AbstractLearning how to solve mathematical problems has been a long withstanding difficulty faced by students and has been given constant focus in mathematical development. Successful problem solvers spend more time analyzing a problem and they will find all the alternatives related to the situation. Therefore many educators (Cai, 2003, National Council of Teachers of Mathematics, 2000; Smith, 1991) agreed that teaching mathematics through problem solving hold great promise. Furthermore, according to the Malaysian Curriculum Development Centre (2003), one of the important objectives in teaching mathematics is to develop student's mathematical problem solving skills. Mathematical problems are not simply computational tasks type but are problems which also require appropriate selection of strategies and decisions that lead to logical solutions. Word problem solving is one of the important components of mathematics problem solving which incorporate real-life problems and applications. However, many researches revealed that students express great difficulties in handling a word or story problem. This study examined the use of visual representations which posed great difficulties in most mathematical tasks. Secondary students were given the 15Mathematical Processing Instrument (MPI) adapted from Hegarty and Kozhevnikov (1999). This study examined 381 students from eight secondary schools in three different districts in Malacca. Results indicated that less than two percent of the problems were solved using pictorial representation and most of the students preferred to used schematics solutions

Ahmad, Azizah

Tarmizi, Rohani Ahmad

Nawawi, Mokhtar

Schmolz, Carl

Digitale Sammlungen - Goethe-Universität Frankfurt am Main

Ueber den derzeitigen Stand der gesetzlichen Schutzbewegung zu Gunsten der Alpenflora unter besonderer Berücksichtigung der Tätigkeit des "Vereins zum Schutze und zur Pflege der Alpenpflanzen" : Nachtrag IV.

Elsevier - Publisher Connector 

Procedia Social and Behavioral Sciences 8 (2010) 356–361
Available online at www.sciencedirect.com
International Conference on Mathematics Education Research 2010 (ICMER 2010) 
Visual Representations in Mathematical Word Problem Solving 
Among Form Four Students in Malacca 
 
Azizah Ahmada,*, Rohani Ahmad Tarmizia,b, Mokhtar Nawawia,b 
 
aLaboratory of  Innovations in Mathematics Education,  Institute for Mathematical Research, 
bFaculty of Educational Studies 
Universiti Putra Malaysia, Serdang, Selangor, MALAYSIA. 
 
Abstract 
Learning how to solve mathematical problems has been a long withstanding difficulty faced by students and has been given 
constant focus in mathematical development.   Successful problem solvers spend more time analyzing a problem and they will 
find all the alternatives related to the situation. Therefore many educators (Cai, 2003, National Council of Teachers of 
Mathematics, 2000; Smith, 1991) agreed that teaching mathematics through problem solving hold great promise.  Furthermore, 
according to the Malaysian Curriculum Development Centre (2003), one of the important objectives in teaching mathematics is 
to develop students’ mathematical problem solving skills. Mathematical problems are not simply computational tasks type but are 
problems which also require appropriate selection of strategies and decisions that lead to logical solutions. Word problem solving 
is one of the important components of mathematics problem solving which incorporate real-life problems and applications.  
However, many researches revealed that students express great difficulties in handling a word or story problem. This study 
examined the use of visual representations which posed great difficulties in most mathematical tasks.  Secondary students were 
given the 15Mathematical Processing Instrument (MPI) adapted from Hegarty and Kozhevnikov (1999).  This study examined 
381 students from eight secondary schools in three different districts in Malacca. Results indicated that less than two percent of 
the problems were solved using pictorial representation and most of the students preferred to used schematics solutions. 
 
© 2010 Elsevier Ltd. 
 
Keywords: Mathematics; Visualization; Word problem 
1. Introduction 
   
Learning how to solve mathematical problems has been a long withstanding difficulty faced by students and has 
been given constant focus in mathematical development.   Successful problem solvers spend more time analyzing a 
problem and they will find all the alternatives related to the situation. Therefore many educators (Cai, 2003, 
National Council of Teachers of Mathematics [NCTM], 2000; Smith, 1991) agreed that teaching mathematics 
through problem solving hold great promise.  Furthermore, according to the Malaysian Curriculum Development 
Centre (2003), one of the important objectives in teaching mathematics is to develop students’ mathematical 
 
* Corresponding author. 
E-mail address: sehanisa@yahoo.com.my . 
1877-0428 © 2010 Published by Elsevier Ltd.
doi:10.1016/j.sbspro.2010.12.050
Open access under CC BY-NC-ND license.
Open access under CC BY-NC-ND license.
Azizah Ahmad et al. / Procedia Social and Behavioral Sciences 8 (2010) 356–361 357
problem solving skills. Mathematical problems are not simply computational tasks type but also require appropriate 
selection of strategies and decisions that lead to logical solutions.  
 
Word problem solving is one of the important components of mathematics problem solving which incorporate real-
life problems and applications.  However, many researches revealed that students express great difficulties in 
handling a word or story problem (Gagne, 1983; Orton, 1992, Ekbia & Alamolhodaei, 2000; Takahashi 2000; 
Alamolhodaei, 2002; Nickson 2004).  Word problem is actually a story problem and the students normally have to 
relate between the known and the unknown.  As Gagne (1983) suggested, in the process of mathematical word 
problem solving, student should be able to translate the concrete to the abstract and the abstract to the concrete.  
Therefore the mathematical word problem exam is more unique and challenging task than the ordinary mathematics 
task (Alamolhodaei, 2002).  Thus, students normally find difficulty in solving word problems initially from 
translating the word representations into mathematical representation. Related problem to this was students’ 
difficulty in deriving ‘mental images’ which then followed by operative actions “transforming in to dynamic images, 
invoking images of formulae from memory” (Dorfler, 1991; Presmeg, 197). Presmeg (1997) pointed out that 
concrete imagery needs be coupled with rigorous analytical thought processes to be effectively used in mathematics 
problem solving. Hence “It is argued that the status of visualization in mathematics education should and can be 
upgraded from that of a helpful learning aid” (Dreyfuss, 1991 pg. 30). Dreyfuss also pointed out that students 
basically were reluctant to use visualization in mathematics and this may be attributed to low status accorded to 
visual aspects of mathematics in the classroom. However, he also posed many effective examples demonstrating the 
powerful use visualization in mathematical reasoning and learning. 
 
The uses of visual representations in mathematical word problem are very useful. Hegarty and Kozhevnikov (1999) 
has approved that the use of schematic spatial representations was associated with success in mathematical problem 
solving, whereas use of pictorial representations was negatively correlated with success.  Lean and Clements (1981) 
also found that there are different types of visual representational strategies used by students when solving 
mathematical problems by separating student-generated imagery into five categories: concrete imagery, pattern 
imagery, kinesthetic imagery, dynamic imagery, and memory of formula.  However Presmeg (1986) argued that 
concrete imagery (vivid pictorial images of objects contained in mathematical problems) may actually focus the 
reasoning on irrelevant details and distract the ‘solver” from the main element of the problem.  
 
This study adapted the method and instruments used by Hegarty and Kozhevnikov (1999).   The research questions 
of this study were as follows: i) What is the level of achievement in mathematical word problem solving among 
Form Four students in Malacca? ii) What visualization strategy do the Form Four students use in attempting 
mathematical word problem solving? iii) What is the relationship between the type of visualization strategy used by 
the students in solving word problems and their mathematical achievement?  
 
2. Methodology 
 
2.1 Population and sample 
 
The target population for this study is Form Four students in the state of Malacca in Peninsular Malaysia. A total of 
381 students took part in this research using the multi-stage cluster sampling. The subjects of study were from eight 
different schools (second cluster) in Melaka chosen from three different districts (first cluster) in Melaka, namely 
Jasin, Melaka Tengah and Alor Gajah.  Table 1 shows the demographic descriptions. 
 
Table 1: Demographic information on the participating students 
 
Gender Male 166 
Female  215 
Race Malay 272 
Chinese 100 
Indian 7 
358  Azizah Ahmad et al. / Procedia Social and Behavioral Sciences 8 (2010) 356–361
Others 2 
Mathematics Grade in PMR A  231 
B 96 
C 54 
 
2.2 The Research Materials 
 
The research material is a set of paper and pencil test named Mathematical Processing Istrument (MPI).  The test 
comprised of 15-items mathematical word problem solving adapted from Hegarty and Kozhevnikov (1999). The 
word problems were also translated into Malay language and posed alongside the English version.  This would help 
the students who have the difficulty in understanding English language.  Table 2 below shows the word problems. 
Forty-five minutes were given to the students to answer the questions (with the maximum of 3 min per question).  
The students’ responses were analyzed.  The first score was the total number of correct answers or work solutions 
shown on the test set.  Each correct answer or work solution were awarded one mark whereas the incorrect answer 
or work solution were awarded zero mark. Further analyses on the type and frequency of visualization used by the 
students for solving the mathematical word problem were conducted.  Four categories of visualizations were used: 
no answer, pictorial, schematic and the combination of pictorial and schematic.  The first category included students 
did not attempt to answer the word problems and the second category included students’ solution involving drawing 
pictures or diagrams as formation of “mental images” related to the word problems.  The third category included 
students’ solutions involving schematics (objects and/or parts of an object described in the problem object) and the 
fourth category included students’ solution involving the combination of pictorial and schematics. 
 
 
Table 2 :  Fifteen mathematical word problems 
 
No Mathematical word problems No of 
words 
No of 
sentences 
1 At the two ends of a straight path, a man planted a tree and then every 5 m 
along the path he planted another tree. The length of the path is 15 m. How 
many trees were planted? 
38 3 
2 On one side of the scale, there is a 1kg weight and half a brick. On the other 
side, there is one full brick. The scale is balanced. What is the weight of the 
brick? 
36 4 
3 A balloon first rose 200 m from the ground, then moved 100 m to the east, 
then dropped 100 m. It travelled 50 m to the east, and finally dropped straight 
to the ground. How far was the balloon from its original starting point? 
44 3 
4 In an athletic race Jim is 4 m ahead of Tom and Peter is 3 m behind Jim. How 
far is Peter ahead of Tom? 
25 2 
5 A square (A) has an area of 1 square meter. Another square (B) has sides 
twice as long. What is the area of B? 
24 3 
6 From a long stick of wood, a man cut six short sticks, each 2 feet long. He 
then found he had piece of 1 foot long left over. Find the length of the original 
stick? 
35 3 
7 The area of a rectangular field is 60 square meters. If its length is 10 m, how 
far would you have travelled if you walked the whole way around the field? 
31 2 
8 Jack, Paul, and Brian all have birthdays on the 1st of January, but Jack is 1 
year older than Paul and Jack is 3 years younger than Brian. If Brian is 10 
years old, how old is Paul? 
38 2 
9 The diameter of a can of peaches is 10 cm. How many cans will fit in a box 
30 cm by 40 cm (one layer only)? 
26 2 
10 Four young trees were set out in a row 10 m apart. A well was located by the 
last tree. A bucket of water is needed to water two trees. How far would a 
gardener have to walk altogether if he had to water the four trees using only 
51 4 
Azizah Ahmad et al. / Procedia Social and Behavioral Sciences 8 (2010) 356–361 359
one bucket? 
11 A hitchhiker set out on a journey of 60km. He walked the first 5km then got a 
lift from a truck driver. When the driver dropped him off he still had half of 
his journey to travel. How far had he travelled in the truck? 
47 4 
12 How many picture frames 6-cm long and 4-cm wide can be made from a 
piece of framing 200-cm long? 
22 1 
13 On one side of a scale, there are three pots of jam and a 100 oz. weight. On 
the other side, there are a 200g and a 500g weight. The scale is balanced. 
What is the weight of a pot of jam? 
44 4 
14 A ship was Northwest. It made a turn of 90 degrees to the right. An hour later 
it made a turn through 45 degrees to the left. In what direction was it then 
travelling? 
34 4 
15 15. There are eight animals on a farm. Some of them are hens and some are 
rabbits. Between them they have 22 legs. How many hens and how many 
rabbits are on the farm?  
33 4 
  
3. Results 
 
Table 3 shows the frequencies and percentage of students’ response to the MPI questions. An overall view of the 
correct answers indicated that a large percentage of students (96.1%) were able to answer MPI-6 question. The most 
difficult question was MPI-10, with only 44 students (11.5%) able to answer the question correctly whilst the other 
184 pupils failed to answer the question correctly.  Questions 4, 6, 8, 11, 13, and 15 were solved correctly by a large 
percentage of students; 95%, 96.1%, 84.8%, 74%, 83.2% and 72.2% respectively.  MPI-3, MPI-7 and MPI-14 
questions were solved correctly by more than half of the students, 58.3%, 62.2% and 63.8% respectively. Overall 
more than half (59.3%) of the questions were answered correctly and less than 8% of the questions left unanswered. 
         
Table 3: Frequencies and percentage of students response to the questions. 
 
Questions Correct answer 
(n=381) 
Incorrect answer 
(n=381) 
Omission (no answer) 
(n=381) 
 F % f % F % 
1 88 23.1 290 76.1 3 0.8 
2 174 45.7 163 42.8 44 11.5 
3 222 58.3 133 34.9 26 6.8 
4 362 95.0 18 4.7 1 0.3 
5 173 45.4 157 41.2 51 13.4 
6 366 96.1 15 3.9 0 0 
7 237 62.2 115 30.2 29 7.6 
8 323 84.8 52 13.6 6 1.6 
9 131 34.4 184 48.3 66 17.3 
10 44 11.5 309 81.1 28 7.3 
11 282 74.0 81 21.3 18 4.7 
12 153 40.2 191 50.1 37 9.7 
13 317 83.2 35 9.2 29 7.6 
14 243 63.8 95 24.9 43 11.3 
15 275 72.2 58 15.2 48 12.6 
 
Table 4 shows the frequencies and percentages of visual representations shown by the students in answering the 15 
MPI questions. Most of the students solved the questions using a schematic representation, with an average of 
66.5% of the students attempted the schematic approach. As low as 38.3% to as high as 81.6% of the students 
showed schematic solution mode hence preferred non-visual mode in solving the word problems. An average of 
1.4% of the students used pictorial representation in answering the 15 MPI problems.  However, nearly a quarter 
(average of 26.4%) of the questions was answered using the combination of pictorial and schematics 
360  Azizah Ahmad et al. / Procedia Social and Behavioral Sciences 8 (2010) 356–361
representations. The following figure shows examples of students’ combination of pictorial and schematics 
representations. 
 
 
Figure 1: Examples of the combination of pictorial and schematics representation for MPI-4 and MPI-5 
 
Table 4: Frequencies and percentages of types of visual representations 
 
Questions No answer 
(n=381) 
Pictorial 
(n=381) 
Schematics 
(n=381) 
Combination 
(n=381) 
 f % f % f % F % 
1 3 0.8 1 0.3 299 78.5 78 20.5 
2 44 11.5 9 2.4 267 70.1 61 16.0 
3 26 6.8 7 1.8 236 61.9 112 29.4 
4 1 0.3 1 0.3 237 62.2 142 37.3 
5 51 13.4 7 1.8 245 64.3 78 20.5 
6 0 0 0 0 306 80.3 75 19.7 
7 29 7.6 6 1.6 201 52.8 145 38.1 
8 6 1.6 1 0.3 311 81.6 63 16.5 
9 66 17.3 8 2.1 230 60.4 77 20.2 
10 28 7.3 8 2.1 193 50.7 152 39.9 
11 18 4.7 0 0 310 81.4 53 13.9 
12 37 9.7 8 2.1 25 65.9 85 22.3 
13 29 7.6 1 0.3 272 71.4 79 20.7 
14 43 11.3 22 5.8 146 38.3 170 44.6 
15 48 12.6 2 0.5 297 78.0 34 8.9 
Total 429 7.5 81 1.4 3801 66.5 1404 24.6 
 
Generally, findings indicated that students preference in solving the word problems were mixed or in combination 
form. This may suggest that this group of students may have formed pictorial images during the problem solving but 
their mental images were not fully concrete or diagrammatize hence does not lend support to learners during the 
problem solving sessions. Ironically, all students took the effort to answer MPI-6 and more than 80% of them used 
schematics representations.  The most frequent used of pictorial representation (22 pupils) was on MPI-16. This may 
be due to the characteristics of question which encouraged or lead students to draw the direction of the ship.  On the 
other hand, students did not show any use of such representations on MPI-6 and MPI-11 hence indicating that 
students preferred to use schematic representation involving arithmetic and algebra solutions.  Overall, these 
findings indicated that learners’ preference in visualization of word mathematical problems is schematic rather than 
pictorial or combination of pictorial and schematic. 
 
Questions MPI-4 and MPI-10 showed weak relationships between visualization and MPI achievement for both 
questions.  The other MPI questions showed low and moderate relationship with correlation index (point-biserial) 
ranging from .34 to .58.  This indicated that students mathematical achievement is associated with ability to 
visualize (pictorial representations) or otherwise (non-visualize). This may suggest that students normally choose 
any strategy that the easiest and less time consuming to solve the problem without prior analyzing the problems 
visually. 
 
Azizah Ahmad et al. / Procedia Social and Behavioral Sciences 8 (2010) 356–361 361
4. Conclusion 
 
The aim of this study is to identify the level of achievement of mathematical word problem solving and the students’ 
representations or visualization of the problems in the process of deriving solutions.  In the two questions, majority 
of the students preferred to use schematics representations rather than pictorial representations.  In conclusion, 
students should be given the opportunity to utilize different types of representations in order to help them to 
understand any mathematical word problem solving.  Students’ preference mostly depends on what their teacher 
have taught them in class.  Therefore teachers also must be alert of the representations that can help them to 
understand and solve the mathematical word problems.  
 
This finding is additional support for past research suggesting visualization or imagery does not really helps the 
processes of problem solving and specifically in word problem solving (Hewitt, 2003; Herman et al., 2004). Much 
has to be done in this area of how visualization can be integrated in teaching and learning of mathematics in the 
classrooms. As Woolner (2004) pointed out, effective pedagogy that can enhance the use and power of visualization 
in mathematics education is perhaps a pressing concern and need further emphases. Ways or strategies to facilitate 
visualization in mathematics teaching and learning should be given concern. 
 
References  
 
Eisenberg, T. and Dreyfus T. 1991, ‘On the reluctance to visualize in mathematics’, in W. Zimmermann and S. Cunningham S. (eds.), 
Visualization in Teaching and Learning Mathematics, Mathematical Association of America, Washington, DC. 
Hegarty, M & Kozhevnikov, M. (1999). Types of Visual-Spatial Representations and Mathematical Problem Solving. Journal of Educational 
Technology Psychology, Vol 91(4), 684-689 
Gagne, R. M. (1983). Some issues in the psychology of mathematics instruction. Journal for  Research in Mathematics Education,14(1), 7–18. 
Presmeg, N. C. (1986a). Visualization in high school mathematics. For Learning of Mathematics, 63, 42-46. 
Van Garderen, D., & Montague, M. (2003). Visual-spatial representations and mathematical problem solving. Learning Disabilities & Research, 
18, 246254. 
Woolner, P. (2004). A comparison of a visual-spatial approach and verbal approach to teaching mathematics. In M.J. Hoines & A.B. Fuglestad 
(Eds.), Proceedings of the 28th PME International Conference, 4, 449-456. 


Visual Representations in Mathematical Word Problem Solving Among Form Four Students in Malacca 

Azizah Ahmad

Rohani Ahmad Tarmizi

Mokhtar Nawawi

Hegarty

Gagne

Presmeg

Van Garderen

Crossref

Visual Representations in Mathematical Word Problem Solving Among Form Four Students in Malacca

Learning how to solve mathematical problems has been a long withstanding difficulty faced by students and has been given constant focus in mathematical development. Successful problem solvers spend more time analyzing a problem and they will find all the alternatives related to the situation. Therefore many educators (Cai, 2003, National Council of Teachers of Mathematics, 2000; Smith, 1991) agreed that teaching mathematics through problem solving hold great promise. Furthermore, according to the Malaysian Curriculum Development Centre (2003), one of the important objectives in teaching mathematics is to develop student's mathematical problem solving skills. Mathematical problems are not simply computational tasks type but are problems which also require appropriate selection of strategies and decisions that lead to logical solutions. Word problem solving is one of the important components of mathematics problem solving which incorporate real-life problems and applications. However, many researches revealed that students express great difficulties in handling a word or story problem. This study examined the use of visual representations which posed great difficulties in most mathematical tasks. Secondary students were given the 15Mathematical Processing Instrument (MPI) adapted from Hegarty and Kozhevnikov (1999). This study examined 381 students from eight secondary schools in three different districts in Malacca. Results indicated that less than two percent of the problems were solved using pictorial representation and most of the students preferred to used schematics solutions

Ahmad Tarmizi, Rohani

Universiti Putra Malaysia Institutional Repository

Visual representations in mathematical word problem solving among form four students in Malacca

MUCC (Crossref)

https://doi.org/10.1016/j.sbspro.2010.12.050

Visual Representations in Mathematical Word Problem Solving Among Form Four Students in Malacca

Abstract

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Digitale Sammlungen - Goethe-Universität Frankfurt am Main

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Universiti Putra Malaysia Institutional Repository

MUCC (Crossref)