an analysis of numbers parts-whole...

Man In India, 96 (1-2) : 397-410 © Serials Publications

Address for communication: Santi Sinnakaudan, School of Educational Studies, Universiti SainsMalaysia, Pulau Pinang, Malaysia, E-mail: [email protected]

AN ANALYSIS OF NUMBERS PARTS-WHOLERELATIONSHIP IN MALAYSIAN YEAR ONEMATHEMATICS TEXTBOOKS

Santi Sinnakaudan and Munirah Ghazali

Teaching mathematics in Malaysian primary schools, invariably involves the use of a set oftextbooks which are loaned by the Ministry of Education to all of the pupils. The purpose of thisstudy was to analyse how numbers parts-whole relationship is presented in Malaysian year onemathematics textbooks. On this account three sets of mathematics textbooks used in SekolahKebangsaan (SK), Sekolah Jenis Kebangsaan Cina (SJKC) and Sekolah Jenis Kebangsaan Tamil(SJKT), the three types of Malaysian primary schools were examined based on the textbookanalysis framework adapted from the TIMSS Study (Valverde et al., 2002) and TIMSS+ study(O’Keeffe & O’Donogue, 2011a). This framework comprises of three key elements: content,structure and expectation. The results indicated that the textbooks are reasonably well alignedwith the year one mathematics syllabus in terms of structure, content and expectation. The widevariety and consistent use of colour and the number of graphics, examples and questions presentin these textbooks contribute to pupils’ comprehension of parts-whole relationship. Howeverthere is lack of general focus on higher order thinking skills (HOTS). Implications are discussedand suggestions are given to accompany the textbooks usage in schools. This analysis seeks toinform teachers about the significance and limitations of using the textbook to teach numbersparts-whole relationship so that they can plan their teaching and learning activities accordingly.

Keywords: Mathematics textbooks, parts-whole relationship, textbook analysis framework,Malaysia

Introduction

In Malaysian primary schools, 12% of schooling time (180 minutes from the totalof 1500 minutes) is allocated for teaching mathematics and develop year one pupils’number sense. One important aspect of number sense is number relationship,especially the parts-whole relationship. In parts-whole relationship a whole quantitycan be decomposed or separated into two or more parts, which in turn can becomposed or combined to the initial whole quantity (Chapin & Johnson, 2006;Fischer, 1990; Jung, 2011; Payne & Rathmell, 1975; Resnick, 1983; Steinke, 2001,2008). Steinke (2008) and Van de Walle, Karp and Bay-Williams (2013) statedthat focusing on a quantity in terms of its parts has important implications fordeveloping number sense.

The explorations of parts-whole relationship allows the pupils to develop anunderstanding of number quantity (e.g., the meaning of ‘4’), place value (e.g., 14is one ‘ten’ and four ‘ones’), addition (combining numerical parts to create a whole),subtraction (partitioning a whole into parts) and fractions (a pizza can be cut into

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six slices). Research indicates that pupils taught parts-whole relationships dosignificantly better with number concepts (Dougherty, Karp, Caldwell, & Kobett,2014; Fischer, 1990; Jung, 2011; Jung, Hartman, Smith, & Wallace, 2013; NCTM,2000; Resnick, 1983; Van de Walle, et al., 2013), problem solving (Cheng, 2011;Fischer, 1990; Fosnot & Dolk, 2001; Hunting, 2003; Riley, Greeno, & Heller,1983; Sophian & McCorgray, 1994; Sophian & Vong, 1995), place value (Baroody,2004; Fischer, 1990; Resnick, 1983; Kamii & Housman, 2000) and fractions(Charlesworth, 2012) than those pupils whose teaching and learning focused oncounting by ones only.

Mathematics textbooks play an important role in introducing parts-wholerelationship to Malaysian year one pupils. In Malaysia every pupil is eligible fortextbook loan scheme as textbooks have been emphasised to be the most importanttool in promoting the curricula. The majority of mathematics teachers depend onthe textbook when planning and implementing the curriculum. Shield and Dole(2009), and O’Keeffe and O’Donogue (2011b) suggested that mathematicstextbooks should be analysed carefully if they are to be used as base for thedevelopment of pupils’ mathematics learning. As there is very little literature aboutMalaysian mathematics textbooks, this paper will report on the presentation of thenumbers parts-whole relationship in year one mathematics textbooks which is thefoundation of number operations and an important aspect of number sense. Therationale for focusing on year one was the importance of early year mathematicsknowledge for future success.

Methodology

Materials

Three sets of year one mathematics textbooks (including activity books) that areused in the three types of Malaysian primary schools, Sekolah Kebangsaan (SK),Sekolah Jenis Kebangsaan Tamil (SJKT), and Sekolah Jenis Kebangsaan Cina(SJKC), published under the recently implemented Kurikulum Standard SekolahRendah (KSSR) curriculum in three different languages, were analysed.Mathematics textbook for SK (TBK) and activity book for SK (ABK) were writtenby Marzita, Wan Yusof and Chan (2010a, 2010b) in Malay, mathematics textbookfor SJKT (TBT) and activity book for SJKT (ABT) were written by Mahandran,Uthamaseelan and Alice Nesamoney (2010a, 2010b) in Tamil, and mathematicstextbook for SJKC (TBC) and activity book for SJKC (ABC) were written by Ooiand Lee (2010a, 2010b) in Mandarin.

Instrument

The author employed an adapted version of the textbook analysis frameworkdeveloped by the TIMSS study (Valverde, Bianchi, Wolfe, Schmidt, & Houng,

AN ANALYSIS OF NUMBERS PARTS-WHOLE RELATIONSHIP... 399

2002) and TIMSS+ study (O’Keeffe & O’Donogue, 2011a) to numbers parts-wholerelationship in year one mathematics textbook sets. This framework focused onthree key elements: content, structure and expectation. The Malaysian year onemathematics curriculum, Dokumen Standard KSSR Matematik Tahun 1 / DocumentStandard KSSR Mathematics Year 1(DSKM) (BPK, 2010) was used as a referentfor analysis and comparison purposes.

Procedure

The lesson on ‘Finding Number Pairs’ from ‘Chapter 2 Number Operations:Addition and Subtraction within 10’ in the three sets of year one mathematicstextbooks were translated into English and back-translated for accuracy. Thedata source consisted of all the items related to numbers parts-whole relationship.All these items were tallied and analysed according to three key elements(structure, content and expectation) of the textbook analysis framework todetermine how numbers parts-whole relationship was presented in year onemathematics textbook sets. A brief description of each criterion and codingprocedure follows.

Structure analysis

Structure analysis includes the physical features of a textbook and how themathematics concepts were delivered, whether through the form of graphics,exercises or narratives (Valverde et al., 2002). Author included number of pages,technical aids, distribution of graphics and items in lesson ‘Finding Number Pairs’for structure analysis. As there were very less material included in the form ofnarrative in Malaysian year one mathematics textbooks, narratives were not includedin this analysis. All the graphics present in this lesson were counted and categorizedinto any one of these two categories: pictures representing objects or real-worldsituations. All the items in this lesson that were related to parts-whole relationshipwere tallied and categorized into any one of these two categories: examples orquestions.

Content Analysis

In content analysis, the subject matter or topic is addressed (Valverde et al., 2002).Author analysed the presentation of numbers parts-whole relationship based onthe Content Standard 2.1 Identify number pairs to construct a relevant wholenumber, of year one DSKM (BPK, 2010). Author also analysed the differentrepresentation forms of the items, the distribution of the terms used, and the contentof the questions given.

For representation criterion, items that incorporated semi-concreterepresentations (e.g., pictures), symbolic representations (e.g., 3 and 2 makes 5),and number bonds were tallied. It must be noted that when an item is represented

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in more than one representation, each was counted separately. All the items werealso analysed for the terms used and categorized into any one of these fourcategories: “number pairs,” “__ and __ makes __” (e.g., 3 and 2 makes 5), “__ is__ and __” (e.g., 5 is 2 and 3) or “__ plus __ is __” (e.g., 3 plus 2 is 5). In analysingthe content of the questions given, any items that focused on a number or numberparts, and involved manipulating, reading, or writing the number parts or whole asa means to reflect on the parts-whole relationship were tallied and categorized intoany one of these three categories: “only whole is known,” “whole and one part areknown,” or “only parts are known.”

Expectation Analysis

The last aspect is expectation analysis, which is what pupils are expected to beable to do with a particular content (Valverde et al., 2002). Author used theLearning Standards (which states that pupils must be able to (2.1.1) state theparts of the number given, (2.1.2) state the total of two numbers, and (2.1.3) listthe combinations of two numbers for the number given) (BPK, 2010) as a criterionin this analysis. All the items were analysed and categorized according to thelearning standards. This was followed by recording the level of questions givenwhich were based on Revised Blooms’ Taxonomy (Anderson, & Krathwohl,2001) concerning the higher order thinking skills (HOTS) that were expectedthat the pupils can acquire (KPM, 2013; Lembaga Peperiksaan, 2013; MOE,2012).

Data Analysis Procedures

All the items related to parts-whole relationship were tallied and analysed manuallyand using the Statistical Package for Social Sciences (SPSS) version 19.0. Acombination of percentage and descriptive analysis was used to report and evaluatethe data. Percentage was computed as the number of agreements divided by thenumber of agreements and disagreements multiplied by 100 (Sood & Jitendra,2007). Mean (M) and Standard Deviation (SD) were computed by using SPSS19.0. The author analysed all items independently. As further check on reliability,three mathematics teachers from SK, SJKT and SJKC, respectively, reanalysedthe items.

Results

Structure analysis

As shown in Table 1, a combination of seven pages was allocated for the numbersparts-whole relationship in addition and subtraction within 10 in all of the yearone mathematics textbook sets. The analysis of technical aids identifies notesfor teachers in all the textbooks, references to websites in TBK and TBC but


none in TBT. All the activity books have no references to any type of technicalaids.

TABLE 1: STRUCTURE ANALYSIS CRITERIA FOR SK, SJKT AND SJKC MATHEMATICSTEXTBOOK SETS ON NUMBERS PARTS-WHOLE RELATIONSHIP

Structureanalysis Textbooks for SK, Activity books for SK,criteria SJKT & SJKC SJKT & SJKC

TBK TBT TBC ABK ABT ABC

n % n % n % n % n % n % M SD

Number of pages 3 4 4 4 3 3

Technical aids

Notes 3 1 2 0 0 0

Websites 1 0 1 0 0 0

Graphics 33 26 14 27 32 10

Objects 31 94 24 92 11 79 27 100 32 100 8 80 90.83 9.35

Real-world 2 6 2 8 3 21 0 0 0 0 2 20 9.17 9.35

Items 19 19 10 39 35 16

Examples 7 37 15 79 9 90 0 0 1 3 3 19 38.00 38.51

Questions 12 63 4 21 1 10 39 100 34 97 13 81 62.00 38.51

Note: n = number of pages, technical aids, graphics and items.

Table 1 shows that distribution of graphics are more emphasised in the formof pictures of objects (M = 90.83, SD = 9.35) rather than real-world situations(M = 9.17, SD = 9.35). The textbook sets of SK and SJKT display the mostgraphics, 60 (TBK: 33, ABK: 27) and 58 (TBT: 26, ABT: 32), respectively,compared to 24 (TBC: 14, ABC: 10) in SJKC. Further analysis of the graphics(Table 1 and Figure 1) shows that even though there are fewer graphics in SJKCmathematics textbook sets, there are more related to real-world situations. Thereare 21% of graphics based on real-world situations in TBC compared to 6% and8% in TBK and TBT, respectively; 20% in ABC whereas none in ABK andABT.

Analysis of the items indicates that examples (M = 38.00, SD = 38.51) andquestions (M = 62.00, SD = 38.51) are not distributed evenly among the textbooksets (Table 1). There are fewer examples in TBK, only 37% compared to 79% and90% in TBT and TBC, respectively (Table 1 and Figure 2). Analysis of the itemsin activity books (Table 1) shows that ABK and ABT display the most questionson numbers parts-whole relationship, 39 in ABK and 35 in ABT, compared to 16in ABC.

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Figure 2: Distribution of items (examples and questions) in SK, SJKT and SJKC mathematics textbooksets on numbers parts-whole relationship.

Figure 1: Distribution of graphics (pictures of objects and real-world situations) in SK, SJKT andSJKC mathematics textbook sets on numbers parts-whole relationship.


Content Analysis

TABLE 2: CONTENT ANALYSIS CRITERIA FOR SK, SJKT AND SJKC MATHEMATICSTEXTBOOK SETS ON NUMBERS PARTS-WHOLE RELATIONSHIP

Content analysis Textbooks for SK, SJKT & Activity books for SK,criteria SJKC SJKT & SJKC


n % n % n % n % n % n % M SD

C. Standard 2.1 19 100 19 100 10 100 39 100 35 100 16 100 100 0Representation 19 19 10 39 35 16Semi-concrete 18 95 19 100 4 40 16 41 0 0 3 19 49.17 40.41Symbolic 15 79 12 63 6 60 26 67 5 14 14 88 61.83 25.69Number bonds 14 74 17 89 3 30 15 38 30 86 4 25 57.00 29.22Terms used 19 19 10 39 35 16Number pairs 16 84 12 63 0 0 39 100 35 100 0 0 57.83 46.81_ and _ makes _ 14 74 12 63 2 20 19 49 0 0 2 12 36.33 29.90_ is _ and _ 3 16 0 0 1 10 4 10 0 0 0 0 6.00 6.92_ plus _ is _ 0 0 0 0 5 50 0 0 0 0 10 88 23.00 37.60Questions 12 4 1 39 34 13Whole 9 75 2 50 1 100 20 51 34 100 4 31 67.83 28.56Whole & part 3 25 0 0 0 0 9 23 0 0 3 23 11.83 12.98Both parts 0 0 2 50 0 0 10 26 0 0 6 46 20.33 23.71

Note: n = number of items and questions. C. Standard 2.1 = Content Standard 2.1 Identify numberpairs to construct a relevant whole number (BPK, 2010).

As shown in Table 2, most of the items in the textbook sets were representedin multiple representations, namely, semi-concrete (M = 49.17, SD = 40.41),symbolic (M = 61.83, SD = 25.69) and number bonds (M = 57.00, SD = 29.22).Though the analysis of SK, SJKT and SJKC textbooks shows that most of theitems were represented in multiple representations but the analysis of SK, SJKTand SJKC activity books shows that only ABK had most of the questions in multiplerepresentation: semi-concrete (41%), symbolic (67%) and number bonds (38%).Whereas, in ABT and ABS, most of the questions were represented in one wayonly, as number bonds (86%) and as symbolic forms (88%), respectively.

Table 2 and Figure 3 shows that the terms used in the textbook sets are varied:number pairs (M = 57.83, SD = 46.81), “__ and __ makes” (M = 36.33, SD =29.90), “__ is __ and __” (M = 6.00, SD = 6.92) and “__ plus __ is __” (M = 23.00,SD = 37.60). The most widely used terms in SK and SJKT textbook sets are “numberpairs” and “__ and __ makes __”. Though there are references to the term “numberpairs” in SJKC textbook sets, but it was not used explicitly. Only SJKC textbooksets used the term and symbol “plus” in parts-whole relationship. SJKT textbooksets were not consistent with the vocabulary used to refer to the parts-wholerelationship. For example, in TBT most of the examples were given based on both

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Figure 3: The terms used in items (examples & questions) in SK, SJKT and SJKC mathematicstextbook sets on numbers parts-whole relationship

Figure 4: Content of the questions in SK, SJKT and SJKC mathematics textbook sets on numbersparts-whole relationship


the terms “number pairs” (63%) and “__and __ makes__” (63%), but in ABT, allthe questions were given based on the term “number pairs” (100%) only.

Table 2 and Figure 4 show that 75%, 50%, 100%, 51% 100%, and 31% of thequestions in TBK, TBT, TBC, ABK, ABT and ABC, respectively, included tasksof finding the parts of a whole number (M = 67.83, SD = 28.56). Questions on thethree variations in parts-whole relationship, finding the parts when only the wholeis known, finding the part when the whole and one part are known, and finding thewhole when the parts are known are only present in ABK and ABC, whereas ABTfocuses only on finding the parts of a given number.

Expectation Analysis

TABLE 3: EXPECTATION ANALYSIS CRITERIA FOR SK, SJKT AND SJKCMATHEMATICS TEXTBOOK SETS ON NUMBERS PARTS-WHOLE RELATIONSHIP

Expectation analysis Textbooks for SK, SJKT Activity books for SK,criteria & SJKC SJKT & SJKC


n % n % n % n % n % n % M SD

L.S. 2.1.1 19 100 19 100 10 100 25 64 35 100 9 56 86.67 20.81L.S. 2.1.2 14 74 17 89 9 90 17 44 0 0 12 75 62.00 34.62L.S. 2.1.3 16 84 19 100 6 60 29 74 18 51 8 50 69.83 19.86RBT:Remembering 19 100 19 100 10 100 39 100 35 100 16 100 100 0Understanding 19 100 19 100 10 100 39 100 35 100 16 100 100 0Applying 1 5 0 0 1 10 2 5 35 100 1 6 21.00 38.83

Note: n = number of items. L.S. = Learning standards: 2.1.1 State number pairs for the given wholenumber, 2.1.2 State the total of two numbers, and 2.1.3 List combinations of two numbers forthe given whole number (BPK, 2010). RBT = Revised Bloom’s Taxonomy.

As shown in Table 3, expectation analysis of the SK, SJKT and SJKC textbooksets identified that the expected three learning standards are entwined and presentin all the items with one exception: Learning standard 2.1.2 was omitted in ABT.

Questions in the first three levels of Revised Bloom’s Taxonomy, remembering(M = 100, SD = 0), understanding (M = 100, SD = 0) and applying (M = 21.00, SD= 38.83) were more common in year one mathematics textbook sets. All the textbooksets provided questions in remembering and understanding level. Only ABTexcelled in providing questions on applying level (100%), in contrast, questionson applying level were present in only 5% and 6%, respectively, in ABK andABC. None of the textbook sets analysed, exhibited any questions in the top threelevels Revised Bloom’s Taxonomy: analysing, evaluating and creating.

Discussion

Results indicated that the year one mathematics textbook sets analysed hadsimilarities and differences in presenting the numbers parts-whole relationship.

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All three sets of textbooks started with parts-whole relationship when introducingaddition and subtraction concepts. Integration of parts-whole relationship is critical,because research (Resnick, 1988; Van de Walle, et al., 2013) indicates thatunderstanding numbers in terms of their different parts in early years is importantfor future mathematical development.

The textbooks analysed, excel in visual aids such as graphics presentation.Though the use of graphics could enhance the presentation of certain mathematicsconcepts and help pupils understand them (Valverde et al., 2002) as in SK andSJKT textbook sets, pictures based on real-world situations can enhance pupil’sawareness of numbers in real life (Bell et al., 2004 as cited in Sood & Jitendra,2007) as in SJKC textbook sets. Activity books for SK and SJKT provided morequestions for pupils to practice the newly introduced parts-whole concepts asresearch (Coyne, Kameenui, & Carnine, 2007; Seabrook, Brown, & Solity, 2005)has proven the need for sufficient practice opportunities to promote acquisitionand retention of newly learned skills and strategies.

Though the number combinations of certain numbers were explored in thetextbook sets, a complete set of number combinations for any number was notfully explored in any of the textbook sets analysed. Research emphasizes theimportance of multiple representations (Rittle-Johnson & Koedinger, 2005;Shiakalli, 2012, 2014; Van Garderen & Montague, 2003) as found in all thetextbooks and SK activity book. However the lessons in the textbook setsanalysed in the current study did not use explicit language to scaffold the connectionbetween the representations as was found by Sood and Jitendra (2007) in theiranalysis.

In Malaysian year one mathematics textbooks, the most important part of theparts-whole relationship is finding the parts of a whole number. Research(Dougherty, et al., 2014; Van de Walle, et al., 2013) indicates that exploring allthe variations in parts-whole relationship can enhance pupils’ parts-wholeunderstanding. Furthermore, pupils’ strategy of finding the missing part, when thewhole and one part is known, can give the teachers’ an insight of pupils’ readinessfor addition and subtraction.

The gaps that are evident from the expectation analysis indicate a lack offocus on higher order thinking skills (HOTS). Though SJKT activity bookprovided questions on applying level in Blooms’ Revised Taxonomy, one of theHOTS, SK and SJKC activity books are lacking on this aspect. All the informationis provided in most of the questions presented, the pupils are only expected tocount the given pictures and fill in the blanks. As suggested by Conklin (2013),pupils need more challenging activities to enhance their HOTS. Problemsolving in context is underrepresented despite this lesson providing chances forposing questions that can have more than one answer which also can enhanceHOTS.


Suggestions

The results and discussions indicate the need to improve the presentation of parts-whole relationship in year one mathematics textbook sets in few aspects.Recommending articles, books and online resources related to part-wholerelationship as technical aids can support teachers in preparing more engagingteaching and learning activities. Emphasizing real-world graphics in SK and SJKTtextbook as well real-world activities or tasks (Dougherty, et al., 2014; Van deWalle, et al., 2013) in all the textbook sets can promote meaningful parts-wholeunderstanding among year one pupils.

Exploring all the number combinations for a few numbers in the textbook setscan help pupils realize the patterns involved when a number is decomposed andcomposed (Chapin & Johnson, 2006) and make generalizations. Makingconnections between and among multiple representations (Van Garderen, 2006),as well as using consistent and explicit language to connect and communicate thatparts make up whole or the whole is made up from parts (Sood & Jitendra, 2007)can enhance the pupils’ understanding of the mathematics process involved innumbers parts-whole relationship.

Rather than providing picture guidance for all the questions on decomposingand composing a number as in the textbooks sets analysed, a few questions shouldbe posed based on Revised Bloom’s Taxonomy to evoke pupils’ higher orderthinking skills (HOTS) as suggested by Malaysian Ministry of Education (KPM,2013; Lembaga Peperiksaan, 2013; MOE, 2012). For example, (a) let the pupils tolist the number pairs for a certain number and ask them to justify / explain whetherthey have listed all the possible answers, (b) ask the pupils whether 2 and 4 makes5, and explain their answer, (c) pose simple story problems that requires pupils tobe active learners, and (d) guide the pupils to create their own numbers parts-whole related stories and solve them as well as share their strategies with the class.

Conclusion

The results indicated that the textbooks are reasonably well aligned with the yearone mathematics syllabus in terms of structure, content and expectation. The widevariety and consistent use of colour and the number of graphics, examples andquestions present in these textbooks contribute to pupils’ comprehension of parts-whole relationship. However the gaps that are evident from the analysis indicateda lack of general focus on pupils’ HOTS. So teachers should consider thesignificance and limitations of using the year one mathematics textbook sets toteach numbers parts-whole relationship and plan their teaching and learningactivities accordingly. Future research can and should explore the other topics inmathematics textbooks as this analysis was limited to numbers parts-wholerelationship to add to the emerging body of literature on mathematics textbookanalysis in Malaysia.

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Reference

Anderson, L.W, Krathwohl, D.R., et al. (2001). A Taxonomy for learning, teaching and assessing:A revision of Bloom’s Taxonomy of educational objectives. New York: Longman.

Bahagian Perkembangan Kurikulum (BPK) [Curriculum Development Centre]. (2010). DokumenStandard Kurikulum Standard Sekolah Rendah (KSSR): Matematik Tahun 1[DocumentStandard Curriculum Standard Primary School: Mathematics Year 1]. Putrajaya, Malaysia:Kementerian Pelajaran Malaysia.

Baroody, A. J. (2004). The role of psychological research in the development of early childhoodmathematics standards. In D. H. Clements, & J. Sarama, Engaging Young Children inMathematics: Standards for Early Childhood Mathematics Education (pp.149-172). Mahwah,NJ: Lawrence Erlbaum Association.

Chapin, S.H, & Johnson, A. (2006). Math matters: Understanding the math you teach, GradesK-8. (2nd ed.). California: Math Solutions Publications.

Charlesworth, R. (2012). Experiences in math for young children. (6th ed.). Boston, MA:Wadsworth.

Cheng, Z.J. (2011). Teaching young children decomposition strategies to solve addition problems:An experimental study. The Journal of Mathematical Behaviour, 31, 29-47. doi:10.1016/j.jmathb.2011.09.002

Conklin, W. (2012). Higher-order thinking skills to develop 21st century learners. California:Shell Education.

Coyne, M.D., Kameenui, E.J., & Carnine, D. (2007). Effective teaching strategies thataccommodate diverse learners. Upper Saddle River, N.J.: Pearson Merril Prentice Hall.

Dougherty, B.J, Karp, K., Caldwell, J., & Kobett, B. (2014). Putting essential understanding ofaddition and subtraction into practice Pre-K-2. USA: NCTM.

Fischer, F.E. (1990). A part-part-whole curriculum for teaching number in the kindergarten.Journal for Research in Mathematics Education, 21(3), 207-215.

Fosnot, C. T., & Dolk, M. (2001).Young mathematicians at work: Constructing number sense,addition, and subtraction. Portsmouth, NH: Heinemann.

Hunting, R.P. (2003). Part-whole number knowledge in preschool children. Journal ofMathematical Behavior, 22(3), 217-235.

Jung, M. (2011). Number relationships in a preschool classroom. Teaching Children Mathematics,17(9), 550-557.

Jung, M., Hartman, P., Smith, T., & Wallace, S. (2013). The effectiveness of teaching numberrelationships in preschool. International Journal of Instruction, 6(1), 165-178.

Kamii, C., & Houseman, L.B. (2000). Young children reinvent arithmetic: Implications of Piaget’stheory. (2nd ed.). New York: Teachers College Press.

Kementerian Pendidikan Malaysia (KPM) [Ministry of Education Malaysia]. (2013). KBAT:Inisiatif Kemahiran Berfikir Aras Tinggi di Sekolah [HOTS: Higher order thinking skillsinitiative in schools]. Putrajaya, Malaysia: Kementerian Pendidikan Malaysia.

Lembaga Peperiksaan [Examination’s Board]. (2013). Pentaksiran kemahiran berfikir aras tinggi[Higher order thinking skills assessment]. Putrajaya, Malaysia: Kementerian PendidikanMalaysia (KPM).


Mahandran, G., Uthamaseelan, R., & Alice Nesomoney, D. (2010a). Matematik Tahun 1: SekolahJenis Kebangsaan (Tamil), Buku teks, Jilid 1 [Mathematics Year 1: National Type School(Tamil), Textbook, Volume 1]. Malaysia: Multi Educational Book Enterprise.

Mahandran, G., Uthamaseelan, R., & Alice Nesomoney, D. (2010b). Matematik Tahun 1: SekolahJenis Kebangsaan (Tamil), Buku Aktiviti, Jilid 1 [Mathematics Year 1: National Type School(Tamil), Activity book, Volume 1]. Malaysia: Multi Educational Book Enterprise.

Marzita, P., Wan Yusof, W.N., & Chan, Y.L. (2010a). Matematik Tahun 1: Sekolah Kebangsaan,Buku Teks, Jilid 1 [Mathematics Year 1: National School, Textbook, Volume 1]. KualaLumpur, Malaysia: Dewan Bahasa dan Pustaka.

Marzita, P., Wan Yusof, W.N., & Chan, Y.L. (2010b). Matematik Tahun 1: Sekolah Kebangsaan,Buku Aktiviti, Jilid 1 [Mathematics Year 1: National School, Activity book, Volume 1].Kuala Lumpur, Malaysia: Dewan Bahasa dan Pustaka.

Ministry of Education (MOE). (2012). Malaysian Education Blueprint 2013-2025. Retrievedfrom http://www.moe.gov.my/en/pelan-pembangunan-pendidikan-malaysia-2013-2025.

National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards forschool mathematics. Reston, VA: The National Council of Teachers of Mathematics, Inc.

Ooi, E.H., & Lee, K.K. (2010a). Matematik Tahun 1: Sekolah Jenis Kebangsaan (Cina), Bukuteks, Jilid 1 [Mathematics Year 1: National Type School (Chinese), Textbook, Volume 1].Malaysia: UPA Press Sdn. Bhd.

Ooi, E.H., & Lee, K.K. (2010b). Matematik Tahun 1: Sekolah Jenis Kebangsaan (Cina), BukuAktiviti, Jilid 1 [Mathematics Year 1: National Type School (Chinese), Activity book,Volume 1]. Malaysia: UPA Press Sdn. Bhd.

O’Keeffe, L., & O’Donoghue. (2011a). A review of school textbooks for Project Maths. Limerick,Ireland: National Centre for Excellence in Mathematics and Science Teaching and Learning(NCE-MSTL).

O’Keeffe, L., & O’Donoghue, J. (2011b, 9 to 13 February). Mathematics Textbook Analysis:The Significance of Textbook Features to Student Learning. Paper presented at the 7thCongress of the European Society for Research in Mathematics Education (CERME 7),Rzeszów, Poland.

Payne, J. N., & Rathmell, E. C. (1975). Number and numeration. In J. N. Payne (Ed.). Mathematicslearning in Early Childhood: 37th Yearbook NCTM (pp.125-160). Reston, VA: NationalCouncil of Teachers of Mathematics.

Resnick, L.B. (1983). A developmental theory of number understanding. In H.P. Ginsburg (Ed.).The Development of Mathematical Thinking (pp. 109-151). New York: Academic Press.

Resnik, L.B. (1988). Treating mathematics as an ill-structured discipline. In R. I. Charles & E.A. Silver (Eds.), The teaching and assessing of mathematical problem solving (pp. 32–62).Hillsdale, NJ: Erlbaum.

Riley, M.S., Greeno, J.G., & Heller, J.I. (1984). Development of children’s problem solvingability in arithmetic. In H. P. Ginsburg (Ed.). The Development of Mathematical Thinking,(pp. 153-196). New York: Academic Press.

Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to supportmathematical problem solving. Cognition and Instruction, 23, 313-349.

Seabrook, R., Brown, G. D. A., & Solity, J. E. (2005). Distributed and massed practice: Fromlaboratory to classroom. Applied Cognitive Psychology, 19(1), 107-122.

410 MAN IN INDIA

Shiakalli, M.A. & Zacharos, K. (2012). The contribution of external representations in pre-school mathematical problem solving. International Journal of Early Years Education.20(4), 315-331.

Shiakalli, M.A. & Zacharos, K. (2014). Building meaning through problem solving practices:The case of four-year olds. Journal of Mathematical Behavior, 35, 58-73.

Shield, M.J., & Dole, S. (2009). An analysis of middle-years school mathematics textbooks. InU.H. Cheah, Wahyudi, R.P. Devadason, K.T. Ng, W, Preechaporn, & J.C. Aligaen (Eds.),Proceedings from CoSMEd 2009: 3rd International Conference on Science and MathematicsEducation (pp. 369-375). Penang, Malaysia: SEAMEO RECSAM.

Sood, S., & Jitendra, A.K. (2007). A comparative analysis of number sense instruction in reform-based and traditional mathematics textbooks. The Journal of Special Education, 41(3), 145-157.

Sophian, C., & McCorgray, P. (1994). Part-whole knowledge and early arithmetic problemsolving. Cognition and Instruction, 12(1), 3-33.

Sophian, C., & Vong, K.I. (1995). The parts and wholes of arithmetic story problems: Developingknowledge in the preschool years. Cognition and Instruction, 13(3), 469-477.

Steinke, D.A. (2001). Does “Part-whole concept” understanding correlate with success in basicmath classes? Paper presented at the Annual Adults Learning Mathematics Conference.Roskilde, Denmark.

Steinke, D. (2008). Using part-whole thinking in math. Focus on Basics: Connecting Research& Practice, 9(A), 1-7.

Valverde, G.A., Bianchi, L.J., Wolfe, R.G., Schmidt, W.H, & Houng, R.T. (2002). According tothe book: Using TIMSS to investigate the translation of policy into practice through theworld of textbooks. New York: Springer Science + Business Media. Doi 10.1007/978-94-007-0844-0

Van de Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle schoolmathematics: Teaching developmentally. (8th ed.). Boston: Pearson.

Van Garderen, D., & Montague, M. (2003). Visual-spatial representation, mathematical problemsolving, and students of varying abilities. Learning Disabilities Research & Practice, 18,246–254.

Van Garderen, D. (2006). Teaching visual representation for mathematics problem solving. InM. Montague & A. K. Jitendra (Eds.), Teaching middle school students with mathematicsdifficulties (pp. 72– 88). New York: Guilford Press.

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