c e p s Journal | V ol.12 | N o 2 | Y ear 2022 119 The Teaching of Initial Multiplication Concepts and Skills in Croatian Textbooks Goran Trupčević* 1 and Anđa V alent 2 • The goal of this paper is to describe the teaching of initial multiplication concepts and skills, up to the multiplication table, in the Croatian edu - cational system. As Stiegler and Hiebert (1999) concluded, teaching is a complex system rooted in a cultural script of a given society. T o describe it without ignoring certain features of it that appear to be self-evident to an insider, it is necessary to step out of this cultural frame. For that reason, we study the teaching of initial multiplication in Croatia by comparing Croa - tian mathematical textbooks with textbooks from Singapore, Japan, and England. For the textbook analysis, we adapt the framework of Charalam - bous, Delaney, Hsu, and Mesa that examines a textbook as an environ - ment for the construction of knowledge of a single mathematical concept. The analysis provides evidence that practice and automation are at the centre of the initial learning of multiplication in Croatia. The meaning of multiplication usually is not clear, and pupils are not provided additional tools for developing understanding, nor they are encouraged to use differ - ent calculation strategies in a flexible manner. The study also indicates that Croatian textbooks present mathematics as a practice that is closed and pre-given, restricted to the one and the only right way through it. Keywords: concept construction, multiplication, textbook analysis 1 *Corresponding Author. Faculty of T eacher Education, University of Zagreb, Croatia; gtrupcevic@ gmail.com. 2 Zagreb University of Applied Sciences, Croatia. doi: 10.26529/cepsj.1303 120 the teaching of initial multiplication concepts and skills in croatian textbooks Pouk začetnih pojmov in spretnosti množenja v hrvaških učbenikih Goran Trupčević in Anđa V alent • Cilj tega prispevka je opisati, kako se poučevanje začetnih pojmov in spretnosti množenja, vključujoče vse do poštevanke, udejanja znotraj hrvaškega vzgojno-izobraževalnega sistema. Kot sta sklenila Stiegler in Hiebert (1999), je poučevanje zapleten sistem, ukoreninjen v kulturnem zapisu dane družbe. Zato da ga opišemo, ne da bi zoperstavili določene značilnosti, ki se sicer njegovim notranjim članom zdijo samoumevne, moramo stopiti izven kulturnega okvirja. Iz tega razloga proučujemo začetno poučevanje množenja na Hrvaškem s primerjanjem hrvaških matematičnih učbenikov z učbeniki iz Singapura, Japonske in Anglije. Za analizo učbenika smo prevzeli ogrodje, ki ga predlagajo Charalam - bous, Delaney, Hsu in Mesa, ki presoja učbenik kot okolje za stvaritev znanja enega samega matematičnega koncepta. Analiza priskrbi z dokazi, da sta praksa in avtomatizem v samem središču začetnega poučevanje množenja na Hrvaškem. Pomen množenja običajno ni jasen, pri čemer učencem niso zagotovljena do - datna orodja za razvoj razumevanja, niti niso spodbujeni, da bi uporabili različne računske strategije na fleksibilen način. Raziskava kaže tudi na to, da hrvaški učbeniki prikažejo matematiko kot prakso, ki je zaprta in vnaprej dana, omejena na en sam in edino pravilen način obravnave. Ključne besede: konstrukcija konceptov, množenje, analiza učbenika c e p s Journal | V ol.12 | N o 2 | Y ear 2022 121 Introduction In 2014, Croatia started a process of curriculum reform for primary and secondary education. New curricula for all school subjects were finished in 2016 and were sent for an expert review. In the new curriculum for mathemat - ics, learning of multiplication and multiplication tables should start in Grade 2, but full mastery was expected in Grade 3, since it requires time and a great deal of practice. In their commentaries on the first draft of the curriculum, many primary teachers called for the expectation of mastery of the multiplication ta - ble by the end of Grade 2 from the old curriculum (MZOŠ, 2007) to be retained. This proposition was adopted in the revised version of the new mathematics curriculum (MZO, 2019). The many recommendations concerning the mastery of the multiplica - tion table lead us to consider the initial teaching and learning of multiplication as an example of what Stiegler and Hiebert (1999) described when they charac - terise teaching as a cultural activity. These kinds of activities are rooted in the cultural script: they reflect attitudes and beliefs of a culture and are manifested in the patterns of behaviour. These attitudes, beliefs, and behaviours are learned implicitly through observation and participation inside the culture. Since they are widely shared, they are difficult to see. This makes them highly stable over time and not easily changed. The old Croatian mathematics curriculum reflects this: ‘In the existing elementary school, mathematics is a subject with a long tradition and well-de - fined content, and no major interventions are required in current programmes. Therefore, the existing programmes formed the starting basis in the design of the mathematics curriculum’ (MZOŠ, 2006, p. 238). To investigate these kinds of practices, one needs to step out of one’s cul - tural circle. In this way, one can become aware of the scripts they are using and, by comparing different cultural scripts, one can see possibilities, not just what is there but also what is not (Stiegler & Hiebert, 1999). One of the ways of doing this for teaching practices is through the comparative international textbook analysis (Haggarty & Pepin, 2002). Multiplicative reasoning That ‘multiplicative thinking appears early and develops slowly’ was the conclusion of Clark and Kamii (1996) based on their analysis involving children in Grades 1 to 5. There are multiple reasons for that. 122 the teaching of initial multiplication concepts and skills in croatian textbooks In studying children’s solutions to multiplication problems, Bell et al. (1984) found that the solutions varied according to the numbers involved, the structures involved, and the context of the problem. They saw this as another manifestation of ‘meaning blindness’. Although traditionally, the teaching of multiplication and division was the teaching of procedures, this showed the necessity of also teaching different kinds of multiplicative structures. Or, in the words of Kaput (1985, as cited in Greer, 1992), mathematics and its ‘applications’ should be taught as being of a piece from the very beginning. In their search for the origins of children’s mistakes in multiplication problems, Fischbein et al. (1985) hypothesised that children link every arith - metic operation to an implicit, unconscious, primitive model that mediates the meaning of operation when solving problems. The model, in turn, imposes its own constraints on the understanding of operations. For multiplication, they argue, children use repeated addition as an implicit model. This hypothesis of repeated addition as the origin of multiplicative thinking is reflected in cur - riculum documents of different countries (Park & Nunes, 2001; MZOŠ, 2006). When considering multiplicative situations or structures from the point of the underlying process or procedure, different kinds of these situations can be identified: equal grouping, rate, array, measure conversion, multiplicative com - parison, Cartesian product (Anghileri, 1989; Bell et al., 1984). Greer (1992) syn - thesises classifications of multiplicative situations into four major classes: equal groups (with equal measure and rate; Ann, Ivy, and Kathy have 5 apples each. How many apples do they have altogether? ), multiplicative comparison ( Mary has 3 times as many chocolates as Steve. If Steve has 5 chocolates, how many chocolates does Mary have? ), rectangular array (and area; Chocolates are sorted in an array of 3 rows and 5 columns. How many chocolates are there? ) and Cartesian product (John has 3 pants and 5 shirts. In how many ways can John dress? ). Vergnaud (1988), in contrast, considers the mathematical relations be - tween quantities in these situations. In this way, he finds two main types of situ - ations. The first he calls ‘an isomorphism of measures’ , which involves two pairs of different quantities. There are two types of relationships in this situation: one between the quantities of a different kind (ratio) and one between values of the same kind of quantities (scalar factor). Another type of situation concerns a product of measures, which involves a third quantity, a factor quantity, con - necting two other kinds of quantities. Different researchers considered children’s solution strategies for multi - plication problems. From the point of the degree of abstraction, these strategies were classified as modelling (and counting), counting (by 1, rhythmic, skip), repeated addition, and multiplicative operations (Anghileri, 1989; Kouba, 1989; c e p s Journal | V ol.12 | N o 2 | Y ear 2022 123 Mulligan & Mitchellmore, 1997). Kouba (1989) also noted that when children used physical objects for solving problems, they either represented individual elements in each set, or they represented only the tallies. For the same multi - plication expression, children’s solution strategies can change if the meaning of multiplication or its representation changes (Fosnot & Dolk, 2001; Kouba, 1989). This phenomenon is not exclusive to multiplication but has been ob - served in other topics (Kolar et al., 2018; Tirosh et al., 2018). Fosnot and Dolk (2001) elaborated multiplicative operations strategies by linking them to the properties of multiplication. The research into children’s solution strategies for multiplication prob - lems put forward a theory of repeated addition as an origin of multiplication into question (Kouba, 1989). Although counting and adding are used to ob - tain the numerical answer, there is something that goes on before the counting (Nunes & Bryant, 1996; Steffe, 1994). These considerations lead to an alternative hypothesis that views the concept of multiplication as defined by an invariant relation between two quan - tities, called ‘ratio’ or ‘rate’ (Piaget, 1965; Vergnaud, 1988). Thus, the develop - ment of a one-to-many correspondence scheme lies at the origin of the un - derstanding of multiplicative situations (Nunes & Bryant, 1996). The works of Piaget (1965) and others (see Nunes & Bryant, 1996) show that children as early as five years of age are able to represent and use one-to-many correspondences. Furthermore, arguments for the repeated addition theory can result from chil - dren’s previous school experience (Nunes et al., 2010). An intervention study by Park and Nunes (2001) showed that pupils who were taught multiplication through correspondence made significantly more progress than pupils who were taught multiplication through repeated addition. This shows that multiplication is not simply a new arithmetic opera - tion to be learned after addition but that there are also new meanings and new situations to be learned. Moreover, although there is a procedural connection between multiplication and addition, there is also a conceptual discontinuity between the two (Nunes & Bryant, 1996). Research on Mathematical Textbooks Mathematics textbook research is a rich and growing area of investiga - tion. In their overview of the literature, Fan et al. (2013) conceptualised four cat - egories of mathematics textbook research: the role of textbooks in mathematics teaching and learning, textbook analysis and comparison, textbook use, and other areas of textbook research. 124 the teaching of initial multiplication concepts and skills in croatian textbooks Studies from different countries have shown that textbooks have a con - siderable impact on the teaching and learning taking place in the classrooms (Fan & Kaeley, 2000; Haggarty & Pepin, 2002; Pepin & Haggarty, 2001). Hag - garty and Pepin (2001, 2002) analysed textbook structure and content, their usage by the teachers, and the access to the textbooks for the pupils. They found that textbooks from different countries differ both in the structure and in the complexities and learning opportunities they offer for the pupils. Furthermore, the way teachers used textbooks varied between different countries, but also between different educational strands in one country. Authors argued that some of the factors that influenced these issues come from the cultural tra - dition, consisting of the organisation of the educational system, educational traditions, values and epistemic beliefs, and the socio-economic conditions of both the pupils and teachers. Studies in Croatia also confirmed the impact that textbooks have on teaching and learning (Glasnović Gracin & Domović, 2009; Glasnović Gracin & Jukić Matić, 2016; Jukić Matić, 2019). These studies also revealed social issues that influence textbook utilisation. A comprehensive TIMSS study of mathematics and science textbooks from 48 countries (Valverde et al., 2002) assumes a four-part model of the cur - riculum; besides the intended, implemented, and attained curriculum, they add the potentially implemented curriculum as the fourth part, and view textbooks as parts of it, as mediators between intended and implemented curriculum. Their description of textbooks considers their macrostructure (size, length, etc.) but also their microstructure for which textbooks are sequenced into blocks of analysis, which are then coded by block type, mathematical content, presentation expectations, and a wider perspective on the subject. This analysis shows variations in textbook structures across different school systems. The study by Jones and Fujita (2013), which uses the adapted TIMSS framework, showed that mathematical textbooks in England and Japan clearly reflect the geometrical component of a national curriculum. Charalambous et al. (2010) developed a framework for investigating learning opportunities and pupils’ expectations in the textbooks, particularly with respect to the presenta - tion of specific content. Their study considered the treatment of the addition and subtraction of fractions in textbooks from Cyprus, Ireland, and Taiwan and found greater variations between textbooks across cultures than within one country, which is in accordance with the findings of the TIMSS video study (Stigler & Hiebert, 1999). This framework was later used in similar comparative studies in different countries and for different mathematical topics (Hong & Choi, 2014; Y ang & Sianturi, 2017). Boonlerts and Inprasitha (2013) analysed the presentation of multiplication in elementary textbooks from Japan, Singapore, c e p s Journal | V ol.12 | N o 2 | Y ear 2022 125 and Thailand through content analysis (sequencing of topics and meaning of multiplication). The meaning given to multiplication was conceptualised using Greer’s (1992) categorisation of classes of situations involving multiplication, and the analysis showed differences in the textbooks from the three countries from this perspective. Analysis of the sequencing of topics showed that text - books from Singapore and Japan offer their pupils increasingly sophisticated strategies to solve multiplication problems. This study aims to describe important features of the treatment of initial multiplication in mathematics textbooks in Croatia. This kind of description could yield important information for the future development of the national curriculum and curricular materials. Also, the findings of this study can be used for informing and changing the teaching practice in primary schools in Croatia. T o describe the treatment of initial multiplication in Croatian textbooks, we compare them with textbooks from Singapore, Japan, and England. The pri - mary reasons for choosing these countries are the good results that students from these countries achieve in international comparative studies and the avail - ability of textbooks in the English language. Furthermore, since these countries are geographically distant, and their tradition is historically and culturally dif - ferent from that of Croatia, one can expect that insight into culturally hidden assumptions and practices of the Croatian teaching tradition could be obtained by comparison with these countries. We attempt to describe features of Croatian textbooks by answering the following research questions: 1. What is the structure of a typical lesson in Croatian textbooks? 2. What possible multiplication constructs and representations are used? 3. What multiplication strategies are promoted? 4. Do Croatian textbooks promote the usage of tools and manipulatives for learning? 5. What features of a mathematics classroom are implicated by Croatian textbooks? 6. To what extent do Croatian textbooks reflect the national curriculum? 126 the teaching of initial multiplication concepts and skills in croatian textbooks General assumptions and initial multiplication in the Mathematics Curricula of Croatia, Singapore, Japan, and England Before the textbook analysis, an overview of national mathematics cur - ricula was made so that it could be used for the interpretation of the results. The Croatian 2006 curriculum is divided into two parts: the introducto - ry part and the list of themes for each year of study, together with pupils’ learn - ing achievements. Themes of beginning multiplication are placed in Grade 2. The Singapore Mathematics Syllabus (MOE, 2012) is organised through a list of mathematical contents and appropriate learning experiences for each year of study are given at the end of the syllabus. The learning of multiplication and multiplication tables takes place in the first three years. The 2009 Japanese curriculum standards (Isoda, 2010) operationalises its objectives for each grade through a list of learning outcomes, a list of rel - evant terms and symbols, a list of mathematical activities to be used, and some specific remarks on the ways of handling the content. The learning of multipli - cation and multiplication tables takes place in Grade 2. The 2013 English Mathematics Curriculum (DfE, 2013) for each year gives statutory requirements but also some notes and guidance. Pupils learn multiplication during the first four years. Table 1 summarises different aspects of the mathematics curriculum of these four countries. One can observe that curricula have much more in com - mon regarding their main emphasis. However, when it comes to the elaboration of the content of initial multiplication and description of the ways of working, the Croatian curriculum is rather concise on that matter. It will be interesting to see whether this will also be reflected in the textbooks. c e p s Journal | V ol.12 | N o 2 | Y ear 2022 127 Table 1 Comparison of mathematics curricula of Croatia, Singapore, Japan, and England regarding multiplication Croatia Singapore Japan England Emphasis of the Curriculum basic mathematical knowledge, development of skills, problem- solving problem-solving, conceptual understanding, skills proficiency, mathematical process, attitudes, metacognition basic knowledge and skills, ability to think and express, attitudes fluency, mathematical reasoning, problem-solving Grades 2 1, 2, 3 2 1, 2, 3, 4 Multiplication constructs addition addition, equal groups, scaling equal groups addition, equal groups, array, scaling, Cartesian product Multiplication strategies commutativity counting, concretisation, patterns, heuristics counting, properties of multiplication, commutativity, adding next, patterns counting, concretisation, doubling, patterns, connections, commutativity, associativity, distributivity Representations and manipulatives pictorial, concrete objects pictorial, concrete objects pictorial, concrete objects Forms of work group work, sharing ideas Method Based on the data of the Croatian Ministry of Science and Education (MZOS, 2014), the three most commonly used Croatian textbook series were chosen for the analysis: Matematika , Moj sretni broj , Nove matematičke priče . In accordance with the Croatian curriculum, all these textbooks are used in the second year of study. Together they comprised a market share of 78.33% of mathematics textbooks in the second grade in Croatia. In addition, the follow - ing textbook series were analysed: Primary Mathematics and My Pals are Here from Singapore, Sansu Math from Japan, and Power Maths from England, used from Grades 1–3, 2–3, and 1–4, respectively. Throughout this paper, the names of these textbooks will be abbreviated as CT1, CT2, CT3, ST1, ST2, JT, and ET, respectively. An analytical framework was based on the framework of Charalambous, Delaney, Hsu, and Mesa (2010) for textbook analysis, which views a textbook as an environment for the construction of knowledge of a single mathematical 128 the teaching of initial multiplication concepts and skills in croatian textbooks concept. All lessons of a textbook concerning initial multiplication learning upon the multiplication table were analysed. Each lesson was divided into smaller blocks that were analysed from several aspects (categories). Based on the results of the pilot study (Baković et al., 2019.), the analytical framework was adapted to include the following categories: block type, social form of work, context, use of concrete materials, images, characters, representations, construct and multiplication strategies. The coding list for each category was created by using the grounded theory approach (Glaser & Strauss, 1967); the initial list was created on the basis of the literature review and was later revised on the basis of the observed data. The full list of codes is given in Table 2. The category block type concerns its function in the text with the way that the block is communicated to pupils. The category context deals with the context of the mathematical problem posed in a block. Authentic context refers to a pupil’s personal experience while realistic to a possible experience. The category construct refers to the meaning of multiplication. For this purpose, Greer’s (1992) classification was revised so that multiplicative com - parison, rate, and equal measure problems were included in a joint class that was named ‘scaling’ . This change was motivated by children’s different ways of representing grouping and matching problems in Kouba’s study (1989). Three categories were concerned with the visual features of blocks. The first one, the use of pictures category registers the presence of the picture and whether it is mathematically relevant or not. The representations category refers to the way that multiplication is rep - resented. The representation usually corresponds to the construct, but some - times the two can differ. For instance, the text can be about four groups of three, while the picture shows 12 objects sorted in a 4×3 array. Finally, the category other graphical objects refers to the function of graphical objects that were used. These were mostly images of pupils or some other characters that communicate some information to pupils, but there were also diagrams, drawings, and images of objects to use. The category use of manipulatives assesses if pupils are instructed to use concrete, manipulative materials and, if so, which ones. Physical manipulatives are divided into standardised, which are used throughout the textbook, and sporadic, which are used only once. The category multiplication strategy assesses both which strategies for carrying out multiplication are promoted in the block, as well as problem-solv - ing and general learning strategies. The category social form of work characterises organisational ways pupils c e p s Journal | V ol.12 | N o 2 | Y ear 2022 129 are supposed to work. Only blocks for which this is explicitly stated were cat - egorised, other blocks were categorised as ‘unclear’ . Table 2 Coding list Block Type Use of manipulatives Situation Standardised manipulative Recap Sporadic manipulative Worked example 10x10 table Definition/rule Games Exercise Multiplication cards Activity/game Hands Self-assessment Strategy Context Commutativity Authentic Multiplications table Realistic Multiplication by 1 and 0 Intra-mathematical Counting Adding/Subt next Construct Distributivity Addition Doubling Equal groups Bar-model Array Modelling/Concretisation/Drawing Scaling/Comparison Metacognition Counting Estimation Cartesian product Mnemonic Without construct Associativity Use of pictures Other graphical elements Absent Instruct/describe Mathematically relevant Solution Context related Explanation Guidance Representation Different solutions Equal groups Additional questions Linear Terminology Array Rule Number line Drawing/diagram Multiplications table Tools/manipulatives Counting Bar/Scaling Social form of work 10×10 table Pair Cartesian product Group Without Representation Family Unclear 130 the teaching of initial multiplication concepts and skills in croatian textbooks We demonstrate the coding procedure on the sample page given in Fig - ure 1. Figure 1 Example of coding a textbook page Results and discussion The total number of blocks that were identified and analysed in the text - books is given in Table 3. Table 3 Number of analysed blocks in the textbooks CT1 CT2 CT3 ST1 ST2 ET JT 139 117 171 197 136 420 101 The results of the analysis of blocks are given below, for each category, in terms of percentages of the blocks in the textbooks. c e p s Journal | V ol.12 | N o 2 | Y ear 2022 131 Block type The distribution of codes inside this category is shown in Table 4. Recaps appear more frequently in Croatian textbooks than in non-Croatian textbooks. In non-Croatian textbooks, they appear only at the beginning of chapters, while in Croatian textbooks each lesson begins with a recap. This can be considered an indicator of a compartmentalised and unconnected view of mathematical knowledge. Also, Croatian textbooks had neither activities/ games nor self-assessment blocks. Table 4 Distribution of block type in the textbooks Block Type CT1 CT2 CT3 ST1 ST2 ET JT Situation 1% 1% 1% 2% Recap 10% 18% 15% 1% 1% Worked example 24% 9% 26% 24% 29% 8% 30% Definition/rule 6% 15% 6% 8% 4% 9% Exercise 59% 57% 53% 57% 48% 89% 53% Activity/game 1% 9% 15% 6% Self-assessment 3% 2% Context and Construct Table 5 Distribution of Context in the textbooks Context CT1 CT2 CT3 ST1 ST2 ET JT Authentic 13% 16% 6% 3% Realistic 32% 24% 30% 63% 71% 62% 47% Intra-mathematical 68% 76% 70% 23% 13% 33% 50% The context of the majority of the blocks in Croatian textbooks is intra- mathematical, and it is never authentic (see Table 5). This is different from non- Croatian textbooks in which there is a prevalence of realistic content, and all of them include blocks with authentic context. 132 the teaching of initial multiplication concepts and skills in croatian textbooks Table 6 Distribution of Construct in the textbooks Construct CT1 CT2 CT3 ST1 ST2 ET JT Addition 17% 27% 19% 8% 7% 14% 4% Equal groups 23% 15% 11% 65% 60% 41% 23% Array 2% 9% 3% 21% 13% 17% 17% Scaling/Comparison 17% 5% 24% 6% 7% 20% 33% Counting 20% 7% 2% 1% Cartesian product 2% Without construct 55% 64% 54% 17% 19% 28% 49% The majority of blocks of Croatian textbooks do not have a construct specified (see Table 6). In non-Croatian textbooks, a construct is evident in almost all blocks. The high rate of blocks without evident constructs in JT text - books is a consequence of the fact that the focus of the last 20% of lessons is to encourage pupils to use different multiplication strategies and to investigate multiplication properties and patterns; thus, a construct in these lessons is not evident. Some non-Croatian textbooks systematically introduce different con - structs and sometimes explicitly ask pupils to make connections between dif - ferent constructs. This is not the case in Croatian textbooks in which new con - structs usually appear ‘out of nowhere’, with the occasional observation that ‘this is also multiplication’ . It can be noted that in Croatian textbooks repeated addition is promoted at a higher rate of blocks than in non-Croatian textbooks. Also, non-Croatian textbooks promote counting as a construct, which is not the case in Croatian textbooks. An explanation for these findings could be rooted in differences between curricula. Unlike other curricula, the Croatian curriculum does not point out different aspects of multiplication and defines multiplication only as repeated addition. c e p s Journal | V ol.12 | N o 2 | Y ear 2022 133 Representations, use of pictures and manipulatives Table 7 Use of pictures in the textbooks Use of pictures CT1 CT2 CT3 ST1 ST2 ET JT Absent 76% 75% 74% 37% 17% 28% 38% Mathematically relevant 24% 25% 24% 60% 79% 66% 55% Context related 2% 3% 1% 5% 7% Almost 75% of blocks in Croatian textbooks are without representation, which is significantly lower than in non-Croatian textbooks (see Table 7). They are usually present in worked examples at the beginning of a lesson, while in exercises a representation is usually absent. Table 8 Distribution of Representation in the textbooks Representation CT1 CT2 CT3 ST1 ST2 ET JT Equal groups 15% 11% 8% 46% 32% 36% 28% Linear 5% 2% 4% 3% 15% 3% 7% Array 1% 10% 8% 20% 29% 20% 27% Number line 3% 12% 8% 14% Multiplications table 7% 11% 6% 6% 13% 3% 20% Counting 1% 7% 20% 10% 2% 2% Bar/Scaling 4% 2% 11% 10% 10×10 table 2% 2% 1% Cartesian product 2% Without Representation 70% 58% 75% 21% 18% 35% 39% Besides pictorial representation, some non-pictorial representations also appear; in Croatian textbooks, these are a multiplication table and count - ing, while in non-Croatian textbooks, we additionally registered a 100-table (see Table 8). There is also a difference in the usage of multiplication tables; in Croatian textbooks, they are simply given without any further instructions. In non-Croatian textbooks, pupils are encouraged to use these tables to observe patterns and properties of multiplication. 134 the teaching of initial multiplication concepts and skills in croatian textbooks Table 9 Use of manipulatives in the textbooks Use of manipulatives CT1 CT2 CT3 ST1 ST2 ET JT Standardised manipulative 15% 10% 5% 3% Sporadic manipulative 2% 1% 10×10 table 2% 1% 1% 8% Games 1% 1% 3% Multiplication cards 2% 2% 2% Hands 8% 1% As shown in Table 9, Croatian textbooks do not promote the use of ma - nipulatives. In non-Croatian textbooks, one can observe the intention to stand - ardise the use of manipulatives: the same manipulatives are used throughout different lessons, giving pupils the opportunity to make connections between mathematical content and its representation with those manipulatives. Strategies Table 10 Distribution of Strategy in the textbooks Strategy CT1 CT2 CT3 ST1 ST2 ET JT Commutativity 6% 15% 13% 23% 8% 10% 16% Multiplications table 7% 10% 12% 5% 13% 8% 24% Multiplication by 1 and 0 7% 6% 8% 2% 1% 3% 4% Counting 1% 7% 21% 10% 6% 1% Adding/Subt next 1% 3% 3% 6% 24% Distributivity 2% 9% 5% 4% 14% Doubling 2% 9% 3% Bar-model 5% 2% 4% 8% Modelling/Concretisation/Drawing 3% 12% 4% 12% 15% Metacognition 1% 5% 6% 15% 22% Estimation 3% Mnemonic 1% 4% 1% 1% Associativity 1% c e p s Journal | V ol.12 | N o 2 | Y ear 2022 135 Croatian textbooks promote only multiplication tables, multiplication by 1 and 0 and commutativity as strategies for carrying out multiplication (see Table 10). Multiplication by 0 and 1 are mostly present inside corresponding lessons listed in the curriculum. In non-Croatian textbooks, this strategy is far less present. Commutativity is also differently treated: in non-Croatian text - books it is a strategy that pupils can use in their calculations, or when looking for patterns or global properties of multiplication. Unlike this, Croatian text - books usually present commutativity as a property that pupils are just supposed to check. Modelling, concretisation, and drawing as strategies that enable pupils to organise their own thinking around multiplication problems are almost never present in Croatian textbooks. This is in accordance with the Croatian curriculum which, unlike non-Croatian curricula, pays no attention to these strategies. The same holds for counting as a strategy. Strategies that enable pupils to calculate flexibly, such as doubling, add - ing/subtracting one and distributivity, are not present at all in Croatian text - books, while non-Croatian textbooks systematically encourage pupils to use these strategies. Non-Croatian textbooks systematically introduce the bar model as a pre-algebraic strategy for solving word problems. This is not present in Croa - tian textbooks where pupils are not given additional tools to help them with modelling or solving problems. Unlike Croatian textbooks, non-Croatian textbooks pay attention to the development of different metacognitive strategies: recognition of boundaries of certain concepts, self-assessment, scaffolding in problem-solving, making con - nections between different parts of knowledge, and similar. 136 the teaching of initial multiplication concepts and skills in croatian textbooks Other graphical elements Table 11 Use of other graphical elements in the textbooks Other graphical elements CT1 CT2 CT3 ST1 ST2 ET JT Instruct/describe 2% 14% 2% 5% 1% 5% Solution 5% 1% 4% 2% 3% Explaining 2% 16% 2% Guidance 11% 8% 11% 14% Different solutions 2% 1% 5% 11% Additional questions 2% 7% 6% 9% Terminology 1% 1% 7% Rule 1% 4% 1% 2% Drawing/diagram 2% 5% Tools/manipulatives 13% 15% The majority of other graphical elements refer to certain characters in textbooks. These are mostly pupils, but some abstract characters also appear (e.g., a magician in CT3). It is interesting to note the difference between the functions of these characters in textbooks (see Table 11). Characters in Croatian textbooks give instructions, solutions, and explanations. The magician in CT3 leads pupils through the story. Besides these functions, characters in non-Croa - tian textbooks give guidance for pupils’ solutions, they ask additional questions and give and seek more than one solution. We believe that the functions of these characters give insight into the way teaching is supposed to unfold in the classroom. Croatian textbooks present a classroom in which the teacher is the central figure, ‘the sage on the stage’ . Pupils in this classroom are expected to give solutions to problems in only one, correct way. The mathematics that this classroom presents is closed and given, restricted to the one and only right way through it. Unlike this, non-Croatian textbooks present classrooms that are pupil- centred. Pupils are the ones that solve problems, guide each other, discuss differ - ent solutions, and open new questions. Mathematics in this case is an open and connected domain of flexible thinking and discussions among practitioners. c e p s Journal | V ol.12 | N o 2 | Y ear 2022 137 Social form of work Table 12 Social forms of work in the textbooks Social form of work CT1 CT2 CT3 ST1 ST2 ET JT Pair 3% 3% 2% 4% Group 7% 2% 4% Family 1% Unlike non-Croatian textbooks, Croatian textbooks do not call for work in pairs or in a group (see Table 12). This additionally supports the idea of a mathematical activity as a solitary endeavour. Furthermore, it gives fewer edu - cational opportunities for pupils (Boaler, 2016). Conclusion This analysis provides evidence that there are only minor variations of teaching multiplication within different Croatian textbooks and at the same time all Croatian textbooks are noticeably different from the observed text - books from England, Japan, and Singapore. The initial learning of multiplica - tion in Croatia puts practice and automation in focus. The majority of content is intra-mathematical, not connected to the real world or authentic problems, and pupils are not encouraged to use manipulatives to help them model real - istic problems or represent abstract mathematical problems. The meaning of multiplication is usually not clear nor is it visually represented. It is expected that the pupils solve exercises, but they are not encouraged to use different strategies and approaches. The study also indicates some general features of teaching and learning mathematics in Croatia. Unlike non-Croatian textbooks, Croatian textbooks do not promote the development of metacognitive skills and strategies. Pupils are not encouraged to find different solutions and to make connections be - tween them, to discuss their solutions, or to pose problems. These findings suggest that the teaching practice should strive to com - pensate for these deficits. This means helping pupils in making sense of multi - plication by situating problems in a familiar context, giving them opportunities to explore multiplication by using pictures and manipulatives, and giving them opportunities to develop different strategies for calculating. 138 the teaching of initial multiplication concepts and skills in croatian textbooks Focusing solely on the textbooks is a limitation of this study. To obtain a fuller picture, one should see what is going on inside the actual classrooms. In fact, to get insight into this, an observational study had been started in March 2020 but was closed soon because of a lockdown due to the Covid-19 pandemic. Since the implementation of the new curriculum in Grade 2 started in Septem - ber 2020, it was no longer possible to continue it. All these features are in accordance with the Croatian curriculum from 2006. The new curriculum, which puts more stress on the use of manipulatives and pictorial representations, on problem-solving and formative assessment, will be fully implemented by 2022. It would be interesting to conduct a follow- up study in the future and see to what extent the changes made in the curricu - lum will be reflected in the textbooks and the teaching practice. References Anghileri, J. (1989). An investigation of young children’s understanding of multiplication. Educational Studies in Mathematics , 20(4), 367–385. https://doi.org/10.1007/BF00315607 Baković, T., Trupčević, G., & Valent, A. (2019). Treatment of initial multiplication in textbooks from Croatia and Singapore. In Z. Kolar-Begović, R. Kolar-Šuper, & Lj. Jukić Matić (Eds.), Towards new perspectives on mathematics education (pp. 216–228). Element. Bell, A., Fischbein, E., & Greer, B. (1984). Choice of operation in verbal arithmetic problems: The effects of number size, problem structure and context. Educational Studies in Mathematics , 15(2), 129–147. https://doi.org/10.1007/BF00305893 Boaler, J. (2016). Mathematical mindsets: unleashing students potential through creative math, inspiring messages. and innovative teaching . Jossey-Bass. Boonlerts, S., & Inprasitha, M. (2013). The Textbook Analysis on Multiplication: The Case of Japan, Singapore and Thailand. Creative Education , 4(4), 259–262. https://doi.org/10.4236/ce.2013.44038 Charalambous, C.Y ., Delaney, S., Hsu, H.-Y ., & Mesa, V . (2010). A comparative analysis of the addition and subtraction of fractions in textbooks from three countries. Mathematical Thinking and Learning, 12 (2), 117–151. https://doi.org/10.1080/10986060903460070 Clark, F. B., & Kamii, C. (1996). Identification of multiplicative thinking in children in grades 1-5. Journal for Research in Mathematics Education , 27(1), 41–51. https://doi.org/10.2307/749196 Department for Education [DfE]. (2013). Mathematics programmes of study: Key stages 1 and 2. National curriculum in England . DfE. https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/ file/335158/PRIMARY_national_curriculum_-_Mathematics_220714.pdf Fan, L., & Kaeley, G. S. (2000). The influence of textbooks on teaching strategies: An empirical study. Mid-Western Educational Researcher , 13(4), 2–9. Fan, L., Zhu, Y ., & Miao, Z. (2013). Textbook research in mathematics education: Development status c e p s Journal | V ol.12 | N o 2 | Y ear 2022 139 and directions. ZDM: The International Journal on Mathematics Education , 45(5), 633–646. https://doi.org/10.1007/s11858-013-0539-x Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education , 16(1), 3–17. https://doi.org/10.2307/748969 Fosnot, C. T., & Dolk, M. (2001). Young mathematicians at work. Constructing multiplication and division. Heinemann. Foong, P . Y ., Chang S.H., Lim L. G. P ., & Wong O.H. (2014). Primary mathematics 1B . Shinglee. Foong, P . Y ., Chang S.H., Lim L. G. P ., & Wong O.H. (2015). Primary mathematics 3A . Shinglee. Foong, P . Y ., Lim L. G. P ., & Wong O. H. (2014). Primary mathematics 2A . Shinglee. Glaser, B., & Strauss, A. (1967). The discovery of grounded theory: Strategies for qualitative research . Aldine. Glasnović Gracin, D. & Domović, V . (2009). Upotreba matematičkih udžbenika u nastavi viših razreda osnovne škole [The use of mathematics textbooks in teaching of upper primary grades]. Odgojne znanosti , 18(2), 45–65. https://hrcak.srce.hr/48441 Glasnović Gracin, D., & Jukić Matić, Lj. (2016). The role of mathematics textbooks in lower secondary education in Croatia: An empirical study. The Mathematics Educator 16(2), 31–58. Greer, B. (1992). Multiplication and division as models of situations. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 276–295). Macmillan. Haggarty, L., & Pepin, B. (2002). An investigation of mathematics textbooks and their use in English, French and German classrooms: Who gets an opportunity to learn what? British Educational Research Journal , 28(4), 567–590. https://doi.org/10.1080/0141192022000005832 Hong, D. S., & Choi, K. M. (2014). A comparison of Korean and American secondary school textbooks: the case of quadratic equations. Educational Studies in Mathematics, 85(2), 241–263. https://doi.org/10.1007/s10649-013-9512-4 Isoda, M. (Ed.) (2010). Elementary school teaching guide for the Japanese course of study mathematics (Grade 1-6) . CRICED, University of Tsukuba. https://www.criced.tsukuba.ac.jp/math/apec/ICME12/ Lesson_Study_set/Elementary_School_Teaching_Guide-Mathematics-EN.pdf Jones, K., & Fujita, T. (2013). Interpretations of national curricula the case of geometry in textbooks from England and Japan. ZDM Mathematics Education , 45(5), 671–683. https://doi.org/10.1007/s11858-013-0515-5 Jukić Matić, Lj. (2019). The teacher as a lesson designer . Center for Educational Policy Studies Journal , 9(2), 139–160. Ttps://doi.org/10.26529/cepsj.722 Kheong, F. H., Ramakrishnan, C., & Choo, M. (2014). My pals are here maths 2A . MC Education. Kheong, F. H., Ramakrishnan, C., & Choo, M. (2015). My pals are here maths 3A . MC Education Kheong, F. H., Ramakrishnan, C., & Wah, B. L. P . (2013). My pals are here maths 1B . MC Education. Kolar, V . M., Hodnik Čadež, T., & Vula, E. (2018). Primary teacher students’ understanding of fraction representational knowledge in Slovenia and Kosovo. Center for Educational Policy Studies Journal , 8(2), 71–96. https://doi.org/10.26529/cepsj.342 140 the teaching of initial multiplication concepts and skills in croatian textbooks Kouba, V . L. (1989). Children‘s solution strategies for equivalent set multiplication and division word problems. Journal for Research in Mathematics Education , 20(2), 147–158. https://doi.org/10.2307/749279 Markovac, J. (2015). Matematika 2 [Mathematics 2]. Alfa. Miklec, D., Prtajin, G., & Jakovljević Rogić, S. (2015). Moj sretni broj 2 [My lucky number 2 ]. Školska knjiga. Ministarstvo znanosti obrazovanja i športa [MZOŠ]. (2006). Nastavni plan i program za osnovnu školu . [Course of study for primary school]. MZOŠ. Ministry of Education Singapore [MOE]. (2012). Mathematics syllabus: Primary one to Six . MOE. https://www.moe.gov.sg/-/media/files/primary/mathematics_syllabus_primary_1_to_6.pdf Mulligan, J. T., & Mitchelmore, M. C. (1997). Y oung children‘s intuitive models of multiplication and division. Journal for Research in Mathematics Education , 28(3), 309. https://doi.org/10.2307/749783 Ministarstvo znanosti i obrazovanja [MZO]. (2019). Odluka o donošenju kurikuluma za nastavni predmet Matematike za osnovne škole i gimnazije u Republici Hrvatskoj [Decision of declaration of the curriculum of Mathematics for primary and secondary school]. Narodne novine 7/2019. https://narodne-novine.nn.hr/clanci/sluzbeni/2019_01_7_146.html Ministarstvo znanosti obrazovanja i sporta [MZOS]. (2014). Konačne liste odabranih udžbenika i pripadajućih dopunskih nastavnih sredstava prema odabiru stručnih aktiva u osnovnim i srednjim školama [Final lists of selected textbooks and related supplementary teaching aids according to the selection of professional boards in primary and secondary schools]. https://mzo.gov.hr/vijesti/konacne-liste-odabranih-udzbenika-i-pripadajucih-dopunskih-nastavnih- sredstava-prema-odabiru-strucnih-aktiva-u-osnovnim-i-srednjim-skolama/1031 Nunes, T., & Bryant, P . (1996). Children doing mathematics . Blackwell. Nunes, T., Bryant, P ., Evans, D., & Bell, D. (2010). The scheme of correspondence and its role in children‘s mathematics. British Journal of Educational Psychology , 2(7), 83–99. https://doi.org/10.1348/97818543370009X12583699332537 Park, J., & Nunes, T. (2001). The development of the concept of multiplication. Cognitive Development , 16(3), 1–11. https://doi.org/10.1016/S0885-2014(01)00058-2 Pepin, B., & Haggarty, L. (2001). Mathematics textbooks and their use in English, French and German classrooms: A way to understand teaching and learning cultures. ZDM: The International Journal on Mathematics Education , 33(5), 158–175. https://doi.org/10.1007/BF02656616 Piaget, J. (1965). The child’s conception of number . Norton. Polak, S., Cindrić, D., & Duvnjak, S. (2014). Nove matematičke price 2 [New mathematical tales 2]. Profil. Sansu Math 2 . (2015). Koyo Publishing. Sansu Math 3 . (2015). Koyo Publishing. Staneff, T. (Ed.) (2017). Power maths 1C . Pearson. Staneff, T. (Ed.) (2017). Power maths 2A . Pearson. Staneff, T. (Ed.) (2018). Power maths 3A . Pearson. c e p s Journal | V ol.12 | N o 2 | Y ear 2022 141 Staneff, T. (Ed.) (2018). Power maths 3B . Pearson. Staneff, T. (Ed.) (2018). Power maths 4A . Pearson. Steffe, L. (1994). Children‘s multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3–40). SUNY Press. Stiegler, J., & Hiebert, J. (1999). The teaching gap: Best ideas from the world‘s teachers for improving education in the classroom . The Free Press. Stručna radna skupina [SRS]. (2016). Nacionalni kurikulum nastavnog predmeta Matematika, Prijedlog [The national curriculum of Mathematics, A proposal]. http://www.kurikulum.hr/wp-content/uploads/2016/03/Matematika.pdf Stručna radna skupina [SRS]. (2016). Odgovori na pristigle priloge stručnoj raspravi Nacionalni kurikulum nastavnoga predmeta Matematika [Answers to the received contributions to the expert discussion of the national curriculum of Mathematics]. http://www.kurikulum.hr/wp-content/uploads/2016/06/komentari-stručna-rasprava-matematika.pdf Tirosh, D., Tsamir, P ., Barkai, R., & Levenson, E. (2018). Engaging young children with mathematical activities involving different representations: Triangles, patterns, and counting objects . Center for Educational Policy Studies Journal , 8(2), 9–30. https://doi.org/10.26529/cepsj.271 Valverde, G. A., Bianchi, L. J., Wolfe, R. G., Schmidt, W . H., & Houang, R. T. (2002). According to the book: Using TIMSS to investigate the translation of policy into practice through the world of textbooks . Kluwer. Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in middle grades (pp.141–161). NCTM. Y ang, D. C., & Sianturi, I. A. (2017). An analysis of Singaporean versus Indonesian textbooks based on trigonometry content. EURASIA Journal of Mathematics Science and Technology Education , 13(7), 3829–3848. https://doi.org/10.12973/eurasia.2017.00760a Biographical note Goran Trupčević , PhD, is an assistant professor in the field of math - ematics at the Faculty of teacher education, University of Zagreb, Croatia. His research interests include representation theory, mathematical textbooks and mathematics teacher noticing. Anđa V alent, PhD, is a master lecturer in the field of mathematics at Zagreb University of Applied Sciences, Croatia. Her research interests include representation theory, history of mathematical education and mathematical textbooks.