Brunei Int.J.of Sci.& Math. Edu., 2010, Vol 2(1), 68-85
ISSN 2076-0868
AN ANALYSIS OF ADDITION AND SUBTRACTION WORD PROBLEMS IN
MATHEMATICS TEXTBOOKS USED IN MALAYSIAN PRIMARY SCHOOL
CLASSROOMS
Parmjit Singh, University of Technology MARA, Malaysia, < parmjit@tm.net.my>
Teoh Sian Hoon, University of Technology MARA, Malaysia
Mathematics textbooks are integral in most classroom-based teaching and learning as both teachers and pupils
use them as a source of mathematical learning. This study embarked on a mission to uncover the types of word
problems associated with addition and subtraction available in the text books used in Malaysian Primary
Schools. Firstly, it examines the distribution of 11 categories of addition and subtraction word problems based
on Van de Walle’s (1998) model, and secondly, it analyses pupils’ achievement in accordance with these
categories. The findings revealed that the textbooks did not adequately distribute or represent all the 11
categories of word problems, and analysis of the pupils’ scores based on tests made up of questions representing
the various categories suggested a relationship with the distributions of the types of problems.
Background
In almost every subject area, at nearly every grade level, students and teachers of
mathematics are expected to use a textbook as a resource (Amit & Freid, 2002; Kluth,
2005).Textbooks play a very important part in the teaching and learning process in schools
as the textbooks provide an important foundation for teachers in assisting pupils to learn
mathematics (Ministry of Education, 2003). Currently in Malaysia, all textbooks are
provided free for pupils in government schools. These textbooks are written by a team of
writers working for a number of publishers and can be considered the best according to the
specifications outlined by the Ministry of Education. In 2003 and 2004, the Ministry of
Education published new mathematics textbooks for Primary 1 and Primary 2, while the
publication of the new textbooks for Primary 3, Primary 4, Primary 5 and Primary 6
occurred from 2005 to 2008. These textbooks are the basis of school instruction and the
primary source of information for schools and teachers. This is because the problems set as
exercises in these textbooks are almost always assigned as homework for the pupils and act
as a platform for discussion of mathematical concepts among pupils and teachers. The
assignments of these exercises as homework for pupils’ conceptual development are sourced
from these textbooks by most teachers (Porter, Floden, Freeman, Schmidt & Schwille,
1988). Schmidt, Mcknight and Raizen (1997) have described in general terms the role of
Paramjit
69
textbooks as “bridges between the worlds of plan and intentions, and of classroom activities
shaped in part by those plans and intentions” (p. 53). In short, one could say that textbooks
may play a significant role in the attempt to achieve an intended learning outcome for
classroom teaching, and for most pupils, they provide the groundwork for the content to be
learned as well as the conceptual understanding that pupils construct during class activities
(Amit & Freid, 2002; Porter et al., 1988).
Various researchers (e.g., Fan & Kaeley, 2000; Fan, & Yan, 2007; Freeman & Porter,
1989; Stodolsky, 1989; Yan & Fan, 2006) have investigated, from different perspectives, the
ways mathematics teachers use textbooks in their classroom settings. Freeman and Porter’s
(1989) study focused on textbook usage by elementary teachers based on the content taught
and textbook content, while other studies (Yan & Fan, 2006; and Fan &Yan, 2007)
compared American, Chinese, and Singaporean school textbooks. Schmidt, Mcknight and
Raizen (1997) analysed textbooks based on depth-of-content-coverage in the textbooks used
in the United States and in other countries. Stodolsky (1989) studied the use and influence
of textbooks in classroom learning and teaching. She proposed that the use and influence of
textbooks should be analyzed with respect to topics, content, and its comparison with
literature on a similar topic. Olkun and Toluk (2003) analyzed the content of school
mathematics textbooks in Turkey. Their study found that textbooks did not adequately
represent all types of addition and subtraction problems and students were less successful on
the problem types underrepresented in textbooks. In echoing this line of investigation, this
current research investigated the distribution of word problems, in the topical context of
addition and subtraction in Malaysian Primary School textbooks. It is significant to
highlight that one of the main thrusts of the Primary Mathematics curriculum in Malaysia is
to develop basic computation skills comprising the four operators of addition, subtraction,
multiplication and division. The usage of the operators of addition and subtraction are
introduced as early as Primary one in Malaysia schools with the objective to develop
computation skills and the ability to use these skills in solving word problems. Furthermore,
the curriculum also places emphasis on problem solving, communication, mathematical
reasoning, and mathematical connections and representations (Curriculum Development
Centre, 2003).
The investigation reported in this paper studied the distribution of word problems
related to the operations of addition and subtraction, in Malaysian primary school
mathematics textbooks. Solving word problems embodying additive structures is an
important aspect of the new curriculum for primary and secondary Schools in Malaysia. The
term ‘word problem’ is often used to refer to any mathematical exercise for pupils stated in
a way that enhances awareness of the semantic structure of the problem in conjunction with
the numerical representation. According to Carpenter, Moses and Bebout (1988), it is
essential for the pupil to think about and analyze a word problem before making an attempt
to solve it. This is because, as stated by Krulik and Rudrick (1982), the word problem is a
situation which demands resolution and that there is no easy approach to solving it. In this
sense, for example, carefully chosen word-problems can provide a rich context for learning
addition and subtraction concepts (Greer, 1997).
The term “additive structures” covers problems involving addition and subtraction
operations, and knowledge of addition and subtraction concepts and skills is a prerequisite
for almost all primary school mathematics topics. Substantial research (Carpenter, Moser &
Bebout, 1988; Clements, 1999; Peterson, Fennema & Carpenter, 1989) has investigated and
found that pupils’ conceptions of word problems demanding addition and subtraction were
70
Addition and subtraction problems
often vague. For young pupils, it is not easy to model problem situations mathematically.
Pupils who have difficulties with reading, computation or both are likely to encounter
difficulties when attempting to solve word problems (Jitendra & Xin, 1997). They are
unable to comprehend the semantics of the word problems and this affects the translation
into mathematical symbolism. The cure for the “I can’t do word problems” syndrome would
appear to be adequate instruction in using mathematics as a language for problem solving in
the curriculum (Parmjit, 2006).
Table 1
Categorizing Additive and Subtractive Word Problems Using Van De Walle’s (1998) Model
SNo
Category
Information
Problem
1.
JRU
Join Result
Unknown
2.
JCU
Join Change
Unknown
3.
JIU
Join Initial
Unknown
4.
SRU
5.
SCU
6.
SIU
Separate Result
Unknown
Separate
Change
Unknown
Separate Initial
Unknown
7.
CDU
8.
CLU
9.
CSU
Hani has 12 flowers in the basket. Sarah gave her 7
more. How many flowers does Hani have
altogether?
Nadzirah had 8 mangoes. Farah gave her some
more. Now Nadzirah has 15 mangoes. How many
did Farah give her?
Tasha had some sweets. Aisha gave her 9 more.
Now Tasha has 20 sweets. How many sweets did
Tasha have at first?
Azhar bought 12 pencils. He gave 5 pencils to
Ranjit. How many pencils does Azhar have now?
Halim catches 18 fishes. He gave some to Ali. Now
Halim has 7 fishes left. How many did he give to
Ali?
Anis baked some cookies. She gave 6 to Chong.
Now Anis has 12 cookies left. How many cookies
did Anis bake at first?
Dinesh has 13 balloons and Lina has 4 balloons.
How many more balloons does Dinesh have than
Lina?
Mira read 6 storybooks. Alya read 12 storybooks
more than Mira. How many storybooks did Alya
read?
Azman has 4 stamps fewer than Lim. Lim has 17
stamps. How many stamps does Azman have?
10.
PWU
11.
PPU
Compare
Difference
Unknown
Compare
Larger
Unknown
Compare
Smaller
Unknown
Part-whole
Whole
Unknown
Part-whole Part
Unknown
Siti has 13 small teddy bears and 6 big teddy bears.
How many teddy bears does she have altogether?
Mimi bought 18 apples from the supermarket. 13 of
them are red and the rest are green. How many
green apples did Mimi buy?
Paramjit
71
Some writers have argued that, from a structural perspective, there are 11 different
categories of questions in the form of word problems for addition and subtraction operations
(Peterson, Fennema & Carpenter, 1989; Van de Walle, 1998). Although outwardly similar,
questions in the 11 categories (see Table 1) can vary greatly in difficulty for pupils (Olkun
& Toluk, 2003; Peterson, Fennema & Carpenter, 1989). While solving different word
problems, pupils are not only challenged to comprehend relationships between language and
mathematical processes, but also to experience sense making and mathematization of
realities (Greer, 1997; Reusser & Stebler, 1997; Wyndhamn & Saljo, 1997).
Parmjit’s (2006) study of pupils’ achievement in addition and subtraction word
problems used Van de Walle’s model. He reported that many pupils found questions in the
CDU, CSU and PPU categories (refer to Table 1) as difficult based on their low scores.
Although one might expect pupils to be able to contextualize problems that relate to realworld settings, often they are unable to move between the semantic structures to the
associated mathematical symbolisms because of the mismatch between their theoretical
knowledge and what they have experienced in the mathematics classroom.
A study by Olkun and Toluk (2003) utilizing Van De Walle’s model, found that the
textbooks used in primary schools in Turkey did not adequately represent all types of
addition and subtraction problems. The JCU, JIU, SCU, SIU, CDU, CLU, and CSU
categories were under-represented. They further argued that this unsystematic distribution of
word problems categories may prevent pupils from developing a rich repertoire of the
addition and subtraction concepts in the categories that are under-represented.
The Purpose of the Study
Both textbooks and word problems occupy an important position in the teaching and
learning process, and as Ball and Cohen (1996) pointed out, “curriculum materials could
contribute to professional practice if they were created with closer attention to processes of
curriculum enactment” (p. 7). Taking into consideration, the importance of mathematics
textbooks used in classrooms, coupled with the difficulty pupils face in solving word
problems, especially in the concepts of additive structures, this research was designed to
analyze the content of additive structures in textbooks used in Malaysian primary schools.
Researchers (Riley, Greeno and Heller, 1983; Van de Walle, 1998) have modeled
addition and subtraction problems into categories based on the kind of relationships
involved. The classification by Riley, et. al., (1983) model was based on the classification of
Change (2 types), Combine (6 types), and Compare (6 types) comprising 14 categories.
While Van de Walle’s (1998) model was classified into Join problems (3 types), Separate
Problems (3 types), Part - Part - Whole Problems/compare problems (2 types) and Compare
or Equalize Problems (3 types) comprising 11 categories. From these two models, Riley, et.
al., (1983) 14 categories model seemed more extensive than Van De Walle’s 11 categories.
This was because problems such as the following were not addressed in Van De Walle’s
model. Both these models comprise similar categories using different names. However, Van
de Walle’s model seemed not able to represent the following three types in his category:
1. There were 4 apples in the basket. Two more apples were added. Now there is the
same number of apples as oranges in the basket. How many oranges are in the basket?
2. There were 12 apples in the basket. 5 of them were removed so there would be the
same number of apples as oranges in the basket. How many oranges were in the
72
Addition and subtraction problems
basket?
3. There were some boys in the team. Four of them sat down so each girl would have a
partner. There are 7 girls in the team. How many boys are in the team?
In general, both Van de Walle’s model and Riley, et. al’s., model were similar besides the
absence of the three types of problems shown above from the latter. However, this study
adopted the former model because it was easier to analyze the content of a textbook based
on 11 categories compared to the 14 categories of the latter. Secondly, as this study focused
on Primary 1 and Primary 2 textbooks, the types of problems as shown above were not
included at this level (primary 1 and Primary 2) based on the Malaysian mathematics
syllabus.
According to Van de Walle’s model, there are 11 different categories of problems in
addition and subtraction; out of which four require addition, while seven require subtraction.
Therefore, the intention of this two-fold study was to analyze:
1. The distribution of the types of word problems with regards to addition and
subtraction concepts available in textbooks used in Malaysian schools in Primary 1 and
Primary 2 using Van de Walle’s (1998) model.
2. Pupils’ relative achievement on word problems in these 11 different categories.
If textbooks play an important role in the teaching and learning of mathematics is it
possible that pupils’ conceptual understanding of addition and subtraction is inhibited due to
inadequate opportunities provided to experience these different types of problems as a result
of imbalance in their school textbooks? Furthermore, is it possible that pupils are facing
these difficulties because they are not exposed to certain types of problems in their
classroom learning? Questions such as these draw attention to the need to investigate which
of the categories cause most difficulty for pupils, and why.
Together with these textbooks, there were also activity books which were to
supplement the textbooks. In other words, at each level, a textbook and two activity books
were supposed to be used in the teaching and learning process in the Malaysian primary
mathematics classroom.
To be noted that these textbooks used in this study were published in the English
language adhering to the policy of teaching Mathematics and Science in English since early
2003. However, this policy has been reversed in August 2009 where effective from 2012 the
teaching of mathematics and science will be reverted back to Bahasa Malaysia, the national
language. The Education Ministry is in the process of translating all the math books into
Bahasa Malaysia. Although the policy has been reversed, the content of the books remains
the same. In view of this, this paper is not affected with the reversal of the policy.
Method
Two modes of methodological analysis were utilized for this research. First, document
analysis was used in analyzing the distribution of the type of word problem categories
involving addition and subtraction operations in Primary 1 and Primary 2 textbooks,
together with the accompanying activity books for each grade. For purposes of comparative
analysis, commercial texts (or workbooks) from an established publisher were also
analyzed. These texts were analyzed according to the benchmark of the eleven types of
Paramjit
73
standard word problems as shown in Table 1, modeled by Van de Walle (1998). The
researcher independently categorized each problem in these textbooks in accordance with
the given categories. In ensuring the validity of the analysis, inter-rater member check was
also undertaken to determine the accuracy of the analysis. Inter-rater member check is a
process whereby another rater, other than the researcher is asked to verify the problem in
accordance to the category. All word problems that could be solved using addition and
subtraction of natural numbers were included, while symbolic expressions such as “8 + 7 =
?” and phrases such as “3 less than 10?” were excluded.
Secondly, an achievement test was administered to 302 pupils from Primary 1 and
Primary 2 in order to quantify the pupils’ achievements according to the eleven categories.
The samples comprised 116 and 186 pupils from Primary 1 and Primary 2 classes,
respectively, with ages ranging from seven to nine years. The actual schools in which the
students were located were selected randomly from five urban schools in a district in the
state of Selangor, Malaysia. Once the schools had been selected, the pupils were selected
from the “top” classes for each of the respective grades, the aim being to measure their
understanding of word problems in each of the 11 Van de Wall categories. These pupils
were selected from the top classes because the researchers want the content knowledge to be
the main issue in this study instead of the language, if the weaker pupils were to be selected.
An instrument was constructed in which the problems were adapted from the 11 categories
(see Table 1). There were 11 problems in this instrument with one representative problem
for each category. Pupils’ responses were categorized based on the following 4-point scale:
3 – All correct; 2 – Minor/careless/silly error(s); 1 – Some attempt but unlikely to lead to a
solution; 0 – No attempt
Table 2
Distribution of Word Problems in the Malaysian Primary 1 Textbook and the
Commercially-Published Workbook
Primary
1School
Textbook (%)
Primary 1
Commercial
workbook (%)
JRU
JCU
JIU
SRU
SCU
SIU
CDU
CLU
CSU
PWU
PPU
25.8
5.2
6.2
39.2
3.1
3.1
5.2
2.1
0
8.3
2.1
25
2
2
24
4
5
5
7
11
11
5
Results
Distribution of Word Problems in the Primary 1 Textbook and in the CommerciallyPublished Workbook, across the 11 Categories
This section details the distribution of word problems in textbooks used in schools across
the 11 categories for each of the respective levels. Table 2 shows the distribution of word
problems according to categories found in the Malaysian Primary 1 mathematics textbook
(together with activity books). It indicates that in the Primary 1 textbook, the SRU category
had the highest representation (39.2% of all problems), followed by the JRU category
(25.8% of all problems). The other categories were, relatively speaking, infrequently
represented (ranging from 0% to 8.3%) – a problem in the CSU category was not to be
found in the Primary 1 text.
74
Addition and subtraction problems
Figure 1a. Percentage of distribution of
word problems in the Primary 1 School
textbook
Figure 1b. Percentage of distribution of
word problems in the Primary 1
commercial workbook.
As shown in Figure 1b, the commercially-published text had reasonably similar
representations, with JRU (25%) having the highest representation and SRU (24%) the next
highest. However, categories such as CLU (7%), CSU (11%), and PWU (11%) have a much
higher representation than in the text used in the schools. Clearly, in both texts the emphasis
is on SRU and JRU questions. However, the more abstract categories, such as CLU, CSU,
and PWU receive greater attention in the commercially-published text. Figures 1a and 1b
show the distributions in a bar chart.
Figure 1c. Percentage of correct answers in the test according to the eleven categories among
Primary 1 pupils
Paramjit
75
Primary 1 Pupil’s Achievements on Questions in the 11 Categories
Figure 1c indicates the percentage of correct responses obtained by Primary 1 pupils for the
questions in the 11 categories. The highest scores were obtained for categories JRU (94.8%
correct), followed by PWU (81%), CLU (73.3%), and SIU (56%). The categories which
posed the greatest difficulty for the pupils were JIU (9.5%), CSU (14.6%), JCU (22.4%) and
CDU (23.2%). It is to be noted that the four categories for which pupils gave the most
correct responses were JRU, SIU, CLU and PWU, and each of these involved the operation
of addition. The questions for other seven categories, which involved the operation of
subtraction, were not as well answered.
Table 3
Distribution of Word Problems in the Malaysian Primary 2 School Textbook (P2ST) and
Commercially-Published Workbook (P2CPW)
JRU
P2ST
P2CPW
31.3
15.2
JCU
3.1
3
JIU
0
0
SRU
SCU
25.0
24.2
3.1
6.1
SIU
4.7
0
CDU
CLU
7.8
0
0
6.1
CSU
1.6
12.1
PWU
PPU
15.6
27.3
7.8
6.1
Distribution of Word Problems in the Primary 2 Textbook and the Commercially-Published
Workbook across the 11 Categories
Table 3 shows the distribution of the categories for questions in the Primary 2 mathematics
textbook used in schools. It indicates that in the Primary 2 textbook, the JRU category had
the highest representation (31.3% of all problems), followed by the SRU category (25.0% of
all problems) and the PWU category (15.6%). The other categories were, relatively
speaking, infrequently represented (ranging from 0% to 7.8% representation, there being no
problem in either the JIU or the CLU categories).
Figure 2a. % of distribution of word
problems in the Primary 2 school textbook.
Figure 2b. % of distribution of word
problems in the Primary 2 commerciallypublished book.
76
Addition and subtraction problems
Figure 2b shows the distribution of the categories in the commercially-produced book.
The highest representations are for the PWU (27.3%), SRU (24.2), JRU (15.2), and CSU
(12.1%) categories. There were no problems in the JIU, SIU and CDU categories. Clearly
Figures 2a and 2b, the emphasis in both of these books is on problems in the JRU, SRU and
PWU categories.
Primary 2 Pupils’ Achievements According to the 11 Categories
Figure 2c shows the percentage of correct responses obtained by Primary 2 pupils for the
questions in each of the 11 categories. Questions that were answered correctly most
frequently were in the JRU (87.6.8% correct), PWU (75.3%), SCU (70.5%), CLU (67.73%),
and SRU (66.7%) categories. The most difficult questions were in the CDU (25.2% correct),
CSU (30.1%), JCU (31.8%), JIU (35%), and SIU (36.5%) categories. Three of the four
categories for which correct responses were most frequently were in the JRU, CLU and
PWU categories, which involved the operation of addition. The questions in the seven
“subtraction” categories were less likely to be answered correctly. Comparison across the
categories for Primary 1 and Primary 2 indicated that results tended to be similar, especially
for questions in the JRU, PWU and CLU categories.
Figure 2c. Percentage of correct answers according to the 11 categories among Primary 2
pupils.
Analyses of Artefacts: Pupils’ Work Based on Computations in Worksheets
This section analyses pupils’ (Primary 1 and Primary 2) incorrect responses. The data from
the worksheets based on pupils’ computations suggested that pupils tended to utilize the
operation of addition instead of subtraction when they faced obstacles in modeling the
situation. The following examples from pupils’ work exemplify this reasoning.
Paramjit
77
Item 3
Tasha had some sweets. Aisha gave her 9 more. Now Tasha has 20 sweets.
How many sweets did Tasha have at first?
Pupils Computation:
Item 5
Halim catches 18 fishes. He gave some to Ali. Halim has 7 fishes left.
How many did he give to Ali?
Pupils Computation:
Item 7
Dinesh has 13 balloons and Lina has 4 balloons. How many more balloons does Dinesh
have than Lina?
Pupils Computation:
Item 9
Azman has 4 stamps fewer than Lim. Lim has 17 stamps. How many stamps does Azman
have?
Pupils Computation:
Item 11
Mimi bought 18 apples from the supermarket. 13 of them are red and the rest are green.
How many green apples did Mimi buy?
Pupils Computation:
78
Addition and subtraction problems
These five items indicate that both Malaysian Primary 1 and Primary 2 pupils used the
addition operation instead of subtraction to solve the given problems. For example, the
response shown for Item 3 seems to indicate that the pupil did not comprehend the
relationship between the quantities 9 and 20. The word problem stated that, “Aisha gave her
one more”. Pupils might have interpreted the word “more” as addition. The recognition and
application of key words during the computational process might have played a role in
pupils’ faulty reasoning thus leading to incorrect responses.
Similarly, for Items 5 and 7, the key words “gave” and “more” probably influenced
pupils when modeling the situation. The word “gave” in Item 5 might have been interpreted
as becoming more as which might have misled them into using the addition operation. The
majority of the students who obtained an incorrect response gave the answer 25 which is the
sum of 18 and 7 or 7 and 18. Again in item 7, the word “more” again seems to have
triggered the pupils to add 13 to 4, which equals 17.
The probable reason that young children are confused is probably because during their
first year at school, children are told by their teachers that you get more by "joining", or
"adding" or "getting", or "finding", etc., and you get less by "taking away" or "separating",
or "giving away", or "losing", or "finding the difference between" (Coombs and Harcourt,
1986, p. xxvii). According to Lean, Clements and Del Campo (1990) analysis, the confusion
which young children experience with questions which specifically contain the words
"more" and "less" often carries over to questions involving giving, taking, finding, losing,
etc., and questions in which two sets are combined or compared. They tend to apply to such
questions the same kind of strategies that they apply to questions which contain "more" and
"less".
Relationship between the Category of Word Problems and Problem Difficulty
According to Tables 2 and 3, the JRU and SRU categories had the highest representation in
the Primary 1 and Primary 2 textbooks. Not surprisingly, pupils in both these levels obtained
a high percentage of success rates of 94.8% and 87.6% in both these categories with
Primary 1 pupils doing better than their counterparts in Primary 2.
Although the JRU category is highly represented in textbooks and activity books, it
was found that both Primary 1 and Primary 2 pupils failed to respond correctly in this
category. Only 47.4% and 66.7% of Primary 1 and Primary 2 pupils, respectably, answered
this question correctly. Category had the lowest representation in the primary mathematics
textbooks. According to Figures 1a and 1b, this category was unrepresented (0%) for
Primary 1, and under-represented (1.6 %) for Primary 2. The results from pupils’
achievement were also unsatisfactory, with only 14.6% and 30.1% of Primary 1 and Primary
2 pupils responding correctly to the question in this category.
Table 4 shows a strong statistically significant correlation (r = 0.85, p < 0.01) between
corresponding item facilities for Primary 1 and Primary 2 pupils. This means that if a
question in a certain category was answered correctly by Primary 1 students, then it was also
tended to be answered correctly by Primary 2 pupils. Likewise, a question answered poorly
by Primary 1 pupils tended to be answered poorly by Primary 2 pupils.
There was also a strong statistically significant correlation (r = 0.86, p < 0.01) between
the distribution categories of word problems between Primary 1 and Primary 2 school
textbooks. This indicated, for example, that where there was a high percentage for a
particular category in the Primary 1 text, the same would be likely to be true in the Primary
2 text. A similar statement would be true if “high” were to be replaced with “low.” A
Paramjit
79
moderately statistically significant correlation (r = 0.64, p < 0.05) existed between scores
item facilities for Primary 2 pupils and the distribution of word problem categories in the
Primary 2 school text. This suggested that if a certain category was well represented in the
Primary 2 textbook, then the pupils tended to answer the question from that category
correctly.. Similarly, if a certain category was unrepresented, or poorly represented, in the
Primary 2 textbook, then the pupils tended to answer the question from that category
incorrectly. Interestingly, though, there was no statistically significant correlation (r = 0.36,
p > 0.05) between scores item facilities for Primary 1 pupils and the distribution of word
problem categories in the Primary 1 school text.
Table 4
Correlations Matrix between Pupils’ Scores and distribution of Word Problems in Primary
1 and Primary 2 Categories
Primary 1
score
Pearson Correlation
Primary 1
1
Sig. (2-tailed)
N
Primary 2
score
Pearson Correlation
Primary 2
.847(**)
Text
Primary 1
.358
Text
Primary 2
.613(*)
.001
.280
.045
11
11
11
1
.508
.641(*)
.111
.034
11
11
1
.863(**)
Sig. (2-tailed)
N
Dist.
Text
Primary 1
Pearson Correlation
Dist.
Text
Primary 2
Pearson Correlation
Sig. (2-tailed)
.001
N
11
1
Sig. (2-tailed)
N
11
** Correlation is significant at the 0.01 level (2-tailed).
* Correlation is significant at the 0.05 level (2-tailed).
Discussion and Conclusions
Textbooks play a crucial role in the Malaysian mathematics teaching and learning process
in schools. The two important pedagogic functions, as succinctly put by Van Dormolen
(1986, cited in Johnson, 2005), are the curricular and conceptual perspectives. The former
relates to the progression of learning materials while the latter relates to the conceptual
development of learners. The role of textbooks in the twin processes of teaching and
learning mathematics has received increasing attention over the last two decades, at least.
As mentioned earlier, the study that has been summarized was a two-fold study. The
distribution of word problem categories found in Malaysian primary schools mathematics
textbooks based on Van de Walle’s (1998) 11 model was identified, and pupils’
achievement on questions representing the various categories were compared. Also, a
comparison of the school textbooks with similar commercially-published workbooks was
carried out.
Entries in Table 5 indicate that the word problem categories based on Van de Walle’s
model were not systematically distributed throughout the Primary 1 and Primary 2
80
Addition and subtraction problems
mathematics textbooks. Three of the categories namely JRU (28.6%), SRU (32.1%) and
PWU (12%) were well represented, but other categories were either under-represented or
not represented at all. In terms of pupils’ achievements across the categories, the findings of
this study were similar to that obtained by Parmjit (2006). Analyses indicated that pupils in
Primary 1 and Primary 2 experienced difficulties with comprehending questions in
categories JCU (27.1%), JIU (22.3%), CDU (24.2%), CSU (22.4%), and PPU (43.6%).
There are at least two reasons that might have been responsible for this.
Table 5
Distribution of Word Problems and Pupils’ Achievement across the 11 Categories
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Categories
JRU
JCU
JIU
SRU
SCU
SIU
CDU
CLU
CSU
PWU
PPU
Pr 1
25.8
5.2
6.2
39.2
3.1
3.1
5.2
2.1
0
8.3
2.1
Pr 2 Mean
31.3
28.6
3.1
4.2
0
3.1
25.0
32.1
3.1
3.1
4.7
3.9
7.8
6.5
0
1.1
1.6
0.8
15.6
12.0
7.8
5.0
Pr 1
94.8
22.4
9.5
47.4
42.2
51.2
23.2
73.3
14.6
81.0
38.8
Pr 2 Mean
87.6
91.2
31.8
27.1
35.0
22.3
66.7
57.1
70.5
56.4
56.5
53.9
25.2
24.2
67.7
70.5
30.1
22.4
75.3
78.2
48.4
43.6
Category 1, 4 and 10 with highest distribution; Category, 1, 4, 8 and 10 utilizing addition operations
First, the comparison between the textbook analysis and pupils’ achievements,
indicated that that pupils performed least well on questions in categories that were underrepresented or not represented at all in the school textbooks. On the other hand, the students
tended to do well on questions for categories (e.g., JRU, SRU and PWU) that were well
represented in the text.
The assumptions here is that questions in those categories that were well-represented in
the textbooks tended to be correctly answered because pupils had considerable experience in
solving such problems in their schools’ mathematics classrooms.
The second point is that four of these eleven categories in the Van de Walle model
involve the operation of addition, but the other seven categories involve the operation of
subtraction. The questions for the four categories (with the success rate in parenthesis)
involving the operations of addition were as follows:
JRU (91.2%)
Hani has 12 flowers in the basket. Sarah gave her 7 more. How many flowers does Hani
have altogether?
Number sentence: 12 + 7 = ___
The operation needed to solve this problem is addition.
SIU (53.9%)
Anis baked some cookies. She gave 6 to Chong. Now Anis has 12 cookies left. How many
cookies did Anis bake at first?
Number sentence: ___ – 6 = 12
Paramjit
81
The operation needed to solve this problem is addition.
CLU (70.5%)
Mira read 6 storybooks. Alya read 12 storybooks more than Mira. How many storybooks
did Alya read?
Number sentence: 12 + 6 = ___
The operation needed to solve this problem is addition.
PWU (78.2%)
Siti has 13 small teddy bears and 6 big teddy bears. How many teddy bears does she have
altogether?
Number sentence: 13 + 6 = ___
The operation needed to solve this problem is addition.
The success rate obtained by pupils in the test were higher in these four categories (91.2%,
53.9%, 70.5% and 78.2%) than the scores in the other categories (e.g. 27.1%, 22.3%,57.1%,
56.4%, 53.9%, 24.2%, 22.4 and 43.6%). There is a strong possibility that the high scores
obtained for these four categories involving addition might not be linked to pupils’ ability in
modeling the situation, but rather through guessing. This is because from these four
categories, it can be said that SIU (53.9%) and CLU (70.5%) involve a higher order
problem-solving skill as compared to that required for the computation of SRU or JIU, and
yet the pupils obtained a higher percentage success rate in the former categories when
compared to the latter. These pupils might have computed the numbers utilizing addition
operations in deriving the answers. It may not have been based on their conceptual
understanding in modeling the situations. This may be because the first arithmetic operation
that pupils learn in early mathematics after counting is addition, and pupils tend to fall back
on their prior experience of construction which they are familiar with in modeling the
situation. This study suggests that when pupils face difficulties in comprehending problems,
they fall back on the operation with which they are most familiar, namely addition. They
tend to commit to the operation of addition if there is a barrier that prevents them from
comprehending or contextualizing the given problem. This was quite prevalent when these
pupils obtained a much higher success rate in CLU (70.5) as compared to CSU (22.4%)
though it involves similar contexts with differences in terms of “more” and “fewer”, where
the former category involves the operation of addition, and the latter subtraction. Another
category that concurs with this line of reasoning is the analysis of SIU as compared to JCU
or JIU. From the problems given, SIU requires a higher thinking mode of contextualization
as compared to JCU or JIU. Yet, the success rate obtained by pupils on the former is much
higher (53.9%) as compared to the latter (27.1% and 22.3%). In other words, the more
difficult category (SIU) which requires the operation of addition produced a higher success
rate as compared to the less difficult categories (JCU and JIU) which produced a lower
success rate. In this respect, a qualitative study ought to be undertaken in assessing pupils’
construction of knowledge based on their conceptual advances in Malaysian settings.
The three categories (JRU, SRU and PWU) that are most frequently represented in the
primary school textbooks seem to represent the lowest cognitive process needed for the
solution, as compared to the other categories (especially CDU, CLU and CSU) which are
under-represented or not-represented at all. The lexicon in the semantic structure such as
“fewer” and “more” in the latter categories (CLU and CSU) require a higher-order
82
Addition and subtraction problems
cognitive operation in order to comprehend and contextualize the problems as compared
with the other categories. The inability of pupils to comprehend and contextualize the
problems compounded with the lexicon used might be the root of the difficulties that the
pupils experienced. The failure of many pupils to answer correctly some of the problems,
especially those for categories CDU, CSU and PPU, appears to have been due to their lack
of experience with certain situations rather than with any difficulty with operating on
numbers (Parmjit, 2006).
Word problems which start with unknowns are relatively difficult for young pupils
(Peterson, Fennema & Carpenter, 1989), since they find it difficult to model the problem
situations mathematically. Textbooks often include only a few of these categories of word
problems, thus limiting children’s ability to learn the operations meaningfully (Greer, 1997;
Peterson, Fennema & Carpenter, 1989). That kind of conclusion concurs with the findings
of this study with Primary 1 and Primary 2 pupils, and their textbooks, in Malaysian
schools.
Both textbooks and word problems occupy an important position in the teaching and
learning process, and as pointed earlier by Ball and Cohen (1996) “curriculum materials
could contribute to professional practice if they were created with closer attention to
processes of curriculum enactment” (p. 7). This simply means that there is a possibility that
the pupils’ inability to solve difficult problems may be due to the fact that they have not
obtained adequate exposure to or experience working with these problems in their classroom
learning. The presumption here is that an analysis of textbooks is a necessary, but not
sufficient condition to understand what really happens in actual classroom teaching (Fan &
Zhu, 2000). When categories of word problems have been rarely experienced by pupils, then
the pupils can be forgiven for having difficulty “translating” the problems into
mathematically correct statements, if, all of a sudden, they are asked to solve them. This
can result in the pupils’ conceptual development in addition and subtraction being inhibited.
Educational Implications
The first implication is that pupils’ ability to link addition and subtraction with additive and
subtractive word problem situations develops slowly throughout pupils’ early stages of
mathematics learning (Greer, 1997; Munirah, 2005; Parmjit, 2004, 2006; Wyndhamn,
1997). It is likely that this shortcoming might be directly linked to the inadequate balance of
content in the mathematics textbooks used in classroom teaching. The result of this research
indicates that Primary 1 and Primary 2 textbooks published by the Ministry of Education
since 2003 (in stages) used in Malaysian classroom teaching, do not adequately represent
the totality of different categories for addition and subtraction word problem questions
which have been well-researched in the literature. Less exposure to these types of problems
might be one reason that many pupils’ faced difficulties in these concepts. In view of this,
textbooks developers should consider including more of the different types of problems in
textbooks.
Secondly, the distribution of the word problem categories in the textbooks has not been
systematically based on research findings. In the Malaysian Primary 1 and Primary 2
textbooks there is an over emphasis on certain categories while other categories are hardly
represented, or indeed not represented at all. The Education Ministry should emphasize the
need for textbook writers to take into account the findings of research outputs.
It is recommended that in-depth research study be carried out by the Ministry of
Education on the content adequacy of current mathematics textbooks used in classrooms,
Paramjit
83
especially for the early primary grades. This is necessary because, at this stage pupils are
being exposed and becoming familiar to word problems. Carefully chosen word-problems
can provide a rich context for learning mathematical concepts (Greer, 1997). Secondly, at
these early grades, pupils who have trouble with fundamental mathematical concepts will be
likely to continue having problems with mathematics throughout their schooling life.
Struggling pupils can quickly cultivate a strong negative attitude towards mathematics.
Further research is clearly essential to explore how knowledge of children’s learning of
mathematics can be applied to the design of instruction via textbooks to bridge that gap that
still exists between teachers’ teaching and children’s learning in mathematics.
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