An ICT Environment To Assess And Support Students Mathematical Problem-Solving Performance In Non-Routine Puzzle-Like Word Problems

An ICT environment to assess and support students’ mathematical
problem-solving performance in non-routine puzzle-like word
problems1
Angeliki Kolovou*
Marja van den Heuvel-Panhuizen*#
Arthur Bakker*
Iliada Eliax
*FIsme, Utrecht University, the Netherlands
#IQB, Humboldt University Berlin, Germany
x
Department of Education, University of Cyprus
Abstract
This paper reports on a small-scale study on primary school students’ problem-solving
performance. In the study, problem solving is understood as solving non-routine puzzle-
like word problems. The problems require dealing simultaneously with multiple,
interrelated variables. The study employed an ICT environment both as a tool to
support students’ learning by offering them opportunities to produce solutions,
experiment and reflect on solutions, and as a tool to monitor and assess the students’
problem solving processes. In the study, 24 fourth-graders were involved from two
schools in the Netherlands. Half of the students who belonged to the experimental group
worked in pairs in the ICT environment. The analysis of the students’ dialogues and
actions provided us with a detailed picture of students’ problem solving and revealed
some interesting processes, for example, the bouncing effect that means that the
students first come with a correct solution and later give again an incorrect solution.
The test data collected before and after this “treatment” did not offer us a sufficient
basis to draw conclusions about the power of ICT environment to improve the students’
problem-solving performance.
Introduction
Problem solving is a major goal of mathematics education and an activity that can be
seen as the essence of mathematical thinking (Halmos, 1980; NCTM, 2000). With
problems tackled in problem solving typically defined as non-routine (Kantowski, 1977),
it is not surprising that students tend to find mathematical problem solving challenging
and that teachers have difficulties preparing students for it. Despite the growing body of
research literature in the area (Lesh & Zawojewski, 2007, Lester & Kehle, 2003,
1
Paper presented in TSG19 at ICME11, 6-13 July, 2008
Theme (d): Research and development in problem solving with ICT technology

176
Schoenfeld, 1985), there is still much that we do not know about how students attempt
to tackle mathematical problems and how to support students in solving non-routine
problems.
In order to get a better understanding of Dutch primary school students’
competences in mathematical problem solving, the POPO study started in 2004. In this
study, 152 fourth-grade students who are high achievers in mathematics were
administered a paper-and-pencil test on non-routine problem solving. In a few items,
students were asked to show their solutions strategies. The results were disappointing.
Students did not show a high performance in problem solving, despite their high
mathematics ability (Van den Heuvel-Panhuizen, Bakker, Kolovou, & Elia, in
preparation). Although the students’ scribbling on the scrap paper gave us important
information about their solution strategies, we were left with questions about their
solution processes. Moreover, after recognizing that even very able students have
difficulties with solving the problems, we wondered what kind of learning environment
could help students to improve their problem solving performance. The POPO study
thus yielded a series of questions. To answer these questions we started the iPOPO
study which – in accordance with the two main questions that emerged from the POPO
study – implied a dual research goal.
First, the iPOPO study aimed at gaining a deeper understanding of the primary
school students’ problem solving processes, and, second, it explored how their problem-
solving skills can be improved. For this dual goal of assessing and teaching, the study
employed ICT both as a tool to support students’ learning by offering them
opportunities to produce solutions, experiment and reflect on solutions, and as a tool to
monitor and assess the students’ problem solving processes. In particular, we designed a
dynamic applet called Hit the target, which is based on one of the paper-and-pencil
items used in the POPO study. Like several of these items, it requires students to deal
with multiple, interrelated variables simultaneously and thus prepares for algebraic
thinking.
This paper focuses on the following two research questions: Which problem-
solving strategies do fourth-grade students deploy in this Hit the target environment?
Does this ICT environment support the students’ problem-solving performance?
Theoretical background
Mathematical problem solving
The term “problem solving” is used for solving a variety of mathematical problems,
ranging from real-life problems to puzzle-like problems. Our focus is on the latter. We
consider problem solving as a cognitive activity that entails strategic thinking, and that
includes more than just carrying out calculations. An episode of problem solving may be
considered as a small model of a learning process (D’Amore, & Zan, 1996). In problem
solving, the solution process often requires several steps. First the students have to
unravel the problem situation. Subsequently, they have to find a way to solve the problem
by seeking patterns, trying out possibilities systematically, trying special cases, and so on.

177
While doing this they have to coordinate relevant mathematical knowledge, organize the
different steps to arrive at a solution and record their thinking. In sum, in our view
problem solving is a complex activity that requires higher order thinking and goes beyond
standard procedural skills (cf., Kantowski, 1977).
An example of a mathematical problem used in the POPO study is shown in Figure 1.
Someone who knows elementary algebra might use this knowledge to find the answer to
this problem by, for example, solving the equation 2x – 1(10 – x) = 8. Fourth-grade students,
however, have not yet learned such techniques, but can still use other strategies such as
systematic listing of possible solutions or trial and error. Grappling with such problems
might be a worthwhile experiential base for learning algebra in secondary school (cf., Van
Amerom, 2002).
Figure 1: Problem used in the POPO Study
Within the complexity that characterizes problem-solving activity, D’Amore and Zan
(1996) identify the involvement of three interrelated discrete variables, as follows: the
subject who solves the task; the task; and the environment in which the subject solves the
task. This study primarily focuses on the third variable, referring to the conditions, which
may help a subject to improve his problem solving abilities.
The research questions stated in Section 1 address two different aspects that are
closely related: monitoring learning and supporting that learning. We have chosen to use
ICT for both of these aspects, because – as Clements (1998) recognized – ICT (1) can
provide students with an environment for doing mathematics and (2) can give the
possibility of tracing the students’ work.
ICT as a tool for supporting mathematical problem solving
A considerable body of research literature has shown that computers can support children in
developing higher-order mathematical thinking (Suppes, 1966; Papert, 1980; Clements &

178
Meredith, 1993; Sfard & Leron, 1996; Clements, 2000; Clements, 2002). Logo
programming, for example, is a rich environment that elicits reflection on mathematics and
one’s own problem-solving (Clements, 2000). Suitable computer software can offer unique
opportunities for learning through exploration and creative problem solving. It can also help
students make the transition from arithmetic to algebraic reasoning, and emphasize
conceptual thinking and problem solving. According to the Principles and Standards of the
National Council of Teachers of Mathematics (NCTM, 2000) technology supports decision-
making, reflection, reasoning and problem solving.
Among the unique contributions of computers is that they also provide students
with an environment for testing their ideas and giving them feedback (Clements, 2000).
In fact, feedback is crucial for learning and technology can supply this feedback
(NCTM, 2000). Computer-assisted feedback is one of the most effective forms of
feedback because “it helps students in building cues and information regarding
erroneous hypotheses”; thus it can “lead to the development of more effective and
efficient strategies for processing and understanding” (Hattie & Timperley, 2007, p.102).
More generally, computer-based applications can have significant effects on what
children learn because of “the computer’s capacity for simulation, dynamically linked
notations, and interactivity” (Rochelle, Pea, Hoadley, Gordin, & Means, 2000, p. 86).
This learning effect can be enhanced by peer interaction. Pair and group work
with computer software can make students more skilful at solving problems, because
they are stimulated to articulate and explain their strategies and solutions (Wegerif &
Dawes, 2004). Provided there is a classroom culture in which students are willing to
provide explanations, justifications, and arguments to each other, we can expect better
learning.
ICT as a window onto students’ problem solving
Several researchers have emphasized that technology-rich environments allow us to
track the processes students use in problem-solving (Bennet & Persky, 2002). ICT can
provide mirrors to mathematical thinking (Clements, 2000) and can offer a window onto
mathematical meaning under construction (Hoyles & Noss, 2003, p. 325). The potential
of computer environments to provide insight into students’ cognitive processes makes
them a fruitful setting for research on how this learning takes place.
Because software enables us to record every command students make within an
ICT environment, such registration software allows us to assess their problem solving
strategies in more precise ways than can paper-and-pencil tasks. Therefore, computer-
based tasks as opposed to conventional paper-and-pencil means have received growing
interest in the research literature for the purposes of better assessment (Clements 1998;
Pellegrino, Chudowsky, & Glaser, 2001; Bennet & Persky, 2002; Burkhardt & Pead,
2003; Threlfall, Pool, Homer, & Swinnerton, 2007; Van den Heuvel-Panhuizen, 2007).
Where early-generation software just mimicked the paper-and-pencil tasks, recent
research shows that suitable tasks in rich ICT environments can also bring about higher-
order problem solving. For example, Bennet and Persky (2002) claimed that
technology-rich environments tap important emerging skills. They offer us the
opportunity to describe performance with something more than a single summary score.

179
Furthermore, a series of studies indicated that the use of ICT facilitates the assessment
of creative and critical thinking by providing rich environments for problem solving
(Harlen & Deakin Crick, 2003).
By stimulating peer interaction we also expect that students will articulate more
clearly their thinking than when working individually. Thus, student collaboration has a
twofold role: it helps them shape and broaden their mathematical understandings and it
offers researchers and teachers a nicely bounded setting in order to observe collaboration
and peer interaction (Mercer & Littleton, 2007).
Method
Research design and subjects
The part of the iPOPO study described in this paper is a small-scale quasi-experiment
following a pre-test-post-test control group design. In total, 24 fourth-graders from two
schools in Utrecht participated in the study. In each school, 12 students who belonged to
the A level according the Mid Grade 4 CITO test – in other words to the 25% best
students according to a national norm – were involved. Actually, the range of the scores
that correspond to level A of the Mid Grade 4 CITO test is between 102 and 151 points.
In both schools, the average mathematics CITO score of the class was A and the
average “formation weight” of the class and the school was 1. This means that the
students were of Dutch parentage and came from families in which the parents had at
least secondary education. First, of each school six students were selected for the
experimental group. Later on, the group of students was extended with six students to be
in the control group. These students also belonged to the A level, but, unfortunately,
their average score was lower than that of the experimental group. The teacher
obviously selected the more able students first.
An ICT environment was especially developed for this study to function as a
treatment for the experimental group. Before and after the treatment, a test was
administered as pre-test and post-test. The control group did the test also two times, but
did not get the treatment in between. The quasi-experiment was carried out in March-
April 2008. The complete experiment took about four weeks: in the first week the
students did the test, in the second week the experimental group worked in the ICT
environment and in the fourth week the students did again the test.
Pre-test and post-test
The test that was used as pre-test and post-test was a paper-and-pencil test consisting of
three non-routine puzzle-like word problems, titled Quiz (see Figure 1), Ages, and Coins.
The problems are of the same type and require that the students deal with interrelated
variables. The test sheets contain a work area on which the students had to show how they
found the answers. The students’ responses were coded according to a framework that was
developed in our earlier POPO study. The framework covers different response

180
characteristics including whether the students gave specific strategy information, how they
represented that strategy and what kind of problem-solving strategies they applied.
Applet used as treatment
The treatment consisted of a Java applet called Hit the target. 2
It is a simulation of an
arrow shooting game. The screen shows a target board, a score board featuring the
number of gained points, and the number of hit and missed arrows, a frame that contains
the rules for gaining or loosing points, and an area in which the number of arrows to be
shot can be filled in. A hit means that the arrow hits the yellow circle in the middle of the
target board; then the arrow becomes green. A miss means that the arrow hits the gray
area of the board; in that case, the arrow becomes red.
Figure 2: Screen view of applet in the computer-shooting mode
The applet has two modes of shooting: a player shoots arrows by him or herself or lets
the computer do the shooting (see Figure 2). In case the player shoots, he or she has to
drag the arrows to the bow and then draw and unbend the bow. The computer can do the
shooting if the player selects the computer-shooting mode and fills in the number of
arrows to be shot. Regarding the rules for gaining points there are also two modes: the
player determines the rules or the computer does this. The maximum number of arrows
is 150 and the maximum number of points the player can get by one shot is 1000.
As the player shoots arrows or lets the computer do so, the total score on the
scoreboard changes according to the number of arrows shot and the rules of the game.
The player can actually see on the scoreboard how the score and the number of hits and
misses change during the shooting. The player can also remove arrows from the target
board, which is again followed by a change in the total score. When the player wants to
start a new shooting round, he or she must click on the reset button. The player can
change the shooting mode or the rules of the game at any time during the game.
2
The applet has been programmed by Huub Nilwik.

181
The aim of the applet is that the students obtain experience in working with
variables and realize that the variables are interrelated (see Figure 3); a change in one
variable affects the other variables. For example, if the rules of the game are changed,
then the number of arrows should be also changed to keep the total points constant.
Figure 3: Variables involved
The 12 students of the experimental group worked for about 30 minutes in pairs with
the applet. The pairs were chosen by the researcher in such a way that all of them would
have about the same average CITO score and consisted of a boy and a girl. The dialogue
between the students and their actions on the applet were recorded by Camtasia
software, which captures the screen views and the sound in a video file. Scrap paper
was also available to the students. Before the students started working, it was explained
to them that they should work together, use the mouse in turns, explain their thinking to
each other, and justify their ideas.
The work with the applet started with five minutes of free playing in which the
students could explore the applet. Then, they had to follow a pre-defined scenario
containing a number of directed activities and three questions (see Table 1). The first
two questions (A and B) are about the arrows while the rules of the game and the gained
points are known. In the third question (C), which consists of two parts, the rules of the
game are unknown.
Table 1: Questions in the pre-defined scenario
Arrows Rules Gained
points
A. How many hits and
misses?
Hit +3 points, miss –1 point 15 points
B. How many hits and
misses?
Hit +3 points, miss +1 point 15 points
15 hits and 15 misses C1. What are the rules?
C2. Are other rules possible to get
the result 15 hits-15 misses-15
points?
15 points
Total points
Rules of the game
Number of arrows
- hits and misses
- total arrows
Total points
Rules of the game
Number of arrows
- hits and misses
- total arrows

182
The directed activities were meant to assure that all the students had all the necessary
experiences with the applet. During these activities, the students carried out a number of
assignments in order to become familiar with the various features of the applet: the
player-shooting mode, the computer-shooting mode, the rules of the game, and the total
score. First, the students had to shoot one arrow, followed by shooting two arrows and
then a few more, in order to get five arrows on the target board. Their attention was then
drawn to the scoreboard; they had five hits and zero misses and their total score was
zero since the rules of the game had been initially set to zero. After that, the rules were
changed so that a hit meant that three points were added. Then, the students had to shoot
again five arrows in both shooting modes, each resulting in a total score of 15 points.
Afterward, the rule was changed again. A miss then meant that one point had to be
subtracted. At this point, Question A was asked, followed by Questions B and C.
Results
The students’ problem-solving strategies in the ICT environment
All pairs were successful in answering the Questions A, B, and C. The solutions were
found based on discussions and sharing ideas for solutions. In all cases, explanations
were provided and the talk between the students stimulated the generation of hypotheses
and solutions. However, some students provided more elaborate explanations and
suggested more successful problem-solving strategies than others.
In order to identify the problem-solving strategies the students applied, we
analyzed all dialogues between the students. In this paper, however, we will only
discuss our findings with respect to Questions C1 and C2, which triggered the richest
dialogues.
Characteristic for Question C is that the number of hits and misses, and the number
of points were given, but that the students had to find the rules. All pairs were able to
answer Questions C1 and C2, and most of them could generalize to all possible solutions
(“It is always possible if you do one less”), albeit on different levels of generalization.
The Tables 2 and 3 show which strategies the pairs used when solving Questions C1 and
C2. Each pair of students is denoted with a Roman numeral. Pairs I, II, and III belong to
school A, while Pairs IV, V, and VI belong to school B.
When answering Question C1 (see Table 2), four out of the six pairs directly came
up with a correct solution. Pair VI found the correct solution in the third trial. The most
interesting strategy came from Pair IV. This pair found the correct solution in the
second trial. The pair started with a canceling-out solution (+1 –1) resulting in a total
score of zero and then changed the solution to get 15 points in total.

183
Table 2: Problem-solving strategies when solving C1
Strategy Pairs
I II III IV V VI
Average CITO score per pair
111 111 114 11
0
111 10
7
1a Directly testing a correct solution (+2 –1 or
+1 +0)
1* 1 1 1
2a Testing incorrect canceling-out solution
(+1 –1)
1
2b Testing other incorrect solution(s) 1
3 Adapting the rules of the game until a
correct solution is reached
2 2
Number of trials 1 1 1 2 1 3
* The numbers in the cell indicate the order in which the strategies were applied
Table 3 shows that having found a correct solution in C1 did not mean that the
students had discovered the general principle (or the correct solution rule) of getting “15
hits-15 misses-15 points”. Even after finding the correct solution rule and generating a
series of correct solutions, some students tested wrong solutions again (we could call
this the “bouncing effect”). Perhaps they were not aware that there is only one correct
solution rule; the difference between the number of points added for every hit and the
number of points subtracted for every miss (or vice versa) should be 1, or the difference
between the number of hit-points and miss-points should be 15 points. The highest level
of solution was demonstrated by Pair VI, who recognized that the difference between
the points added and the points subtracted should be 15 (and that explains why the
difference between the number of points added for every hit and the number of points
subtracted for every miss – or vice versa – should be 1). A clever mathematical solution
came from the Pairs I and II. These students just used the correct solution to C1 in the
reverse way to get the required result of 15 points in total.
Besides strategies that directly or indirectly lead to a correct solution or rule, some
other characteristics were found in the solution processes (see Table 4). Four pairs
altered or ignored information given in the problem description. It is noteworthy that
during subsequent attempts to answer Question C2, some students insisted on keeping
the rules constant and changing the number of hits and misses in order to get a total of
15 points. Pair V, for example, changed the problem information (15 hits and 15 misses)
and started C2 with trying out the solution 1 hit is 15 point added and 1 miss is 15
points subtracted. The total score then became zero; subsequently, they set the number
of hits to 30 and the number of misses to 15, which resulted into a high score. Even
though at that point the researcher repeated the correct problem information, the
students ignored it persistently. In their third attempt, they changed the number of hits
and misses to 1 and 0 respectively and the total score became 15 instead of the reverse
(15 hits and 15 misses resulting in 15 points). Only when the researcher repeated the

184
question they considered the correct information and tried out the solution +4 –2 with
15 hits and 15 misses.
Table 3: Problem-solving strategies when solving C2
Strategy Pairs
I II III IV V VI
Average CITO score per pair
111 111 114 11
0
111 10
7
4a Repeating the correct solution to C1 2
4b Reversing the correct solution to C1 to
find another correct solution (–1 +2 or –0
+1/+0 +1)
1* 1/3
5a Generating a correct solution rule based
on testing of (a) correct solution(s) for
which the difference between the number
of points added for every hit and the
number of points subtracted for every
miss (or vice versa) is 1
2 4 6 1 4
5b Generating a correct solution rule based
on understanding that the difference
between hit-points and miss-points is 15
1
5c Generating a general correct solution rule
(“the difference of 1 also applies to 16-16-
16”)
8
6 Testing more correct solutions from a
correct solution rule
3 7 2 2
2b Testing other incorrect solution(s) 4 2 1/3/
5
1/3/
5
7 Generating an incorrect solution rule
(keeping ratio 2:1 or using rule +even
number –odd number) based on correct
solution(s)
2/4
* The numbers in the cell indicate the order in which the strategies were applied
However, the total score was 30 points and they suggested doubling the number of
misses to 30 so that the number of total points would be halved. This is clearly an
example of a wrong adaptation. Another example is from Pair VI. After having +3 and –
1 as the rule of the game, resulting in a total of 30 points, the students change the
number of hits into 10 in order to get 15 points as the result but forgetting that the
number of hits should be 15.

185
Table 4: Other characteristics of the solution processes
Characteristics C1 C2
Pairs Pairs
I II III IV V VI I II III IV V VI
Altering or ignoring
information
X X X X
Exploring large
numbers (≥1000)
X X X X X
Another characteristic of the solution processes was testing rules including large numbers.
Four of the six pairs tried out numbers bigger than 1000. These explorations all took place
when answering the second part of Question C. The students found this working with
large numbers quite amusing, since they then could get a large amount of total points.
That the students worked with numbers larger than 1000 was quite remarkable, because it
was not possible to fill in numbers of this size in the applet. Consequently, the students
had to work out the results mentally. It is also worth noting that some students understood
that one could go on until one million or one trillion (Pair IV). This means that several
students knew that there are infinite solutions, as it was made explicit by one pair (see
Pair II). Furthermore, most of the students used whole numbers and no one used negative
numbers. In one occasion, a student (from Pair II) suggested adding 1½ points for a hit,
but the applet does not have the possibility to test solutions with fractions or decimals.
Observing the students while working on the applet revealed that the students
demonstrated different levels of problem-solving activity. For example, there were
students that checked the correctness of their hypotheses by mental calculation, while
others just tried out rules with the help of the applet. None of them questioned the
infallibility of the applet; when they used the applet after they had found out that a rule
was wrong, they did this to make sure that they were really wrong. Furthermore, the
students also showed differences in the more or less general way in which they
expressed their findings. One of the students articulated that the general rule “a hit is
one point more (added) than the number of points (subtracted) by a miss” also applies to
other triads such as 16 hits-16 misses-16 points and in general to all triads of equal
numbers.
To conclude this section about the ICT environment, we must say that observing
the students while working with the applet gave us quite a good opportunity to get
closer to the students’ problem-solving processes.
Does the ICT environment support the students’ problem-solving performance?
In this section, we discuss the results from the pre-test and the post-test in the
experimental and control group. Figure 4 shows the average number of correct answers
per student in both groups in school A and school B.

186
Figure 4: Average number of correct answers per student in the pre and the post-test in
both groups
Figure 5: Average CITO score of the experimental and control group
As can been seen in Figure 4, if the group of students is taken as a whole, the
experimental group gained slightly from the treatment. However, we have too few data
to give a reliable answer to the research question. Only 12 students from school A and
12 students from school B were involved in this study and among these schools, the
results were quite different. Only in school A, there is a considerable improvement in
the scores of the post-test. Another issue is the mismatch between experimental and
control group (see also Section 3.1). In both schools, the control group scored lower
than the experimental group. This mismatch was more evident in school A. A plausible
explanation for these differences could be that although all students had an A score in
mathematics, the average CITO scores of the experimental group and the control group
were different in school A and school B (see Figure 5).
In fact, the differences between the average CITO score of the experimental and
control group in each school, presented in Figure 5, are similar to the differences
between the average scores of these groups in the paper-and-pencil test. In school A, the
control group has a lower CITO score than the experimental group. The same holds for
school B, but there the difference is smaller than in school A.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Exp Contr Exp Contr Exp Contr
School A School B Total
Pre-test
Post-test
Average
number
of
correct
answers
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Pre-test
Post-test
Average
number
of
correct
answers
102
104
106
108
110
112
114
Average
CITO-score
102
104
106
108
110
112
114
Average
CITO-score

187
Discussion
We started this study with two questions that emerged from the earlier POPO study. To
investigate these questions, we set up, as a start, a small-scale study in which an ICT
environment played a crucial role. The dialogues between the students and their actions
when working in the ICT environment gave us a first answer to the first research
question. The collected data provided us with a detailed picture of students’ problem
solving and revealed some interesting processes, for example, the bouncing effect and
the making of wrong adaptations.
Our second question is difficult to answer. The sample size, and the number of the
test items were not sufficient to get reliable results and the time we had at our disposal
was not enough to gather and analyze more data. Moreover, the time that the
experimental group worked in the ICT environment was rather limited to expect an
effect. Despite these shortcomings, we decided to carry out a small-scale study in order
to try out the test items and the ICT environment with a small group of students first.
Clearly, more data (more students, more schools and more problems) are needed
to confirm or reject our conjecture that having experience with interrelated variables in
a dynamic, interactive ICT environment leads to an improvement in problem solving
performance. For this reason, we will extend our study to larger groups of students,
involving students of higher grades and different mathematical ability levels as well.
Moreover, to see more of an effect we will enlarge the working in the ICT environment
substantially. In addition, we will extend the study by analyzing the students’ problem-
solving strategies when solving paper-and-pencil problems. Our experiences from the
present study will serve as a basis for doing this future research.
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