Paramjit 1
An Analysis of Addition and Subtraction Word Problems in
Mathematics Textbooks Used in Malaysian Primary SchoolClassrooms
Parmjit Singh, University of Technology MARA, Malaysia,
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 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 rolein the attempt to achievean 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 otherstudies (Yan Fan, 2006;and FanYan, 2007) comparedAmerican, Chinese, and Singaporean school textbooks. Schmidt, Mcknight and Raizen (1997) analysed textbooks based on depth-of-content-coverage in the textbooks used in theUnited 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 MalaysianPrimary 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 importantaspect ofthe 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”coversproblems 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 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 wordproblems (Jitendra & Xin, 1997). They are unable to comprehendthe semantics of the word problemsand this affects the translation into mathematical symbolism. The cure for the “I can’t do word problems” syndrome would appear to beadequate 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 / Problem1. / JRU / Join Result Unknown / Hani has 12 flowers in the basket. Sarah gave her 7 more. How many flowers does Hani have altogether?
2. / JCU / Join Change Unknown / Nadzirah had 8 mangoes. Farah gave her some more. Now Nadzirah has 15 mangoes. How many did Farah give her?
3. / JIU / Join Initial Unknown / Tasha had some sweets. Aisha gave her 9 more. Now Tasha has 20 sweets. How many sweets did Tasha have at first?
4. / SRU / Separate Result Unknown / Azhar bought 12 pencils. He gave 5 pencils to Ranjit. How many pencils does Azhar have now?
5. / SCU / Separate Change Unknown / Halim catches 18 fishes. He gave some to Ali. Now Halim has 7 fishes left. How many did he give to Ali?
6. / SIU / Separate Initial Unknown / Anis baked some cookies. She gave 6 to Chong. Now Anis has 12 cookies left. How many cookies did Anis bake at first?
7. / CDU / Compare Difference Unknown / Dinesh has 13 balloons and Lina has 4 balloons. How many more balloons does Dinesh have than Lina?
8. / CLU / Compare Larger Unknown / Mira read 6 storybooks. Alya read 12 storybooks more than Mira. How many storybooks did Alya read?
9. / CSU / Compare Smaller Unknown / Azman has 4 stamps fewer than Lim. Lim has 17 stamps. How many stamps does Azman have?
10. / PWU / Part-whole Whole Unknown / Siti has 13 small teddy bears and 6 big teddy bears. How many teddy bears does she have altogether?
11. / PPU / Part-whole Part Unknown / Mimi bought 18 apples from the supermarket. 13 of them are red and the rest are green. How many green apples did Mimi buy?
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 real-world 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 inTurkey 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 modelscomprise 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 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 subtractionconcepts available in textbooks used in Malaysian schools in Primary 1 and Primary 2 usingVan 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 investigatewhich 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 standard word problems as shown in Table 1, modeledby Vande 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 anotherrater, 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 wereselected 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,theaimbeing to measure their understanding of word problems in each of the 11Van 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 11categories (see Table 1). There were 11 problems in this instrument with one representativeproblem 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