that the computer offers means of representation that are powerful but limited. With the computer it is possible to do many things, some of them quite extraordinary. But computers are limited in what they allow to represent, and they may prove to be unsuitable for some purposes. Social interactions. One of most common features of the use of computers in mathematics education is a change towards group work. Investigations tend also to be suggested to be performed in groups preferably to individually. However, there are satisfying and less than satisfying situations of group work. Another important partner in the learning process is, of course, the teacher. Therefore significant questions are for example: -How does the relationship with the teacher and the colleagues interfere (positively, negatively) with the development of the task? -How far is carried the process of arguing? Do the students articulate arguments or just statements? Is there listening to others' arguments? -Is there a search for group consensus or one takes the lead and determines the course of the group? -What seem to be the implications of the situation of social interaction among students (what seem to be positive effects? negative effects?) -Why do some students seem to have more initiative than others? Why do some students seem paralysed? Why are some students apparently not able to take profit from the fact that they are in a social interaction situation? The Investigation on Inspirals In this activity we may distinguish 8 different segments, in which there was a significant turnover in the course of the events. All of the transitions between segments are characterized by a change in the composition of the group. Segment 1. The task begun with two students, Maria and Nuno, and the researcher, who handed the sheet with the situation, presented it in general terms, formulated the objective, and gave a suggestion to get them started. This segment lasted for about 2:30 minutes. The objectives stated in the sheet concerned the nature of the figures that it is possible to get and asked for a theory about the number and the kind of enrollments. These objectives were
- 256. 245 rephrased orally by the researcher as "Let us see what happens" and "Try to understand the actual functioning of the procedure". A fIrst experience was made with the input values of 5 0 5. The students commented on the appearance of the shape: "It looks like a spring!". Then the researcher introduced one trick: how to slow down the procedure introducing a waiting instruction. He focused the attention of the students in "Why does the turtle seem to tum left?", which was meant as a more specifIc investigational objective. The students made several comments about what they were seeing on the screen, specially around the "enrolment points". It appeared that the turtle was drawing "something like a square". We can say that the intervention of the researcher was dominant in this segment. He stated the objectives, general and specifIc, made a fIrst experiment, recommended the recording and showed a specifIc strategy. The students were quite intrigued with the behaviour of the turtle. Segment 2. In a second segment the students worked for themselves, following the suggestions given. They started exploring the procedure, giving values, and making changes in the fIrst parameter. Having arrived at some conclusion they called the researcher. The segment took 9:30 minutes (1 :30 ofjust waiting time). The students wanted a larger fIgure to analyse it better. Following a suggestion of Maria they decided to try out with 10 (therefore introducing the values 10 0 5) and realised that "the fIgure does not change". Nuno commented that such could be because "they are multiples of 5". At the same time Maria tried to give an explanation for what she was seeing: "The turtle comes back because she does not have a way out". New attempts were made with the inputs 12 0 5, 240 5, 28 0 5, 40 0 5. These produced larger fIgures with the same shape. The students soon realised that modifying the fIrst parameter had an effect on the size but not on the shape. It did not matter if the values were or not multiples of 5. From some point on making the fIgure larger and larger just became a strategy to see it better and try to understand the behaviour of the turtle. However, at this point the students become much less communicative. They took turns at the keyboard, performed the experiences, registered and carried on with very little or no discussion. Since apparently the effect of the parameter length was understood and nothing else was happening just by varying it, the students called the researcher. We enter in a third segment of the activity in which he interacts with the students. In the second segment the students carried out the investigation and successfully discovered the role of one of the parameters of the procedure. At this point they had no idea of what to do further. The discussion that occurred next revealed that they had so far no understanding of the mechanic of the Logo procedure with which they were working. Segment 3. This segment lasted for about 5:00 minutes. The conclusions reached were briefly presented to the researcher. It should be noted that the students did not address the issue of turning right and turning left. The reason why the turtle comes back still puzzled them.
- 257. 246 Maria repeated the fonner idea: "The turtle comes back because she does not have a way out". The researcher felt that clarifying the working of the procedure was of importance for the pursuing of the investigation and introduced another trick: how to write the successive angles that the turtle was turning, so to give a trace of what it was doing. The students were really surprised to see that such was possible. An experience was made with 20 0 20. The researcher asked "What is the angle that she is doing when she turns back?" Nuno responded incorrectly taking the increment for the angle "20!" but Maria stated correctly: "180!". This response of 180 seemed to indicate some grasp of the situation but the course of the discussion showed how they were far from a clear understanding. Maria was intrigued: "Why is it adding up the angles?" The students realised that the angle was varying but did not relate it to the mechanic of the procedure. They were really surprised to see written on the computer that the angle was then becoming larger than 180 degrees. Angles larger than 180 seemed a strange thing to them. Logo was certainly a familiar environment, but in common tasks one gets well along with angles between 0 and 180 degrees, taking both left and right turns. At this point for them RT 200 did not have any meaning. The researcher made additional questions and comments to try to clarify the role of the increment in the procedure. The point did not come across with the note that "the angle is always increasing". Maria still replied: "And why is it increasing?" The researcher attempted in another way: "It turns 200 right". He asked Maria to perfonn a body experience which made her finally understand then that RT 200 is equivalent to some left tum. The question posed by Maria, "Why is it adding the angles?", prompted the researcher to draw her attention again to the instructions specified in the procedure. At this point the functioning of the increment was apparently finally understood by her. However, she added a strange comment: "The turtle goes by the most difficult side". The discussion also considered the effects of different increments, realising that increment 20 gives a more pronounced enrolment to the shape than increment S. A new objective was then proposed by the researcher: "Maintaining the length and the increment, vary the value of the angle, starting with 5 1 5, and see the kind of figure that arises". Although there were already some discoveries made about the situation, the procedure was still largely not understood. Feeling that, the researcher attempted a clarification. His presence was again quite important, conducting the dialogue, which turned out to be much more intense with Maria than with Nuno. Segment 4. The students were let to work by themselves, following this suggestion. This constituted a new segment that lasted for 18:00 minutes.
- 258. 247 A first experience was made with inputs 5 1 5. The students showed their surprise as they counted 10 enrollments. Nuno introduced the waiting instruction. The idea seemed to be: if it worked while ago let us try it again. Maria commented on the situation of 10 enrollments. She was thinking aloud but she did not articulate any sensible idea. Nuno suggested further "Let us do wait 10 and write the angles". In fact. writing the angles could be of some help. but slowing down the procedure just made it take more time to be performed and Maria did not agree with putting wait 10 ("if 5 is already slow!"). Her attention got concentrated in the initial conditions "I would like to know where it started". She did not recall that such information was given by herself to the computer. "This does not go from 5 to 5... It always appears 1.6.1.6.1.6" (reading the last digit of the numbers written on the screen 11.16.21•...). She looked at the numbers, not at the differences, which constituted a factor for further confusion. Maria realised that what made a difference was the new initial angle. Nuno took the waiting out and. following a suggestion of Maria. a new experiment was performed with 5 2 5. She commented: "With angle 2 it does not go until there", that is, the figure looks "closed". They counted the enrollments. which turned out to be 10. Maria asked Nuno to make it smaller "so that we can see" and a new experiment was made with 222. With 2 they got a much different figure and immediately returned to an increment of 5. That was the increment that they were studying. The idea to use 2 was to make it smaller so that the figure would fit on the screen (avoiding the effect of the wrap mode) in order to see it better. But they realised that changing the increment implied a big change. and come back to were they were before. An experience was done with 2 2 5 but registered as 5 2 5. They knew well by then that the form was the same. Nuno described these enrollments as "The turtle coming out in a different way she went in". He advanced a quite complicated (but not thoughtfulness) explanation on why she behaved that way: "Look. this is as like the positive and negative numbers. The turtle begins here as if this was the zero... It goes on enrolling, enrolling. gets to 180 and comes back by the same way... And then it is like getting back to zero... 180. 178. 176. etc. And when she finishes she needs to be at 180... 180 and -180 are numbers... how you say it... symmetric". But Maria, who did not follow very well his explanation conjectured herself: "I think it always will do 10 enrollments. But they will be different". Nuno suggested a comparison of two different experiences with different inputs on the same screen. but a misplaced command erased the first one before starting the second. Maria said to herself. but loudly "the researcher said to count the number of enrollments". giving an indication of how she was interpreting the objectives of the task, and pursued "she is always doing the same number of enrollments. but they are different". And then she stated with determination "I will discover something!"
- 259. 248 Nuno at some point in this segment appeared to have some intuition about the situation, which he tried to explain without much success for two occasions. But he did not take any consequences of his intuition and it was abandoned. Segment S. A new student, Victor, arrived to the group. He was given a very brief explanation by his colleagues about what they were registering (but not about the Logo procedure or their former conclusions) and got seated observing with some attention. This fifth segment lasted for 5:30 minutes. A new experience was carried with inputs 5 3 5. Maria registered 10 enrollments. Them 5 4 5 with the corresponding recording. Them 5 6 5 and Maria commented "It is exactly the same as 4 and 2". (she referred to experiments 4 and 2 not to initial values for angles 4 and 2, what would be a wrong observation.) Maria conjectured that it should exist a rule for the pairs, giving similar shapes, and the researcher arrived again at the group (now by his initiative). Victor, the new student who joined the group was not really integrated. By the contrary, his arrival led to a growing distraction of Nuno, who so far had been in second plane but even so was participating in the work along with Maria. Segment 6. This was a very short segment, lasting for 45 seconds. The researcher arrived and noticed in a glance the experiences already performed by the students. He suggested 7,8,9,10 to be tried for initial angles, and then come back to 5. And he immediately leaved. Segment 7. In this long segment, that took 35:00 minutes, the students continued their experiences just by themselves. They begun with 5 7 5. Maria said with confidence "It is going to give the same. I bet they are 10 enrollments. I do not need to say." And in fact the experience confirmed what she had predicted. A new experience was made with 5 8 5. Maria felt the result strange: "The even numbers should give similar figures and they do not". She showed difficulty in given up her conjecture about the pairs. The objective of the investigation was them reformulated by Maria: "To me it does not matter the number of enrollments. It matters the way they look like" and she added: "Stupid thing! This breaks all my plans!" A new experience was made with 5 10 5. Maria became excited again: "I am getting to it now! 2 enrollments." And new experiments were made in turn with 5 11 5,5 125,5 13 5, and 5 145. It became almost a mechanical activity ofexperimenting and registering. They got to try 5 15 5 and Maria commented: "I bet that 15 is going to give the same as 10... You saw, it did!... When they are multiples of 5 it always gives 2 enrollments".
- 260. 249 And she added: "The angles (or increments) do not matter", what in fact is not true; they only had tried increment 5 (and increment 2 with what they regarded as a strange result). Maria gave finally up of the conjecture on the pairs. She was sure that for multiples of 5 it should give 2 enrollments and that the other numbers have always 10 enrollments in two different families. She announced: "Let us try 16 to see if it is not also like this. 16 and 19". The experiences, of course, confirmed her prediction. Maria indicated: "The numbers which end in 1,4,6,9 make an enrollments like this". She went on: 20 1920, with the expected comment "That is the same thing, only it is larger". In this segment there was an interesting reformulation of the objective, made just by Maria. She was making predictions and testing them eagerly. The other two students became less and less involved in the activity. Segment 8. In this last segment, that took 32:00 minutes, the researcher joined the group to discuss the activity. In the beginning, a couple of minutes were spent talking about some issues related to the video-taping and then to ecology that were raised by Victor. Then the conversation focused on the activity. Things were summarized. "Length does not matter. The turtle does not tum left. With the multiples of 5 there are 2 enrollments..." The discussion got tricker as the researcher asked for the justification of these results, the reasons for the different behaviour of the turtle, and other increments than 5 were considered. The students had a class to go, however, and the activity was left with several questions still open. Discussion The proposed situation is a quite complex one. There are three parameters to investigate. The role of one of them is fairly simple but the role of the other two is quite complex since their effects are interrelated. These students had a reasonable previous contact with Logo. But even so they did not understood just by themselves the recursive mechanism of the procedure. They even did not had understood that RT 200 is equivalent to some left tum, a fundamental consequence of the fact that the angle measure works with modules of 360. Objectives. This situation yields to the formulation of many possible research objectives. Let us regard some of them: a) Determine the role of each of the parameters (it should be noted that parameters 2 and 3 can not be understood isolated but just together). Students considered this objective as they verified that the first parameter was irrelevant but for size, and then froze the third parameter to study the effects of variations on the second. b) Understand specific aspects about the working of the procedure: In what point does the turtle tum back? Why does it tum back? Why do enrollments exist? Why does it seem tum left
- 261. 250 when the procedure just says turn right? What is the relationship between the "coming out angle" (whatever that may be) and the number of enrollments? These aspects were mostly considered by the researcher and did not seem to catch great attention from students, except when he raised specifically that issue. c) Identify the different kinds of figures that we can get. How can they be classified? What is the reason for each figure? This was just partially pursued by the students for increment 5, and with no search for reasons. d) In a later stage, when a good grasp of the possible figures begins to emerge, one might also ask: When do figures "tend to infinity"? And when are they "auto-superimposed"? In these, when are 2,3,4,5,6... enrollments? When do we have a figure of kind "open" or of the kind "crossed"? Such questions were raised by the researcher in the final discussion but were left unanswered. e) State a rule that allows, given a triple of numbers (a,b,c), to say what are the figures that appear, preferably with a proof of such rule. This objective was not considered. In this activity the first formulation of objectives was made in segment 1 by the researcher. The general objectives in fact pursued by the students followed his suggestions, except that they disregarded the issues related to the working of the procedure - they were concentrating themselves on the behaviour of the turtle on the screen - and focused more in the influence of all the parameters. The second formulation of objectives was made in segment 3 also by the researcher. These were pursued by the students who, however, made them more precise. And there was a slight precision of the objectives by Maria when she said in segment 7: "To me it does not matter the number of enrollments; it matters the way they look like". There was general agreement in the interpretation of the task by students and the researcher, although we can see this one much more concerned with aspects of the functioning of the procedure than the students. This was most clearly apparent in the discussion on the equivalence of left turns and right turns. Strategies and conducting of experiences. By its own nature, and giving the suggestions made, the determination of the sequence of experiments was not a major difficulty in the activity. The students were quite organised in following a natural sequence (they jumped over angle value 5 in segment 5 because of the conjecture about the pairs). The idea of taking notes in a systematic way was given by the researcher in the beginning and reinforced during the work. It was taken up by Maria. The other students did not get involved in that task although they followed the recording. Specific strategies used were to make the figure larger in one case and smaller in another. This was done in order to see it better, to figure out what was going on, and was used when the effect of the parameter length was readily understood. As previously noted, the students showed willingness to try specific strategies (or tricks) that they were shown by the researcher. One of the students, Nuno, was not much critical about
- 262. 251 them. Other, Maria, in contrast, appeared much more independent, and accepted them when they seemed to be useful. Reflections, conjectures and tests. The students were puzzled by a number of things that occurred in this activity. They had real surprises with a number of aspects. Some concerned specific features of the situation: The strange behaviour of the turtle, the resulting figure with 10 enrollments, the accumulation of the angle values. They were amazed with the tricks used by the researcher to make sense of the working of the procedure. But the most striking surprise was with the turtle doing angles larger than 180 degrees. Apparently the usual thing to do is just left turns and right turns with angles smaller than 180 and there was no idea that the angles could be larger. There were "magical explanations" advanced by Maria about the behaviour of the turtle. She used a metaphor of animal behaviour such as "she come backwards because she does not have a way out" and "does not have another way to follow", etc, which appeared to be a strategy to try to make sense of the situation. Some discoveries were made by the students in conversation with the researcher: (a) 180 is the angle that the turtle is doing when it returns back, (b) the working of the procedure concerning the angle increments (segment 3), (c) the relation between left turns and right turns (segment 3), and (d) larger values for the increment give a more pronounced enrolment, and change the forms (segment 3). Other discoveries were made by the students themselves. One concerns the parameter "length forward" that has no effect of the form, other than in its size. Other refers to the fact that there are three main kinds of forms for increment 5: -multiples of 5 give 2 enrollments -numbers ended in 1,4,6,9 give 10 enrollments "open" -numbers ended in 2,3,7,8 give 10 enrollments "crossed" A conjecture about pairs was made by Maria. Is was dismissed with difficulty, only with the accumulation of contrary evidence. Later on she was able to make predictions about the number of enrollments and kinds of shapes, and verify them. By the end of the activity she was strongly confident in her conjectures. Students' involvement and interactions. Maria was uniformly highly involved throughout the session. She took the task quite seriously and with determination. As she exclaimed at some point: "I will discover something!". She knew that her role in this activity was to try hard to make discoveries. Nuno was participative, although in second plan, until Victor came. Then both of them had very little participation, becoming more and more distracted. The interaction between the students was not very productive in this activity. In the beginning Maria and Nuno had several interchanges. As the time went on, most of the work was carried out by Maria or under her direction. The dialogue became less and less effective. For several occasions, we could observe one of the students saying one thing, the other saying
- 263. 252 a completely unrelated thing, giving rise to no discussion among them, and then they just moving on. Finally, even these interactions became less frequent. In general, Maria tended to assume that Nuno was making an interpretation similar to her's, or she did not even try to understand his view on the successive results obtained at the computer. Victor had no relevant intervention in the development of the activity. Maria led the investigative process, making most of the suggestions which were then acted upon as collaborative decisions, or by making the decision on her own. From segment 4 on, until the end of the activity, Maria took an even more important role, taking decisions and reflecting on their results. Nuno was accompanying her activity, but not really intervening in the decision making. And from time to time he was speaking with Victor of subjects unrelated to the activity. The absence of collaborative work was identified since the beginning of segment 4. With the progress of the investigation Maria seemed to become more confident about what she was doing and assume that the other students would not be of much help. In this way, the attempt to an explanation of the behaviour of the turtle by Nuno in segment 4 was lost as a discussion opportunity. Personality factors may constitute the main reason for the nature of the students' interactions. Additional reasons may have to do with their increasing awareness of the complexity of task and also with the fact that the researcher, in his moments with the group, had more interchanges with Maria. Although absent for most of the time, the researcher (who was supervising the work of other two groups) had an important role on this activity, mostly in the definition of the objectives, in the adoption of general strategies, and in the suggestion of specific strategies (in this case "computer tricks"). Conclusion One should not underestimate the difficulties of the students in investigating complex situations. We know that making significant discoveries in mathematics is difficult enough for mathematicians [6, 7] and we should not forget that they are strongly motivated for their subject. Rich environments, like this one, entail many complexities and students are likely to find many embarrassments with them and are not necessarily highly motivated for mathematics [12]. But, by the other side, such difficulties have their reverse. They provide good opportunities for discussion and reflection, reveal misconceptions, and promote an awareness of global issues that may become significant for the progress of the students. What happened in this episode with rotations over 180 degrees is quite illustrative in that respect.
- 264. 253 Mathematical investigations may be important educational activities. They prove to be useful in the development and consolidation of specific concepts and mathematical ideas. They bear on important thinking skills. They may promote a broader vision of mathematics, much closer to the actual practice of the mathematician. The development of this activity seems to indicate that two fundamental characteristics appear to be necessary to deal successfully with mathematical investigations: sharpness and flexibility. Sharpness is vital in the formulation of objectives, so that they correspond to essential features of the situation and are amenable to a description in mathematical terms. Flexibility is important in the choice and evaluation of strategies (that is, the ability to set up and modify approaches that do not look promising anymore). What may be the principal difficulties and obstacles of this kind of activity? There may be problems involving content knowledge, reasoning processes, or general attitudes and appreciation. Students may not be able to figure out any sensible way of starting an investigation. They may do not know relevant background 4:ontent, or not be able to evaluate a given result. Many other questions need to be addressed regarding mathematical investigations. What may be the criteria for the assessment of an activity undertaken by the students? Content leaming? Development of cognitive skills? Development of appreciation of mathematics? What are good investigation proposals? What is the role of the teacher? This episode shows that the development of mathematical investigations may involve unexpected difficulties for teachers. Research on this topic, mapping the cognitive processes and the social interactions of the students, is necessary to bring new constructs and provide support for teaching practice. References 1. Abelson, H. & DiSessa, A.: Turtle geometry. Cambridge, MA: MIT Press 1980 2. Anderson, J.: Coin-turning: Anatomy of an investigation. Mathematics Teaching, NOs 131-132, 8-11 and 38- 42 (1990) 3. Barclay, T., Martin, K., & Riordon: A nodal-land investigation. The Computing Teacher, Vol. 13, ng 6, 20- 22 (1986) 4. Bell, A. W., Costello, J., & Kucheman, D.: Research on learning and teaching. Windsor: NFER-Nelson 1983 5. Cockcroft, W. H.: Mathematics counts. London: HMSO 1981 6. Davis, P. J. & Hersh, R.: The mathematical experience. Boston: Birkhauser 1980 7. Hadamard, J.: The psychology of invention in the mathematical field. Princeton: Princeton University Press 1945 8. Kissane, B.: Mathematical investigation: Description, rationale and example. Mathematics Teacher, 81, 520- 528 (1988) 9. Lawler, R. W.: Extending a powerful idea. MIT Logo Memo 58,1980 10. NCTM: Curriculum and evaluation standards for school mathematics. Reston, VA: NCI'M 1989 11. Papert, S.: Mindstorms: Children, computers and powerful ideas. New York, NY: Basic Books 1980
- 265. 254 12. Ponte. 1.• & Carreira. S.: Spreadsheet and investigational activities: A case study of an innovative experience (In this volume). 1991 13. Ridgway. 1.: Assessing mathematical attainment. Windsor: NFER-Nelson 1988
- 266. Aspects of Computerized Learning Environments Which Support Problem Solving Tommy Dreyfus Institute of Mathematics, University of Fribourg, Switzerland1 Abstract: Examples of students' problem solving processes using computerized learning environments are described in some detail. These descriptions serve as basis for abstracting features of learning environments that support specific aspects of problem solving, mainly control processes such as planning, switching one's point of view, or deciding to work backward; flexible approaches during conjecturing or search for solution paths are also focussed on. The following software features were identified as particularly relevant: tool character, non- evaluative feedback, and operational structure; this structure should include a limited number of operations, predominantly transformations modelling the underlying mathematical domain. Keywords: computerized learning environments, problem solving, problem posing, planning, control, conjectures, flexibility, software tools, operational structure, Stereometrix, Triple Representation Model Introduction2 According to Schoenfeld [4], a problem-solving situation is one where the solver does not have easy access to a solution for solving a problem but does have an adequate background with which to make progress on it. Schoenfeld investigated students' behavior while they thought about problems with substantial mathematical content. He asked questions such as how the problem solver decides which mathematical knowledge to access, and how to use it. These are questions of heuristics and control in problem solving, but also questions of resources - in particular the mathematical resources available. Students often have a hard time with anyone of these aspects: heuristics, control, and resources; but difficulties become compounded, when all three need to be handled simultaneously. Computerized learning environments can help students handle such situations by providing mathematical resources in a directly accessible form and by making available to the students tools that support their heuristic steps and their control of the problem solving process. In this paper, three examples of problem solving processes with IOn leave from the Center for Technological Education, Holon, Israel 21 would like to acknowledge the contributions of Nurit Hadas and Baruch Schwan to this paper. Our interaction during the time we collabprated on the development of and research with Stereometrix and TRM has greatly influenced the formation ofthe ideas presented here.
- 267. 256 computerized learning environments will be described; these descriptions will then serve as a basis for discussing general features of the environments that are supportive of heuristics and control during problem solving. Spatial Geometry The Stereometrix Software Stereometrix is a computerized learning environment for spatial geometry, in particular for the geometry of standard solids such as cubes, pyramids, and prisms. Work within Stereometrix is operational, i.e., it is organized in operations to be performed by the student. Two main types of operations are available in the environment: Constructions and transformations. Constructions change a given solid, e.g. by adding points or lines to it. The most important construction which can be carried out by means of the software is that of the perpendicular from a point to a line or a plane. Others are copying segments, finding midpoints, connecting or disconnecting points, bisecting angles and intersecting lines. A rather particular "construction" enables one to change the shape of a solid by moving a point in such a way that incidences are preserved. Transformations do not change the solid but its position in space and thus its representation on the screen; in particular, it is possible to rotate the solid around either of three fixed axes; but it is also possible to rotate it by specifying the desired final position, namely specifying a plane that shall be parallel to the screen plane. Stereometrix also has a replay feature, which allows one to save the sequence of operations one carries out and then reuse this sequence on the same or on a different solid. Stereometrix has been used with some success to help students visualize three-dimensional solids, to support them in forming basic concepts of spatial geometry such as the angle between a line and a plane, to encourage them to formulate and check conjectures about solids, and to assist them in solving problems of varying complexity. Some of these problems are computation problems as they might occurin a standard curriculum; others go beyond that level, and should be considered as rather extensive projects. A more detailed description of Stereometrix and its possible uses has been published elsewhere [2], Here, we will concentrate on the description of two episodes that highlight the support that Stereometrix gives during problem solving processes. Example 1: Problem of the elusive volume: Given a right prism ABCDEF with an isosceles triangular base ABC, compute its volume from the following data: The base AC of the base triangle ABC, the base angle a=CAB of the same triangle and the angle g between one of the two equal "walls", ADEB, and the diagonal AF of the third wall, ACFD (see Figure I),
- 268. 257 D A c Figure 1 This problem is taken from the spatial geometry unit of a 12th grade mathematics curriculum in Israel. The problem appears toward the end of the unit, after students have (hopefully) become familiar with the basic notions, in particular with the definition of the angle between a line and a plane. At this stage, students are not expected to have great difficulty in visualizing the given triangular prism, nor in realizing that the area of the base triangle can be calculated from AC and a and that therefore the altitude of the prism would complete the information necessary to compute the required volume. Then, however, they might well be stuck. They enter a stage where they need to search for a solution path: They have some data, they have a well specified aim - compute the altitude AD = CF - but no path that leads from the data to the altitude. This is a problem solving situation according to Schoenfeld: Students have an adequate background but no easy access to a solution. Building the bridge from the given angle g to the required altitude involves the management of large amounts of predominantly visual information, including the establishment of explicit relationships between the involved data within the three- dimensional solid. Work with the learning environment can support the design of a plan at this stage in an essential manner. It is by then standard procedure for the students to use the environment to visualize the solid with the given data; an essential aspect here is that they can use the automated features of the software to construct the angle g according to its definition: g is the angle between AF and AG, where G is the footpoint of the perpendicular from F to the plane ABED. Properly carrying out this definition already helps with an analysis of the relationships; for example, G will be on DE because the prism is right (Le., AD, BE and CF are all perpendicular to the plane ABC). The fact that the students need not actually execute the constructions and draw the lines, enables them to concentrate their attention on the elements essential to the problem. Paper and
- 269. 258 pencil drawings made by students at such a stage are often inexact and thus confusing; moreover, such drawings are fixed, whereas the computer screen image can easily be dynamically turned to be viewed from an angle from which the most important details can be seen best. This is the region within which the relationship between the given data and the missing altitude has to be built; and by implication, the importance of this region goes far beyond: this is the region, within which possible plans for solving the problem have to be established and their feasibility checked. The students are thus given a tool which enables them to take control of their actions and to think at a level of planning: For instance, if I am given g, and g is an angle in triangle AFG, I could possibly compute other quantities in that triangle; but the missing altitude DA=FC belongs to triangle AFC, of which I know only AC. But wait: AF is common to the two triangles... Above, specific features of the Stereometrix environment and their importance for planning and control have been pointed out; Stereometrix also incorporates some general features of learning environments which have a bearing on problem solving: Tool character of the software, students' control over their actions, advantages offered and limitations imposed by the operational character of the environment, and type of feedback. These will be discussed below. Example 2: Problem of the intersecting altitudes: Do the four altitudes of a tetrahedron meet in a single point? This problem is non-trivial by most standards. In fact, Davis [1] cited Coolidge to the effect that most highly educated mathematicians do not know the answer. While it is not difficult to convince oneself that the altitudes of a regular tetrahedron intersect in a single point, anything beyond that is anybody's guess. In other words, whereas the "elusive volume" problem is a textbook problem with a guaranteed answer, the tetrahedron problem puts the solver in a situation where heuristics will necessarily playa central role: Not only is the answer unknown, the problem might not even be properly fonnulated. This is a problem solving situation, in which one first has to come up with a conjecture. Some experimenting might help to find such a conjecture, but for experimenting one needs tools which are adapted to the problem without being too constraining. Stereometrix provides construction and transfonnation tools in spatial geometry without either evaluating or severely constraining the user's actions. It is not difficult to decide what to do frrst: One wants to check many tetrahedra, construct the altitudes in each one of them and check whether they intersect. Two features of Stereometrix help in this endeavor: The replay feature which enables one to do the lengthy construction of the four altitudes one single time and have it repeated automatically on all the following tetrahedra; and the special rotation which enables one to tum a specific plane of the tetrahedron into the screen plane and check whether the suggested intersection point exists there. But more than that: It would be nice if this experimental heuristic stage were organized in a systematic way. And again, like any well organized learning environment, Stereometrix provides the structure, the
- 270. 259 scaffolding, against which to change parameters or data in a systematic manner: In this case, Stereometrix includes a set of pre-programmed solids, among them tetrahedra with equilateral, isosceles, right or general triangular base. It also makes it possible to move one of the vertices, e.g. the top vertex, in a systematic manner in a given plane or on a given line, and observe what happens to the intersection point of the altitudes as the vertex is moved. The learning environment, through its tools and structure, thus guides the problem solver's heuristic demarche without imposing a specific approach. One person who was observed working on the tetrahedron problem with Stereometrix, soon tried a right pyramid with a right triangle as base; the four altitudes of such a tetrahedron do not intersect in a single point. She thus reformulated the problem as Modified Problem: In which tetrahedra do the four altitudes meet in a single point? In this formulation, it becomes clear that the problem is not a straightforward "compute" or "prove that" exercise. So far, there is nothing to compute, nor to prove; a situation has to be explored; one needs to systematically investigate different classes of tetrahedra in order to come up with a conjecture; the subject we observed decided to start with a tetrahedron whose base was an equilateral triangle. Although there is no direct evidence for this, it can be hypothesized that this choice was influenced by the ready availability of a tetrahedron with an equilateral triangular base in Stereometrix. After discovering that for this tetrahedron the four altitudes do meet in a single point, she constructed a plane through the vertex parallel to the base plane and moved the vertex in that plane, an action which destroyed the coincidence. More judicious experimentation made her realize that the altitudes were still intersecting in pairs if the vertex was moved on a parallel to one of the edges of the base triangle. A dynamic view of the problem was generated by means of the dynamic aspects of the software: The question whether a particular motion of the vertex will destroy the coincidence or not, came within the range of questions with which the subject felt it was possible to deal. From here, through several more stages, some with and some without using the tools offered by the software, she arrived at the conjecture that the four altitudes intersect, if the projection of any vertex onto the opposite face is the intersection point of the altitudes of that face. She then used the tools of the learning environment for checking this conjecture: A point can be moved on a line; thus, a vertex can be moved "up and down" on the corresponding altitude (the perpendicular from the vertex to the opposite face). Conjecturing and testing conjectures, two central components ofproblem solving heuristics, are thus supported by the design of the learning environment. Again. this has been shown here for a particular environment and a particular problem; however. in addition to the specific mathematical tools realized in this environment, some general design features of the environment are conducive to such heuristics; these general features will be discussed below.
- 271. 260 Functions The Triple Representation Model Microworld The Triple Representation Model (TRM) is a computerized learning environment about functions. The environment as well as an associated curriculum and several possible activities have been described in some detail elsewhere [7]. Its description here will therefore be limited to what is necessary in order to describe its use as a problem solving tool. Work within TRM is possible in three modes: Table, Graph and Algebra; each mode corresponds to a functional representation. The work within any mode is operational, like in Stereometrix. The most important operations are Find-Image in the tabular representation, Draw in the graphical one, and Search in the algebraic one. Once a function f is defined, Find-Image yields the value off(x) for any given x in the domain that the user chooses to specify, and lists (x, f(x» in a table. Draw generates a graph of y=f(x) on the screen; before using Draw, the student has to specify a window of values for x and y within which the graph is to be drawn. The Search operation enables the student to check algebraic conditions for a large number of equidistant values of x, for example: "From a to b step d: if f(x»C print x", where the student has to fix the lower bound a and the upper bound b of the search, the step d, the type of comparison (>, <, = or ;t), and the goal value C. The Search operation prints on the screen the values of x for which the condition is true. Passage between representations in TRM is semi-automatic: The student has to specify the mode (representation) in which (s)he wants to work but is free to switch to another representation at any time; the link to other representations is established by operations named Read, which allow one to consult results previously obtained in one representation while working in another one. Thus information may be consciously transferred from anyone representation into any other one, but such transfer is not made automatically by the software. The features that the student has to explicitly specify the representation and explicitly transfer information between representations are based on pedagogical design considerations: Beginning students should be consciously in control of their choices and actions. Below, it will be shown that these same features are directly related to corresponding control over problem solving with TRM. Example 3: Solving a third order equation: Find a solution of 3x-x3=1 with an accuracy of 10-4. Students were presented with this problem when they were about three months into a TRM based introductory functions curriculum; at that stage, they were fairly familiar with the relevant features of TRM and the multi-representational notion of function as underlying TRM, but they had not yet solved any equations beyond linear ones. Many of them quickly wrote a Search
- 272. 261 operation of the fonn "From a to b step d: if 3x-x3=1 print x", with different choices of a, b, and d. Search conditions of this type did not yield any answers. Although they did not understand the real reasons for this (namely that the solution of the equation is irrational whereas all values of a+nd are rational), they were thus forced to face the problem whether there is a solution to the equation at all. The possibilities of TRM gave them concrete ideas how to proceed at this stage. The structure of TRM is important here: The three separate representations of TRM imply the suggestion to use another representation, or, in tenns of the problem to be solved, to rather radically change the point of view. From the point of view of the problem to be solved, this is an action of control. Such actions are explicit in TRM: The student has to consciously choose the representation to be used. Such actions at the control level are supported through the design of the software; they were also strongly used in the accompanying curriculum. Many students thus switched representations at this stage, and decided to draw a graph of y=3x-x3• From the graph, they could learn that there are three solutions to the equation, one of them between x=O and x=1. At the same time, they had the opportunity to experience the power of using several representations in conjunction when solving a problem. On a more remote level, and from the mathematics educator's point of view, they had the opportunity to experience the heuristic efficiency of actions at the control level. Obviously, the problem was not solved for them yet. One way to progress toward a solution was to zoom in graphically until a solution was located with some accuracy, say between 0.3 and 0.4, and then to return to the algebraic representation; then there was an opportunity to discuss why the original Search operation did not produce the solution, and to suggest a more sophisticated one such as "From 0.3 to 0.4 step 0.0001: if 3x- x3>1 print x". The first x which satisfies this condition, x=0.3473, is the solution with the desired accuracy. But it is more important here, to stress the combined effect of the curriculum and the software on students' problem solving behavior. Students were, over a period of several months, regularly presented with carefully chosen and sequenced problems which exposed them to situations in which actions of control were necessary. They were supported, by the tremendous power of TRM to use these opportunities to take action at the control level; although we have not directly examined to what extent they became conscious of such action, we did observe 42 students, one by one, at the end of the instructional period in a problem solving situation; of these, 32 successfully solved a maximum problem of rather high difficulty [6]. Relevant Features of Learning Environments One of the most frequent phenomena observed when students are asked to solve complex mathematical problems is that they are inactive, they seem to be stuck; two possible reasons for being stuck are that they do not have any idea what they could do next or that they do not dare to try an idea they might have. The examples described above point to the potential of computerized
- 273. 262 learning environments to prevent students from getting stuck. Such environments put at their disposal tools, more specifically, a set of mathematical operations, with which they have presumably become familiar during earlier, introductory sessions. The limited number and the concreteness of the tools makes it possible for students to envisage using them, and ask themselves what would happen if such and such operation was carried out. Students' thinking about problem situations thus becomes more flexible. The tools might suggest, how to change a certain parameter or which other point of view may be taken. The ease of use of the tools - often only a single keystroke - renders their use feasible technically; if it is easy to try an operation, why shouldn't I? Students are becoming less reticent, to use an experimental approach, because the appropriate tools are conveniently laid out in front of them. Specifically, these tools can have a beneficial effect on planning, conjecturing, and changing one's point of view. Planning How planning occurs during problem solving sessions with learning environments has been exemplified by means of the elusive volume problem. In that case, planning had to do with the search of a procedure to compute the volume. It was necessary to build a bridge between the given data and the goal. The planning process in that example is very explicit. The learning environment supports that process by two features: the visual support and the list of possible operations (this list may also be seen as a constraining factor - see remark (ii) in the conclusions below). The visual support makes the solid to be acted upon more concrete for the students; the list of operations makes their field of action more concrete, i.e. the choice of possibilities at the student's disposal. The student is not put in front of a completely open field (which may possibly disable him(her) for lack of direction) but in front of a well defined, delimited array of possibilities, ofoperations (constructions and transformations), which may be carried out with ease. In order for the students to be in full control of their actions and thus to give them full responsibility over the planning, the environment must offer a set of tools but not direct them which tools to use. It must give them as much feedback as possible on what happens if a particular tool is used; but this feedback must not be evaluative; the evaluation of the feedback is again the student's responsibility; it is dependent on the answer to the question: Did the tool or operation do what I wanted and/or expected it to do? Planning as described here is closely connected to control: Decisions on choice of tools and operations, and thus control over the problem solving process are entirely with the student; and the situation of the student in front of the computer makes this explicit. The limited number of possibilities, and the fact that the operations are implemented in a concrete model, render this decision making process manageable for the students; therefore, they are less likely to get stuck.
- 274. 263 A similar discussion could be made on the basis of the third order equation example with TRM, specifically at the moment the students saw that there seems to be no solution. This is typically a moment when students could easily give up, unless they have some convenient, and possibly useful investigative tools at their disposal. Conjecturing Some aspects of heuristics are involved in all three examples discussed above: A solution path has to be searched for in the elusive volume problem, and a zero of a function has to be found in the third order equation problem. But in the intersecting altitudes problem, the heuristic aspects were central, because the problem statement soon turned out to be inadequate. The students are, therefore, in front of a problem posing situation, in which they need to first define the problem. An experimental attitude, an approach which encourages trying out various transformations, changing relationships and parameters, is necessary in such a situation. And again, the tools put at the students' disposal are crucial: The fact that Stereometrix is designed to include certain transformations not only makes it easy to execute them, but even suggests their use to the student, again under the assumption that this problem is only presented after familiarization with the learning environment. In order to encourage such an experimental approach, similar software characteristics as described above are helpful: Operational organization of the environment, large but manageable choice of operations (transformations and constructions in the case of Stereometrix), and non- evaluative feedback. As a consequence, control over their actions remains fully with the students. Two additional features, are particularly important for an activity that is intended to lead to conjectures; one is the possibility to change one's point of view; in Stereometrix, this is implemented by a considerable choice of transformations, which allow one to visualize the three- dimensional solid from different points of view. The other feature is the possibility to check and test conjectures, once they have been made. For this purpose, the replay feature enables one to repeat the same sequence of operations on different solids. Similar features have been implemented in other geometric "conjecturing environments", such as the Geometric Supposer, Cabri-geometre or Geometric Sketchpad. Flexibility: Changing One's Point of View Being stuck can be due to a lack of flexibility, an impossibility on the part of the student to imagine the situation (s)he is facing from a different point of view. The importance of flexibility in this sense has been pointed out in the two Stereometrix examples, but it takes on added importance in the third order equation example: The changes of point of view offered by TRM
- 275. 264 are more radical than those offered by Stereometrix: TRM not only allows one to choose a completely different representation for the same function but, in fact, has been designed specifically with this purpose in mind; such change ofrepresentation is the most important single operation offered by TRM; and systematic interviews have shown that students familiar with TRM have used this feature extensively in solving complex problems [5]. It is through such radical change of point of view that students were able to progress toward solving the third order equation; and it is because of the easy availability of such change of representation through TRM that even beginning students had access to powerful problem solving techniques. Again, the operational organization of the environment, the availability of transformation operations are obviously essential here. It is an open question whether for questions of flexibility, it is essential that control be with the student. One could imagine an environment similar to TRM, in which several representations would always be presented on the screen and all information would be transferred automatically from one representation to the others. It is an interesting question how such a change would affect students' problem solving behavior, in particular their concurrent use of several representations. Conclusion In summary, a number of features of learning environments support students in the problem solving skills of planning (control), conjecturing (heuristics), and flexibility (control and heuristics). These are the following features: the software is a tool, organized into operations; often, such operations have the character of transformations which help changing one's point of view. The number of operations is limited (in accordance with students' age and ability). The students thus can control their actions, and take decisions within a manageable framework. The feedback of the software is only in response to student action and is neutral: It does not evaluate the students' actions. Thus, it does not directly influence their planning; it gives "objective" mathematical information, and this information supports the student in decision making, without prejudicing the decision. Several aspects critical for problem solving in computerized learning environments have not been discussed in this paper; some of them will now be mentioned in brief remarks that do not do justice to their complexity. (i) It has been stressed above that the feedback is objective in the sense of not evaluating the students' actions. Some care in interpreting the word "objective" needs, however, be taken: the feedback is necessarily influenced by choices made in the design of the software, in particular by how mathematical concepts are modelled, which representations are implemented, how a solid is represented. The model in the learning environment will always be a model of the mathematical
- 276. 265 concepts, and not more than that. This problem is often neglected; one notable exception is Cabri-gwmetre (see [3]). (ii) The argument for having a limited number of operations in a learning environment can be inverted: Because some transformations and constructions have not been included in Stereometrix, they will most probably not be used by the students. Similarly, because some operations on functions are not available in TRM, students working with TRM are unlikely to even think of them. Thus, each learning environment, in addition to offering a choice of possibilities, also imposes limitations and constraints. Obviously, this is unavoidable; therefore the designer of any learning environment has to take certain decisions which are likely to be influenced by considerations of learning goals, student population and curriculum. The solution of non-routine problems may but need not be among the goals. (iii) It has been repeatedly stressed above that students are supposed to be familiar with a learning environment before attempting to use it for solving complex, non- standard problems. While this seems a fairly trivial requirement, it has some far- reaching implications: Because of their character of not imposing any action on the student, learning environments by themselves do not teach. They only make sense within a curriculum. This may be a problem based curriculum [8]. Such a problem based introductory functions curriculum has been developed for TRM [7]. Heuristics and control during problem solving activity demand the availability to the problem solver of a set of appropriate mental tools. Often, students are weak problem solvers, because they lack such tools. Several cases have been discussed, in which the availability of concretely modelled mathematical operations within a familiar computerized learning environment helped students solve complex non-standard problems. The mathematical operations of the learning environment served as tools for the student which they were able to shape into the necessary mental tools. The operations are concrete and accessible to the students, the mental tools are abstract and removed, but apart from this they are often closely related: For example, the operation of switching a representation is a concrete action in the environment, and a parallel but rather removed mental tool in the abstract formulation. In many cases, the students succeeded to map the ones to the others and thus to construct meaning (see also [5]). Operations in a learning environment become cognitive tools for the student problem solver. Thus the fact that the software puts at their disposal a limited number of well defined, specific operations from which to choose (like from a set of tools in a toolbox) leads to the students being stuck less often and less deeply: they have means to extricate themselves from being stuck and to progress toward solving problems in a meaningful way.
- 277. 266 References 1. Davis, P.: Are there coincidences in mathematics? American Mathematical Monthly 88, 311-320 (1981) 2. Dreyfus, T. & Hadas, N.: STEREOMETRIX - A learning tool for spatial geometry. In: Visualization in mathematics (W. Zimmermann & S. Cunningham, eds.). Notes Series, Vol. 19, pp. 87-94. Providence, RI: Mathematical Association of America 1990 3. Laborde, C.: L'enseignement de la ｧｾｯｭｾｴｲｩ･＠ en tant que terrain d'exploitation de pMnomenes didactiques. Recherches en Didactique des MatMmatiques 9(3), 337-363 (1989) 4. Schoenfeld, A.: Mathematical problem solving. New York, NY: Academic Press 1985 5. Schwarz, B.: The use of a microworld to improve ninth graders concept image of a function: The triple representation model curriculum. PhD thesis, Weizmann Institute of Science, Rehovot, Israel 1989 6. Schwarz, B. & Dreyfus, T.: Assessment of thought processes with mathematical software. In: Proceedings of the 15th International Conference on the Psychology of Mathematics Education (F. Furinguetti, ed.). Italy 1991 7. Schwarz, B., Dreyfus, T. & Bruckheimer, M.: A model for the function concept in a three-fold representation. Computers in Education 14(3),249-262 (1990) 8. Thompson, P.: Experience, problem solving and learning mathematics: Considerations in developing mathematics curricula. In: Teaching and learning mathematical problem solving: Multiple research perspectives (E. Silver, ed.), pp. 189-236. Hillsdale, NJ: Lawrence Erlbaum 1985
- 278. A General Model of Algebraic Problem Solving for the Design of Interactive Learning Environmentsl ｊ･｡ｮＭｆｲ｡ｮｾｯｩｳ＠ Nicaud LRI, CNRS URA 410, BAt 490, ｕｮｩｶ･ｲｳｩｾ＠ de Paris XI, F-9140S Orsay, France Abstract: A general model for a class of algebraic problems is presented as a framework for the design of Interactive Learning Environments. This model enables us to consider several levels for the reference knowledge of a learning environment. It allows us to represent knowledge for the control of the student's problem solving activity without the model tracing constraint which requires the student to follow the behavior of the reference knowledge. The APLUSIX system is an Interactive Learning Environment in the domain of factorization of polynomials which has been developed in that framework. Experiments have been conducted in France and protocols have been collected in order to study human learning process in that domain. Keywords: algebraic problem domain, knowledge state, strategic knowledge, plans, heuristics, tasks, APLUSIX The design of Interactive Learning Environments (ILE) in problem solving requires the modelling of the knowledge domain for the production of examples and explanations and for the control and guiding of the student's activity. Many problems are met in this work: some of them, like rmding precise definitions of words, objects and concepts of the domain, are classic problems in Artificial Intelligence (AI) but need a didactic approach in this context; others, like taking into account an evolution of the reference knowledge (objects, concepts, strategies, etc.) are more specific to learning contexts. In this paper, we consider problems that are solved by successive transformations of algebraic expressions, like simplification of expressions, factorization ofpolynomials, equation solving, calculus of derivatives, calculus of primitives, etc. We propose a general model for this class of problems as a framework for the design of ILEs in algebra. The APLUSIX system, an ILE in the domain of factorization of polynomials, is described as an example of development within that framework. 1I would like to thank N. Balacheff and M.e. Rousset for their comments during the elaboration of this paper
- 279. 268 General Model We consider an autonomous agent that possesses knowledge and is capable of applying a part of this knowledge to try to solve problems. An agent can be a person (a student, a teacher), an idealized person (like the Anderson's ideal student) or computer software (or a part of software). Algebraic Problem Domains, Behaviors Definitions. An algebraic problem domain2 is an n-uple ( 'll' , lB , lP , 18 ) in which: 'll' is a problem type (like equation solving), lB is a set of well formed expressions, lP is a set of problems which is a subset of lB, 18 is a set of possible behaviors. A behavior is a search tree developed by the reasoning process of an agent in which nodes are expressions. The root is an element of lP (the problem to be solved), the other nodes are generated by the application of transformations to nodes of the tree according to strategies. Each transition of the tree is labelled by the applied transformation. A successful behavior is a behavior in which at least one node is labelled by the indication solved (which means that the problem is solved and the nodes labelled by solved contain solutions). A failed behavior does not include any solved node and is labelled by the sentence the problem is unsolvable (which means that the agent has the knowledge to recognize some states as unsolvable) or the sentence I abandon (which means that the agent does not want to continue searching for a solution). A partial behavior is a part of a behavior. Given a behavior, there are many classic ways to generate partial behaviors: transitions may be not labelled, the tree may be reduced to one of its branches leading to a solution, it may have not termination indication, etc. Remarks. This definition of behavior includes the productions usually erased or struck off in a traditional paper-pencil resolution by a person, but it does not involve all the elements that can be seen on a sheet of paper, in particular remarks about properties of expressions like this is a polynomial ofthird order or remarks about strategies like now I will try to reduce are not included in this definition. The nature of the links in the tree is succession: step A generates step B. Different meanings can be associated to these successions: production of equivalent problems by the application of transformations, generation of a subproblems, etc. 2 In comparison with Dillenbourg and Selfs definition [9] of domain problem, we introduce the problem type and we make explicit the concept of behaviour for algebra.
- 280. 269 B indicates the syntax of the expressions that can be manipulated. lP and B can be identical, for example in equation solving physicists try to solve any equation using computer software that finds approximations of the solutions. lP and B can be very different. For example, during a learning stage in mathematics, the algebraic problem domain equation solving can be defined in this way: B is the set of the syntactic forms ofpolynomial equations with rational numbers (the student is able to manipulate such expressions), P is composed of polynomial equations of degree I or 2, which are not too complex (limited size, few parenthesis levels) and eventually have some good properties like being solvable with some degree ofknowledge. Example. We describe here the algebraic problem domain currently implemented in APLUSIX: T is factorization of polynomials, B is the set of well-formed expressions that can be generated with variables, integer numbers, addition, subtraction, multiplication and power operators, parenthesis. P is a set ofproblems generated by human teachers. Behaviors are trees in which nodes contain expressions. All the expressions are implicitly supposed to be equivalent (the equivalence relationship being to be expressions representing the same polynomial). Figure I is an example of behavior. FRCTORIZE 1 QX4-6X3+X2_16 I fr -1 Is a root of 9X4_6X3+X2_t6 I 2 CX+I)C9X3_15X2+16X-16) abandon I re.arkable square 9X4 -6X3 +X2 3 C3X2_X )2-16 I difference bet.een 2 squares C3X2_X 4 C3X2_X-4)C3X2_X+4' I -1 Is a root of 3X2_X-4 :5 C3X-4)(X+I)C3X2_X+4) solved I Figure l. An example of behavior as it appears at APLUSIX interface. Here, abandon at step 2 means abandon of that search direction and not abandon of the reasoning process. This problem domain corresponds more or less to the problem domain for French 16 year old students. It differs in the following points: (1) at the moment B is limited to integer numbers. (2) P includes more difficult problems at strategic level.
- 281. 270 Knowledge State Definition. We consider Balacheffs definition [3] of knowledge state of an agent in an algebraic problem domain in which: (1) we exhibit the behavioral knowledge component as Dillenbourg and Self [9] do; (2) we abandon the temporal indexation. A KS (knowledge state) is described by a n-uple ( C , B , L , P ) in which C is the conceptual knowledge, B is the behavioral knowledge, L a set of signifiers, P a set of problems. C , B , L are effective tools for solving the problems of P. B is decomposed in ( T , M , Q, S ) where: T is a set of transformations, M is a set of matching procedures, Qis a set of calculus procedures, S is the strategic knowledge, C contains the concepts underlying B and a solution predicate s. The existence of pieces of knowledge described here is necessary for any agent capable of solving problems: as he produces a behavior, he has to apply transformations to expressions, which involves knowledge in transformations (T), knowledge to determine when transformations are applicable (M), knowledge to apply transformations (Q), and knowledge to choose transformations (S). The existence of the solution predicate s is necessary to evaluate new nodes and to stop the reasoning process. The form of pieces of knowledge and the connections between them can be modelled in many ways. Some will be proposed later. Any KS can include correct and incorrect knowledge at any level: correct behavioral knowledge, bugs in procedural knowledge (bugs in Buggy [6], mal-rules in LMS [16, 17], bugs in ACT [1], misconceptions [18], etc.). A KS can be deterministic or not. In the second case, the strategic knowledge includes a random component, two resolutions of the same problem can generate different behaviors. Links between a KS and its associated algebraic problem domain. W e consider an algebraic problem domain (T ,18, P ,18) and a KS, KSI=( C, B ,L, P) for an agent, in this domain. KSI has the basic function of being applicable to the problems of P. The application of KS I to a problem of P starts a reasoning process which generates a behavior with success or failure. P is composed of problems on which KS I is effective. L contains signifiers usable for the agent, 18 contains signifiers usable for the problem domain. Example. For factorization of polynomials defined above, here is a partial description of the KS of an imaginary agent IA. BEHAVIOURAL KNOWLEDGE Transformations: factor A out of B
- 282. A2_B2 -> (A-B)(A+B) A2+2AB+B2 -> (A+B)2 standard developments standard reductions Matching: 271 capacity to recognize squares for simple numbers direct matching for factorization standard matching for developments standard matching for reductions Calculus: capacity to apply the above transformations Strategies: classify applicable transformations ofeach node according to priorities give the highest priority to reductions give an intennediate priority to factorizations concerning the entire expression give the lowest priority to other transformations choose the node which has the transformation of highest priority in case ofconflict in the choice between nodes, prefer the current one apply only reduction on reducible expressions CONCEPTUAL KNOWLEDGE About expressions variable integer monomial, degree ofmonomial polynomial, degree ofpolynomial Aboutsolving a problem a problem is solved by the application of transformations until a factorized form is found it is possible to abandon the current expression and to return to a previous one About transformations a transformation can be applied to any sUbexpression when it matches this subexpression transformations can be classified according to their effect: factorization, development or reduction About strategies concepts underlying the behavioral strategies Solution predicate: a 0 or 1 order polynomial is factorized a 2 or higher order polynomial is factorized if it is a product of at least 2 non constant polynomials
- 283. 272 EXAMPLE OF BEHAVIOUR Given the problem <factor (X-2)(X-I)+(X-2)(X-3)-18> an example of partial behavior of this KS is: (X-2)(X-l)+(X-2)(X-3)-18 factoring out X-2 in (X-2)(X-l)+(X-2)(X-3) (X-2)(2X-4)-18 factoring out 2 in 2X-4 2(X-2)(X-2)-18 reduction of (X-2)(X-2) 2(X-2)2_18 factoring out 2 in 2(X-2)2_18 2[(X-2)2-9] factoring (X-2)2-9 as a difference between two squares 2(X-2-3)(X-2+3) reduction ofX-2-3 and X-2+3 2(X-5)(X+l) solved This tree is a one branch tree; links between nodes are implicit. Organization of KSs Behavioral knowledge has to be organized in order to be efficient. It can be structured in different ways by general concepts like heuristics, goals, tasks, or plans. We present below different organizations (in a nonexhaustive form). All the strategic knowledge in heuristics. We define heuristics as small pieces of knowledge used in making choices. In an all the strategic know/edge in heuristics organization, we consider that T and M contain no strategic component and that all the strategic knowledge has the form of heuristics. Heuristics are small pieces of knowledge. This means that no heuristic envisages the entire situation and that a choice is the result of the application of several heuristics. The KS of agent IA has this characteristic if we suppose that there is no strategic aspect in the matching knowledge M. This organization is well structured according to separation into strategic/non-strategic knowledge. Strategic knowledge can be described easily because of the small size of its elements, strategic knowledge is not structured. This organization is not very efficient: transformations are envisaged on many subexpressions and heuristics have to deal with more information. A part of strategic knowledge in transformations. Conditions can be attached to transformations. For example, the transformation A2_B2 -> (A-B)(A+B) is applicable to 9-4 according to some matching knowledge. This application is not interesting for many problem
- 284. 273 domains, the transfonnation may be associated with the condition the expression is not constant. Goals. Strategic knowledge can use explicit goals and subgoals. This is familiar in AI conceptions in many domains with reasoning processes that realize decompositions of a goal into subgoals and construct trees (for example and/or trees). In algebra, goals are generally associated with other strategic feature, in particular with plans. Static plans. We define static plans as predefined combinations of actions to realize one goal. A static plan is a recorded piece of knowledge and not the result of a planning process (which is a dynamic plan). Static plans can have different structures: combinations of actions can be limited to sequences or can be more complex, including alternatives. For example plans can be generated with or without the constraint ofinvoking only actions that succeed. CAMELIA [19] is a system using plans for solving problems. In CAMELIA plans can be very complex and can invoke actions that fail. Actions consist of immediate actions or subgoal choice. An example of a CAMELIA plan in the domain of primitive calculation is: FOR the calculation of the primitive of F in a variable R IF the main operator ofF is + DO I) generate a variable R1 2) instantiate R with 0 3) FOR EACH term T ofF DO a) calculate the primitive ofT in RI b) add RI to R 4) free RI This plan is an executable piece of knowledge for the calculation of the primitive of a sum. It can be applied to any sum (which requires the use of a FOR EACH operator in the plan). It invokes subgoals: calculation of primitives of tenns, addition of expressions. It can fail (if a primitive cannot be obtained). The ALGEBRA TUTOR [2]3 is another system using plans and goals for tutoring a student solving an equation. Plans are sequences of actions which consist of immediate actions or subgoal choices. An example of an ALGEBRA TUTOR plan is: IF the goal is to rewrite an equation with a subexpression distributed THEN set as subgoals (1) find the coefficient associated with the subexpression (2) multiply the parenthesized part by the coefficient (3) replace the subexpression by the production. 3 See also the TEACHER'S APPRENTICE [12] which is the previous name of the same system.
- 285. 274 Dynamic plans. Dynamic plans are plans which are elaborated during the resolution process, taking into account the search space and the goals. They are generated by strategic knowledge. They can generally be seen as skeletons of the search space (or of a part of the search space) that will be developed afterwards. As far as I know, this kind of plans is not implemented in the current ll.Es in algebra. Tasks. We define a task" as an executable set of knowledge to realize one goal. The difference between static plans and task is the following: a task involves all the knowledge concerning one goal, it can be seen as a sub-KS; a static plan involves only a part of the knowledge concerning one goal, this knowledge is organized in a procedural way. In the current version, APLUSIX has an organization with tasks, static plans and heuristics for the knowledge used when examples of resolution are generated. Each task has plans and heuristics; plans invoke tasks or immediate actions. For the factorization of polynomials, the following constraints have been chosen: plans involve only sequences of actions; actions always succeed. An example of an APLUSIX plan in the domain of factorization ofpolynomials is: IF AND AND AND TIlEN a subexpression E of the problem is a sum an expression U can be factored out in a part ofE the partial factorization of U in E produces a new expression V V is a possible factor of another part of E factoroutU arrange the result of the factorization factor out V arrange the result of the factorization This plan invokes four successive tasks. When the plan is applied, factor out performs the selected factorization, arrange realizes some usual developments and reductions after a factorization. Each task may be realized in several steps. Control in static plans and tasks. Static plans and tasks correspond to a proceduralization (or compilation) and structuration of knowledge. The natural way to install control in static plans is to oblige their execution, i.e., when a static plan is started, no control is applied, the next step is always executed when no failure occurs. This is the case of the examples presented below. This kind of control is not psychologically plausible without certain constraints: if a subtask involving complex reasoning is started, nothing can stop it (it may succeed or fail within too much time and may even enter an infinite loop). ALGEBRA TUTOR and APLUSIX solve this problem by using only small subtasks. 4 We use the AI meaning of task [8] which is the description of a process or a process which accomplishes a rask in the general meaning of wk.
- 286. 275 An alternative is to introduce control in static plans. A priori, this is in contradiction to the concept of static plan and heuristics seem the best way to evaluate the developed tree at each step, however heuristics are less structured. A way to combine these two objectives is to introduce other control features like complexity or time and to use them according to principles such as the following: (1) when a task or a plan takes too much time, then abandon it; (2) when a task or a plan generates complex expressions, then abandon it. CAMELIA, uses an estimation of the cost of a plan before starting it, but it does not manage the cost during the application of the plan, so a plan that seems cheap before starting is applied until the end (with success or failure), even ifit is in fact expensive. For further developments of APLUSIX in other domains of algebra, we will introduce expensive plans and control them by giving information to them. This information probably will be: a level of complexity (if the expression developed in the plan oversteps this level, then the plan is stopped); a credit given to plans and tasks according to some heuristic reasoning (each calculus decrease this credit and the plan/task is stopped if its credit becomes null). A Conglomerate of KSs A KS can be seen as a structure containing pieces of knowledge assembled according to the criterion being the pieces ofknowledge ofa modelled agent. We define a KS* (conglomerate of knowledge states) as a structure of the same sort in which pieces of knowledge are assembled according to some criterion. For example, with the KSs KS(A) and KS(B) of two agents A and B, we can generate KS*(AB) that assembles the knowledge of A and B5. Generally, a KS* is not applicable: in the previous example, if KS(A) and KS(B) are different, there are no means to choose between KS(A) and KS(B) strategies. The elaboration of a KS* allows the evaluation of pieces of knowledge in terms of being, or not being, elements of this KS*. Reference Knowledge in ILEs In ILEs, we call reference knowledge the knowledge of the domain. This knowledge is used in different situations; we particularly consider the production of examples and explanations as well as the control, guiding and help of the student. KS and KS* are suitable for modelling a part of the knowledge involved in these functions. 5:1(,8" (AB) can be seen as the union of the knowledge of :1(,8(A) and :l(,8(B). It is different to the union of :J<,8(A) and :J<,8(B) which is a set of two :J(,8s.
- 287. 276 Production of examples. Many ILEs envisage a short learning stage. Then a reference KS is a good model for the production of examples. When a wider learning stage is to be treated, the reference knowledge generally evolves so that a unique reference KS is maladapted. This evolution can be realized by considering a sequence of KSs. It is this way we used in APLUSIX. This method is not very suitable because on one hand using few KSs implies important differences between them and on the other hand using a lot of KSs is difficult to manage. The genetic graph [11] uses an evolutionary reference KS: a unique representation implements several KSs with links allowing to find what knowledge can be learnt at any time. This model has been used in the WUSOR software in a domain involving a small set of knowledge in a monotonous context. It is an interesting idea, but the elaboration of evolutionary KSs for large sets of knowledge in non monotonous contexts is a complex problem6. Production of explanations Explanations can be generated using reference KSs. These are model-based explanations. First, a behavior of a KS is an explanation. More explanations can be given during a problem resolution, at factual and strategic level, using explanation knowledge. Second, explanations can be produced in describing reference KSs. We can separate explanation models according to their means and their goals: - With one reference KS and explanation knowledge, one gets an epistemic explanation that can exhibit features of a reasoning process according to general characteristics (like importance of a piece of knowledge in a KS) or describe a reference KS. - With an evolutionary reference KS, it is possible to take into account the differences between the current KS and the previous ones, giving priority to recently learned knowledge for the production of explanations. This is a genetic explanation. It needs to know the previous KSs used or to form hypothesis on them. - With a student's model, one can get a personalized explanation in which information on the student, differences between the reference KS and the student's model are used to select pertinent components of explanationsfor that particular student. What is the correct model for an ILE? It depends on the context in which explanations have to be elaborated. For example, if a teacher uses an ILE in a classroom, epistemic explanation is pertinent when the reference KS fits the classroom level; genetic explanation ought to be better if the evolutionary reference KS can be adapted to the evolution of the classroom. Of course, when a student learns alone with an ILE, a personalized explanation is the best model; however its quality depends on the quality of the process building the student's model. 6 The learning ofalgebra involves a nonmonotonous evolution of the references knowledge, examples: (1) when only rational numbers are known, X2-2=O is an equation without a solution; when irrational numbers are known, it has solutions; (2) in factorization of polynomials, when the discriminant is not known, developing second order polynomial is a mediocre strategy; when the discriminant is known, it is a good strategy.
- 288. 277 Control of the student. When a tutor has to evaluate the behavior of a student, he has to show what is correct/incorrect, legaVillegal, authorized/forbidden according to the didactic contract [4]. In a traditional context, the didactic contract is mainly implicit; in an ILE, it must be entirely specified. Its task is to accept or refuse the action (or the request) of the student, and to give appropriate feedback. One way to control the student's actions is the model tracing methodology [2]. Model tracing uses a deterministic reference KS and a set of bugs (which is a KS*) to evaluate the student's action. With this methodology, the student is not allowed to use a path different from the reference KS. In fact model tracing is suitable for learning skills in problem domains in which skills are sufficient. When no deterministic reference KS is available or when a discovery approach is used, the control of the student's actions requires another methodology. We propose to model the control with two KS*. The first is the epistemic control which verifies the syntax of expressions, the correctness of the transformations, the applicability of transformations on expressions, etc. The second is the didactic control which executes controls not directly related to the scientific knowledge. For example: Factor out X-2 in (X-2)(X-5)+(X-3)(X-I) is judged as an incorrect request at epistemic level. Factor out X-2 in (4X-8)(X-5)+(3X-6)(X-I) is judged as a correct request at epistemic level; it can be judged as an incorrect one by the didactic control if the didactic contract requires all factors to be identical for the factor out operator (which is a plausible didactic contract at some learning stage). Guiding the student. Guiding the student can be envisaged according to several principles. Using a deterministic reference KS, the model tracing methodology achieves strong guiding. At the opposite, controlling the student's actions according to a didactic contract without adding guiding, allows the student to evolve in a sort of algebraic microworld. Given a problem, a reference KS is basically capable of generating its own behavior with, at any node, the best action to proceed. If the reference KS is an extended KS i.e. it has in addition the capability of evaluating any action in any behavior, a parametrized model based guiding can be designed as follows: at any time, the student's action is compared to the reference KS action in terms of distance, the student's action is accepted if this distance does not overstep a chosen level (the guiding parameter), otherwise it is refused. Help. We consider here the help asked for by a student when solving problems in an ILE. A model based help consists in using a reference KS to provide this help. When no strong guiding is realized, an extended KS is required for providing this help because the KS has to evaluate a behavior different to his own. Help also can be based on performance by using a reference KS to try to solve the problem from the current node without taking care of the others. This kind of help can be effective, however its effect is in finishing the current resolution; a model based help is supposed to produce knowledge.
- 289. 278 The APLUSIX Project APLUSIX is an IACI project in algebra, currently developed in the domain of factorization of polynomials. This domain is treated at three levels and involves complex strategies. Currently, the software appears as a learning environment including two learning modes: the learning-by- example mode in which the system shows how it solves exercises and can be asked for explanations at a factual or strategic level, and the learning-by-doing mode in which the student solves exercises. Experiments have been conducted in France in March 1990 and in April 1991. APLUSIX is implemented on a Macintosh. It uses a LISP environment (Le_Lisp) and the inference engine SIM [13]. Other learning environments have been developed in algebra during the last few years. Most of them are concerned by equation solving at a rather low level of knowledge in which few context-free heuristics [7] are sufficient for solving problems. This is the case of ALGEBRALAND [5] and ALGEBRA TUTOR [2,12], two systems involving two different orientations. In ALGEBRALAND the student uses operators in a menu to transform and solve equations (this interaction mode is close to the learning-by-doing mode of APLUSIX). The student can perform backtrack when he wants. The search space is clearly represented as a tree at the interface, reifying the resolution process of the student. Strategies have to be discovered by the student. In ALGEBRA TUTOR the student activity is controlled by a tutor which uses an ideal student model and buggy rules. The main purpose of the work with this system is to learn skills according to the ACT theory [1]. The model tracing methodology is used for controlling and guiding of the student. The Learning-by-Example Mode The model in factorization of polynomials. Factoring polynomials is modelled as a general task in which a reference KS organization based on static plans and tasks has been chosen. Plans are sequences of actions and have, in that particular context, the following properties: when a plan is chosen, it cannot fail; a plan generally does not lead to the solution; no strategic reasoning is processed between two actions of a plan. Actions are calls to general tasks or well known subtasks like development ofA,factoring A out ofB, and are executed when the possibility of being executable has been remarked. Subtasks develop steps of calculus according to the granularity required in the KS. Plans are classified and chosen by heuristics. Three reference KSs have been defined by high school teachers corresponding to beginners, intermediate and high-level reference knowledge. A first example of a plan at a high-level has been presented above. Second example of a plan at a high-level: IF a subexpression E of the problem is a second order polynomial THEN develop E factor the result
- 290. 279 This plan implements the following knowledge: when the concept of discriminant is known, second order polynomials can be factorized by developing and factoring with discriminant. However, having been developed, they can sometimes be factorized with simpler process than the discriminant. Third example of plan at beginner-level: IF AND AND TIffiN E is a subexpression of the problem a number N can be factored out of E factoring NinE brings a factorizable expression factor N out ofE factor the result This plan implements the following knowledge: when one has seen some factorization through factoring a number but cannot execute this two transformations altogether, one uses a plan to execute them in sequence. Remark that all these plans model a mental process executed during the analysis of a node of the tree: they correspond to a limited look ahead. First example of heuristic (at every level): IF the plan P executes a "factoring out" and leads to having a constant as the only developed term in a sum, TIffiN discard this plan For example the following instantiated plans: PI: factor out X in X2+4X+4 P2: factorize X2-4 then factor out X-2 in X(X2-4)+(X-2)(2X2+3)+4 are discarded by this heuristic, because the constant 4 will be the only developed term in the expression. Remark that this heuristic can be seen as an advanced heuristic (example P2) but one has to remember that a reference KS is used to show examples: this KS must not be too complex but must not be poor. If this heuristic is discarded at the beginner level, the system would present the application of PI which is very inefficient. Indeed, at beginner level, plan P2 does not exist. Second example of heuristic (at high level): IF AND AND AND AND AND TIffiN current step CS concerns expression CE CE has N factors the best plan CP of CS has quality CQ, with CQ <== weak a non current step S concerns expression E E has N-I factors the best plan P of S has quality Q, with Q >== fair-well prefer step S and plan P
- 291. 280 This heuristic proposes to perform a backtrack to a less factorized step when the current step is weak. Of course, in that case, if there are other steps with an N factors expression and some interesting plans, the backtrack will be done to one of these other steps in order to maintain the factorization already obtained. An example of the use of this heuristic is shown in figure 1 (backtrack after step 2). The problem solving process. At each cycle, the problem solving process first studies the current step. Heuristics associate a quality to each envisaged plan (according to a symbolic scale (very-weak, weak, medium ,fair-well, well, very-well)) in order to find the best one. According to a cognitive economy principle, the problem solVIng process tries to find the best plan for the current step without studying all possible plans. This has been implemented by giving each plan a maximum quality (which is the highest quality the plan can reach) and by considering them as follows: the problem solving process first envisages the plans having maximum-quality = very-well and applies heuristics on them. Then, if one envisaged plans has quality = very-well (real quality), no other plan is envisaged for the moment, at this step. IT not, plans having maximum-quality = well are envisaged, heuristics are applied to them and the system looks for a plan with quality =well among all the plans envisaged at this step. This process continues until the best plan has been found. When the best plan of the current step is found, heuristics are used to choose between the current step and other steps (between going on and backtracking). Stability rules are first applied: if the best plan of the current step is good according to some concepts, the current step is chosen and no backtrack is envisaged. In the other case, a backtrack is envisaged, other steps are updated (as some plans have been applied on most of them, their best remaining plans have to be found again) and heuristics choose between all these steps. This reasoning process has been achieved by M. Sardi [15]. The interaction mode. In the learning-by-examples mode, APLUSIX uses the behavioral knowledge of the chosen reference KS to generate, step by step, a search tree and to display at each step a transformation. APLUSIX allows the leamer to search for explanations about the matching and the strategy used. This helps learners to verify their conjectures, if they have any, or to formulate explicitly the reasons for choosing an action. As the advance in the solving process is under the learner's control, he can ask questions about the matching and about the strategies at every step of the resolution. Explanations concerning the matching. Questions about the matching are asked when the learner does not understand how an extended matching is performed. APLUSIX gives explanations by presenting a description of the rule and the critical intermediate transformations. Here is an example of explanation (with an intermediate level):
- 292. 281 3X2_12=3(X2_4) I applied the rule A2_B2 --+ (A+B)(A-B) to X2_4 with A=X , B=2 Explanations concerning the strategy. Questions about strategy may be either of the type why is it a good action? (applying a rule or backtracking), or of the type I would propose another rule I will indicate, explain to me why it is not the best way to proceed? Figure 2 shows an example in which the learner wants to know why APLUSIX has decided to generate the state 2 using the rule difference between 2 squares. The status of each plan is determined by heuristics implemented as meta-rules. A plan may be: (1) instantiated, chosen and applied, (2) instantiated and kept in memory, (3) instantiated and discarded, (4) not yet instantiated. As several heuristics may be applied to determine the status of a plan, the trace of the heuristics used to solve a given problem is not necessarily the best way to explain how the solution was reached. This problem arises for most expert systems at the factual level [10] and at the strategic level [8]. In APLUSIX it concerns basically the strategic level: explanations are epistemic explanations, they are produced by extracting the most significant information from the reasoning process. FACTORIZE Uhy this backtrack? Uhy HOT a backtrack? difference bet.een 2 squares X2 -4 2 (5-2X)(X-2)(X+2)+8(2-X'+4X2_12X+9 fach'r,zatlon of Xl-4 is very ir,teresting • fact(,rizat,on of X.!-4 produces the factor X-2 X-2 is also a factor in 8(2-X) Figure 2: Answer to "why this action?" The explanation consists here in indicating the chosen plan. In the first version of the system (which had no plan), explanations were generated by a reasoning process which operated on a knowledge base of 90 rules, and on the trace of the strategic reasoning [14]. In the current version, the explanation module has not yet been implemented. It will be a reasoning process and will give information depending on the
- 293. 282 question and according to several explanation concepts. Outputs will include presentation of the applied plan and of the main heuristics, or heuristic concepts, that lead to the choice or non- choice of a plan. Some of them will take into account the optimization process and give answers like: I chose the plan P because I considered that it is of a good quality. I did not consider the development you propose because developments can only reach a fairly-good qUality. The Learning-by-Doing Mode The learning model. In the learning-by-doing mode of APLUSIX, the student solves exercises by choosing the transformations and the subexpressions; the system executes the calculations. The system memorizes the transformations for the student (they are presented in menus). The student can ask for help when he wants. Learning strategic knowledge is an important part of the teaching object of APLUSIX. The goal is to build a strategic knowledge that is highly efficient in the current problem domain. This knowledge has not to be the most efficient for each problem. Its average effectiveness on the problem domain is only expected to reach a level fixed by the teaching object. FACTORIZE R2_B2 I 1!IIII+(X-3H2X+1) Ideselect II select all I ICRnCEL I Figure 3: The student chooses the step 3, chooses a transformation in a menu and marks out a subexpression with the mouse.
- 294. 283 The control of the student's request involves an epistemic and a didactic level. Example of feedback for an incorrect request at epistemic level: X2+2X+I is not a difference between two squares. Example of feedback for an incorrect request at didactic level: show me how to/actorize 3X2_12 as a difference between two squares. The control module has been realized by J. M. Gelis. No guiding is implemented now. The help module. The help module has been designed as a model based help. The goal of this module is: To help the student to reach a/airly good effectiveness on P The help menu in the APLUSIX learning-by-doing mode includes two submenus: the fIrst one gives access to the applicable transformations of a given node and the second is a general help. When the student asks for the applicable transformations on a node, all the applicable transformations that can be generated by the system (that are not discarded by the strategic knowledge of the system and that have not been used by the student) are presented. When the student asks for general help, the main strategic concepts are used for giving advice. Here is a part of them, at a beginner's level: - reduction transformations have to be applied, - when factorization transformations are not discarded for some particular reason they are good transformations, - when no reduction and no factorization can be applied, it is necessary to perform development, - backtracking has sometimes to be envisaged. Example of general advices: - you can reduce the expression of the current step, - factorization transformations remain in other steps, - you have now to develop. Some of these advices give access to complementary information, for example,factorization transformations remain in other steps is completed by a which button giving access to the factorization transformations remaining in the search tree. The help module has been completed by M. Sardi. Experimentation A. Nguyen-Xuan, C. Aubertin, and P. Wach conducted an experiment in March 1990 with 24 students (tenth grade). Each student spent four hours with APLUSIX. The exercises were more difficult than those usually given at school. 40 exercises had been designed, with increasing levels of difficulty: 20 exercises associated with beginner level, 20 exercises associated with intermediate level. The student interacted with APLUSIX only in the learning-by-doing mode,
- 295. 284 and was not allowed to abandon an exercise without trying at least twice to follow the advice given by the system. Before the experiment, the students were given a paper and pencil test composed of exercises of the same level of difficulty. This test was collected and not corrected. A month after the experimentation they were given the same paper and pencil test. These tests showed that the students were making progress: during the fIrst test, 10% exercises were solved; during the second one, 34% were solved. Between these two tests, progress were made: in matching, in the use ofremarkable identities and in the use of normalizations (factor out a number in order to have a normalized polynomial). A new experiment took place in April 1991 with the new version of the learning-by-doing interaction presented in this paper. Protocols have been recorded again and we will try to analyze them by combining human and software analysis. Conclusion The general model presented in this article addresses a wide variety of algebraic problems. It allows us to consider different knowledge states and different activities. Because of its general approach, this model does not lead to a precise architecture of ILE, but several functions of an ILE can be based upon it. This model can be instantiated more when particular subclasses of problems and particular activities are considered; then precise architectures can be elaborated. An important issue for the design of algebraic ILEs is the representation of evolutionary reference knowledge states involving evolution of objects, concepts, transformations, matching, strategies, etc. With such a representation, an ILE can achieve another dimension. The existence of general features for evolutionary reference knowledge states in algebra that can help the design of ILEs is an open question. Researchers in AI and didactics have to join forces to try to answer this question. References 1. Anderson, J. R.: The architecture of cognition. Cambridge: Harvard University Press 1983 2. Anderson, J. R., Boyle C. F., Corbett A. T., & Lewis M. W.: Cognitive modeling and intelligent tutoring. Artificial Intelligence,42(1) (1990) 3. Balacheff, N.: Nature et objet du raisonnement explicatif. In: Actes du colloque "L'explication dans I'enseignement et l'EIAO" (M.G. Sut et A. Weil-Barais, eds.), Ecole Normale Superieure de Cachan, Avril 1990 4. Brousseau, G.: Fondements et ｭｾｴｨｯ､･ｳ＠ de la didactique des ｭ｡ｴｨｾｭ｡ｴｩｱｵ･ｳＮ＠ Recherche en Didactique des ｍ｡ｴｨｾｭ｡ｴｩｱｵ･ｳＬ＠ 7(2), 33-115 (1986) 5. Brown, J. S.: Process versus product: a perspective on tools for communal and informal electronic learning. Journal of Educational Computing Research, 1 (1985)
- 296. 285 6. Brown, J. S. & Burton, R. R.: Diagnostic models for procedural bugs in basic mathematical skills. Cognitive Science, 2,155-191 (1978) 7. Bundy, A.: The computer modeling of mathematical reasoning. New York: Academic Press 1983 8. Chandrasekaran B., Tanner, M. C., & Josephson, J. R.: Explaining control strategies in problem solving. IEEE Expert, 9-20 (1989) 9. Dillenbourg, P., & Self, J.: A framework for student modelling. Technical Report TR-A1-49, Computing Department, University ofLancaster 1990 10. Gilbert, G.: Forms of exp1anation. In: Proceedings of AAAI'88 Workshop on Explanation, Saint Paul, 72- 75 (1988) 11. Goldstein, I. P.: The genetic graph: A representation for the evolution of procedural knowledge. In: Intelligent tutoring systems (Sleeman & Brown, cds.). London: Academic Press 1982 12. Lewis, M. W., Milson, R., & Anderson, J. R.: The teacher's apprentice: designing an intelligent authoring system for high school mathematics. In: Artificial intelligence and instruction, application and methods (G. Kearsley, ed.). New York: Addison Wesley 1987 13. Nicaud, I.F.: APLUSIX: un systeme expert en resolution pMagogique d'exercices d'algebre. These de ｬＧｕｮｩｶ･ｲｳｩｾ＠ de Paris XI 1987 14. Nicaud, I.F. & Sardi, M.: Explanation of algebraic reasoning: the APLUSIX System. Lecture Notes in Artificial Intelligence, 444. Berlin: Springer-Verlag 1989 15. Sardi, M.: Evaluation optimisoo et planification en resolution d'exercices d'algebre. 2eme Journ6es EIAO de Cachan. Ecole Normale Su¢rieure de eachan, Ianvier 1991 16. Sleeman, D. H. & Smith, M. I.: Modelling student's problem solving. Artificial Intelligence, 171-187 (1981) 17. Sleeman, D. H.: An attempt to understand student's understanding of basic algebra. Cognitive Science, 8, 387-412 (1984) 18. Stevens, A., Collins, A. M & Golding, S. E.: Misconceptions in students' understanding. In: Intelligent tutoring systems (Sleeman & Brown, eds.). New York: Academic Press 1982 19. Vivel, M.: Expertise ｭ｡ｴｨｾｭ｡ｴｩｱｵ･＠ et informatique: CAMELIA, un logiciel pour raisonner et calculer. These ､Ｇｾｴ｡ｴＬ＠ Paris VI 1984
- 297. Problem Solving: Its Assimilation to the Teachers's Perspective Paul Emest School of Education, University ofExeter, SL Luke, Heavitree Rd, Exeter EXt 2LU, England Abstract: It is argued that "problem solving" has multiple meanings. according to the teacher's mathematics-related belief-system or perspective. A model of teacher belief-systems is proposed. It is argued that the central feature is a personal philosophy or conception of mathematics. and that this determines the teacher's understanding of the nature of problem solving. Also. mediated by other factors. it influences classroom behavior. Some supporting evidence is cited. Keywords: problem solving. multiple meanings. teacher beliefs, philosophy of mathematics. absolutism, fallibilism Introduction It could be said that there is a problem-solving bandwagon roIling. Many influential reports published in the USA, UK and elsewhere, such as the Agenda/or Action [25], the Cockcroft Report [6], the Standards [26] and the National Curriculum in Mathematics [10], all strongly endorse a problem solving approach to school mathematics. These recommendations have been made for over a decade - three decades, if we include the 'discovery learning' of the 196Os. There are powerful arguments behind these endorsements, which I will not rehearse here. But the fact is that most mathematics teaching remains routine and 'instrumental'. More often than not, children are given a method for carrying out a type of task, and then many graded exercises to practice and reinforce the method. Each task has a unique correct answer. Of course such procedures do generate some 'relational' and well as 'instrumental' understanding on the part of the learner, and perhaps some strategic skills. But the primary focus of such exercises is the successful acquisition and deployment of procedures, and the acquisition ofrelational understanding or strategic skills is incidental. So what is going on? Why the disparity between the problem solving recommendations and the routine and convergent nature of much of school mathematics? There are many
- 298. 288 reasons, including institutional resistance to change, vested interests behind the status quo, individuals' resistance to change, be they teachers, learners, parents or others, and so on. In this paper I want to pick out one strand to explore, which in my view has received insufficient attention. That is the assimilation of problem solving to the teacher's perspective, by which I mean the teacher's mathematics-related belief-system. Teachers have different beliefs about the nature of mathematics and its teaching and learning, which powerfully affect their classroom practices. In some form or other, this relationship is now widely accepted, and I elaborate on it below. But one unremarked consequence of this, I wish to claim, is that problem solving is understood differently by different teachers, in accordance with their beliefs. A demonstration of the lack of unique meaning and the diversity of interpretations given to the term 'problem- solving' by teachers and others might go some way towards explaining how widespread espousal of problem-solving can co-exist with its widespread rejection in practice. The Meaning of Problem Solving Problem solving has been a widespread part of the rhetoric of British mathematics education since Cockcroft [6]. Worldwide, problem solving can be traced further back, at least to Brownell [3] and Polya [28], and probably earlier. (In fact Descartes' Rules for the direction of the mind of 1628 constitute a remarkably complete set of problem solving heuristics.) One of the difficulties in discussing problem solving is that the concept is ill defined and understood differently by different authors. However, there is agreement that it relates to mathematical inquiry. Thus, there are a number of preliminary distinctions which can be usefully applied, for it is possible to distinguish the object or focus of inquiry, the process of inquiry, and an inquiry based pedagogy. In this section, I shall explore these different aspects of problem solving. The Object of Inquiry The object or focus of inquiry is the problem itself. One definition of a problem is Ita situation in which an individual or a group is called upon to perform a task for which there is no readily accessible algorithm which determines completely the method of solution... It should be added that this definition assumes a desire on the part of the individual or the group to perform the task." [22, p. 287]. This definition indicates the non-routine nature of problems as tasks which require creativity for their completion. This must be relativized to the solver, for what is routine for one person may require a novel approach from another. It is also relative to a mathematics curriculum, which specifies a set of routines and algorithms. For a problem is only non-routine if it belongs to the complement of the scope of a given set of routines. The definition also
- 299. 289 involves the imposition of a task on an individual or a group, and willingness or compliance in perfonning the task. The relationship between an individual (or group), the social context, their goals, and a task, is complex, and the subject of the Soviet Activity Theory which has developed from the work of Vygotsky. The Process of Inquiry In contrast with the object of inquiry is the process of inquiry itself. If a problem is identified with a question, the process of mathematical problem solving is the activity of seeking a path to the answer. However this process cannot presuppose a unique answer, for a question may have multiple solutions, or none at all, and demonstrating this fact represents a higher order solution to the problem. The formulation of the process of problem solving in tenns of finding a path to a solution, utilizes a geographical metaphor of trail-blazing to a desired location. Polya elaborates this metaphor. "To solve a problem is to fmd a way where no way is known off-hand, to find a way out of a difficulty, to find a way around an obstacle, to attain a desired end that is not immediately attainable, by appropriate means." [17, p. I]. This metaphor has been represented spatially [12, figure 8]. Since Nilsson [27] it has provided a basis for some of the research on mathematical problem solving, which utilizes the notion of a "solution space" or "state-space" representation of a problem [which] is a diagrammatic illustration of the set of all states reachable from the initial state. A ?state? is the set of all expressions that have been obtained from the initial statement of the problem up to a given moment." [22, p. 293]. The strength of the metaphor is that stages in the process can be represented, and that alternative 'routes' are integral to the representation. However a weakness of the metaphor is the implicit mathematical realism. For the set of all moves toward a solution, including those as yet uncreated, and those that never will be created, are regarded as pre-existing, awaiting discovery. Thus the metaphor implies an absolutist. even platonist view ofmathematical knowledge. Inquiry Based Pedagogy A third sense of problem solving is as a pedagogical approach to mathematics. Cockcroft [6] endorsed such approaches under the heading of 'teaching styles', although the tenninology employed does not make the distinction between modes of teaching and learning. In considering problem solving as an inquiry based pedagogy,number of further distinctions can be made. A range of different approaches can be generated by distinguishing the roles of the teacher and the learner, as in Table 1. In the literature a number of terms are used across the whole range of
- 300. 290 Table 1: A comparison of inquiry methods for teaching mathematics ME1HOD GUIDED DISCOVERY PROBLEM SOLVING PROBLEM POSING Teacher's role Poses problem, or chooses situation with goal in mind. Guides towards solution or goal Poses problem Facilitates solving Frames initial context, (or sanctions student initiative) Facilitates posing and solving Student's role Follows guidance Finds own way to solve problem Poses own problem Attempts to solve and extend in his or her own way. inquiry approaches to teaching mathematics, including 'discovery', and 'investigations' (or 'investigational approach'), as well as 'problem solving'. To fix the meanings of the terms more precisely I distinguish 'guided discovery', 'problem solving' and 'problem posing' pedagogies according to the openness of the approach. Table 1 illustrates that the shift from a guided discovery, via problem solving, to a problem posing approach involves more than mathematical processes. It also involves a shift in power with the teacher relinquishing control over the answers, over the methods applied by the learners, and over the choice of content of the lesson. The learners gain control over the solution methods they apply, and then finally over the content itself. Thus the shift to a more open inquiry orientated approach involves increased learner autonomy and self-regulation. If the overall classroom ethos is to be consistent with this shift, a necessary accompaniment is increased learner self-regulation over classroom movement, over access to learning resources, and over initiating and participating in learner-learner and learner-teacher interactions (discussion). Evidently problem solving and posing as teaching approaches force a consideration of the social context of the classroom, and its power relations. Problem solving allows the learner to apply his/her learning creatively, in a novel situation, but the teacher still maintains a great deal of control over the content and form of the instruction. Where the problem posing approach allows the learner to pose problems and questions for investigation relatively freely, it becomes empowering and emancipatory. However the characteristics that have been specified are necessary but not sufficient for such an outcome. What is also needed, I wish to claim, is the communication of a fallibilist philosophy of mathematics in the classroom experience, which de-emphasizes the uniqueness and correctness of answers and methods.
- 301. 291 Theorizing the Teacher's Mathematics-Related Belief-System In this paper I want to stress the importance of the teachers mathematics-related belief-system, their perspective on mathematics and its teaching and learning, which the title of the paper refers to, with regard to the implementation of problem solving in the classroom. In the past few years there has been a growing recognition of the importance of teachers' beliefs and conceptions [4], and in particular on their mathematics-related belief systems [8, 13, 30]. What underlies this new emphasis is the hypothesis, by now reasonably well confirmed, that teachers' belief systems are crucial determinants of their classroom practice. Teachers' conceptions of the nature of mathematics and their personal theories about its teaching and learning are very important factors in determining how they teach mathematics in the classroom, and in particular, the place given to problem solving in their teaching. This is not to deny the vital importance of many other factors, such as teachers' attitudes and confidence, the extent of their knowledge of mathematics, and of course the social context in which they work. Figure 1 provides a simplified model of some of the relationships involved. It represents one central belief, the teacher's conception of the nature of mathematics, as underpinning two pedagogical components of the teacher's belief system, the theories of teaching and learning mathematics. These in tum have an impact on practice in the form of the models of learning and teaching mathematics which are enacted. One use of resources, namely that of mathematics texts, is singled out as important, for texts embody an epistemological approach to mathematics, and the extent to which their presentation and sequencing of school mathematics is followed is a crucial determinant of nature of the implemented curriculum [8,15]. In particular, texts which embody an algorithmic approach to mathematics teaching can represent a major obstacle to the introduction of a problem solving pedagogy in the classroom. The downward arrows in the figure show the direction of influence on behaviour. The content of the upper positioned belief components is reflected in the nature of the lower positioned components. Because the enacted models are interrelated, as are the espoused theories of teaching and learning, this is represented in the figure as horizontal links drawn between them. As the figure illustrates, the impact of the espoused theories on practice is mediated by the constraints and opportunities provided by the social context of teaching (Clark and Peterson, 1986). The social context has a powerful influence, as a result of a number of factors including the expectations of others, such as students, their parents, fellow teachers and superiors. It also results from the institutionalised curriculum: the adopted text orcurricular scheme, the system of assessment, and the overall national system of schooling. The social context leads the teacher to internalise a powerful set ofconstraints affecting the enactment of the models of teaching and learning mathematics. According to Lacey [18] the influence of the social context and significant others in it leads to the 'socialisation' of teachers. If the espoused beliefs of the teacher are at
- 302. 292 PHILOSOPHY OF MATHEMATICS ESPOUSED MODEL OF LEARNING MATHEMATICS ESPOUSED MODEL OF TEACHING MATHEMATICS : CONSTRAINS AND OPPORTUNITIES PROVIDED : BY THE SOCIAL CONTEXT OF TEACHING I .. _- --------------- ---_...------------------- ENACTED MODEL OF LEARNING MATHEMATICS ENACTED MODEL OF TEACHING MATHEMATICS USE OF MATHEMATICS TEXTS Figure I: The relationship between beliefs, and their impact on practice odds with those of an authority figure the teacher becomes 'strategically compliant', or in more extreme cases makes an internalized adjustment to the belief-system. Thus newly prepared mathematics teachers who are eager to employ a problem solving approach often need to adapt in their classroom practices to comply with the prevailing ethos of a traditional algorithmic approach. This may be a temporary 'strategic compliance' until they are able to influence and change the ethos. It may cause conflict and tension, resulting in the withdrawal of the teacher or in internal adjustments to the teacher's belief system. Naturally the model illustrated in Figure 1 is greatly simplified, since the relationships are far more complex and far less mechanistic than they appear. Thus, for example, although the enacted beliefs in are shown separate from the social context, they are embedded in it. Furthermore, all the beliefs and practices are part of an interactive system, and pressures at any point, such as in classroom practices, or from authority figures such as heads of departments, as we have seen, will feed-back and may influence all the other components temporarily or permanently. Teachers' Philosophies of Mathematics A central component in the model of mathematics-related belief-systems sketched above is the teacher's personal philosophy of mathematics. This is his or her conception of the nature of
- 303. 293 mathematics as a discipline. Three groupings or types of philosophies of mathematics can be distinguished: absolutist, progressive absolutist and conceptual change or social constructivist philosophies of mathematics [7, 14]. Absolutism The most widespread philosophies of mathematics are absolutist, which view it as a body of fixed and certain, objective knowledge. Logicism, fonnalism and to a large extent platonism are absolutist [1]. Most of these views are foundationist, regarding the truths of mathematics as being based on certain logical foundations. One consequence is that the structure of mathematical knowledge is seen to be hierarchical, building upwards from its logical foundations by chains of logic and definition. Another consequence is that mathematics is seen as objective, neutral, culture and value free. Progressive Absolutism The progressive absolutist position also views mathematics as made up of certain, objective knowledge. But in addition, it accepts that new knowledge is continually being created and added by human creative activity. Confrey [7] distinguishes this view, which underpins Popper's [29] epistemology. It also describes the intuitionist philosophy of mathematics, which places human activity as central in the creation of mathematics, and which argues that its logical foundations are never complete [16]. Thus a key feature of this view of this position is that it emphasizes the human processes of knowledge getting in mathematics, as much as their product. Progressive absolutism can be identified as the philosophy of mathematics implicit in the ideology of progressive education [14]. From this perspective, the certainty of mathematical knowledge is not questioned, but the creative role of human activity in extending it is acknowledged. This is partly why this position emphasises the process and creative human aspects of school mathematics. (It also stems from the ideological model of childhood adopted by this position.) Fallibilism The fallibilist philosophy of mathematics is largely due to Lakatos [19, 20], but is also to be found in Davis and Hersh [9] and Tymoczko [32]. This is a social view of mathematics which
- 304. 294 for all its rigour sees it at base as fallible and corrigible, the ever-changing product of social human creative activity. The claim of fallibilism is that the concepts and propositions of mathematics, as well as the logic on which its system of proof rests, are tentative human creations, rather than objective absolutes, and remain eternally open to revision and change. Fallibilism points to the history of mathematics in which theoretical structures rise and fall, like waves in the sea. The foundations of mathematics, that which is taken to be its firmest or most basic part, varies from generation to generation, and has included such things as geometry, arithmetic, logic, set theory, and category theory, in tum. Given these three main groupings with regard to the philosophy of mathematics from the literature, there is the issue of their occurrence in teachers' of mathematics belief-systems. The claim I wish to make is: Thesis 1 Absolutism, Progressive Absolutism and Fallibilism are the three main philosophies of mathematics held by school and college teachers of mathematics, although they may be implicit and unarticulated philosophies. The Assimilation of the Problem Solving to the Teacher's Perspective In this section I put forward the the main thesis of this paper, which is reflected in its title. Thesis 2 Problem solving is assimilated to the teacher's mathematical perspective. In other words, what a teacher understands by problem solving in mathematics and in the teaching and learning of mathematics is largely a function of that teacher's personal philosophy of mathematics. Relating this to the three different philosophies of mathematics distinguished above (Thesis 1), we have the following specific theses. Thesis 2.1 A teacher with an absolutist view of mathematics will view problem solving as the carrying out of non-routine teacher imposed tasks with determinate right answers. Problem-solving is thus
- 305. 295 an activity which follows on from the transmission of mathematical content, and provides the means to apply previously learned knowledge and skills. Associated with this perspective one might expect to find the view that there is a single best method for solving any problem. In addition, problems may be viewed as the means of motivating students through challenge or 'relevance'. The teacher's primary role is that of transmitter and communicator of knowledge, and problems are a secondary means of applying, reinforcing and motivating learning. Given that the emphasis of this perspective is on objective truth in mathematics, and hence on the determinate answer to any problem, there is a tendency to identify problem solving with the object of inquiry, the problem itself. Thesis 2.2 A teacher with a progressive absolutist philosophy of mathematics will view problem-solving as the means to develop and employ the strategies and the processes of mathematics, and to uncover the truths and structures of mathematics. This perspective can be expected to emphasise the role of human i.e. learner activity in the uncovering of mathematical knowledge, as well as valuing the process aspects of mathematics for themselves. Thus carefully chosen and contrived environments, contexts and problems are given to the learners to experience and explore, and they are guided to solve the problems implicitly or explicitly contained in them. Knowledge is expected to emerge from the learner's experiences, and the teacher's role is that of manager and facilitator. Given that the emphasis of this perspective is on human activity in creating or at least uncovering new mathematical knowledge, there is a tendency to identify problem solving with the process. However, strategic compliance with an ethos which is not conducive to an inquiry based classroom approach may lead to problem-solving being treated as another area of content to be added on to the mathematics curriculum. This may, in practice, result in the identification of problem solving with the object of inquiry, the problems themselves. Thesis 2.3 A teacher with a fallibiJist view of mathematics will view problem solving as the appropriate pedagogy to employ in the classroom. In particular, it is seen as a socially mediated process of problem posing and solution construction, requiring discussion for the negotiation of meanings, strategies and proofs. In an appropriate environment, this perspective can be expected to emphasise the autonomy
- 306. 296 of the learner in choosing and posing problems for exploration and solution. These problems can be expected to arise from mathematical puzzles and relationships, from issues in the learners' cultural and social environment, as well as engagement with the kinds of tasks that will be used (usually externally imposed) for assessment and certification. The learners can be expected to question the choice and nature of classroom activity, pedagogy, classroom organisation and the teacher's authority. By challenging some of the authority structures and status quo in education, such an approach can be expected to be particularly vulnerable to social pressures and expectations. Thus there may be a tendency to adapt a problem solving rather than problem posing pedagogy, with learner control over methods, but not over content. However this can still admit substantial discussion and learner negotiation over meanings and answers. Thus strategic compliance may lead to problem posing being treated as classroom based inquiry, and to a problem-solving pedagogy. (Really powerful dissonance may lead to the type ofcompliance discussed under Thesis 2.2.) As a summary, Table 2 indicates the distinct theories of teaching and learning mathematics associated with the different philosophies of mathematics (following Theses 2.1 - 2.3). Table 2: Three ｃｯｮｴｲ｡ｳｴｩｮｾ＠ Teacher Mathematics-Related Belief Systems PlllLOSOPHY OF THEORY OF TEACHING THEORY OF LEARNING MATHEMATICS MATHEMATICS MATHEMATICS ABSOLUTIST TRANSMISSION UNDERSTANDING A fixed structured Clear explanation Acquisition of body of applicable Illustration meaning pure knowledge Motivation Application PROGRESSIVE-ABSOLUTIST LEARNER-CENTRED DISCOVERY An absolute body of knowledge Facilitation of exploration Active search for pattern uncovered by human activity Management of inquiry Creativity Process emphasis Experience FALLIBILIST NEGOTIATION EMPOWERMENT A social construction: Discussion Problem-posing challenged, changing Negotiation Discussion and reformulated Overt discussion Collaboration of teacher's role Development of Autonomy
- 307. 297 Empirical Evidence Supporting Thesis 2 A number of studies offer partial confirmation of Thesis 2 and its sub-theses. A growing number of these are referred to in the literature, including recent surveys of research on teacher beliefs in mathematics, such as those of Thompson [31] and Brown, Cooney and Jones [2]. The empirical studies quoted below are representative selections, chosen on the basis of their explicit treatment ofconceptions of mathematics and problem solving in the classroom. Lerman [21] describes how open-ended problem posing and solving work has been introduced into school mathematics in Britain as a consequence of the new GCSE examination for 16 year olds with coursework assessment of 'investigations'. He reports how the intentions of this innovation have been subverted by some teachers' view that there is a unique correct outcome to these tasks, betraying an underlying absolutist philosophy of mathematics. This supports Thesis 2.1. Marks [23] provides a detailed case study of a single high school mathematics teacher Sandy. Sandy views problem solving, by which he means non-routine inquiry into received problems, as central to mathematics. "It is not...Problem Solving, a topic to be squeezed in somewhere between Fractions and Quadratic Equations. It is problem solving with a small "p"...woven into the fabric of mathematics rather than stamped on top" (p. 2). He also states that it is central to the teaching of mathematics; high school algebra in his case. Sandy's classroom practice bears this out, with his frequent use of high-level questioning and his emphasis on heuristic methods. The case study gives every indication of that Sandy has a progressive absolutist conception of mathematics. Consequently, in view of his pedagogical beliefs and practices, it supports Thesis 2.2. A number of further studies offer evidence supporting both sub-theses 2.1 and 2.2. Thompson [31] provides a case study of three teachers which support these theses. In only two of the cases (Lynn and Kay) is problem-solving explicitly mentioned. Lynn has an absolutist conception of mathematics described in terms such as: "cut and dried", and "exact, predictable absolute and fixed" (p. 116). She sees classroom problem solving as "recall[ing] the appropriate method or procedure...correctly apply...and obtain the right answer." (p. 117) This fits perfectly with Thesis 2.1. In contrast Kay sees mathematics as certain but "continuously expanding...and undergoing changes to accommodate new developments" (page 113), a progressive absolutist view. Her conception of problem solving in the classroom andher observed practice was to "encourage the students to guess and conjecture...allow them to reason on their own rather than show them how to reach a solution or an answer" (p. 114).During teaching Kay made explicit reference to the heuristic processes of mathematics. All inall, she exemplifies Thesis 2.2 very well. Dougherty [11] studied II teachers of grades 4-6. In brief, ten of these were found to have some sort of absolutist conception of mathematics (emphasising mathematics as rules and
- 308. 298 procedures, as a tool, or as step by step methods}. Each of these teachers viewed problem solving as the more or less routine application of learned methods and procedures. Thus these ten teachers confirm Thesis 2.1. The remaining teacher in the sample viewed mathematics as "experiential and not a static body of knowledge" (p. l23), indicative of progressive absolutism. This teacher also viewed classroom problem solving as 'solving thinking problems', that is as a creative activity. Classroom observation also showed an appreciation of divergent student thinking, and students taking responsibility for contributing meaningful explanations and presentations. This teacher is confirmatory of Thesis 2.2. It is harder to find empirical evidence in favour of Thesis 2.3, because of the infrequent realisation of a problem-posing pedagogy in practice. Mellin-Olsen [24] implicitly endorses a fallibilist, socially based view of mathematics and gives convincing examples of a problem- posing mathematics curriculum as realised in Norway. This fits well with Thesis 2.3, but is a carefully argued academic case with practical illustrations, rather than an empirical case study. Elsewhere I have referred to the combination of a fallibilist philosophy of mathematics with a problem-posing pedagogy as indicative of a 'public educator' ideology, and illustrated its adherents, support and implementations, from the literature [14]. One example of classroom practice based on a fallibilist view of mathematics is the constructivist teaching experiments carried out by Paul Cobb, Terry Wood and Ema Yackel (see for example [5]). These experiments strategically complied with the expectations of the school context, such as teaching the normal Grade 2 curriculum. They adopted a small-group problem-solving pedagogical approach in which children discussed and contrasted their own solutions and methods, rather than relying on the authority of the teacher for arbitration. Thus this example is similar to the strategic compliance discussed under Thesis 2.3. I should close this section by admitting that having pieced together some confirmatory evidence, as I have done, does not constitute a scientific test of my theses. For I have not sought to falsify them. However, what I offer is a plausible but conjectured explanation, with some theoretical and empirical support. Conclusion The key point I have tried to make in this paper is that the terms 'problem' and 'problem- solving' vary in meaning according to the perspective of the speaker. In particular, I have offered a theory based on the assumption that the teacher's personal philosophy of mathematics is the major determinant of what the teacher means by problem solving in school mathematics. I would like to suggest this as an area of research that could be fruitful in the future. What do teachers (and student teachers) mean by problem solving? How do their meanings relate to the mathematics-related belief-systems, especially their philosophies of mathematics? If my
- 309. 299 conjectures and theories are borne out by further research, there are serious implications for problem solving in schools. Namely, that if we wish to promote a problem solving pedagogy in schools, it is not enough to gain sincere agreement from teachers and others. For the concepts and understandings underpinning our agreements will vary great!y, according to the perspective of the person involved. One teacher's agreement to implement problem solving may involve word problems, another may mean the occasional non-routine exercise, a third may carry some practical applications. None of these reflect a process-orientated classroom, let alone a problem- posing pedagogy. References 1. Benecerraf, P. & Putnam, H. (Eds): Philosophy of Mathematics: Selected Readings. Englewood Cliffs, NI: Prentice-Hall 1964 2. Brown, S.I., Cooney, T.I. and lones, D.A.: Research in Mathematics Teacher Education. In: Handbook of Research on Teacher Education (R. Houston, ed.). New York: Macmillan 1988 3. Brownell, W.A.: Problem Solving, The Psychology of Learning (41st NSSE Yearbook, Part II). Chicago: National Society for the Study of Education 1942 4. Clark, C.M. and Peterson, P.L.: Teachers' thought Processes. In: Handbook of Research on Teaching (3rd edition), (M. C. Wittrock, ed), pp. 255-296. New York: Macmillan 1986 5. Cobb, P.: A Year in the Life of a Second Grade Class: Cognitive Perspective, In: Proceedings of PME 11 Conference (J. C. Bergeron, N. Herscovics, and C. Kieran, eds), Vol. 3, pp. 201-207, Montreal, 1987 6. Cockcroft, W.H. (chair): Mathematics Counts. London: HMSO 1982 7. Confrey, I.: Conceptual Change Analysis: Implications for Mathematics and Curriculum, Curriculum Inquiry, 11 (5),243-257 (1981) 8. Cooney, T.I.: The Issue of Reform, Mathematics Teacher, 80, 352-363 (1988) 9. Davis, PJ. and Hersh, R.: The Mathematical Experience. Boston: Birkhauser 1980 10. Department of Education and Science: The Parliamentary Orders for Mathematics. London: Department of Education and Science 1989 11. Dougherty, B.I.: Influences of teacher cognitive/conceptual levels on problem-solving instruction. In: Proceedings of PME-14 (G. Booker, P. Cobb, and T. N. de Mcndicuti, eds), Vol. 1, pp. 119-126, Mexico, (1990) 12. Ernest, P.: The Problem-solving Approach to Mathematics Teaching, Teaching Mathematics and its Applications, 7 (2), 82-92 (1988) 13. Ernest, P.: The Knowledge, Beliefs and Attitudes of the Mathematics Teacher: A Model, Iournal of Education for Teaching, 15 (1),13-33 (1989) 14. Ernest, P.: The Philosophy of Mathematics Education, London: Falmer Press 1991 15. Goffree, F.: The Teacher and Curriculum Development, For the Learning of Mathematics, 5(2), 26-27 (1985) 16. Heyting, A.: Intuitionism: An Introduction. Amsterdam: North-Holland 1956 17. Krulik, S. and Reys, R.E. (Eds): Problem Solving in School Mathematics (1980 Yearbook). Reston, VA: National Council of Teachers of Mathematics 1980 18. Lacey, C.: The Socialization of Teachers. London: Methuen 1977 19. Lakatos, I.: Proofs and Refutations. Cambridge: Cambridge University Press 1976 20. Lakatos, I.: Mathematics, Science and Epistemology (Philosophical Papers Vol. 2). Cambridge: Cambridge University Press 1978
- 310. 300 21. Lennan, S.: Investigations: Where to Now? In: Mathematics Teaching: The Slate of the Art. (p. Ernest ed.), pp. 73-80. Basingstoke: Falmer Press 1989 22. Lester, F.K.: Research on Mathematical Problem Solving, In R. J. Shumway (Ed.), Research in Mathematics Education. Reston, Virginia: National Council ofTeachers of Mathematics, 286-323 (1980) 23. Marks, R.: Problem Solving with a Small up": A Teacher's View. Paper presented at AERA, Washington DC, April 1987 24. Mellin-Olsen, S.: The Politics of Mathematics Education. Dordrecht: Reidel 1987 25. National Council of Teachers of Mathematics: An Agenda for Action. Reston, Virginia: National Council ofTeachers of Mathematics 1980 26. National Council of Teachers of Mathematics: Curriculum and Evaluation Standards for School Mathematics. Reston, Virginia: National Council of Teachers of Mathematics 1989 27. Nilsson, N.: Problem Solving Methods in Artificial Intelligence. New York: McGraw-Hill 1971 28. Polys, G.: How to Solve it. Princeton, New Jersey: Princeton University Press 1945 29. Popper, K.R.: Objective Knowledge (Revised Edition). Oxford: Oxford University Press 1979 30. Thompson, A.G.: The Relationship Between Teachers Conceptions of Mathematics and Mathematics Teaching to Inslructional Practice, Educational Studies in Mathematics, IS, 105-127 (1984) 31. Thompson, A.G.: Teachers' Beliefs and Conceptions: A Synthesis of the Research. In The Handbook of Research on Mathematics Teaching and Learning 1990 32. Tymoczko, T. (Ed.): New Directions in the Philosophy of Mathematics. Boston: Birkhauser 1986 33. Yackel, E.: A Year in the Life of a Second Grade Class: A Small Group Perspective. In: Proceedings of PME II Conference (J. C. Bergeron, N. Herscovics, and C. Kieran, cds), Montreal: University of Montreal, Vol. 3,208-214 (1987)
- 311. Computer Spreadsheet and Investigative Activities: A Case Study of an Innovative Experience loao Pedro Ponte, Susana Carreira Departamento de ｅ､ｾＬ＠ Faculdade de Ciencias, Universidade de Lisboa, Campo Grande, Lisboa Abstract: This paper analyses an experience undertaken by a group of teachers, who introduced the computer in a 10th grade mathematics classroom for carrying out problem solving and investigational activities. The main question of interest is the discussion of significant issues and unexpected situations that emerged within this innovative process, specially regarding the reactions of the students. Using a case study methodology [1], the sources of data were interviews with the teachers, visits to the school, and the learning and dissemination materials produced. Keywords: computers in mathematics education, problem solving, investigations, teachers' beliefs, innovation, new information technologies The MINERVA Project - Node DEFCUL The experience described in this study was carried in one of the schools in the MINERVA Project. This national Project, in operation since 1985, was designed for the introduction of New Information Technologies (NIT) in primary, middle and secondary schools, both as tOpIc of interest in itself and related to the teaching of all school subjects. The Node DEFCUL is one of its groups, based at the Department of Education of the College of Sciences of the University of Lisbon. It carries out development, research and extensive teacher training activities. A detailed description of the work of this Pole of the Project was given elsewhere [2] and here we will just review some aspects of its educational approach, based in the principles of active learning and in a concern with broad cultural and educational changes. The concept of New Information Technologies (NIT) implies far reaching ideas than the mere use of electronic hardware. These technologies assume an important role in many social activities, bringing about significant structural changes in ways of working and thinking. They have influenced scientific research, economic planning, goods production, management, communication media, etc. The NIT mean the possibility of automatic information processing and communication. They represent new ways of looking at information and at the process of knowledge development and dissemination, and touch on important values. In practice, most of the work carried out in the schools has developed around the computer (with or without peripherals), but the emphasis is in the general concept of NIT.
- 312. 302 One major pedagogical idea about the educational use of computers is that they are essentially regarded as something for the students to work with. Of course, teachers need also to be able to deal with computers in many ways, in particular for demonstrations. But the most interesting and innovative aspects of the educational practices that can be expected to develop in schools may have origin on the intensive interactions between students and computers. The computer is also regarded as a working tool. There are other roles that it may fulfil, and they may be of significant educational value. However, it is through its use as a general or subject specific tool, for carrying out investigations and undertaking sophisticated projects, or to perform quite simple tasks, that the computer is expected to find an important place as a cultural artifact influencing human thinking and interpersonal relationships. In an information-driven society, it becomes necessary to be able to use with ease the computer and other equipment associated with NIT. New educational objectives need to be considered. The youngsters need to develop, since quite early, the ability of knowing where and how to locate information, how to select, interpret, process it and evaluate the results. It is also necessary to reflect about what is its role in social, economical, and cultural processes and what may be misuses and unethical procedures. Quite specially, it is important to be aware of its role in scientific thinking, project design, and more generally, in the development of knowledge. It should not be surprising, therefore, that project work, usually undertaken in groups, has been one of the most important activities within the schools. Students get an opportunity to establish working goals, set strategies and methodologies, collect data and analyse it, and get used to present their results and ideas to large audiences. In mathematics, a common activity is also carrying out investigations, in which students explore the possible relationships between concepts and try to get specific understandings or arrive at broad generalizations (for another example of this kind of activity, see [3]). The NIT, as a new subject, rise the issue of curricular articulation. It has been felt that instead of setting a new discipline apart to deal with these matters, they should inform deeply in the teaching of all existing school subjects, being integrated with them. These technologies, besides their curricular implications, are also, and more generally, an important factor of transformation of the school, yielding the setting of new objectives, new learning situations, new activities, and values. They point towards the reorganization of the spaces, working methods and teacher/student relationships. The NIT put new demands on the teachers. They need to know how to use these technologies in an effective and confident way. Teachers need to develop new attitudes and abilities, have access to specialised information, receive formal training, and have the possibility of informal interaction with other teachers with similar interests and concerns. The role of the teacher is changing in a number of aspects. Knowledge is becoming more and more of a dynamic nature, and the active involvement of the learner is critical in the process of its acquisition and development. Teachers have to become experts in supporting this kind of
- 313. 303 learning, fostered by the dissemination of information and the availability of sophisticated processing and communication technologies. The concepts of active involvement and recurrent reflection in the learning/training process is both relevant to students and teachers. In this Project teachers are not viewed as software developers, but as curriculum developers and project leaders, both at classroom and school level. This requires the ability to make appropriate use of existing software, both specific and of general purpose. It suggests the need of new approaches to preservice and inservice teacher training, getting them involved in working in their own projects, together in groups at school or local level, and in many instances in cooperation with higher education institutions such as the group that concerns us in this study. Context of the Experience The School. This experience took place in a relatively small secondary school located in one of the largest suburbs of Lisbon. Its students are from grades 7 to 11. The number of teachers is about 80. It is a quite recent school, which just opened 7 years ago. It is governed by a Directive Board offive elements who have been in charge since the creation of the school. The students came from the surroundings and their socioeconomic level is slightly below average, with a great diversity of social backgrounds. The students in this experience were all in the same 10th grade class. This school has attracted an increasing number of qualified mathematics teachers. At the present time, they constitute a very dynamic group who has taken several initiatives to promote a positive view of the discipline. In general, they have the support of the Directive Board. However, some of the teachers still complain about resistances and lack of pedagogical concerns in some of their colleagues. The school was involved in the MINERVA Project six years ago. This has been recognized as a decisive factor to rise new pedagogical concerns and attitudes among teachers. It also promoted an important setting to stimulate a bigger participation of students in project work within the school activities and provided stimulus to the up rising of other school projects. The teachers and their motivations. In the school year of 1989/90, the school had for the first time a group of four mathematics teachers who were doing preservice teaching to complete their graduation and certification requirements. We will refer to them as "practising teachers". Their work at the school was supervised by a senior teacher of the school staff. The four practising teachers were aged between 24 and 28 years old and had completed four years of academic studies at the University. Three of them were teaching for the first time and only one had some previous experience as a Religion and Moral teacher. As the supervisor teacher stated, the group had a very enjoyable way of working. They all liked to discuss their points of view and committed themselves to the activities they decided to
- 314. 304 carry out. They had very different personal characters and also distinct opinions on education matters, but they respected each others' ideas. The four practising teachers and the supervisor decided to get involved in a project of introducing calculators in this mathematics class. In a subsequent stage of the work computers were also included. As they reponed, the group was, from the first moment, interested in establishing a setting which could help to promote a different son of mathematical experience to the students. The calculator and then the computer were viewed as good carriers of the innovative strategies that they were looking for. The work of this group was supported by a training programme on spreadsheets and calculators in mathematics education. Within this programme they had several meetings with practical activities, discussions and exchanges of experiences with teachers of other schools and were visited several times by the project coordinator, often at their request [4]. The practising teachers felt that their lack of teaching experience was a good reason to be gladly open to new ideas and teaching methodologies. They were much concerned with the picture they conceived of the current situation in the teaching and learning of mathematics. They were also very enthusiastic about doing problem solving activities and enhancing students ability to investigate mathematical questions in rich content situations. They have taken these concerns in their university studies. especially in the previous year, in the mathematics methods course. Other important motivations and inspirations were received through their participation in a mathematics teachers national meeting (PROFMAT). The supervisor always supponed their initiatives and was ready to help them to go forward with their projects. She had the caution of making the group reflect upon their actions and understand the possible advantages and risks involved. but she never wanted to dictate their decisions. The practising teachers emphasized the need for many changes in mathematics education. They believed that the teachers' practices are too often a straight routine consisting in teacher presentation of mathematical concepts. followed by a series of standard exercises where students are supposed to acquire procedural behaviors intended to make them apply the previous information presented. With this kind of exhaustive practice, they found very plausible that students would be deprived of the real opportunity to understand the meaning of mathematical ideas and would possibly perform many calculations without figuring out the reasons for them. With this in mind and the conviction that plain recepies were not available to create a rich educational environment, the group of teachers embraced some pedagogical guidelines and started out the search for new challenges to themselves and their students. They wished to give students a less statical view of mathematics and to stress that many mathematical questions demand creative strategies and can be explored in many different ways. They began with problem solving activities which were supposed to encourage students' work and discussion in groups. This and the opportunities for students to engage in mathematical investigations created the general framework for the introduction of curricular topics.
- 315. 305 The first idea of using computers occurred in a discussion among teachers about the possible ways of treating some algebraic issues like equations and inequalities. In the context of their pedagogical concerns, the computer was seen as another ingredient to be added in a natural way. They trusted that it would be a learning facilitator and a powerful tool to shift the emphasis from the calculus to the interpretation of graphs and the translation between different representations of mathematical concepts. The teachers also alluded to the particular reluctance they felt about the usual way this subject is presented to the students, especially in their text books, because it leads to a heavy and boring use of computations. They thought that the computer could give a good contribution to avoid all that and so they decided to bring it into the classroom. The students. The students involved in the experience were all in one 10th grade class. This class was under the responsibility of the supervising teacher who shared it with the four practising teachers. Their work was organised in such a way that each of them was alternately the responsible for the class during a sequence of lessons. Nevertheless, for the majority of lessons, the teachers planned the activities together and were frequently all present in the classroom. The 10th grade class, which had 33 students, was viewed by the teachers as a quite special class. They were above average achievers in mathematics who had a great concern with their school marks in mathematics, having in mind their academic future, namely their access to university. Good results in tests and examinations were some of their utmost objectives. These students were used to work hard and were persistent. They always made their homework and some of them even gave the impression of having private lessons in extra school time. Resources and class organization. The computer activities were performed during class time and took place in the computer centre where six computers and two printers were available. As the number of students was quite large, they were divided in eleven groups of three. Six groups were working with the computers and five were doing parallel activities without them. On the subsequent lesson groups would interchange roles and as soon as they finished both kinds of tasks there would be a discussion with all the class to expose the most important conclusions and results. Occasionally one data-show was used to help this synthesis of students' findings with the computer. The software used throughout the experience was an electronic spreadsheet, the SUPERCALC 4. The teachers chose it having in mind their small experience with computers and the goals they pursued. They examined some other computer graphing utilities as alternative options and decided that the spreadsheet was quite easy to manipulate with a minimum of training. Furthermore, they judged it as an appropriate software to deal with algebraic expressions, calling for a decisive intervention of the students in organizing and creating tables and graphs. According to their reports, students were reasonably capable of interacting with the spreadsheet after one introductory lesson of fifty minutes.
- 316. 306 As the teachers described a typical lesson with computers, students were indicated their exact seats in the beginning of the class and had already on their tables a worksheet that would guide their investigations with and without the spreadsheet. They would start working in their groups and the teachers (usually the five of them) would circulate, monitoring and observing the students' progresses. They would answer students' questions, having always in mind never to give the solutions they were looking for. Instead, they would pay attention to them, making suggestions or pointing out possible clues, but sometimes they would just encourage students ta think a little more. After a sequence of classes with computers, there would be a summary lesson with the teachers leading the discussion. Students were asked to present written reports on their work and these were given to the teachers before the plenary lesson. Some of the issues brought to this large group discussion came from such students' reports. The computer activities. Computers were associated with three main topics of the national 10th grade curriculum: (a) domains of rational and algebraic expressions, (b) polynomial equations and inequalities, including second degree expressions, (c) algebraic expressions representing straight lines, including algebraic conditions for parallel and perpendicular lines. All of these topics came at a different time. The first was in November, taking about two weeks; the second was in February/March and lasted for a month; the third was in May/June and took two weeks. The teachers recognize the difficulties they had in creating good problem solving activities allowing the students the desired freedom in their own computer experimentations. They also admit their failure in the design of some of them and in their efforts to tum them into less directive formats. A brief analysis over some examples of the worksheets, may contribute for a clearer idea of their nature. Starting with the first topic, domains of rational and algebraic expressions, the activities were aimed to illustrate how the spreadsheet "detected the forbidden values". In fact, the spreadsheet was used to reveal error messages for some values of the variable. Students introduced formulas in some cells as translations of the expressions they were asked to analyse. Then, they freely assigned values to the variable and observed the results obtained with each formula. Nothing was said in the worksheet about ranges of values to consider and students had to make many trials. Some of them used one single cell to allocate the values of the variable and others conducted a more systematic procedure by means of creating a column of values for the variable. In this way they generated a table to have a better view of what was happening to the transformation of a set of values. The students did not feel the need to explore any possible graphs for their analysis. With this kind of activity, students became aware of the meaning of error messages in such cases as the assignment of zero to a denominator or negative values under a square root. The students began to think in terms of finding the values that did not belong to the domain and, from there, they were able to get the desired domain. The second set of activities concerned solving quadratic equations and inequalities and the teachers found in it a good starting point to study the parabola in its several aspects: zeros,
- 317. 307 extrema, monotony in intervals, upwards and downwards concavities, etc. All this was sustained by the analysis of graphs created with the spreadsheet. The first approach to the understanding of different expressions for different parabolas was based on a worksheet where students had six graphs selected by the teachers to represent typical examples. They expected to have covered all the possible cases with those few typified examples and hoped that students would be capable of making correct generalizations from that. Instead, students were driven to false generalizations due to unforeseen limitations of the material provided. They concentrated their attention on certain details of the parameters and related them to some irrelevant aspects of the parabolas, therefore jumping to conclusions that were definitely uncorrect. The teachers next expedient was to create a simple spreadsheet file so that students only needed to introduce and modify the values to assign to each parameter of a quadratic expression in order to visualize the effects produced in the consequent graphs. This was seen as a much better strategy to make students understand the relation between the coefficients in the expression and the type of parabola generated. Later on, students also learnt to solve quadratic inequalities in a non standard way, that is, making the immediate sketch of the appropriate parabola. Other activities included in this sequence of lessons were of a different nature and involved a problem solving task. Students had to discover the right value for the length of an edge of the square face of a box so that the box had its maximum volume. This problem and a similar one were solved in the spreadsheet and were a motivation to study polynomial inequalities of third degree, using graphs as a resource. Finally, for the third topic, the activities were, once again, devoted to guide students in investigative processes. On the study of straight lines from an algebraic point of view, students used a spreadsheet file previously prepared by the teachers. They looked for the effects of changing the parameters in a linear expression over the position of the straight lines shown in the graphs. Students had to find the conditions for parallel and perpendicular straight lines, as well as to relate the coefficients with their slope and points of intersection with the axes. As teachers pointed out, computer activities were intended to develop students' awareness of graphical information and to support new strategies for the solution of algebraic problems traditionally treated in formal ways requiring paper and pencil computations. Students' reactions. The reactions of the students to the introduction of computers in the classroom were not uniform. There were clearly two major kinds of attitudes expressed by them. Some accepted the idea very well, apparently enjoyed their work and even found, in dealing with computers, ways of showing hidden capacities of intuition, discussion of ideas, conjecturing and reasoning. As the teachers pointed out, those were the students who had already manifested in class a flexibility to deal with more open questions. Moreover, those students were not the most brilliant ones that usually got everything correct in their written tests. On the other hand, many other students were very suspicious about the computer activities suggested by the teachers. In the beginning, they showed a cooperative attitude, having in mind
- 318. 308 to please and even to help the practising teachers, assuming that this experience was a constraint of their own training program. But after some time, they started manifesting a clear rejection of the work being developed with computers, claiming that they needed a different type of teaching, a more serious one, in order to be prepared for future examinations. They wanted more practical exercises, more teacher exposition and more individual work in the classroom. Some of the students demonstrated they were not prepared for achieving their goals in the tasks involving computers. Their failure brought out some revolted feelings from the moment they realized they were having troubles in keeping up with their standards of academic achievement in mathematics. Things did not go very smoothly in terms of evaluation either. Indeed, teachers explained a certain uneasiness in conciliating the kind of performance students revealed in their work on computers with the difficulties they showed later on, when asked to elaborate on some questions in a more consistent and rigorous form, without the computers. They also mentioned the confusing way of students expressed themselves in a written test situation. Although a few students got very interested in going beyond the objectives of some activities - as it happened with the domains of algebraic expressions when a group of students decided to investigate the sense of dividing by zero and shared their discoveries with the class - the students' reports on their work were very disappointing. They turned out to be a strict list of their answers to the questions from the worksheets. The supervisor teacher who is this year teaching the same class (now on 11th grade) showed her sorrow about the fact that students are not interested in taking again the possibility of working with computers in mathematics. She assured that students seem to be more cooperative and more likely to question their ideas but still persist on asking for detailed explanations from the teacher and valuing their academic results above everything. In a word, they continue to be quite reluctant to accept deep changes in their structured way of learning mathematics. Teachers' reactions. For the teachers, and overall, this was a very positive and rewarding experience. They enjoyed working together, raising questions about current content and methods of teaching mathematics. They became very sensitive about the power of graphical representations for mathematical concepts, namely for functional ideas. The four practising teachers discovered the computer as an educational tool and became much more articulate about their sought goals and more aware of possible pitfalls in their approaches. The teachers constituted a group with an excellent rapport. They had lively discussions in which they planed work and produced proposals and materials. Their personal and working relationship with the supervisor was also very good. As the experience went along, the teachers felt the need to explain to the students what was the purpose and the meaning of the proposed activities, in terms of their mathematical experience, but they found difficulties in having their messages accepted. The teachers commented on the huge effort they had to do to make students communicate and share their ideas, and in the end they expressed some doubts about having succeeded on it.
- 319. 309 However, they were confident that some positive changes were induced in their students and pointed that these are hard phenomena to detect and need time to be revealed. They were concerned with the poor nature of the reports produced by the students. They explained it by the insufficient orientation and lack of explicitness they provided for their organization as it was an activity that students were not used to produce in mathematics. In spite of the obstacles they found, the practising teachers felt that the experience changed their own way of seeing mathematics and became even more convinced that experimentation is an important feature in the process of doing mathematics. Questions This innovative process rises a number of issues that certainly deserve an in-depth discussion. It is impossible to state with absolute confidence what went right and what went wrong and we will not assume the role ofjudging the teachers. Rather, in our approach, we will try to present what appears to be the ways in which both the teachers and ourselves try to make sense out of what happened in this experience. Different perspectives between teachers and students. It is quite apparent that different perspectives about the value of the computer-based activities developed in the teachers and in the students during this activity. As the experience proceeded the teachers got more and more excited about the computer and discovered many of its possibilities for the exploration of mathematical concepts and supporting problem solving activities. The students, or at least many of them, increasingly found the computers an embarrassment, a device that, instead of promoting, hindered "real learning". As the teachers perceived that the reactions of the students become not very favorable to the proposed activities, they developed possible explanations for this phenomenon. It resulted, in their opinion, of a multitude of factors, all of them stressing the special nature of this 10th grade class: -The students' former learning habits and assumed roles in the discipline of mathematics; this let them to feel uncomfortable with classroom work that required a lot of discussion; -Their lack of familiarity with this kind of activities; it is something very different of what they have in mathematics classes and in their home study as they get prepared for the tests; -Their strong competitive spirit; the students were used to compete against each other for the highest marks in the tests and courses and not to cooperate (this competitive spirit has originated discipline problems with other teachers); -Their great concern in getting good grades and the fear that this kind of activities made their performance decrease; as university entrance partially depends on these grades this could threaten some of their personal expectations; -Their need for feeling of success in their work; the students appeared to need clear feedback mechanisms on the success of their work and felt uncomfortable as these mechanisms were not clear or pointed towards increasing failures;
- 320. 310 -Their cognitive styles, which could be classified in two different kinds - the analytical and the intuitive; the analytical students were used to work with fonnalised mathematics and had the greatest difficulties in the proposed tasks; the intuitive adjusted themselves better to the computer work, although they still had difficulties in integrating this learning in more formalised tasks; -The special arrangement for this class, with 4 practising teachers, plus the supervisor teacher, sharing responsibilities, teaching in a rotative way a sequence of lessons, created an ambiance in which students lacked reference points. It should be noted, however, that all these reasons pointed to preexisting factors, and do not include things that were done in this experience. Nature of the proposed activities. The kind of activities that were proposed to the students may have promoted in unexpected ways their negative reactions. The teachers themselves felt uneasy with some of them and considered that they should be improved. Some aspects in which this might be done include for example (and the teachers agree with these comments): -A broader exploration in the case of domains of expressions; the activity carried was quite narrow, following a strictly algebraic perspective (with no functional or graphical ideas), well in the spirit of the current Portuguese syllabus; -A more gradual exploration of the second degree polynomial; the activity as carried begun with a complex task which was quite difficult to handle; the students were required to make some generalizations from this situation, but they did the "wrong" generalizations, which were false, and there was no clear way for them to proceed correcting their conjectures; However, there is something more difficult to describe about the tasks that might also be of importance. The proposed activities do not provide the sort of "closure" that most teachers feel necessary to give confidence to the learner. After doing them the students were left with some feeling of dispersion, of lack of structure, which may have contributed to their anxiety in a very significant way. What Else? Some other factors, besides the current attitudes of the students and the design of the activities may have contributed to the verified mismatch between the views of teachers and students. They have to do with the pedagogical relation and the classroom climate. We mention very tentatively: -The way teachers decided to respond to the questions of the students; to require the students to struggle with the things that they do not understand may appear to be a good instructional practice to arrive at but not be a good strategy to start with - it may inhibit the establishment of an affective relationship between teachers and students without which the students do not respond effectively to the teachers' demands; -The lack of a clear attempt to integrate the students as responsible partners in this experience;
- 321. 311 -The organization of the classroom in two main groups, one on and the other off the computer, taking turns, which has worked well in previous experiences with younger students, may have promoted a pace of work that is not suitable for students at this grade level and with these expectations; -The fact that no way was found to overcome the special situation of this class (the four practising teachers plus the advisory teacher), providing strong reference points to the students; -Eventually there was an insufficient conceptualization of the experience: What are our objectives? Why do we want the computer? What should be its role? What is the role of the students? What shall we do to let them assume it? -The lack of concern with possible limitations or undesirable features of the software; the spreadsheet - as any other general purpose program - has limitations as well as potentialities and it is necessary to be aware of them to create the appropriate educational activities and learning environment. In fact much as been said about the potentialities of the spreadsheet. Let us mention just a few limitations: (a) there is only one way for reasoning to proceed: from a formula (or given values) to a table to a graph; (b) "misbehaved" points may exist in the domain of a function and the spreadsheet does not indicate them (specially if they do not appear on the constructed table); and (c) non required lines may appear on the graphs such as pseudo-asymptotes and other connecting lines. What did the teachers learn in this process? For the teachers this was a strong and positive experience that they enjoyed. It will certainly leave a mark for many years in their professional styles. However, the teachers were aware that their experience had several limitations. They were not sure how to regard the difficulties of the students, and specially how to overcome them. They and we can draw several lessons from what happened (hopefully these will not appear as contradictory): -Innovation is not always easy to implement; sometimes things develop in unexpected ways; -Innovations may result or not result; there is not a special axiom saying that all the innovations are necessarily successful as we fIrst conceived them; -Innovations are not valuable just by themselves; their value depends on the measure they contribute towards students' learning and growth; -The fact that things become difficult is not a reason for giving up; it is a reason to let us think on the sources of problems and on how to overcome them; -Group dynamics is a factor of creativity and personal confIdence in carrying out an innovation project; it brings a completely different experience from the teacher working in isolation; -The spreadsheet, with all its limitations, can be indeed a very useful tool for mathematics teaching and to promote a different approach to many mathematics topics; -There are many ways of working in mathematics and of what means to learn mathematics;
- 322. 312 -The activities usually made in the classroom tend to leave out some students which do not adapt themselves easily to formalised mathematics but can have success with different approaches; -Also, students with facility in formalised mathematics may have great difficulties in investigational tasks or in applying mathematics to real world situations; -Conceptions and deeply established habits in the students may pose quite difficult problems to innovation processes; That these lessons were taken seriously is best shown by the fact that this year one of the practising teachers, now in a different school, is doing a similar experience, but with improved activities. The supervisor and another teacher are working on innovative activities for 9th grade geometry. Conclusions The students started as being cooperative and sympathetic towards the experience. They gradually changed their attitudes as they felt threatened in their expectations and securities. These attitudes and the factors that underline them should be viewed as a given that needs to be taken into account in an innovation process instead of an explanation for its difficulties. We will end with three further conclusions: -In innovation processes the desire of innovate and the willingness to make mistakes and run risks are not sufficient; it is necessary to complement them with hard work of conceptualization ofobjectives and strategies and reflection on the outcomes; -New and promising ideas such as the computer, problem solving, investigational tasks and other pedagogical proposals are always multi-sided and may carry with them unexpected issues to which attention should be paid; -The pedagogical relationship which is established between teachers and students is always a key factor in the learning process, and is a particularly critical condition for the development of an innovation. References 1. Merriam, S.: Case study research in education: A qualitative approach. San Francisco, CA: Jossey-Bass 1988 2. Ponte, J.: Computers as biggers for educational change. International Conference on Teacher Training, Research and the Role ofUniversities, ViIamoura, Portugal 1991 3. Ponte, J., & Matos, J.: Cognitive processes and social interactions in mathematical investigations (In this volume), 1991 4. Veloso, M. G.: Novas tecnologias de informal;iio: Um programa de formalrAo de professores de Matematica, unpublished master thesis, Universidade de Lisboa 1991
- 323. Examining Effects of Heuristic Processes on the Problem-Solving Education of Preservice Mathematics Teachers Domingos Fernandesl Instituto de ｉｮｯｶ｡ｾｯ＠ Educacional, Trav. Terms de Sant'Ana 15, 1260 Lisboa, Portugal Abstract: This eight-week study analyzed effects of two heuristic models of instruction in mathematical problem solving on preservice teachers' problem-solving performance, on their awareness of the problem-solving strategies they employ, and on their perceptions about specific problem-solving issues. Both models taught four problem-solving strategies and employed P6lya's four-step model of problem-solving; each subject solved or saw the solutions to the same 24 experimenter-selected process problems. Study findings suggest that both models of instruction significantly improved preservice teachers' problem-solving performance; the explicit model appeared to be more effective in promoting organization of problem solutions and description of problem-solving procedures. Study results and conclusions yielded recommendations on the use of the focused holistic scoring, on the study design, and on teacher education in mathematical problem solving. Keywords: mathematical problem solving, problem-solving performance, problem-solving strategies, problem-solving behavior, teacher awareness, teacher perceptions, heuristic models of instruction, focused holistic scoring It has been suggested that preservice teachers should be exposed to formal instruction in mathematical problem solving. This recommendation has been made based upon the assumption that if teachers are to teach problem solving then they must be exposed to formal instruction which develops their abilities as problem solvers, their knowledge about problem solving, and their skills to plan, implement, and evaluate problem-solving activities. Mathematics educators have proposed models to meet these goals, and recommendations have been made concerning the need to investigate the efficiency and effectiveness of those models. Most research efforts on mathematical problem solving have focused at the precollege level and have sought answers to questions such as (a) What kind of strategies do students use while they are engaged in problem solving? [e.g., 12, 30]; (b) What methods of instruction are appropriate to mathematical problem solving? [e.g., 4, 29, 23, 30]; c) What is the influence of the processes children use on their problem-solving performance? [e.g., 12, 29]. These research efforts identified strategies and processes used in problem solving as well as IOn leave from Escola Superior de ｅ､ｵ｣｡ｾｯ＠ de Viana do Castelo.
- 324. 314 instructional procedures that seem to be effective in teaching problem solving to precollege students. As a consequence of demands to implement problem-solving programs in precollege mathematics instruction, mathematics educators and professional associations have proposed programs with goals of better teacher preparation and greater instructional effectiveness in promoting problem solving in the classroom [4, 7, 10, 11, 14, 15,24,26]. However, there is no research evidence in direct support of conjectures that the use of instructional approaches which implicitly or explicitly emphasize problem solving will contribute to the improvement of (a) teacher's performance in problem solving; (b) teachers' awareness about the problem- solving skills that they employ; or (c) teachers' effective use of models of instruction in problem solving. Teacher education in mathematical problem solving is a natural consequence of the problem- solving emphasis that has been recommended. The vision for the improvement of the precollege mathematics curriculum set forth by the National Council ofTeachers of Mathematics (NCTM) is based upon a teaching method which models and encourages problem-solving behavior. In fact. NCTM's An Agenda for Action [25], Guidelines for the Preparation of Teachers of Mathematics [26], and its Curriculum and Evaluation Standards for School Mathematics [27] specify that (a) teacher training programs must be re-designed to meet the problem-solving emphasis; (b) teachers must be competent problem solvers; and (c) teachers must be able to teach problem-solving techniques. These goals have widespread support within the mathematics education community. A practical result of this support has been the development and implementation of models aimed at educating teachers in mathematical problem solving [4, 11, 14, 15, 16]. The present investigation was based on the premise that teachers, if they are to become teachers of problem solving, must be exposed to instruction which is directed toward improving their abilities as problem solvers, their knowledge about problem solving, and their skills to plan, implement, and assess problem-solving activities (R.I. Charles, personal communication, October 25, 1986). The major goal of this study was to examine the effects of two heuristic models of instruction in mathematical problem solving on preservice teachers' problem-solving performance, on their awareness of the problem-solving strategies they employ, and on their perceptions about specific problem-solving issues Method SUbjects. The subjects were 68 preservice teachers enrolled in two classes of a three- semester hour Mathematics Methods Course. More than 75 percent were in the 20-22 year old age span and had no formal teaching experience. About halfof them claimed that they had taken four or fewer mathematics courses in high school whereas the remaining half had taken five or more. About 65 percent took four or fewer mathematics courses at the college level. All study subjects were females.
- 325. 315 Treatments. Subjects in each class were randomly assigned to one of two treatments. Two Treatment 1 subgroups and two Treatment 2 subgroups were obtained. Treatment 1 subjects were exposed to an implicit method of instruction in problem solving which made organized use of specific problem-solving strategies to solve problems but did not overtly identify or reflect upon the selection or application of those strategies. Treatment 2 subjects were exposed to an explicit model of instruction which named, discussed, purposefully applied, and reflected upon the organized use of specific problem-solving strategies to solve problems. Both treatments taught four problem-solving strategies (Check Some Guesses, Make an Organized List or Table, Find and Test a Pattern, and Make and/or Use a Drawing or Other Model) and employed P6lya's four-step model of problem solving; each subject solved or saw the solutions to the same 24 experimenter-selected process problems.(A sample of the problems is displayed in Appendix 1.) Procedure. Both experimental treatments were implemented in one 6O-minute laboratory session per week during a period of eight weeks. Subjects were assigned at random to small groups of three or four. The groups' composition remained constant throughout the treatment sessions. For each problem-solving strategy, six related problems were solved during two 6O-minute sessions. In the first of the two sessions the instructor demonstrated the solution to an introductory problem which elicited utilization of the targeted problem-solving strategy. This problem was solved by following P6lya's four phases of understanding the problem, devising a plan, implementing the plan, and looking back [28]. Afterwards, subjects working in small groups solved a similar problem. Two homework problems were distributed at the end of the first session; one was similar to the one solved in class, and the other was dissimilar but could be solved using the same problem-solving strategy. In the second session, the same small groups of subjects discussed their solutions to the homework problems until a consensus was reached. Instructor-selected students then presented and discussed their groups' solutions to the problems. Two more problems were distributed and solved. Their discussion was conducted at the chalkboard by instructor-selected students. Instruments. In order to assess the effects of the two treatments, three investigator- developed instruments were used. The Problem-Solving Test (PST), a paper-and pencil instrument, was used to gather pretest and posttest data on subjects' problem-solving performance and on their awareness of the strategies they used. The Problem-Solving Observation Guide (PSOG) was utilized to conduct 48 systematic observations of subjects' small-group problem-solving activities in each treatment group; these 96 observations focused on nine selected problem-solving behaviors. The PSOG also served to collect anecdotal data on students' problem-solving activities. The Strategies Interview Protocol (SIP) structured the collection of data on three issues: 1) subjects' perceptions of their ability to solve process problems; 2) subjects' professed willingness to teach problem solving; and 3) subjects' ability to identify problem-solving skills that should be taught to precollege students. The SIP also permitted validation of subjects' self-reported uses of problem-solving strategies on the PST. SIP data were gathered through interviews with random samples of eighteen treatment 1
- 326. 316 subjects and sixteen treatment 2 subjects; interviews were conducted immediately following completion of the posttesl The data were analyzed quantitatively and qualitatively. Analysis ofcovariance, a two-tailed i-test for independent samples, and chi-square tests were used to test three null hypotheses. Qualitative analysis was applied to the data gathered with the SIP and the PSOG. Results Four questions were addressed by the study. Responses to the first three employed quantitative data; responses to the fourth made use ofqualitative data. Quantitative data were gathered from the subjects' solutions to the eight-item pretest and posttest measures of problem-solving performance, from validation of the subjects' self-reported uses of strategies on the PST, and from the observation of subjects' small-group problem-solving behavior. Qualitative data were collected in the form of subjects' responses to three open-ended questions on the SIP and in the form of observers' comments recorded on the PSOG during in-class observations. Quantitative Findings The first research question dealt with preservice teachers' problem-solving performance as measured by the PST. Analysis of covariance, using the pretest score as the covariate, was performed to ascertain if there was a difference between the posttest means of the two treatments. It was found that there was no significant difference between the two experimental groups on problem-solving performance (p > .05). The second research question was concerned with preservice teachers' awareness of the problem-solving strategies they employed. A two-tailed t-test for independent samples was used to investigate if there was a difference between the two groups in the number of validated self-reported uses of problem-solving strategies. Test results demonstrated that there were no significant differences between the two experimental groups concerning that issue (p > .05). The third research question focused on preservice teachers' participation in small-group problem-solving activities and asked if that participation was independent of the treatment (i.e., instructional model) used to teach problem solving. Chi-square tests of independence for eight of the nine behaviors did not permit rejection of the hypothesis that behavior was independent of the experimental treatment (p > .05). Results indicated that the behavior Names a Strategy to Be Used or Being Used was not independent of the experimental treatment (p < .05); that behavior was exhibited more frequently by treatment 2 subjects. Qualitative Findings The fourth research question was concerned with preservice teachers' perceptions of their ability to solve problems, their professed willingness to teach problem solving, and their ability
- 327. 317 to identify problem-solving skills that should be taught to precollege students. The findings are summarized as follows. Perceived Ability to Solve Problems. Most of the subjects in either group saw themselves as good problem solvers. Actually, 25 of the 34 subjects who were interviewed rated their ability as six or above in a IO-point scale. These subjects asserted that they had learned how to read and to understand a problem and/or that they had gained confidence in solving problems. Eighteen of the 25 subjects commented that their ability to solve problems was directly attributable to the instruction in problem solving that they have received during the study. Of the nine subjects who rated their ability in solving problems as five and below, six claimed that they did not like mathematics, that they have never been good in the subject, or that they had lack ofconfidence in doing mathematics and in solving problems. Subjects in both groups seemed to relate their ability to solve problems to the time they had to solve the problems. Most reported that they could do well if they were given enough time to work on the problems. Professed Willingness to Teach Problem Solving. Twenty-seven preservice teachers rated their willingness to teach problem solving as six or above on the to-point scale. Eighteen indicated that problem solving should be taught because it develops students' thinking skills. The other nine reported that they were willing to teach problem solving because they now felt confident in their own ability to solve mathematics problems. All subjects with ratings of five or below asserted that they were not confident that they could teach problem solving. Ability to Identify Problem-Solving Skills. Most of the subjects who were exposed to the explicit model of instruction referred to the four steps of P6lya's model as important to teach to precollege students; no subjects who were exposed to the implicit model referred to all steps of that model. All subjects in treatment 2 and thirteen of the subjects in treatment 1 indicated that problem-solving strategies of the kind presented during the study should be taught to precollege students. Seven treatment 2 subjects and three treatment 1 subjects referred to metacognitive skills and attitudes as important skills to be taught to precollege students. Discussion Problem.Solving Performance While problem-solving performance, as measured by the PST, improved under both treatments, neither treatment was more effective. An instructional model which simply demonstrates the use of appropriate and effective problem-solving procedures and then engages small groups of students in problem-solving activities is as productive as an instructional model which explicitly names, discusses, and purposefully applies problem-solving procedures as well as prompts students use of those procedures during small-group problem-solving activities. The finding that preservice teachers' problem-solving performance improved under both instructional models is consistent with the findings of research on the teaching of problem
- 328. 318 solving to precollege students [17,30]. It supports assertions that, despite current limitations on knowledge of how problem solving is learned, it is possible to design and implement instruction that has positive effects on students' problem-solving perfonnance [18, 19,20,21]. Study conditions did not pennit a design which included a control group in which subjects would be exposed to the problems of treatments I and 2 but would not be given implicit or explicit instruction in problem-solving procedures. This model of instruction is titled Osmosis by Kilpatrick [13] Thus, while the design supports the conclusion that treatments 1 and 2 are equally effective, it does not address the significance of the pretest-posttest gains of either group. Did the subjects show significant improvement in their problem-solving perfonnance as measured by the PST? If so, can that improvement be attributed to the experimental treatments (i.e., to instruction in problem solving)? The pretest-posttest gains of the subjects, separated by treatment, are shown in Table 1. Two-tailed l-tests for paired measures show that these gains are statistically significant (p < .0005) for treatment I, treatment 2, and the combined group of 68 students. Since all pretest to posttest difference scores were nonnegative, it is reasonable to infer that the preservice teachers' problem-solving performance did improve during the eight-week period. While the absence of a control group makes it impossible to attribute the gains recorded in this study to instruction, studies which have included such a control group suggest that growth in problem-solving performance requires direct instruction [30,31]. Thus, there is reason to believe that the large gains in problem-solving performance observed in this study are a consequence of instruction. Based upon the t-test results summarized in Table I, it is concluded that preservice teachers can learn to solve process problems in mathematics. It is further suggested that such teachers' problem-solving perfonnance can be improved substantially and equally well by instruction which employs either an implicit or explicit model of mathematical problem solving. The data gathered call for the note that success in applying problem-solving procedures seems to begreater when a new problem is similar to the problems employed during instruction (Le., is a near-transfer problem); perfonnance on far-transfer problems is less marked. Subjects' explanatory work on the PST problems revealed parallel growth in the number of problems solved and in the evidence that problem-solving strategies were used. Analysis of treatment 1 subjects' work on corresponding items of the pretest and posttest illustrated that overall organization and clarity of the explanatory work improved greatly from pretest to Table 1. Paired t-Test Summary of Problem-Solvin& Test Gains by Treatment Groups and All Subjects Pretest Posttest Group M* SD M* SD D <f t Treatment 1 10.889 5.476 17.556 6.801 6.667 35 9.77- Treatment 2 10.719 5.299 18.375 5.912 7.656 31 10.97- All Subjects 10.809 5.354 17.941 6.364 7.132 67 14.61- *Maximum possible value=32; ..p<.OOO5
- 329. 319 posttest. Moreover, the organization and clarity of the work of treatment 2 subjects was generally superior to that of treatment 1 subjects. Generally speaking, use of the language of problem solving and systematic application of P6lya's four-step model differentiated treatment 1 explanations from treatment 2 explanations. It is concluded that explicit naming of strategies and explicit reference to problem-solving procedures does improve the problem-solving organization and explanatory power of preservice teachers. Awareness and Utilization of Problem.Solving Strategies Validation of subjects' self-reported uses of strategies in solving the problems of the posttest led to the finding that preservice teachers exposed to either model of instruction were aware of the four basic problem-solving strategies taught. The validation process also revealed that the subjects frequently failed to indicate strategies that they had used. During the validation phase (Le., administration of the SIP after the problem-solving posttest), subjects could justify almost every strategy marked during the testing. Hence, the validation process found that subjects did not mark strategies that they did not use. However, several subjects responded to the SIP question "Did you use any other strategy on this problem that you would like to check now?" by indicating strategies that they had overlooked while taking the test. Some had used a model (diagram or sketch) but said that they were not certain that what they had used could be called a 'model'; others had made an organized list of guesses but questioned whether it could be called a list or table since it did not have headings. Such uncertainties in identifying strategies were frequent among treatment 1 students; lack of a common language of problem solving appeared to be one source of their failure to indicate all of the strategies used on the test. A second probable source was the subjects' test-taking priorities. Since the problem-solving test had a time limit, the subjects' attention was focused upon solving problems; reflection on the problem-solving process during testing received less attention. Posttest interviews are, then, necessary to the accurate assessment of problem-solving strategies use during a test. Subjects' written work on the posttest and their verification of strategy use during the SIP interviews showed that they purposefully applied problem-solving strategies during the test. This behavior was typical of problem solvers identified as "expert" by Shoenfeld [32]. Confidence in the efficacy of the strategies Table and Pattern are evident in the written work of posttest papers. Whereas students abandoned such strategies after testing three or fewer cases on the pretest, perseverance in testing seven or more cases produced solutions on the posttest. Subjects' response to the SIP request to list several specific things to be taught to a student in Grades 1-9 during a unit on problem solving provided another measure of their awareness of the strategies and processes of problem solving. Although neither treatment included explicit discussion of the problem-solving skills that should be taught to precollege students, subjects in both treatments were able to identify those skills. Such identification indicates not only an awareness of the skills but also a belief that they are understandable to elementary and middle
- 330. 320 school students. Overall, responses of treatment 2 subjects gave evidence of a more organized view of the problem-solving process. The problem-solving vocabulary of treatment 2 students may be at the foundation of that apparently greater organization, since treatment 2 students frequently referred to P6lya's four-step model and used it to structure their solutions to problems; treatment 1 students referred only to the step which calls for understanding the problem. The findings of this study that preservice teachers are able to learn to apply problem-solving strategies to solve process problems is consistent with results obtained in studies of precollege students [e.g., 17J and college students [e.g., 22J. This study also found that preservice teachers taught by either model of instruction are aware of the strategies that they employ, but that teachers having access to the language provided by the explicit model of instruction seem to identify more strategies than do teachers exposed to the implicit model. Small.Group Problem·Solving Behavior Generally speaking, the nine behaviors defining small-group problem-solving participation were independent of treatment. Only the behavior Names a Strategy to Be Used or Being Used appeared to be treatment related; its more frequent exhibition by treatment 2 subjects probably is a result of their introduction to the use of a common language of problem solving. Each of three behaviors (Seeks to Clarify the Problem, Selects an Appropriate Solution Strategy, and Participates in Executing a Plan) were displayed in from 71 to 90 percent of observations of members of either treatment group. This indicates that most of the subjects observed sought to understand the problem before attempting a solution, to select appropriate problem-solving strategies, and to participate in the small-group solution of the problems. Problem solving was a shared activity; it was not a task "assigned" to a few able subjects within the total treatment group. Anecdotal data from the PSOG notes many situations in which small-group members tutored each other. Another common PSOG observation was that students engaged in discussions of the phases of the problem-solving process or in discussions of a specific strategy. This suggests that small-group problem-solving activities extended and reinforced whole-class instruction by providing a forum for student dissection and elaboration of problem- solving procedures. These discussions also appeared to foster the development of language with which to communicate problem-solving procedures. Moreover, subjects' participation in the small-group sessions was relaxed and continuous; anxiety related to mathematical problem solving was reduced or eliminated during those sessions. Observations of subjects' performances in the small-group problem-solving activities of this study support the recommendations for their use at both the precollege and college levels [1, 34J. Small groups also appeared to encourage the metacognitive activity recommended by
- 331. 321 Shoenfeld [33] even when metacognition is not taught explicitly. Furthennore, the results of the use of small-group activities in this study support recommendations to employ instructional models which focus on the problem-solving process [3] and help students to recognize that problem solving requires time and perseverance [8]. Teachers' Perceptions: Problem.Solving Ability and Willingness to Teach Problem Solving Analysis of data from the SIP led to the conclusion that the majority of preservice teachers exposed to either treatment perceived themselves as able to solve process problems of the kind presented in the experimental sessions. Subjects' experiences in solving the problems presented during the study is one possible source of their volunteered profession of an increase in confidence. A second source may be found in the fact that both treatments sought to maintain an open and relaxed atmosphere during the problem-solving sessions. These results support the recommendation of Stacey and Southwell [34] that students should be put at ease and that teachers should acknowledge that problem solving is a time-consuming activity. Based upon data from the SIP, it was concluded that subjects in either treatment were of the opinion that it was very important to teach problem solving to precollege students; they also professed a willingness to do such teaching. Related comments recorded on the SIP and PSOG suggest that the subjects' willingness to teach problem solving is linked to their perceived ability to solve problems. During early problem-solving sessions of the study, subjects expressed doubt that the kind of process problems by which they were confronted could be presented to their students. This position was supported by the subjects' observations that they experienced difficulties in solving the problems. However, this assessment reversed as the subjects successfully solved problems and became more confident of their problem-solving ability. This supports an assumption common to models for educating teachers in mathematical problem solving: teachers must be taught to be problem solvers if they are to be expected to teach problem solving to precollege students. Summary The preceding discussion sought to link the main findings of this study to the results of related studies at the college and precollege levels and to the recommendations of mathematics educators who have proposed models for instructing teachers in problem solving. In making those connections, it has made use of a post hoc analysis of subjects' gain scores on the PST and of qualitative data provided by subjects' responses to open-ended items of the SIP, observers' comments recorded on the PSOG. and subjects written explanatory discourse on the
- 332. 322 PST. The synthesis of infonnation from those several sources is presented in the following summary observations. 1. Preservice teachers can learn to solve process problems. 2. Models of instruction in problem solving that implicitly or explicitly identify appropriate strategies and general procedures of mathematical problem solving are equally effective in improving the process problem-solving perfonnance ofpreservice teachers. 3. Preservice teachers can be taught to consciously apply problem-solving strategies to solve mathematical problems. An explicit model of instruction that names, discusses, purposefully applies, and reflects upon the organized use of problem-solving strategies enhances the organization of problem solutions and the description of problem-solving procedures. 4. Both models of instruction used in this study may have been more effective in teaching preservice teachers to solve near-transfer problems that to solve far-transfer problems. 5. In this study, subjects' participation in small-group problem-solving activities was, generally speaking, independent of the model of instruction. More frequent reference to strategies and procedures by treatment 2 subjects probably was a result of the fact that the treatment 2 provided a language ofproblem solving whereas treatment 1 did not. 6. Both models of instruction appear to have improved preservice teachers' self-perceptions of their ability to solve process problems. 7. Both models of instruction appeared to produce preservice teachers who were willing to teach problem solving to precollege students and who were aware of specific problem-solving skills that should be taught. There is evidence to suggest that both models contribute to the reduction of students' misconceptions regarding the nature of mathematics and problem solving. 8. Explicit instruction in problem solving is necessary to instruction of preservice teachers in the conscious use of P6lya's four-step model. Implicit instruction seems to produce an awareness of only one step in the model: the need to understand the problem. 9. Systematic instruction in mathematical problem solving can be incorporated into a mathematics methods course for preservice teachers without introducing major changes in the structure or content of the course. Recommendations The findings of this study and the experience of conducting it suggest three kinds of recommendations. Recommendations regarding test-scoring and study design are directed to persons conducting research into mathematical problem solving. Recommendations for classroom practice are directed to teacher educators seeking to develop the mathematical problem-solving skills of elementary and middle school school teachers.
- 333. 323 Procedural Recommendations: Focused Holistic Scoring The scoring of the paper-and-pencil instrument used to measure problem-solving performance employed a five-point focused holistic scoring scale based upon the one developed by Charles, Lester, & ODaffer [5]. This scale was chosen because it focused on the process of problem solving rather than its end product. In the word of its developers, such a scale is appropriate when the evaluator's interest is "in a general rating of the processes used and explicit criteria are needed or wanted to guide the assigning of points" [5, p. 38]. This focus on process was consistent with the goals of the study and is judged to be an important tool in the assessment of problem-solving performance. However, experience in using the scale to score pilot versions of the PST dictated that the scale's criteria be modified in order to make it applicable to all items of the test. In particular, the test items for the strategy Check Some Guesses were difficult to evaluate with the original criteria. Refinement of the criteria of the focused holistic scoring scale in order to evaluate a particular set of process problems is not viewed as a violation of the scoring procedure. In fact, Charles, Lester, and ODaffer speak not of the focused holistic scoring scale but rather of a focused holistic scoring scale. They point out that it is likely that revisions will be called for and that the need for such revisions will become apparent only when the scale is put to work. This assertion of a need to customize the scale was confirmed by this study. This study also conftrmed the adaptability of the scoring procedure to the process problems involved. The scoring of the several versions of the PST also amplified the issue of scoring consistency. Experience confirmed the need to identify "anchor papers" for each point category of the scale [5, p. 38]. In fact, it is necessary to identify anchor papers which are both item- specific and criterion-specific. That is, for each item of the test and for each criterion of each point category of the scale it is necessary to identify a paper which exemplifies satisfaction of that particular criterion for that particular point category on that particular item. Thus, it is recommended that pilot testing or prereading of all responses to a given item take place before data collection begins. Such a scoring procedure will guide the refinement of the scale and will support its consistent application. Finally, it is recommended that each item be scored by at least two persons. Even when operating with the modified scale and exemplary anchor papers, the two evaluators of the tests used in this study found it necessary to negotiate the point assignment of approximately 20 percent of the items. In several cases, differences in the scoring of an item were revealed to be due to an evaluator's overlooking critical evidence which was written on the back of a page or in an area other than that designated as the work space. In other cases, an unusual method of solution judged as "not understandable" by one evaluator was deciphered by the other evaluator. In still other cases, the scoring disagreement was the result of a mismatching of a student's work with the appropriate anchor paper. While each of these scoring conflicts was easily resolved by the evaluators, it is clear that a single evaluator would have improperly scored at least 10 percent of the items.
- 334. 324 Study Design Recommendations Broad guidelines for needed research in mathematical problem solving have been set forth by, among other researchers, Hatfield [9], Lester [18, 21], and Shoenfeld [31]. The dimensions of needed research regarding the teaching of mathematical problem solving have been defined by LeBlanc [16], Kilpatrick [13], and others. The specific research recommendations which follow are directly related to the findings and limitations of this study. 1. The assertion that the subjects of this study improved their problem-solving performance under either treatment is based upon the results of precollege studies and upon the large jump in mean scores on the PST. There is a need to test that assertion with a design that includes a control group that is exposed to the same set of process problems but is not exposed (implicitly or explicitly) to instruction in problem-solving procedures. 2. Subjects' performances on the PST suggest that their pretest to posttest gains were grounded in the five near-transfer items on the tests; gain scores on the far-transfer items are less impressive. Since extended application of generic strategies is a goal of instruction in problem solving, there is a need to focus on the study of far-transfer performance. 3. Failure to detect differences between the two treatment groups with respect to problem- solving or strategy-use awareness could be a function of the limited time frame of the study. Studies are needed that extend instruction over a longer period of time. Such studies might address additional strategies as well. 4. This study assessed, among other things, preservice school teachers' problem-solving performance and strategy-use awareness immediately following eight weeks of instruction and concludes that such awareness/performance can be taught. Studies are needed to determine if it is retained. 5. Subjects in this study professed a willingness to teach problem solving and exhibited an ability to identify problem-solving skills that should be taught to precollege students. Clinical studies of such preservice teachers should be conducted during their student-teaching experience to determine whether that willingness is expressed in teaching which reflects the problem- solving procedures that the teacher had been taught. 6. The principal assumption of this study was that teaching teachers to solve problems will improve the problem-solving performance of their students. Since the findings of this study suggest that teachers can be taught to solve problems, a next study is needed to measure the effectiveness of such teachers in teaching problem solving to precollege students. 7. During the execution of this study, both the investigator and the observers noticed a change in classroom atmosphere. Subject anxiety evident during the first two weeks quickly gave way to confidence and optimism. A clinical study should be conducted to systematically assess students' beliefs and attitudes during instruction in problem solving. Such research could reveal the effects of models of instruction upon student's belief systems regarding mathematics and problem solving.
- 335. 325 Classroom Practice Recommendations: Teacher Education The review of the literature on precollege problem solving identified research-based recommendations for precollege classroom practices [1,2,3,6, 16,20,34,35,36]. While the search also identified recommendations for classroom practice in the instruction of preservice teachers in problem solving, proposals at this level were not based on findings of studies involving preservice teachers. The following recommendations are related to the findings of this single study. 1. Preservice teachers are mathematically anxious and have little confidence in their ability to do mathematics. Data gathered through formal and informal interviews during this study indicate that anxieties appear to diminish and confidence appears to increase when a non- threatening atmosphere for problem solving is established. It is recommended that such an atmosphere be pursued by emphasizing that problem solving takes time and that perseverance rather than innate ability is the key to the solution of the process problems presented. 2. Preservice teachers appeared to enjoy solving problems of the kind presented in this study. They found the problems challenging (but within their grasp); they recognized the problems as being adaptable for use with precollege students. Since the latter two conditions were design criteria of the problems, it is recommended that problems used in the preparation of teachers meet those criteria. 3. Students who entered this study expressing low confidence in their ability to do mathematics reported that the use of small groups both reduced their anxiety and contributed to their growth in problem-solving performance. Therefore, it is recommended that small-group work be the primary mode of classroom problem-solving activity. 4. Chalkboard presentation of problem solutions by preservice teachers placed them in the role ofdemonstrating solution procedures; it generated discussion both of the solution presented and of alternative solution procedures. Presentation techniques improved and students became aware that there truly was more than one way to solve a process problem. It is recommended that such chalkboard presentations be a component of instruction model for teaching problem solving to teachers. 5. Both treatments of this study were based upon heuristic procedures, procedures found to be effective in teaching precollege students [17, 23, 29, 30]. That is, instructional features common to the two methods of instruction were dialogue, insistence upon active participation by the learners, and the systematic use of learners' suggestions (even when the instructor was aware that the suggestion would not lead to a solution). Based upon the findings, it is recommended that instruction of preservice teachers in problem solving should utilize heuristic procedures. 6. Although this study found that the problem-solving performance of preservice teachers exposed to the implicit model of instruction (treatment 1) did not differ from that of preservice teachers exposed to the explicit method of instruction (treatment 2), the data from small-group observations and the written work of students revealed that treatment 2 subjects did realize an
- 336. 326 advantage. Specifically, treatment 2 students had greater ease in discussing proposed solutions orally (in the small groups) or on paper. The written work of treatment 2 students had superior organization and clarity. The "common language" provided by the explicit model was evident in this improved communication and may contribute to retention of the procedures learned. Therefore, it is recommended that the explicit model of instruction be used in teaching problem solving. Appendix 1 PROBLEM 1. LaBelle Telephone & Telegraph company (LT&1) installs party lines with various numbers of customers on a given line. After much research, LT&T found that it costs $2 per month to service a party line that has exactly one phone, $4 per month to service a party line having exactly two phones, $7 per month to service a party line having exactly three phones, $11 per month to service a party line having exactly four phones, etc. It charges a fee of $10.50 per month for each phone on such a party line. How many phones should LT&T put on a party line in order to make the greatest monthly profit from that line? PROBLEM 2. In order to protect his chickens from foxes, a farmer has chained a guard dog to a water faucet located outside his chicken house and near the base of its only door. The door is kept closed; all windows are covered with heavy wire mesh. The chicken house is a rectangle 28 feet long and 16 feet wide. The faucet is located in one of the longer sides of the building and is 7 feet from a corner. If the dog is on a chain 40 feet long, what (to the nearest square foot) is the outside area over which it can roam? PROBLEM 3. The Prisoner of Zenda lay staring at the ceiling of his cell. It was made of square blocks of stone that measured one metre on each side. The cell was 6 metres long and 4 metres wide. To pass the time, he counted every square measuring 2 metres on a side that he could see in the ceiling. How many such squares should he have found? PROBLEM 4. The figure below shows a large triangle (the outermost one); its vertices were assigned the numbers 3, 5, and 7. A second triangle was created by connecting the midpoints of the sides of the original triangle and a RULE was used to assign the numbers 8, 12, and 10 to its vertices. The rule is: "Add the numbers at the two endpoints of the segment for which the new vertex is a midpoint." A third triangle was nested within the figure by connecting the midpoints of the sides of the second triangle; its vertices were assigned numbers (18, 20,22) using the same rule as was used for the second triangle. The resulting figure is a 3-triangle "nesttf • 5 ｾｾＭＭＭＭＭＭＭＭＭＭＭＭＭＭｾｾＭＭＭＭＭＭＭＭＭＭＭＭＭＭｾＷ＠ Suppose that the nesting process were continued for the triangle shown to produce a 1987-triangle nest. What would be the sum of the numbers at the vertices of the innermost triangle (Le., the 1987th triangle) of that nest? PROBLEM 5. While doing her mathematics homework, Charlene discovered that some whole numbers have exactly 3 factors. (For instance, she found that the number 49 had only the factors 1,7, and 49.) She even found numbers greater than SOOO that had exactly 3 factors. What is one of those numbers greater than SOOO that has exactly 3 factors?
- 337. 327 PROBLEM 6. The Rudloe's built their mountain cabin in the middle of a large meadow. The cabin. a rectangle measuring 36 feet by 28 feet, has a single electrical outlet located in the center of one of its 28-foot walls. The Rudloe's electric Iawnmower has a 50-foot cord. About what is the area of meadowland that can be kept trimmed with that Iawnmower? PROBLEM 7. An Australian wombat is happiest living in a rectangular field having a length that is three- halves of its width. Also. the wombat must have an area of at least 216 square metres in which to find enough food to remain healthy. What are the dimensions (length and width) of the smallest rectangular field that will keep a wombat both happy and healthy? PROBLEM 8. A pair of newborn mice got loose in a pet shop. Mice take one month to reach maturity; mature mice will produce another pair of mice at the end of their second month of life and will produce a pair of mice every month thereafter. Each new pair of mice will mature and reproduce at the same rate. If no mouse dies. how many pairs of mice will be loose in that pet shop at the end of 12 months? PROBLEM 9. The mathematicians attending a national convention arrived at their motel in a very orderly way. The first time the motel office door opened. 1 mathematician entered; the next time it opened. 3 mathematicians entered. Each time the door opened after that, the number of mathematicians entering the motel was 2 more than in the preceding group. If the motel office door opened a total of 50 times to admit groups of mathematicians, how many mathematicians stayed at the motel? References 1. Bums, M.: How to teach problem solving. Arithmetic Teacher, 29(6),46-49 (1982) 2. Charles, R. : Get the most out of "Word Problems". Arithmetic Teacher. 29(3),39-40 (1981) 3. Charles. R.: The role of problem solving. Arithmetic Teacher. 32(6),48-50 (1985) 4. Charles, R. & Lester, F.: Mathematical problem solving. Springhouse, PA: Learning Institute 1986 5. Charles. R., Lester. F. & O'Daffer, P.: How to evaluate progress in problem solving. Reston, VA: The National Council of Teachers of Mathematics 1987 6. DeVault, M. V.: Doing mathematics is problem solving. Arithmetic Teacher, 28(3), 40-43 (1981) 7. Dossey, J.: Preservice elementary mathematics education. Arithmetic Teacher, 31(7), 6-8 (1984) 8. Garofalo, J.: Metacognition and school mathematics. Arithmetic Teacher, 34(9), 22-23 (1987) 9. Hatfield, L.: Heuristical emphases in the instruction of mathematical problem solving: Rationales and Research. In: Mathematical problem solving: Papers from a research workshop (L. L. Hatfield & D. A. Bradbard, eds.), pp. 21-42. Columbus, OH: ERIC/SMEAC 1978 10. Jacobs, J. E.: One point of view: Preparing teachers to teach problem solving. Arithmetic Teacher, 31(4), 1 (1983) 11. Kansky, R. J.: Problem solving in mathematics education: A missing component of the teacher education curriculum. In: Proceedings of the Sino-American on secondary mathematics education seminar. Taipei, Taiwan: National Science Council of the Republic of China 1987 12. Kantowski, E. L.: Processes involved in mathematical problem solving (Doctoral dissertation, University of Georgia,1974). Dissertation Abstracts International, 36, 2734A (1975) 13. Kilpatrick, J.: A retrospective account of the past twenty five years of research on teaching mathematical problem solving. In: Teaching and learning mathematical problem solving: Multiple research perspectives (E. A. Silver, ed.), pp. 1-15. Hillsdale, NJ: Lawrence Erlbaum 1985 14. Krulick, S. & Rudnick, J.: Teaching problem solving to preservice teachers. Arithmetic Teacher, 29(6), 42- 45 (1982) IS. LeBlanc, J.: A model for elementary teacher training in problem solving. In: Mathematical problem solving: Issues in research (F. K. Lester & J. Garofalo, eds.), pp. III-lIS. Philadelphia, PA: The Franklin Institute Press 1982 16. LeBlanc, J.: Teaching textbook story problems. Arithmetic Teacher, 29(6), 52-54 (1982)
- 338. 328 17. Lee, S. L.: An exploratory study of fourth graders' heuristic problem-solving behavior (Doctoral dissertation, University of Georgia. 1977). Dissertation Abstracts International, 38, 4004A (1977) 18. Lester, F. K.: Mathematical problem solving in the elementary school: Some educational and psychological considerations. In: Mathematical problem solving:Papers from a research workshop (L. L. Hatfield & D. A. Bradbard, eds.), pp. 53-87. Columbus,OH: ERIC/SMEAC 1978 19. Lester, F. K.:. Building bridges between psychological and mathematics education research on problem solving. In: Mathematical problem solving: Issues in research (p. K. Lester & 1. Garofalo, eds.), pp. 55-85. Philadelphia, PA: The Franklin Institute Press 1982 20. Lellter, F. K.: Issues in teaching mathematical problem solving in the elementary grades. School Science and Mathematics, 82(2), 93-98 (1982) 21. Lester, F. K.: Trends and issues in mathematical problem-solving research. In: Acquisition of mathematics concepts and processes (R. Lesh & M. Landau, eds.), pp. 229-261. New York, NY: Academic Press 1983 22. Lucas, 1. F.: An exploratory study on the diagnostic teaching of heuristic problem-solving strategies in calculus (Doctoral dissertation, University of Wisconsin, 1972). Dissertation Abstracts International, 32, 6825A (1972) 23. Marcucci, R. G.: A meta-analysis of research on methods of teaching mathematical problem solving (Doctoral dissertation, The University of Iowa, 1979). Dissertation Abstracts International, 41, 2485A (1980) 24. Mathematical Association of America.: Recommendations on the mathematical preparation of teachers. Washington, D. C.: Author 1983 25. National Council of Teachers of Mathematics.: An agenda for action: Recommendations for school mathematics of the 1980's. Reston, VA: Author 1980 26. National Council of Teachers of Mathematics.: Guidelines for the preparation of teachers of mathematics. Reston, VA: Author 1981 27. National Council of Teachers of Mathematics.: Curriculum and evaluation standards for school mathematics. Reston, VA: Author 1989 28. P6lya. G.: How to solve it. Princeton, Nl: Princeton University Press 1945 29. Proudfit, L.: The examination of problem-solving processes by fifth-grade children and its effects on problem-solving performance (Doctoral dissertation, Indiana University, 1980). Dissertation Abstracts International, 41, 9, 3932A (1980) 30. Putt, 1.1.: An exploratory investigation of two methods of instruction in mathematical problem solving at the fifth grade level (Doctoral dissertation, Indiana University, 1978). Dissertation Abstracts International, 39, 5382A (1979) 31. Shoenfeld, A. H.: Explicit heuristic training as a variable in problem-solving performance. 10urnal for Research in Mathematics Education, 10(3), 173-187 (1979) 32. Shoenfeld, A. H.: Teaching problem-solving skills. American Mathematical Monthly, 87, 794-805 (1980) 33. Shoenfeld, A. H.: Cognitive sCience and mathematics education: An overview. In: Cognitive science and mathematics education (A. H. Shoenfeld, ed.), pp. 1-31. Hillsdale, Nl: Lawrence Erlbaum 1987 34. Stacey, K. & Southwell, B.: Teaching techniques for problem solving. Australian Mathematics Teacher, 40(1),5-7 (1984) 35. Suydam, M. N.: Update on research on problem solving: Implications for classroom teaching. Arithmetic Teacher, 29(6), 56-60 (1982) 36. Suydam, M. N.: Research report: Problem solving. Arithmetic Teacher, 31(9), 36 (1984)
- 339. Mathematics Problem Solving: Some Issues Related to Teacher Education, School Curriculum, and Instruction Randall!. Charles San Jose Slate University, San Jose, California 95192-0103, USA Abstract: The massive number of teacher education efforts underway throughout the United States related to problem solving is one source of evidence of the significant role that problem solving is playing in school mathematics. Unfortunately, the leaders of these teacher education activities have little to guide their work beyond their own intuitions and experience. The first part of this paper is a description of a course on mathematics problem solving designed for elementary and middle school teachers of mathematics. The major components of this course are described together with some of the theoretical underpinnings from which the experiences contained in it evolved. The second part of the paper identifies issues and questions relative to school curriculum, instruction, and teacher education vis-a-vis problem solving. Keywords: mathematics education, problem solving, teacher education, curriculum, teaching /'m not worried about the current emphasis on problem-solving; it will go away in afew years just like the Metric system! Several years ago I was about to begin a workshop on problem solving when I overheard a teacher make the above comment to one of her colleagues. Although it's true the massive attention given to metrification in the United States has subsided, it is not true that problem solving has gone away! Indeed, evidence abounds that problem solving is thriving in our schools and is on its way to becoming the focus of mathematics education. The massive number of teacher education efforts underway throughout the United States related to problem solving is one source of evidence of the significant role that problem solving is playing in school mathematics. The first part of this paper is a description of a course on mathematics problem solving designed for elementary and middle school teachers of mathematics. The major components of the course are described together with some of the theoretical underpinnings from which the experiences contained in it evolved. The second part of the paper identifies issues and questions concerning school curriculum, instruction, and teacher education vis-a-vis problem solving.
- 340. 330 A Mathematics Problem Solving Course There are two reasons for sharing a description of a problem-solving course. First, a great deal of experience with this course suggests that it has potential to bring about change in the classroom teaching of problem solving, and second, clear descriptions of successful teacher education efforts may lead to the identification of questions amenable to research. I begin by briefly discussing goals for teacher education and problem solving. Then I discuss the general organization of the course. For the remainder of this section I focus on two aspects of the course, the school curriculum and instruction. Goals for Teacher Education vis-a-vis Problem Solving Shulman [17] suggests that a teacher education program needs to construct answers to two questions: (1) What do we want teachers to know?, and (2) What do we want them to be able to do? Drawing on work with teachers and research on students' problem-solving processes and effective teaching, the following goals were established for the problem solving course. 1. Teachers should be able to solve problems of at least the same level of difficulty that they will use with their students. 2. Teacher's content and curricular knowledge and abilities. 2.1 Content knowledge - A teacher should know: 2.1.1 the meaning of a "problem" and "problem solving," 2.1.2 why problem solving is important, 2.1.3 what factors influence success, and 2.1.4 the thinking processes involved in solving problems. 2.2 Curricular knowledge - A teacher should know: 2.2.1 types of problems and problem-solving experiences appropriate for different grade/age levels, and 2.2.2 the roles problem solving has played in the mathematics curriculum. 2.3 Curricular abilities - A teacher should be able to: 2.3.1 select and create problem-solving experiences appropriate for a given population of students, and 2.3.2 evaluate problem-solving curriculum material. 3. Teachers' pedagogical knowledge and abilities: 3.1 Pedagogical knowledge - A teacher should know: 3.1.1 different roles for the teacher in the classroom, 3.1.2 teaching actions that facilitate problem solving, 3.1.3 ways to manage instruction, 3.1.4 assessment altematives,and 3.1.5 factors that influence the classroom climate. 3.2 Pedagogical abilities - A teacher should be able to: 3.2.1 implement a model for teaching problem solving, 3.2.2 implement classroom management practices, 3.2.3 develop and use assessment techniques, and 3.2.4 build a classroom climate conducive for problem solving.
- 341. 331 General Organization of the Course This course has been taught in many configurations. The most common is a I-semester 15- week course through a university. Also, it has been presented as a series of workshops over time. Regardless of the configuration or time available, the course moves through three phases with approximately one-third of the total amount of time devoted to each phase. Phase One focuses on developing teachers' problem-solving abilities and shaping their beliefs and attitudes about themselves as problem solvers and about problem solving (see Goal 1). Teachers need some level of competence as problem solvers before they not only teach problem solving but before they begin to learn about problem solving. Also, the importance of addressing affect at the beginning of a teacher education program on mathematics problem solving cannot be overemphasized. Many prospective and inservice teachers have considerable anxiety related to problem solving. Beginning the course by solving problems and discussing feelings, attitudes, and beliefs has been very successful; it is one way to give teachers permission to admit their anxieties and begin to deal with them. In Phase Two teachers continue to solve problems to further improve their problem-solving abilities and develop helpful attitudes and beliefs. However, the emphasis here is on developing the teachers' content and curricular knowledge related to mathematics problem solving (see Goal 2). Phase Three focuses on developing the teachers' pedagogical knowledge and abilities (see Goal 3). Models for teaching problem solving, techniques for assessing progress in problem solving, and classroom management issues are explored. School Curriculum and Instruction There are many topics addressed in this course related to a teacher's knowledge about problem solving, a teacher's knowledge and abilities related to curriculum, and a teacher's knowledge and abilities related to instruction. In Phase 2, particular emphasis is given to examining types of problem-solving experiences that can be used in school curriculum and the purpose or purposes each serves in developing student thinking. In Phase 3, emphasis is given to the teacher's role in facilitating the development of thinking. Because these topics are emphasized in the course, they are discussed below in some detail. Curriculum Identifying the kinds of thinking experiences appropriate and useful for an elementary-middle grades mathematics curriculum is a task partially amenable to research (see [18], for a historical
- 342. 332 look at problem solving in the curriculum). Unfortunately, beyond the work of the Mathematical Problem Solving Project at Indiana University in the 1970s, the nature of the curriculum as it relates to problem solving has not been the subject of much research; it seems that most suggestions for curriculum have emerged from popular opinion. Two aspects of the school mathematics curriculum vis-a-vis problem solving are examined in the course: • problem-solving skills and strategies • ways to integrate thinking to develop concepts, operations, and skills. Problem Solving Skills and Strategies: A Stand Alone Approach Extensive space in the elementary and middle school grades mathematics curriculum is devoted to experiences aimed at developing students' use of strategies and skills for solving problems. Explicit instruction is given to helping students learn how to make organized lists, how to search for patterns, how to simplify problems, how to decide if an answer seems reasonable, how to use deductive reasoning processes and so on. These experiences reflect a stand alone approach to developing thinking [12]. In a stand alone approach to developing thinking, emphasis is given to the thinking processes elicited in the experiences used; few mathematics content demands are made of the student in completing them and no new mathematics concepts typically emerge from working with them. Experiences in existing curriculum that reflect the stand-alone approach usually appear as special problem solving or thinking skills lessons such as can be seen in [9]. They appear as lessons separate from those that focus on developing mathematics operations, concepts, and skills. Much work over the past 10 years has been devoted to stand-alone lessons and their use is now quite common in most classrooms. However, for many teachers these lessons are the "extra ones" taught (if there's sufficient time) after they teach "the regular math." Integrating Thinking Experiences: An Immersion Approach The immersion approach for developing thinking is an emerging one that Prawat [12] suggests does not have great support. However, in mathematics education many calls for reform in the teaching of mathematics, most notably the Curriculum and Evaluation Standards from the National Council ofTeachers of Mathematics [11], strongly encourage an immersion approach. The immersion approach has its roots in a constructivist view of learning (see [3] for a summary of the implications of constructivism for teaching). In an immersion approach,
- 343. 333 highest priority is given to the mathematics content understandings associated with students' thoughts, not to the thinking processes involved in completing the instructional tasks. Figure 1 shows how the immersion approach can be reflected in organizing curriculum. Student Exploration and Discussion -.. Questions Example. rules. tenns. definitions . generalizations. and so on Figure 1: An alternative curriculum organization based on an immersion approach. In traditional curriculum, lessons begin with rules, examples, definitions, generalizations, and so forth. In an immersion approach, a lesson begins with a mathematical task that calls for higher-level thinking. These tasks can be of various types; it may be a problem to be solved, it may call for the use of manipulatives, or it may simply be an interesting question. The duration of the experience can vary; it may be a brief activity or it may extend over several days. Although many tasks can be completed alone, cooperative group work requiring students to communicate orally and/or through writing with each other and with the teacher to articulate their thoughts and understandings is desirable. After students have explored a task and constructed their initial understandings, key tenns, rules, definitions, examples, and so forth can then be introduced. Figure 2 gives an example of a middle grades experience that can be used with an immersion approach. Students can work in small groups using blocks to grapple with the task. They can write about which rectangles are possible and how they decide they have all possible rectangles. After students explore this task, the language of factors, prime numbers, composite numbers, and square numbers can be introduced. A curriculum organized to reflect an immersion approach, unlike a stand alone approach, requires many teachers to reexamine the manner in which they have been teaching mathematics. It requires a shift from teaching by telling students key ideas to one of facilitating the development of students' understandings as they explore mathematical tasks. Because curriculum experiences designed to promote an immersion approach to developing thinking are new to many teachers, considerable emphasis in the course is given to this manner of curriculum design.
- 344. 334 How many different rectangular solids one layer thick can be made where each consists of 60 same-size cubes? What are the dimensions of the rectangles that can be traced? What ifthere are 25 cubes? 17 cubes? You can trace a 5 by 4 rectangle from this solid. Figure 2: A sample experience that can be used with an immersion approach. Instruction The stand alone and immersion approaches to developing thinking have implications for instruction as well as curriculum. A teacher implementing stand alone instructional activities such as those shown above could direct all questions and discussions to the thinking skills used in an experience, not to the mathematics content. For the immersion approach the reverse could be true. The questions and discussion could emphasize the mathematics understandings students might gain from an experience with little attention given to the thinking skills exhibited during the experience. In the problem solving class, discussions are held as to the merits and limitations of these approaches. However, an argument is presented that the instructional implications of the stand alone and immersion approaches when carried to the extreme are inappropriate. Instead, an embedded thinking skills approach is encouraged [12]. In an embedded thinking skills approach, explicit attention is given both to the thinking processes used in completing a task and to the mathematics content involved; a balance between the two is attempted. It uses modeling and coaching by both the teacher and students. The reciprocal teaching strategy of Palinscar and Brown developed initially for reading instruction and later extended to mathematics [4] and Schoenfeld's [15] method for teaching problem solving are two prominent examples of this approach discussed in the course. The reciprocal teaching model as adapted to mathematics and Schoenfeld's method for teaching problem solving emphasize the instructional milieu. In the reciprocal teaching model, students work on three boards; Figure 3 shows a sample [4, p. 104]. In this approach, "learning leaders [teacher or students] guide the group in working on three successive chalkboards designed to help students proceed systematically...these procedures generate an external record of the group's problem solving which can be monitored, evaluated, and reflected upon" (p. 103).
- 345. 335 Text ofProblem: Harry ate a hamburger and drank a glass of milk which totaled 495 calories. The m1lk contained half as many calories as the sandwich. How many calories were in the sandwich and how many were in the milk? PLANNING BOARD DRAWING BOARD DOING BOARD A hamburger aNS a gla" 495 I I of milk total.d 495 caloriu. H + H =495 Tlw milk contaiMd halt a, I I I I tNlny caloriu than tlw L-.-.J hamburger. H H H =caloriu of hamburger H =calori.. of milk Figure 3: Sample boards from a reciprocal teaching approach. Schoenfeld's [15] approach is based on a list of questions displayed in the classroom that the teacher initially draws to the students' attention. With experience, students should learn to ask themselves these questions as they solve problems. Schoenfeld's approach is aimed at developing the student's metacognitive abilities for solving problems. Figure 4 shows the questions Schoenfeld would display. Questions to help you control your work. • What (exactly) are you doing? (Can you describe it precisely?) • Why are you doing it ? (How does it fit into the solution ?) • How does it help you? (What will you do with the outcome when you attain it ?) Figure 4: Schoenfeld's questions for student work. The reciprocal teaching method and Schoenfeld's method are ones that reflect an embedded thinking skills approach because the discussions of student work can address both the mathematics content of the task and the thinking used to complete it. Another method that's based on an embedded thinking skills approach also explored in the course is one developed by
- 346. 336 Charles and Lester [5]. In this approach, a bulletin board is displayed in the classroom. Figure 5 shows a middle grades version of the bulletin board. Similar to Schoenfeld's method, the teacher can initially draw students' attention to elements of the board and later, with experience, students can utilize the board on their own. UNDERSTANDING THE PROBLEM • Read the problem. • Decide what you are trying to find. • Find the important data. SOLVING THE PROBLEM • Look for a pattem. • Draw a picture. • Guess & check • Make an organized list. • Write an equation • Make a tabie. • Use logical reasoning • Use objects or act out. • Work backwards • Simplify the problem. ANSWERING THE PROBLEM & EVALUATING THE ANSWER • Be sure you used all of the important information. • Check your work. • Decide whether the answer makes sense. • Write the answer in a complete sentence. Figure 5: A problem-solving bulletin board for the Charles and Lester method. For the reciprocal teaching method and Schoenfeld's method, little specific direction is provided for the teacher beyond suggesting that he or she facilitate student thinking. Work with the Charles and Lester bulletin board suggests that most teachers interested in teaching problem solving need specific suggestions for their roles during instruction beyond a bulletin board and beyond being told that they should act as a facilitator. Based on work with teachers and drawing from research on students' problem-solving processes and effective teaching, a teaching strategy for problem solving was developed.
- 347. 337 The Charles and Lester teaching strategy was designed to reflect four major roles a teacher might play in facilitating student thinking and a sequence in which those roles are typically played out during a lesson (see Figure 6). Promote Student Understanding Monitor & Assess Promote Student .. Student ThinkIng .. Reflection and Work Extend .. Understanding Figure 6: Some major roles a teacher can play in facilitating student thinking. These four roles were translated into "teaching actions" and presented in a lesson plan format (see Figure 7). The three phases of the lesson (Before, During, and After) were selected to reflect the four roles a teacher might play to facilitate thinking. (Two of the roles are contained in the After part of the plan.) A complete discussion of the teaching actions can be found elsewhere (see [5]). Also, recommendations for research on this teaching strategy can be found elsewhere [7]. The remainder of this section contains the rationale for this teaching strategy. The idea of a lesson plan was conceived as a vehicle for inviting teachers to begin teaching problem solving. There may be concern when a teaching strategy is given in the form of a lesson plan. One possible concern is that the strategy becomes a rigid sequence of behaviors that does not allow for individual differences of either teachers or students. Another possible concern is that a lesson plan may promote the false belief that an algorithm exists for teaching problem solving. Although these concerns are legitimate, experience with this lesson plan and research on teachers' planning behaviors and thought processes suggest that a problem-solving lesson plan is not only helpful in getting teachers started teaching problem solving but may indeed be a necessary ingredient in changing teachers' instructional behaviors. In a study by Charles and Lester [6], interviews with teachers who used this lesson plan revealed that all of the teachers found it helpful in getting started teaching problem solving. Also, the teaching strategy was considered flexible enough to allow teachers to adapt it to their own teaching styles but structured enough to help less confident teachers feel comfortable teaching problem solving. Research on teachers' planning behaviors and thought processes provides an explanation for the value of the teaching actions. A major component of teachers' planning is the selection and sequencing of instructional activities [16]. In general, teachers do not begin their planning for a lesson by thinking about specific instructional techniques they will use to achieve an objective [8]. Rather, they begin by establishing activity chunks sequenced in a particular way. Each activity chunk may contain several sub-activities. A sequence of activity chunks, provides an agenda [10] or a script [I, 14] for the lesson. The value of the agenda is that it structures the
- 348. Teachino Actions Before 1. Read the problem. 2. Ask questions for undemanding the problem. 3. Discuss possible solution strategies. ram. Understanding Teachino Actions During 4. ObselVe studelis. 5. Give hints as needed. 6. Require students to check back and a/lSl/erthe problem. 7. Give an extension as needed. Idon.r and Assess 338 [Insert problem statement here] Questions for Understanding the Problem (fA 2) (Write questions here related to understanding the problem.] Possible Hints for Solving the Problem (fA 5) [Write clusters of hints related to particular solution approaches] Teacbino Actions After Possible Solution (fA 8) 8. Discuss solutions. Name strategies. [ShOW one or more possible solutions here.] 9. Discuss related problems and the extension. 1O. Discuss special features as needed. 11. Connect math conteli. Promote Rellection E*nd Understandinos Related Problems: (Reference similar problems solved previously.] Problem Extension: (Write here an extension of the original problem.] Figure 7: A lesson plan fonn for teaching problem solving. activities of a lesson, making both the teacher's and students' actions predictable and reducing the complexity of the information teachers encounter during instruction. With experience, the
- 349. 339 agenda becomes internalized or routinized relieving the teacher from having to consciously develop a mental road map for each lesson. Leinhardt [10] suggests that, "The use of routines [agendas] also reduces the cognitive processing for the teacher and provides them with the intellectual and temporal room needed to handle the dynamic portions of the lesson" (p. 27-8). Experience with this lesson plan shows that teachers initially follow the teaching actions quite closely. With experience, most teachers internalize the teaching actions and refer to the lesson plan only on occasion. Observations and interviews with teachers also showed that only after teachers internalized the teaching actions were they (cognitively) ready to focus on the "dynamic portions" of teaching problem solving (e.g., ways to improve their skills giving hints or ways to improve their discussions with students about problem solutions). Until teachers internalized the teaching actions, they did not feel in control of the lesson and they were not interested in exploring ways to improve individual teaching actions. The Charles and Lester method described here is another example of an embedded thinking skills approach to developing thinking. It is similar to the reciprocal teaching method and Schoenfeld's method in that a board is used to draw attention to thinking processes and discussions about student work solving a problem can focus both on the mathematics content involved and on the thinking processes exhibited. However, unlike the reciprocal teaching method and Schoenfeld's method, Charles and Lester's method provides specific ideas for ways the teacher can facilitate student thinking and work. Needed Research: Curriculum, Instruction, and Teacher Education Reflecting on this course and the many instances in which it has been implemented suggests several issues and questions that need investigation. These issues and questions can be grouped into three areas: curriculum, instruction, and teacher education. Curriculum Building a school curriculum with a focus on problem solving or thinking is a challenging activity; an activity for which there are many unanswered questions. For example, there are unanswered questions about the nature of the instructional tasks that might be used in a thinking-based curriculum. Which instructional tasks contribute most toward the development of thinking abilities, and how do we evaluate tasks for inclusion in the curriculum? Much attention in the 1980s was given to teaching problem-solving skills and strategies. Should these continue as elements of the curriculum? Furthermore, are the problem-solving strategies typically introduced in the elementary school grades the ones that should be included? A look at
- 350. 340 the research on critical thinking makes it unclear as to which strategies should receive explicit attention [13]. Curriculum tasks can be presented in a variety of environments. Many of the papers in this book explore the nuances of computer environments, but there are indeed other environments that should be considered (e.g., manipulatives). Should there be concern about the balance among these environments in building curriculum? Scope and sequence have always been key building blocks for school curriculum. Are issues of scope and sequence no longer relevant when problem solving is the foundation of the curriculum? How should a problem-solving based curriculum evolve across the grades? Where should the curriculum fallon a fixed-dynamic dimension; fixed meaning the traditional scope and sequence models and dynamic meaning that scope and sequence evolve situationally? Perhaps the time has come for drastically different definitions and conceptualizations of curriculum. Finally, an issue seldom discussed is how much higher-level thinking can students and teachers tolerate? Much human activity is aimed at reducing complexity to routine. Is there value in routine? Does it provide time for mental and physical resources to rebuild themselves for the next complex task to be encountered? Instruction An important issue related to teaching is how to invite teachers to begin making thinking/problem solving the focus of instruction. Knowing how we want teachers to think and act as facilitators of student thinking and knowing what advice will help them move toward that vision are separate issues. Boyle e Ponder [2] identified four conditions necessary for teachers to act on advice. The advice must (i) have administrative support, (ii) not conflict with the teacher's role definition, (iii) be cost effective in terms of the teacher's time and energy, and (iv) suggest specific behaviors. All of these need to be considered as ways to invite teachers to become facilitators of thinking are explored. Finally, four roles for a facilitator were identified in this paper; are they appropriate? Are these roles generic? That is, do they apply to all tasks and for all task environments? Teacher Education It was mentioned at the beginning of this paper that a massive number of teacher education activities related to problem solving are taking place throughout the United States. As issues of curriculum and instruction continue to unfold and be resolved, issues of teacher education need to be explored. One of the most important is articulating goals for teacher education. Several
- 351. 341 goals were established for the program described above. Are these reasonable goals? Were imponant ones orrrlned? Defining goals, of course, will lead to the articulation of subsequent issues and questions for investigation. However, in spite of not having agreement on goals for teacher education, there are some issues that seem reasonable to investigate. Among the most important are which background variables have the greatest impact on teacher education and teaching performance? What roles can new technologies play in teacher education? What are the implications for teacher education of the growing diversities in schools and teacher education programs? How does preservice education compare to inservice education? What roles do field experiences play? And finally, can teacher education really occur outside of real classrooms? The time may be right to take a serious look at traditional models of teacher education, to challenge the assumptions of these models, and to explore a variety of alternatives. References 1. Abelson, R.: Script processing in attitude formation and decision making. In: Cognition and social behavior (J. Caroll & I. Payne, eds.), pp. 33-45. Hillsdale, N.J.: Lawrence Erlbaum 1976 2. Boyle, W. & Ponder, G.: The practicality ethic in teacher decision making. Paper presented at the Milwaukee Curriculum Theory Conference, Milwaukee, WI 1975 3. Brooks, J.: Teachers and students: Constructivists forging new connections. Educational Leadership, 47(5), 68-71 (1990) 4. Campione, I., Brown, A. & Connell, M.: Metacognition: On the importance of understanding what you are doing. In: The teaching and assessing of mathematical problem solving (R. Charles & E. Silver, eds.), pp. 93-114. Reston, VA: NCTM 1988 5. Charles, R. & Lester, F.: Teaching problem solving: What, why and how. Palo Alto, CA: Dale Seymour Publishing Company 1982 6. Charles, R. & Lester, F.: An evaluation of a process-oriented instructional program in mathematical problem solving in grades 5 and 7. Iournal for Research in Mathematics Education, 15(1), 15-34 (1984) 7. Charles, R.: Some directions for research on teaching problem solving. Paper presented at the Research Presession of the Annual NCTM Conference, Austin, TX, 1985 8. Clark, C. & Dunn, S.: Second generation research on teachers' planning, intentions, and routines. In: Effective teaching: Current research (H. Waxman & H. Walberg eds.), pp. 183-201. Berkeley, CA: McCutchan 1991 9. Eicholz, R.. O'Daffer, P., Charles, R., Young, S., Barnett, C., Fleenor, C, & others: Mathematics (Book 5), Addison-Wesley 1991 10. Leinhardt, G.: Routines in expert math teachers' thoughts and actions. Paper presented at the annual meeting of the AERA, Montreal 1983 11. National Council of Teachers of Mathematics: Curriculum and evaluation standards for school mathematics. Reston, VA: NCTM 1989 12. Prawat, R.: The value of ideas: The immersion approach to the development of thinking. Educational Researcher, 20(1), 3·10 (1991) 13. Resnick, L. & Klopfer, L. (eds.).: Toward a thinking curriculum. Alexandria, VA: Association for Supervision and Curriculum Development 1989
- 352. 342 14. Schank, & Abelson, R.: Scripts, plans, goals, and understanding: An inquiry into human knowledge struCtures. Hillsdale, NJ.: Lawrence Erlbaum 1977 15. Schoenfeld, A.: Metacognitive and epistemological issues in mathematics understanding. In: Teaching and learning mathematical problem solving: Multiple research perspectives (E. Silver, ed.) pp. 361-379. Hillsdale, NJ: Lawrence Erlbaum 1985 16. Shavelson, R. & Stem, P.: Research on teachers' pedagogical thoughts, judgments, decisions, and behavior. Review of Educational Research, 51 (4),455-498 (1981) 17. Shulman, L.: On teaching problem solving and solving the problems of teaching. In: Teaching and learning mathematical problem solving: Multiple research perspectives (E. Silver, ed.), pp. 439-450. Hillsdale, N.J.: Lawrence Erlbaum 1985 18. Stanic, G. & Kilpatrick, J.: Historical perspectives on problem solving in the mathematics curriculum. In: The teaching and assessing of mathematical problem solving (R. Charles & E. Silver, eds.), pp. 1-22. Reston, VA: NCTM 1988
- 353. Subject Index A-validation 156,161 Absolutism 289, 293-4, 297-8 ACT theory 278 Active thinking 222 Algebra 65, 69, 73, 155-64, 268, 270 Algebra Tutor 263, 265, 278 Algebreland 278 Anchored instruction 69-70, 72 Aplusix 267,269,274-83 Applications 45-6 Arithmetic 155-64 Assessment 45,51,56-7,62,68-71 Awareness 19-20,23-8,314,316,320,325 Basic level 95 Behavior 268-9, 272, 276 failed- 268 partial- 268 sucessful- 268 Bugs 9-11,270,277-8 ｃ｡｢ｲｩＭｧｯｯｭｾｴｲ･＠ 180-7 Camelia 273, 275 Category theory 94 Classroom climate 310 culture 65 pratice 65-6 organization 310 processes 7 tasks 141, 145 work 308-11 Cognition 6-8 10-2, 42 social- 98-9 situated- 37,42,71 Cognitive activity 5 behavior 11 conflict 142, 146-7, 149-53 Cognitive knowledge 4 processes 8, 62, 64, 243 psychology 65,68-9 strategies 2424,251-2,271,276,281 structure 142 Cognitive models idealized 99 image-schematic 101 metaphoric 1m, 107-8 metonymic 101, 103, 107 propositional 101 Collaterallearning 17-8 Communicating solutions 42-3 Communication 79, 122 Complex thinking processes 173-5 Computer assisted instruction 167, 171 environment 177-8,181, 186-7 feedback 205-6,2114,217 tools 224 Computerized learning environments 267 Computers in mathematics education 90, 126, 193- 4,205-17,219-37,244,255-65,301-11 Concept definition 106 images 106 Conceptual field 81 Condictional forms 127-35 Conjectures 20,23,334, 168-74,243,247-8,251, 258-9, 2634 Context 62,66-7,71-2,77-83,86,88, 128-9, 132, 139, 141, 143, 151 sensitivity 77-9,"87-8, 113, 117-21 Contextual conditions 5 Control 85, 235, 255, 258, 260-1, 264-5, 267, 270,274-7,280,282
- 354. 344 Control Heuristic models of instruction 314, 318, 322, didactic 277, 282-3 326-7 epistemic 277,282-3 Hewet project 45,52,57 Convincing 23 Hypen:arre 187 Cultural models 98-9 Hypothetical reasoning 84,129-31, 133 Curriculum 10,22,224,232,235-6,329-34,339- Innovation 309-12 40 Insights 18,20,25,28 Data analysis 193-4,200-1 Diagnostic teaching 142 Didactical contract 121,276-7 Discovery approach 277 Discussion 142,145-7,150-2,240,245-50,252 Division story problems 114-7 Domains 62-6, 68-70, 73 ill-structured 63-4 well-structured 62 Epistemological obstacles 155-6 style 156, 162 Equation solving 268 Ethnomathematics 77 Explanation 276 genetic 276 personalized 276 Euclide 182 Fallibilism 290, 293, 295-6, 298 Family resemblance 94 Feedback 178-9,181,183-4,187,263-5,283 Frameworks 24 Field ofexperience 80-1, 87, 89, 186-7 Final examination 50-6 Flexibity 263-4 Frames 97, 100 Generalizing 20,22-3,27,29-30,302,307,310 Geometric Sketchpad 263 Geonetric Supposer 171-5,179-84 Geometrical figure 178-82 Geometry 63,65, 177-87 learning 104-10 Goals 68, 272-4 Guided discovery 290 Hawex project 45,52,57 Hedges 103 Heuristic 4, 19,27,85,88, 128, 168, 181-4,206- 7,217,255-65,272,274-5,278-81 Instructional outcomes 8-10, 13 Intellectual mirror software 170-9 Interactive learning environments 267,274-7,284 Investigation 21, 139, 141, 205, 217, 223-4, 227- 36,239-44,252-3,302,304-8,312 Knowledge acquisition 177 behavioral 269-70, 272, 280 conceptual 186,269,271 conditional 62, 188 contextua1ized 211 declarative 62,66,71,167, 189 domain-specific 62, 66-70 geometrical 179 implicit 188 interactive 68-9 mathematical 255 non-strategic 272-3 procedural 62-3,69,71, 187, 189,270 production 206,208-9,211,217 reference 275-9,284 stategic 62, 65-6, 68-70, 208, 270, 272-4, 282 theorical 187-9 Knowledge state 269-70, 275-7, 284 deterministic- 270, 277 extended- 187-9 Language 103-4 Learning 138, 150,230,233,238 environment 179 situations 302 Logical connectives 127,131-2,134-5 Logo 182,209-15,220-37,240 Mal-rules 270 Mathematical activity 137-8,141 Mathematical modeling 45-56, 83 Meaning 6, 80, 85 Measurement theory 38-9,42, 69-71 Mental tools 265
- 355. Metacognition 4,6-8,10-2,19,27,64,66,335 Metaphors 100-1 Metonymy 100 Minerva project 240,301-3 Misconceptions 64-5,67,69,71,270 Modelofstudent'sknowledge 188 Modeling 168,171 New Infmnation Technologies 219,237,301-2 Noticing 20, 25, 27 Objective assessment 38-39 Operational structure 256, 258 Philosophy of mathematics 291-2 Planning 84, 255, 258, 262, 272-4, 278-81 dynamic 274 static 273-5,278-9 Pre-algebra 155-61 Primitives 171-3 Problem domain 268 formulation 38 posing 168-70, 174, 195, 242, 263, 290, 296-7 situations 177-8, 187 types 39-40 Problem solving as composition 42-3 assessment 5, 37-43 behavior 321 ｾｴｩｶ･＠ 22,235-6 embedded thinking skills approach 334 immersion approach 33 instruction 1-3 perfomance 313,316-9,323-7 skills 9,76-7 stand alone approach 332 strategies 125-35, 168, 195, 197, 207-9, 211-3,215,314-7,319-21 Procedural polarity 159-61 Procedure 84,157-61,171-4 Programming 171-5,22,224-5,235 Progressive absolutism 293,295,297-8 Project work 302 Prototypes 94-5,97, 100-1, 103,107-8 Rea1istics mathematics education 45-6, 52, 57 345 Reasoning deductive 183-5 inductive 183-4 Reflection 152,242-4,251-2 Relational polarity 159-61 Remainders 114-20 Repair theory 69 Representations external 79-80, 89 internal 86-7 mental 96,100 social 98, 100 Role of theory 3, 13 Schemata 97, 100 Schemes 80 Scoring analytic 40 focused holistic 313,324 process 40 Scripts 99-100, 108-10 Search space 274,278 Self perception 68 Semantic field 79-86 processing 113, lIS, 122 space 97 Sense-making 114, 116, 118, 120-1 Situation-based reasoning 113,117-8,120 Specializing 21,23,26,33 Social cognition 98-9 interactions 80, 244, 253 stereotypes 100 Software subject specific 178 tools 177-8,255,258-9,261-5,302 Statistical laboratory 194 Statistics 193-203 Stereometix 256-9 Student's affects 4,6-9, 12 behaviors 8, 68 beliefs 9,206,215-7,305,307-10,312 outcomes 8-9
- 356. Systemic 62. 71 Task 272.274.278 features 5. 10 validity 36-7 variables 129. 195. 197-9 Teacher affects 4.6-11 346 Teaching 6-11. 17-27.89.162.330-1.334-41 actions 2. 24 skills 6 Terminals 98 Theoretical frame 78 Triple Representation Model 260-1 Validation 156.160-1 beliefs 4.6-7.288.291.304-5.309-12 van Hiele theory 106.110 behaviors 8 Verbal reports 128. 132 education 10.70.221-2.225-6.237. 302-4. Verbalizations 79.88 309.315.327.329-31.340-2 knowledge 10 perceptions 314.316-7.322-3 planning 6-7 role 2.183.220.228-30.232.235-7 strategies
- 357. NATO ASI Series F Including Special Programmes on Sensory Systems for Robotic Control (ROB) and on Advanced Educational Technology (AET) Vol. 22: SoftwareSystem Design Methods. TheChallengeofAdvanced ComputingTechnology. Edited by J. K. Skwirzynski. XIII, 747 pages. 1986. Vol. 23: Designing Computer-Based Learning Materials. Edited by H. Weinstock and A. Bork. IX, 285 pages. 1986. Vol. 24: Database Machines. Modern Trends and Applications. Edited by A. K. Sood and A. H. Qureshi. VIII, 570 pages. 1986. Vol. 25: Pyramidal Systems for Computer Vision. Edited by V. Cantoni and S. Levialdi. VIII, 392 pages. 1986. (ROB) Vol. 26: Modelling and Analysis in Arms Control. Edited by R. Avenhaus, R. K. Huber and J. D. Kettelle. VIII, 488 pages. 1986. Vol. 27: Computer Aided Optimal Design: Structural and Mechanical Systems. Edited by C. A. Mota Soares. XIII, 1029 pages. 1987. Vol. 28: Distributed Operating Systems. Theory und Practice. Edited by Y. Paker, J.-P. Banatre and M. Bozyigit. X, 379 pages. 1987. Vol. 29: Languages for Sensor-Based Control in Robotics. Edited by U. Rembold and K. Hormann. IX, 625 pages. 1987. (ROB) Vol. 30: Pattern Recognition Theory and Applications. Edited by P. A. Devijver and J. Kittler. XI, 543 pages. 1987. Vol. 31: Decision Support Systems: Theory and Application. Edited bv C. W. Holsapple and A. B. Whinston. X, 500 pages. 1987. Vol. 32: Information Systems: Failure Analysis. Edited by J. A. Wise and A. Debons. XV, 338 pages. 1987. Vol. 33: Machine Intelligence and Knowledge Engineering for Robotic Applications. Edited by A. K. C. Wong and A. Pugh. XIV, 486 pages. 1987. (ROB) Vol. 34: Modelling, Robustness and Sensitivity Reduction in Control Systems. Edited by R.F. Curtain. IX, 492 pages. 1987. Vol. 35: ExpertJudgment and ExpertSystems. Edited byJ. L. Mumpower, L. D. Phillips, O. Renn and V. R. R. Uppuluri. VIII, 361 pages. 1987. Vol. 36: Logic of Programming and Calculi of Discrete Design. Edited by M. Broy. VII, 415 pages. 1987. Vol. 37: Dynamics of Infinite Dimensional Systems. Edited by S.-N. Chow and J. K. Hale. IX. 514 pages. 1987. Vol. 38: Flow Control of Congested Networks. Edited by A. R. Odoni, L. Bianco and G. Szego. XII, 355 pages. 1987. Vol. 39: Mathematics and ComputerScience in Medical Imaging. Edited byM. A. Viergever and A. Todd-Pokropek. VIII, 546 pages. 1988. Vol. 40: Theoretical Foundations of Computer Graphics and CAD. Edited by R. A. Earnshaw. XX, 1246 pages. 1988. Vol. 41: Neural Computers. Edited by R. Eckmiller and Ch. v. d. Malsburg. XIII, 566 pages. 1988.
- 358. NATO ASI Series F Including Special Programmes on Sensory Systems for Robotic Control (ROB) and on Advanced Educational Technology (AET) Vol. 42: Real-Time Object Measurementand Classification. Edited byA. K. Jain. VIII, 407 pages. 1988. (ROB) Vol. 43: Sensors and Sensory Systems for Advanced Robots. Edited by P. Dario. XI, 597 pages. 1988. (ROB) Vol. 44: Signal Processing and Pattern Recognition in Nondestructive Evaluation of Materials. Edited by C. H. Chen. VIII, 344 pages. 1988. (ROB) Vol. 45: Syntactic and Structural Pattern Recognition. Edited by G. Ferrate, T. Pavlidis, A. Sanfeliu and H. Bunke. XVI, 467 pages. 1988. (ROB) Vol. 46: Recent Advances in Speech Understanding and Dialog Systems. Edited by H. Niemann, M. Lang and G. Sagerer. X, 521 pages. 1988. Vol. 47: Advanced Computing Concepts and Techniques in Control Engineering. Edited by M. J. Denham and A. J. Laub. XI, 518 pages. 1988. Vol. 48: Mathematical Models for Decision Support. Edited by G. Mitra. IX, 762 pages. 1988. Vol. 49: Computer Integrated Manufacturing. Edited by I. B. Turksen. VIII, 568 pages. 1988. Vol. 50: CAD Based Programming for Sensory Robots. Edited by B. Ravani. IX, 565 pages. 1988. (ROB) Vol. 51: Algorithms and Model Formulations in Mathematical Programming. Edited by S. W. Wallace. IX, 190 pages. 1989. Vol. 52: Sensor Devices and Systems for Robotics. Edited by A. Casals. IX, 362 pages. 1989. (ROB) Vol. 53: Advanced Information Technologies for Industrial Material Flow Systems. Edited by S. Y. Nof and C. L. Moodie. IX, 710 pages. 1989. Vol. 54: A Reappraisal of the Efficiency of Financial Markets. Edited by R. M. C. Guimaraes, B. G. Kingsman and S. J. Taylor. X, 804 pages. 1989. Vol. 55: Constructive Methods in Computing Science. Edited by M. Broy. VII, 478 pages. 1989. Vol. 56: Multiple Criteria Decision Making and Risk Analysis Using Microcomputers. Edited by B. Karpak and S. Zionts. VII, 399 pages. 1989. Vol. 57: Kinematics and Dynamic Issues in Sensor Based Control. Edited by G. E. Taylor. XI, 456 pages. 1990. (ROB) Vol. 58: Highly Redundant Sensing in Robotic Systems. Edited by J. T. Tou and J. G. Balchen. X, 322 pages. 1990. (ROB) Vol. 59: Superconducting Electronics. Edited by H. Weinstock and M. Nisenoff. X, 441 pages. 1989. Vol. 60: 3D Imaging in Medicine. Algorithms, Systems, Applications. Edited by K. H. Hahne, H. Fuchs and S. M. Pizer. IX, 460 pages. 1990. Vol. 61: Knowledge, Data and Computer-Assisted Decisions. Edited by M. Schader and W. Gaul. VIII, 421 pages. 1990. Vol. 62: Supercomputing. Edited by J. S. Kowalik. X, 425 pages. 1990. Vol. 63: Traditional and Non-Traditional Robotic Sensors. Edited by T. C. Henderson. VIII, 468 pages. 1990. (ROB)
- 359. NATO ASI Series F Including Special Programmes on Sensory Systems for Robotic Control (ROB) and on Advanced Educational Technology (AET) Vol. 64: Sensory Robotics for the Handling of Limp Materials. Edited by P. M. Taylor. IX, 343 pages. 1990. (ROB) Vol. 65: Mapping and Spatial Modelling for Navigation. Edited by L. F. Pau. VIII, 357 pages. 1990. (ROB) Vol. 66: Sensor-Based Robots: Algorithms and Architectures. Edited by C. S. G. Lee. X, 285 pages. 1991. (ROB) Vol. 67: Designing Hypermediafor Learning. Edited byD. H. Jonassen and H. Mandl. XXV, 457 pages. 1990. (AET) Vol. 68: Neurocomputing. Algorithms, Architectures and Applications. Edited by F. Fogelman Soulie and J. Herault. XI, 455 pages. 1990. Vol. 69: Real-Time Integration Methodsfor MechanicalSystemSimulation. Edited by E. J. Haug and R. C. Deyo. VIII, 352 pages. 1991. Vol. 70: Numerical LinearAlgebra, Digital Signal Processing and Parallel Algorithms. Edited by G. H. Golub and P. Van Dooren. XIII, 729 pages. 1991. Vol. 71: Expert Systems and Robotics. Edited by T. Jordanides and B.Torby. XII, 744 pages. 1991. Vol. 72: High-Capacity Local and Metropolitan Area Networks. Architecture and Performance Issues. Edited by G. Pujolle. X, 536 pages. 1991. Vol. 73: Automation and Systems Issues in AirTraffic Control. Edited byJ. A. Wise, V. D. Hopkin and M. L. Smith. XIX, 594 pages. 1991. Vol. 74: Picture Archiving and Communication Systems (PACS) in Medicine. Edited by H. K. Huang, O. Ratib, A. R. Bakker and G. Witte. XI, 438 pages. 1991. Vol. 75: Speech Recognition and Understanding. RecentAdvances, Trends and Applications. Edited by P. Laface and Renato De Mori. XI, 559 pages. 1991. Vol. 76: Multimedia Interface Design in Education. Edited by A. D. N. Edwards and S. Holland. XIV, 216 pages. 1992. (AET) Vol. 77: Computer Algorithms for Solving Linear Algebraic Equations. The State of the Art. Edited by E. Spedicato. VIII, 352 pages. 1991. Vol. 78: Integrating Advanced Technology into Technology Education. Edited by M. Hacker, A. Gordon and M. de Vries. VIII, 185 pages. 1991. (AET) Vol. 79: Logic, Algebra, and Computation. Edited by F. L. Bauer. VII, 485 pages. 1991. Vol. 80: IntelligentTutoringSystemsforForeign LanguageLearning. Edited byM. L. Swartzand M. Yazdani. IX, 347 pages. 1992. (AET) Vol. 81: Cognitive Tools for Learning. Edited by P. A. M. Kommers, D. H. Jonassen and J. T. Mayes. X, 278 pages. 1992. (AET) Vol. 82: Combinatorial Optimization. New Frontiers in Theory and Practice. Edited by M. AkgOI, H. W. Hamacher and S. TOfekQ. XI, 334 pages. 1992. Vol. 83: Active Perception and Robot Vision. Edited by A. K. Sood and H. Wechsler. IX, 756 pages. 1992.
- 360. NATO ASI Series F Including Special Programmes on Sensory Systems for Robotic Control (ROB) and on Advanced Educational Technology (AET) Vol. 84: Computer-Based Learning Environments and Problem Solving. Edited by E. De Corte, M. C. Linn, H. Mandl and L. Verschaffel. XVI, 488 pages. 1992. (AET) Vol. 85: Adaptive Learning Environments. Foundations alld Frontiers. Edited by M. Jones and P. H. Winne. VIII, 408 pages. 1992. (AET) Vol. 86: Intelligent Learning Environments and Knowledge Acquisition in Physics. Edited by A. Tiberghien and H. Mandl. VIII, 285 pages. 1992. (AET) Vol. 87: Cognitive Modelling and Interactive Environments. Edited by F. L. Engel, D. G. Bouwhuis, T. B6sser and G. d'Ydewalle. IX, 311 pages. 1992. (AET) Vol. 88: Programming and Mathematical Method. Edited by M. Broy. VIII, 428 pages. 1992. Vol. 89: Mathematical Problem Solving and New Information Technologies. Edited by J. P. Ponte, J. F. Matos, J. M. Matos and D. Fernandes. XV, 346 pages. 1992. (AET) Vol. 90: Collaborative Learning Through Computer Conferencing. Edited byA. R. Kaye. X, 260 pages. 1992. (AET) Vol. 91 :NewDirectionsfor IntelligentTutoring Systems. Edited by E. Costa. X, 294pages. 1992. (AET)