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Institutional Relativity of Reduction within Arithmetical Fractions: An Analysis of Japanese, Indonesian, and Malaysian School Textbooks

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Title: Institutional Relativity of Reduction within Arithmetical Fractions: An Analysis of Japanese, Indonesian, and Malaysian School Textbooks
Language: English
Authors: Sani Sahara (ORCID 0000-0001-5426-4296), Didi Suryadi, Turmudi Turmudi, Agus Hendriyanto, Lukman Hakim Muhaimin
Source: International Journal of Science and Mathematics Education. 2025 23(7):2397-2425.
Availability: Springer. Available from: Springer Nature. One New York Plaza, Suite 4600, New York, NY 10004. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-460-1700; e-mail: customerservice@springernature.com; Web site: https://link.springer.com/
Peer Reviewed: Y
Page Count: 29
Publication Date: 2025
Document Type: Journal Articles
Reports - Research
Descriptors: Foreign Countries, Mathematics Instruction, Teaching Methods, Arithmetic, Fractions, Textbook Evaluation, Mathematical Concepts, Epistemology
Geographic Terms: Japan, Indonesia, Malaysia
DOI: 10.1007/s10763-025-10554-x
ISSN: 1571-0068
1573-1774
Abstract: Didactic transposition transpose knowledge relatively across various institutions, or the place where knowledges live within. In this study we analyse the position and the role of reduction of fraction arithmetic, specifically on how the piece of knowledge takes account, in the mathematics school textbooks of Japan, Indonesia, and Malaysia. We follow the institutional analysis by the Anthropological Theory of the Didactic (ATD) as framework for our examination. Moreover, we also seek to contribute to both theoretical discourse and didactical practice, as the study also takes a part in the prospective study in the Didactical Design Research (DDR). As our model of analysis, in this paper, we propose a Reference Epistemological Model (REM) as an objective basis for our interpretation to the knowledge to be taught. Our findings show the varieties of types, positions, and roles of reduction within the praxeological organisations of each textbook. The roles in which reductions perform are distinctively unique for each textbook, as they are positioned differently in terms of technique and technology. Additionally, the cases in the arithmetical fractions are also discussed.
Abstractor: As Provided
Entry Date: 2026
Accession Number: EJ1492869
Database: ERIC
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  Value: <anid>AN0189055851;[3d0g]01oct.25;2025Nov05.04:00;v2.2.500</anid> <title id="AN0189055851-1">Institutional Relativity of Reduction within Arithmetical Fractions: An Analysis of Japanese, Indonesian, and Malaysian School Textbooks </title> <p>Didactic transposition transpose knowledge relatively across various institutions, or the place where knowledges live within. In this study we analyse the position and the role of reduction of fraction arithmetic, specifically on how the piece of knowledge takes account, in the mathematics school textbooks of Japan, Indonesia, and Malaysia. We follow the institutional analysis by the Anthropological Theory of the Didactic (ATD) as framework for our examination. Moreover, we also seek to contribute to both theoretical discourse and didactical practice, as the study also takes a part in the prospective study in the Didactical Design Research (DDR). As our model of analysis, in this paper, we propose a Reference Epistemological Model (REM) as an objective basis for our interpretation to the knowledge to be taught. Our findings show the varieties of types, positions, and roles of reduction within the praxeological organisations of each textbook. The roles in which reductions perform are distinctively unique for each textbook, as they are positioned differently in terms of technique and technology. Additionally, the cases in the arithmetical fractions are also discussed.</p> <p>Keywords: Anthropological theory of the didactic; Didactic transposition; Fractions; Fractions arithmetic; Praxeology</p> <p>Copyright comment Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</p> <hd id="AN0189055851-2">Introduction</hd> <p>People do not drink water in the same way everywhere. Bosch et al. ([<reflink idref="bib6" id="ref1">6</reflink>]) asserted that illustration as she addresses that knowledge changes when moving from one institution to another. It followed the discussion in the anthropological theory of the didactics (ATD), in which Chevallard ([<reflink idref="bib12" id="ref2">12</reflink>], [<reflink idref="bib13" id="ref3">13</reflink>], [<reflink idref="bib15" id="ref4">15</reflink>], [<reflink idref="bib16" id="ref5">16</reflink>]) had started developing the framework in the didactics of mathematics. Furthermore, in the belief of ATD, different versions of knowledge exist within institutions, the place where knowledges live. In this case, we need to acknowledge the version that lives in the school and the didactics as another version of knowledge, a scholarly version, as always two almost different versions (Artaud & Bourgade, [<reflink idref="bib4" id="ref6">4</reflink>]). Every institution, therefore, imports knowledge from other institutions and has to adapt to the conditions prevailing in the previous place. Classically, this process should 'copy' knowledge to 'paste' it. However, it is not merely a copy-and-paste process—it entails many distortions. The process is called a transposition, and since it occurs to be a didactic institution, it is labelled as a didactic transposition (Bosch & Gascón, [<reflink idref="bib5" id="ref7">5</reflink>]; Chevallard, [<reflink idref="bib12" id="ref8">12</reflink>]; Gascón, [<reflink idref="bib22" id="ref9">22</reflink>]; Kang & Kilpatrick, [<reflink idref="bib26" id="ref10">26</reflink>]).</p> <p>In the discipline of mathematics, the 'scholars' (or 'savants') of today are unquestionably the professional mathematicians (Bosch et al., [<reflink idref="bib6" id="ref11">6</reflink>]; Chevallard, [<reflink idref="bib16" id="ref12">16</reflink>]). They are responsible for generating mathematical knowledge and, directly or indirectly, warrant the body of knowledge of the discipline. Nonetheless, distortions can occur when this knowledge is transposed across institutions, sometimes in unprecedented ways (Artaud, [<reflink idref="bib3" id="ref13">3</reflink>]; Schneider, [<reflink idref="bib41" id="ref14">41</reflink>]). For instance, arithmetic education in schools often starts with an emphasis on whole numbers before progressing to integers and then to more complex number types such as fractions, real numbers, and even imaginary numbers at higher levels of education. This sequence is seemingly logical—beginning with basic numeracy and the addition, subtraction, multiplication, and division associated with whole numbers—but sets a foundational understanding specifically tied to whole numbers. It is only at a later stage that students are introduced to other categories of numbers, each requiring a set of additional rules for the aforementioned arithmetic operations to be applicable. Consequently, the initial comprehension and familiarity with the properties of binary operations within the context of integers often become deeply embedded in students' conceptual frameworks. This established mindset poses challenges when students attempt to apply these foundational concepts to different types of numbers, such as fractions, which inherently differ from whole numbers in their operational logic.</p> <p>Several researchers have reported the direction of effects generated by arithmetic operations, where such knowledge has been explored using whole numbers (e.g. Braithwaite & Hall, [<reflink idref="bib7" id="ref15">7</reflink>]; Ni & Zhou, [<reflink idref="bib36" id="ref16">36</reflink>]) or number line (Lesner et al., [<reflink idref="bib29" id="ref17">29</reflink>]). They pointed out an epistemological obstacle where students developed the notion that the act of adding or multiplying natural numbers invariably increases the magnitude of each number or that subtracting a natural number could not result in as number greater than the one being subtracted—a similar understanding applies to division. This understanding arises from the linear processing of arithmetic operations on whole numbers—both are directly engaged in the binary operation. However, when students engage in arithmetic operations with fractions, a new, more complex scenario unfolds, leading to numerous errors, not limited to a few students. For instance, treating fraction numerators and denominators as independent whole numbers, such as assuming <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></math> </ephtml> equals <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mo stretchy="false">/</mo><mn>4</mn></mrow></math> </ephtml> , has been documented by various researchers (e.g. Hecht & Vagi, [<reflink idref="bib25" id="ref18">25</reflink>]; Ni & Zhou, [<reflink idref="bib36" id="ref19">36</reflink>]; Van Hoof et al., [<reflink idref="bib53" id="ref20">53</reflink>]). Additionally, when a large, nationally representative sample of US students from the United Kingdom and the United States was tested, only <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>50</mn><mo>%</mo></mrow></math> </ephtml> of 8th graders could correctly sequence from smallest to largest the fractions <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mo stretchy="false">/</mo><mn>7</mn></mrow></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>5</mn><mo stretchy="false">/</mo><mn>9</mn></mrow></math> </ephtml> , and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn><mo stretchy="false">/</mo><mn>12</mn></mrow></math> </ephtml> (Martin et al., [<reflink idref="bib33" id="ref21">33</reflink>]).</p> <p>Upon revisiting textbooks related to the knowledge presented within the educational system, we understand that mathematics textbooks deliver pieces of knowledge from the body of knowledge. These pieces comprise a larger part of mathematical knowledge that aligns with and adapts to the curriculum. The content within these books undergoes a process known as didactic transposition, which transforms mathematical knowledge previously developed and discussed by scholars (<emph>savoir savant</emph>) into the knowledge to be taught (<emph>savoir enseigné</emph>). This transposition process is inseparable from the concept of institutional relativity. The transposed knowledge is organised into what is called a praxeology, a sub-theory of the anthropological theory of the didactic (ATD) (Chevallard, [<reflink idref="bib15" id="ref22">15</reflink>], [<reflink idref="bib16" id="ref23">16</reflink>]). The elements of a praxeology may serve different roles depending on the specific praxeology under consideration. One study taking account on this is how Aoki ([<reflink idref="bib2" id="ref24">2</reflink>]) investigate how the directives for the teaching of fractions, set by the Japanese Ministry of Education, are transposed to the Japanese supplementary school in Denmark. From this perspective, we seize the opportunity to conduct an institutional analysis that explores how scholarly knowledge about fractions, particularly their arithmetic, is transposed into educational content across different institutions.</p> <p>This study is therefore motivated by the aforementioned phenomenon in teaching arithmetical fractions and the relatively understudied area of reduction within fractions. Our focus centres on how the concept of fractions extends into its arithmetic. A key aspect of this transition towards extension is the concept of reduction, which involves the orderly transformation of numbers from one form to another. Our analysis adopts an institutional perspective, drawing on ATD, and concentrates on the institutional relationships that define this introduction.</p> <p>Every process we undertake serves as a means to contribute to both scholarly discourse and practical objectives. Thus, this study is part of an extensive series of research within the Didactical Design Research (DDR) framework for research established by Suryadi ([<reflink idref="bib48" id="ref25">48</reflink>], [<reflink idref="bib49" id="ref26">49</reflink>], [<reflink idref="bib50" id="ref27">50</reflink>]). By this step, we are not yet addressing how the arithmetic of fractions should be developed. Instead, our institutional analysis aims for a transcendental perspective that contributes to the future development of didactical designs. After examining the phenomena through which textbooks construct their reality of knowledge. The outcomes of this analysis will, in time, serve as a foundation and justify the developed empirical didactic design—a design wherein the hypothetical learning trajectory is developed based on the didactic potential identified through a prospective study—to contribute and challenge the current status quo, thus joining both theoretical discourse and didactical implementation.</p> <hd id="AN0189055851-3">Theoretical Foundation</hd> <p></p> <hd id="AN0189055851-4">ATD for the Institutional Analysis Framework</hd> <p>Yves Chevallard (Chevallard, [<reflink idref="bib12" id="ref28">12</reflink>], [<reflink idref="bib13" id="ref29">13</reflink>], [<reflink idref="bib14" id="ref30">14</reflink>], [<reflink idref="bib15" id="ref31">15</reflink>]) introduced the Anthropological Theory of the Didactic (ATD) as a framework for research in the didactics of mathematics that has been evolving since the 1980s, starting with initial explorations into the concept of didactic transposition. This theoretical framework emerged from the ambition to establish a 'science of the didactic,' a vision proposed by Guy Brousseau and kick-started through his Theory of Didactic Situations (TDS) (Brousseau, [<reflink idref="bib10" id="ref32">10</reflink>], [<reflink idref="bib11" id="ref33">11</reflink>]). In the exposition of the Anthropological Theory of the Didactic (Chevallard, [<reflink idref="bib15" id="ref34">15</reflink>], [<reflink idref="bib16" id="ref35">16</reflink>]), the processes of didactic transposition highlight the institution-dependent nature of knowledge and position didactic challenges within an institutional context. The didactic transposition process pertains to the alterations a 'content' or knowledge body undergoes from its inception and application to its dissemination and assimilation within a specific educational institution.</p> <p>Therefore, this study considers the scholarly knowledge produced by mathematicians or other scientists recognised as experts in their fields as a foundation for legitimizing the knowledge to be taught. It is emphasised that through the primary texts embodying the knowledge to be taught, the objective of didactics as a discipline is to clarify the processes through which knowledge is spread across institutions and among individuals within a society (Chevallard, [<reflink idref="bib15" id="ref36">15</reflink>], [<reflink idref="bib16" id="ref37">16</reflink>]). For instance, in the context of arithmetic involving fractions, for students to comprehend fraction and decimal arithmetic, they must first understand fractions and decimals themselves. Indeed, difficulties in mastering fraction and decimal arithmetic often indicate a more profound underlying confusion regarding the fundamental components of fractions.</p> <p>ATD suggests that any activity related to the creation, dissemination, or acquisition of knowledge should be viewed as ordinary human activity and thus proposes a model of human activities in the notion of praxeology.</p> <p>The notion of praxeology was introduced as an essential means of analysing human actions, whether in mathematics or otherwise. The theory is termed "anthropological" as it views the cognitive universe as a product of activity. Chevallard ([<reflink idref="bib15" id="ref38">15</reflink>], [<reflink idref="bib16" id="ref39">16</reflink>]) asserted that the interaction with an object arises from our need to engage in an activity where the object has some role. Furthermore, our actions—whether handling, thinking, loving, or discarding something—shape our cognitive universe. This theory therefore allows us to reckon textbooks in terms of how they encourage students to build their knowledge through engagement in specific designed activities (Godino, [<reflink idref="bib23" id="ref40">23</reflink>]).</p> <p>From the viewpoint of ATD, human activities can be described and interpreted using praxeologies, which consist of the praxis block—encompassing the tasks that form the activity and the techniques for executing them—and the logos block, which helps justify the activities. Thus the praxis component consists of 'types of tasks' <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mtext>T</mtext></mfenced></math> </ephtml> and an assortment of 'techniques' <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mi>τ</mi></mfenced></math> </ephtml> , that is justified by 'technology' <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mi mathvariant="normal">θ</mi></mfenced></math> </ephtml> employing the original meaning of discourse of a legitimate 'theory,' <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mi mathvariant="normal">Θ</mi></mfenced></math> </ephtml> whose principal role is to furnish foundational support for the technological discourse. The quartet <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>p</mi><mo>=</mo><mfenced close="]" open="["><mi>T</mi><mo stretchy="false">/</mo><mi>τ</mi><mo stretchy="false">/</mo><mi>θ</mi><mo stretchy="false">/</mo><mi mathvariant="normal">Θ</mi></mfenced></mrow></math> </ephtml> being ['capital tau' / 'tau' / 'theta' / 'capital theta'], thus referred to as a praxeology, is articulated into two blocks: the praxis block <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Π</mi></math> </ephtml> (<reflink idref="bib1" id="ref41">1</reflink>) and the logos block <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Λ</mi></math> </ephtml> (<reflink idref="bib2" id="ref42">2</reflink>), both of which identify the pinpoint praxeology (<reflink idref="bib3" id="ref43">3</reflink>).</p> <p>1 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Π</mi><mo>=</mo><mo stretchy="false">[</mo><mtext>T</mtext><mspace width="0.333333em" /><mo stretchy="false">/</mo><mi mathvariant="normal">τ</mi><mo stretchy="false">]</mo></mrow></math> </ephtml></p> <p>Graph</p> <p>2 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Λ</mi><mo>=</mo><mo stretchy="false">[</mo><mi mathvariant="normal">θ</mi><mo stretchy="false">/</mo><mi mathvariant="normal">Θ</mi><mo stretchy="false">]</mo></mrow></math> </ephtml></p> <p>Graph</p> <p>3 <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Π</mi><mo>⊕</mo><mi mathvariant="normal">Λ</mi><mo>=</mo><mo stretchy="false">[</mo><mtext>T</mtext><mspace width="0.333333em" /><mo stretchy="false">/</mo><mi mathvariant="normal">τ</mi><mo stretchy="false">]</mo><mo>⊕</mo><mo stretchy="false">[</mo><mi mathvariant="normal">θ</mi><mo stretchy="false">/</mo><mi mathvariant="normal">Θ</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mtext>T</mtext><mspace width="0.333333em" /><mo stretchy="false">/</mo><mi mathvariant="normal">τ</mi><mo stretchy="false">/</mo><mi mathvariant="normal">θ</mi><mo stretchy="false">/</mo><mi mathvariant="normal">Θ</mi><mo stretchy="false">]</mo></mrow></math> </ephtml></p> <p>Graph</p> <p>A praxeology may transform when transposing between institutions, and its components can serve different roles depending on the praxeology considered. Therefore, an institutional analysis can help us to discuss the existence and evolution of praxeologies within specific institutional settings and ecologies. To conduct an institutional analysis is necessary to include both praxeological and didactic analysis (Bosch & Gascón, [<reflink idref="bib5" id="ref44">5</reflink>]). The praxeological analysis seeks to understand the praxeologies involved in creating didactic knowledge, while didactic analysis focuses on identifying the elements within the didactic system and the praxeologies it employs (Bosch et al., [<reflink idref="bib6" id="ref45">6</reflink>]). Furthermore, questions about interpreting the mathematics being taught emerge when examining mathematical content's teaching and learning processes. Different institutions involved in the didactic processes offer varying explicit answers to these questions. There are studies reported the examination of the consequences of transposition (see Lundberg & Kilhamn, [<reflink idref="bib32" id="ref46">32</reflink>]; Takeuchi & Shinno, [<reflink idref="bib51" id="ref47">51</reflink>]). If researchers accept these answers without critical analysis, there is a risk of not adequately and biasedly addressing the empirical facts observed. Therefore, ATD proposes developing a REM to provide the necessary elements to formulate didactic problems and make epistemological assumptions about the subject matter that is taught and learnt explicitly (Gascón, [<reflink idref="bib21" id="ref48">21</reflink>], [<reflink idref="bib22" id="ref49">22</reflink>]).</p> <hd id="AN0189055851-5">Reduction in the Discourse of Arithmetical Fraction</hd> <p>Upon a time in an old study, there was a term initially used in the study of fraction addressed by Recorde ([<reflink idref="bib38" id="ref50">38</reflink>]) in his book "The Ground of Arts", that is "reduction", to explain the steps for simplifying fractions and other numbers to perform arithmetic operations. The book also serves as a reference for our understanding in the process of didactic transposition in our institutional analysis, as the scholarly knowledge—the knowledge which is validated and accredited by the supreme authority in the field (Chevallard, [<reflink idref="bib15" id="ref51">15</reflink>]).</p> <p>Furthermore, studies on arithmetic involving fractions have undergone various perspectives, revealing a common thread in issues related to conceptual and procedural. They usually start with highlighting that most theories of numerical development have predominantly focused on whole numbers.</p> <p>Arithmetic generally discusses numeration, followed by its arithmetic operations such as addition, subtraction, multiplication, and division (Saidan, [<reflink idref="bib40" id="ref52">40</reflink>]). Numeration is the arithmetic skill through which we accurately value, express, and read any number proposed or represent any known or named number using digits and positions (Rips et al., [<reflink idref="bib39" id="ref53">39</reflink>]). Moreover, arithmetic operations for whole numbers and fractions follow distinct procedures. For whole numbers, the process starts with numeration, determining each digit's value based on its position. Subsequent addition, subtraction, multiplication, and division are executed directly. This linear process does not necessitate altering the numbers before arithmetic operations.</p> <p>However, in the case of fractions, after numeration, a critical step of reduction precedes any arithmetic operations (Fazio et al., [<reflink idref="bib19" id="ref54">19</reflink>]; Lortie-Forgues & Siegler, [<reflink idref="bib30" id="ref55">30</reflink>]). Textbooks often present reduction as simplifying fractions with large numerators or denominators to smaller numbers (e.g. Alajmi, [<reflink idref="bib1" id="ref56">1</reflink>]; Yang et al., [<reflink idref="bib56" id="ref57">56</reflink>]), but reduction entails more complex scenarios. The most common form of reduction is simplifying fractions to their lowest terms or finding a common denominator to ease the complexity inherent in fraction calculations (Lortie-Forgues & Siegler, [<reflink idref="bib30" id="ref58">30</reflink>]; Lortie-Forgues et al., [<reflink idref="bib31" id="ref59">31</reflink>]). It ensures uniformity and simplifies addition or subtraction, where a common denominator is required (Braithwaite & Hall, [<reflink idref="bib7" id="ref60">7</reflink>]; Braithwaite & Siegler, [<reflink idref="bib8" id="ref61">8</reflink>]; Braithwaite et al., [<reflink idref="bib9" id="ref62">9</reflink>]).</p> <p>Fraction arithmetic requires learning a large number of distinct procedures, probably more than for any other mathematical operation taught in elementary school. Lortie-Forgues et al. ([<reflink idref="bib31" id="ref63">31</reflink>]) categorised several fraction problems related to how reduction takes place in the addition, subtraction, multiplication, and division, based on whether denominators were the same or different. All of the arithmetic procedures require skills as well as mastery of procedures for finding equivalent fractions, simplifying fractions, converting fractions to mixed numbers and mixed numbers to fractions, knowing whether to invert the numerator or denominator when dividing fractions, and understanding when equal denominators are maintained in the answer (addition and subtraction) and when the operation in the problem should be applied to the denominator as well as the numerator (multiplication and division).</p> <p>There are also studies on how to promote approaches to support students developing their understanding. Simon et al. ([<reflink idref="bib44" id="ref64">44</reflink>], [<reflink idref="bib45" id="ref65">45</reflink>]) conducted a teaching experiment to promote reinvention of the multiplication-of-fractions algorithm, while Kara et al. ([<reflink idref="bib28" id="ref66">28</reflink>]) studied the concept of recursive partitioning (anticipation of the results of taking a unit fraction of a unit fraction). Simon et al. ([<reflink idref="bib45" id="ref67">45</reflink>]) found a solution for that students whose concept of fractions is limited to part-whole have difficulty with advanced fraction concepts. It is followed by the several approach to accommodate the discussions, particularly in reduction (see also Braithwaite & Siegler, [<reflink idref="bib8" id="ref68">8</reflink>]; Mostert & Hickendorff, [<reflink idref="bib35" id="ref69">35</reflink>]; Siegler et al., [<reflink idref="bib43" id="ref70">43</reflink>]).</p> <p>Regarding these studies, including some mentioned in the introduction, the focus on the difficulties and errors caused by bypassing the reduction process and assuming arithmetic operations on fractions are the same as those on whole numbers. As such, in this study we focus on how the reduction is addressed in textbooks to discern how the scholarly knowledge on the arithmetic of fractions differs across various institutions.</p> <hd id="AN0189055851-6">Research Questions and Scope</hd> <p>In regard to the rationale and theoretical foundation, we consider that both praxis and the corresponding logos are utmost to the development of knowledge to be taught, as also asserted by Bosch et al. ([<reflink idref="bib6" id="ref71">6</reflink>]). Thus, the processes of didactic transposition highlight the institutional relativity of knowledge, wherein its status, function, and the discourse of justification may shift. To constitute such an analysis regarding the relativity, this study addresses the following research questions:</p> <p></p> <ulist> <item> How are praxis and the corresponding logos of reduction presented within mathematics textbooks?</item> <p></p> <item> How does the relativity of the status and function of the knowledge vary across institutions?</item> </ulist> <p>To address these questions, we examined three textbooks representing three different institutions that incorporate reduction in the process of fractions arithmetic to reveal the positions and roles, in the form of topographic praxis and the justifying logos of reduction in each textbook, using the research framework discussed earlier. This study therefore aims to seek the reality of the relativity of piece of knowledge reduction in its position and authority and how its praxis-logos takes influence the developing of knowledge to be taught.</p> <hd id="AN0189055851-7">Methodology</hd> <p></p> <hd id="AN0189055851-8">Research Design</hd> <p>This qualitative study developed under two perspectives. We began by examining the phenomena by analysing textbooks and interpreting these findings within Van Manen's ([<reflink idref="bib54" id="ref72">54</reflink>]) philosophical school of hermeneutic phenomenology. The analysis focuses on three mathematics textbooks through institutional structures, categorising our investigation as an institutional analysis. Throughout this process, we worked in the framework of the ATD to underline our praxeological and didactic examination. Furthermore, this research also contributed to a further study within the DDR framework, positioning this study within the phase of prospective analysis. In this perspective, curricula undergo a thorough evaluation. Using textbooks as a case study enables us to delve into and debate topics deemed universally significant in the worldwide didactical design of fraction arithmetic.</p> <hd id="AN0189055851-9">Educational Backgrounds and Textbooks Sampling</hd> <p>We gathered empirical data from textbooks used in primary school mathematics in Japan, Indonesia, and Malaysia. The selection of these countries was based on the following criteria:</p> <p></p> <ulist> <item> given that institutional relativity sheds light on differences in transposition, we selected countries with both differing and nearly similar achievements, choosing Japan based on its Programme for International Student Assessment (PISA) scores (Organisation for Economic Co-operation and Development [OECD], [<reflink idref="bib37" id="ref73">37</reflink>]), followed by Indonesia and Malaysia with closely related scores;</item> <p></p> <item> the textbooks are officially approved and certified by the respective Ministry of Education or equivalent authority, ensuring they reflect the curricular documents; and</item> <p></p> <item> the textbooks are accessible free of charge at least to students in the respective countries.</item> </ulist> <p>The educational framework in Japan is defined by a national curriculum known as the Course of Study, updated approximately every ten years (Ministry of Education, Culture, Sports [MEXT], [<reflink idref="bib34" id="ref74">34</reflink>]). The latest version was published in 2018 and implemented in elementary schools starting in 2020, with Japanese students receiving 235 min of mathematics instruction yearly. Textbook selection is conducted by local education boards, albeit under strict authorisation from the Japanese Ministry of Education, which approves textbooks in line with the Course of Study. Following the established availability and standards, we selected the 'Fun with Math for Elementary School' series by Keirinkan for Grades 3 to 6 to analyse Japanese mathematics textbooks.</p> <p>In Indonesia, the education system is overseen and regulated by the Ministry of Education, Culture, Research, and Technology, which introduced 'kurikulum merdeka' or emancipated curriculum, as the latest national curriculum in 2021. This curriculum prescribes 180 min of yearly mathematics instruction for students. In response to this curriculum, the ministry issued textbook standards and, following their publication, provided free online access to textbooks that comply with these standards through its website and the 'platform merdeka mengajar.' Established publishers like Erlangga and Yudhistira have also published textbooks, but the 2022 edition of the ministry's textbooks remains the primary source. Therefore, we focused on the ministry's 'Matematika SD/MI' textbooks for Grades 4 to 6.</p> <p>Meanwhile, in Malaysia, the Ministry of Education, alongside state Education Departments and district Education Offices, oversees education. The Kurikulum Standard Sekolah Rendah, or Primary School Standard Curriculum, revised from its 2011 version, was implemented in 2017, with students receiving 210 min of mathematics lessons yearly. The Resource and Educational Technology Division ensures textbook alignment with curriculum standards, inviting renowned publishers to contribute (Solis & Isoda, [<reflink idref="bib46" id="ref75">46</reflink>]; Tan et al., [<reflink idref="bib52" id="ref76">52</reflink>]). We selected the 'Matematik Tahun 4 Sekolah Kebangsaan' series for Grades 4 to 6 for our study of Malaysian mathematics textbooks, also available through a free online system.</p> <hd id="AN0189055851-10">Analysis of the Textbooks</hd> <p>We carried out our analysis through a two-phase process. The initial phase was mapping tasks from each textbook to examine how knowledge of fraction arithmetic was structured and presented. This phase concentrated on the tasks outlined in the textbooks, explicitly identifying any reductions in the techniques presented, whether they were stated openly or implied. For every task, we documented the suggested techniques for resolution and the technologies employed to explain these techniques.</p> <p>In the subsequent phase, considering the systemic framework, we examine the types of reductions observed and their functions within the tasks. Throughout this examination, we relied upon the REM to maintain an objective viewpoint, adhering to the principle that scholarly knowledge should not dominate as the only benchmark for all educational mathematical practices (Bosch & Gascón, [<reflink idref="bib5" id="ref77">5</reflink>]). According to ATD, REMs are defined through a combination of local and regional praxeologies and a series of interconnected praxeologies of escalating complexity (Bosch & Gascón, [<reflink idref="bib5" id="ref78">5</reflink>]; Bosch et al., [<reflink idref="bib6" id="ref79">6</reflink>]). Additionally, we aligned with Gascón's ([<reflink idref="bib21" id="ref80">21</reflink>]) viewpoint that no singular reference system should be elevated above others when analysing various bodies of knowledge through the stages of didactic transposition. Therefore, our analysis offers an interpretative rationale for employing reductions in terms of techniques and technology to fulfil task objectives and provide a basis for the chosen techniques.</p> <p>To develop such a model, we included scholarly knowledge produced by esteemed mathematicians and scientists, acknowledging it as a credible foundation for the knowledge to be imparted. Nonetheless, it is crucial to recognise that scholarly knowledge alone should not be the unique reference for all scholastic mathematical endeavours. It is equally important to consider the corresponding institutions: the mathematical community, the educational system, and the classroom setting. In this light, we engaged in focus group discussions involving academic mathematicians, curriculum assessment experts, and reviews of published research on the challenges of learning and teaching fractions. The insights from these discussions are presented as REM in the ensuing results section.</p> <hd id="AN0189055851-11">A Prospective Study Position of DDR</hd> <p>Alongside the ATD framework, in the series, this study also takes a part of the framework established by Suryadi ([<reflink idref="bib47" id="ref81">47</reflink>], [<reflink idref="bib48" id="ref82">48</reflink>], [<reflink idref="bib49" id="ref83">49</reflink>]) as the DDR framework for research. The research within the DDR framework is grounded in two main research paradigms: the interpretive and the critical paradigm. The interpretive paradigm is concerned with analysing the subjective experiences and the meanings individuals assign to them. Subsequently, the critical paradigm engages researchers in contributing to change by developing knowledge in the form of didactical designs.</p> <p>To vision the prospect into future implementation, we take the position of this study in a transcendental way in which after this, the result of our institutional analysis will justify the epistemological basis for the developed didactical design (Fig. 1). This approach is motivated by how García et al. ([<reflink idref="bib20" id="ref84">20</reflink>]) concluded a reflection about the necessary dialectic between research, design and teaching practices. In perceiving and interpreting the phenomenon of the textbooks, our purpose is oriented within the interpretivism paradigm (Crotty, [<reflink idref="bib17" id="ref85">17</reflink>]), to discern the meanings in text interpretation, that is, the interpretation of the meaning of the knowledge to be taught in the textbooks constructed by noospheres. The noosphere is addressed within the context of didactical transposition, which contribute to the one of the types of knowledge transposed, that is, the knowledge to be taught. Here, noosphere means the sphere of those who "think" (noos) about teaching of those piece of knowledge (Bosch & Gascón, [<reflink idref="bib5" id="ref86">5</reflink>]).</p> <p>Graph: Fig. 1 This study's position in the DDR framework</p> <p>We take its stance within this interpretive study in a prospective study as one of the bases to use the critical paradigm to generate new designs whose hypotheses are measurable and based on the current study as one of them, with the orientation of producing new knowledge for learners (Fig. 2).</p> <p>Graph: Fig. 2 The didactic transposition diagram (from Bosch & Gascón, [<reflink idref="bib5" id="ref87">5</reflink>])</p> <hd id="AN0189055851-12">Results and Discussion</hd> <p></p> <hd id="AN0189055851-13">Reference Epistemological Model for School's Fractions Arithmetic</hd> <p>Following the ATD framework, to examine how fractions as the knowledge to be taught, we need to avoid a very scholarly biased viewpoint (Bosch & Gascón, [<reflink idref="bib5" id="ref88">5</reflink>]). Therefore, in providing a legitimate point of view, we elaborate our reference model as a basis of the mathematical field of activity that serve as a reference framework for interpreting the model in the institution under study, referred to as reference epistemological model (REM) (Bosch et al., [<reflink idref="bib6" id="ref89">6</reflink>]). This reference is as necessary as Siegler et al. ([<reflink idref="bib43" id="ref90">43</reflink>]) asserted that understanding fractions requires recognising that many properties of natural numbers are not properties of numbers in general. We illustrate these perspectives in Fig. 3 with a diagram.</p> <p>Graph: Fig. 3 The proposed REM for arithmetical fractions</p> <p>Initially, students are expected to be used to fractions as part of their understanding of numeration. When encountering fraction arithmetic problems, they are expected to have grasped the foundational concepts and the procedural steps required. In the second phase, students explore the various arithmetic operations and the elements entail. They assess the need for reduction and the method to accomplish it. The third phase involves performing arithmetic operations with fractions for each concept. These three educational approaches are displayed in Fig. 3 as an REM.</p> <p>The REM constitute the definition for understanding fractions and highlighting six types of reductions labelled <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>1</mn></msub></math> </ephtml> to <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>6</mn></msub></math> </ephtml> , following the work of Recorde ([<reflink idref="bib38" id="ref91">38</reflink>]) and presented in Table 1 within our REM. This framework aims to fulfil several purposes. It is an analytical instrument for examining the spectrum of mathematical practices in educational contexts and assessing their effects on other mathematical domains—both the opportunities and constraints. For instance, it facilitates an in-depth investigation into the aspects of elementary algebra excluded from school syllabi, the likely reasons behind their exclusion, and the origins of these reasons, as explored by (Bosch & Gascón, [<reflink idref="bib5" id="ref92">5</reflink>]). Moreover, the framework allows for a comparative study with different research methodologies, such as the 'structural approach' described by Kaput et al. ([<reflink idref="bib27" id="ref93">27</reflink>]). In the subsequent sections, we analyse the tasks in various textbooks and, as per the REM, pinpoints where reductions are introduced and their relevance in the context of the arithmetic operations.</p> <p>Table 1 The common six varieties or forms of reduction (from Recorde, [<reflink idref="bib38" id="ref94">38</reflink>], pp. 272–273) and the interpretations</p> <p> <ephtml> <table rules="groups"><thead><tr><th align="left"><p>Code</p></th><th align="left"><p>Form of fraction(s)</p></th><th align="left"><p>Reduction suggested</p></th></tr></thead><tbody><tr><td align="left" rowspan="3"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>r</mi><mn>1</mn></msub></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq20.gif" /></p></td><td align="left"><p><italic>Sundry fractions of one entire unit</italic></p></td><td align="left"><p><italic>Reduce into one denomination, and also into one fraction</italic></p></td></tr><tr><td align="left" colspan="2"><p>Interpretation:</p></td></tr><tr><td align="left"><p>Various fractions of a whole unit</p></td><td align="left"><p>Simplify into one common denominator and combine them into a single fraction</p></td></tr><tr><td align="left" rowspan="3"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>r</mi><mn>2</mn></msub></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq15.gif" /></p></td><td align="left"><p><italic>Fraction of fraction</italic></p></td><td align="left"><p><italic>Reduce likewise into one fraction, for otherwise they cannot be brought into one denomination</italic></p></td></tr><tr><td align="left" colspan="2"><p>Interpretation:</p></td></tr><tr><td align="left"><p>Fraction of fraction</p></td><td align="left"><p>Also combine them into a single fraction, because otherwise, they cannot be converted into one common denominator</p></td></tr><tr><td align="left" rowspan="3"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>r</mi><mn>3</mn></msub></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq16.gif" /></p></td><td align="left"><p><italic>An improper fraction, that is to say, a fraction in form, which indeed is greater than a unit</italic></p></td><td align="left"><p><italic>Reduce into a form expressing the unit or units of it, and the proper fraction distinctly</italic></p></td></tr><tr><td align="left" colspan="2"><p>Interpretation:</p></td></tr><tr><td align="left"><p>An improper fraction is a fraction where the fraction form is greater than one unit</p></td><td align="left"><p>Simplify it into a mixed number, separating the whole number part from the fraction</p></td></tr><tr><td align="left" rowspan="3"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>r</mi><mn>4</mn></msub></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq17.gif" /></p></td><td align="left"><p><italic>A mixed numbers of units with fractions</italic></p></td><td align="left"><p><italic>Convert such into the form of a fraction, that is, into an improper fraction</italic></p></td></tr><tr><td align="left" colspan="2"><p>Interpretation:</p></td></tr><tr><td align="left"><p>A mixed number includes both a whole number and a fraction</p></td><td align="left"><p>Change it into an improper fraction, where the fraction represents the whole number too</p></td></tr><tr><td align="left" rowspan="3"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>r</mi><mn>5</mn></msub></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq18.gif" /></p></td><td align="left"><p><italic>Fraction written in great numbers, which might be written in lesser numbers</italic></p></td><td align="left"><p><italic>Reduce such great numbers into their smallest number</italic></p></td></tr><tr><td align="left" colspan="2"><p>Interpretation:</p></td></tr><tr><td align="left"><p>A fraction with large numbers can often be written with smaller numbers</p></td><td align="left"><p>Simplify the fraction by reducing the numbers to their smallest form</p></td></tr><tr><td align="left" rowspan="3"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>r</mi><mn>6</mn></msub></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq21.gif" /></p></td><td align="left"><p><italic>The parts of a whole thing, which has by common partition certain parts, but none of like denomination with that fraction</italic></p></td><td align="left"><p><italic>Reduce the said fractions into another whole denomination shall express the common parts of that whole thing</italic></p></td></tr><tr><td align="left" colspan="2"><p>Interpretation:</p></td></tr><tr><td align="left"><p>A fraction that represents parts of a whole but does not share a common unit with another fraction</p></td><td align="left"><p>Reduce the fractions so they have a common unit, making them easier to compare or add</p></td></tr></tbody></table> </ephtml> </p> <hd id="AN0189055851-14">Task Mapping and Reduction Identification</hd> <p>To map the position of reduction, we firstly identified a total of eight types of tasks across all textbooks, hereafter referred to by their respective country codes: <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> for Japanese textbook, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> for Indonesian textbook, and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> for Malaysian textbook. Within these tasks, there exist reductions and are categorised into three distinct types, which may differ from Mostert and Hickendorff's ([<reflink idref="bib35" id="ref95">35</reflink>]) categorisation that discuss every arithmetic operation. However, in here, each task employs a unique technique, and it is observed that not all textbooks present the same variety of task types. According to Table 2, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> includes three types of tasks <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub><mo>-</mo><msub><mi mathvariant="normal">T</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mrow></math> </ephtml> , whereas <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> covers <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub><mo>-</mo><msub><mi mathvariant="normal">T</mi><mn>2</mn></msub></mrow></math> </ephtml> , and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> only presents <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub></math> </ephtml> . All tasks utilise techniques supported by technologies grounded in a well-established theory, as detailed in Table 3.</p> <p>Table 2 Task mapping on reduction among three textbooks</p> <p> <ephtml> <table rules="groups"><thead><tr><th align="left"><p>Type of task</p></th><th align="left"><p>Issued in</p></th><th align="left"><p>Technique</p></th></tr></thead><tbody><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq31.gif" /> To express or manipulate proper fractions and improper fractions </p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>J</mi><mi>P</mi><mo>,</mo><mi>I</mi><mi>D</mi><mo>,</mo><mi>M</mi><mi>Y</mi></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq32.gif" /></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>1</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq33.gif" /> Directly using the proper fraction algorithm; <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>2</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq34.gif" /> Using visual interpretation of mixed numbers; <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>3</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq35.gif" /> Using the long-division method to find the remainder; <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>4</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq36.gif" /> Representing the size; <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>5</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq37.gif" /> Using visual of numbers line </p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi mathvariant="normal">T</mi><mn>2</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq38.gif" /> To compare fractions </p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>J</mi><mi>P</mi><mo>,</mo><mi>I</mi><mi>D</mi></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq39.gif" /></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>4</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq36.gif" /> Representing the size; <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>5</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq37.gif" /> Using visual of numbers line; <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>6</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq42.gif" /> Comparing the numerators; <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>7</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq43.gif" /> Comparing the integers; <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>8</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq44.gif" /> Using visual of part of a whole or part of a unit; <math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>9</mn></msub></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq45.gif" />: Expressing equivalent fractions </p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi mathvariant="normal">T</mi><mn>3</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq46.gif" /> To order fractions </p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi mathvariant="italic">ID</mi></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq23.gif" /></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>5</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq37.gif" /> Using visual of numbers line; <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>6</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq42.gif" /> Comparing the numerators; <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>7</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq43.gif" /> Comparing the integers </p></td></tr></tbody></table> </ephtml> </p> <p>Table 3 The technologies and theory</p> <p> <ephtml> <table rules="groups"><thead><tr><th align="left"><p>Technology</p></th><th align="left"><p>Theory</p></th></tr></thead><tbody><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mn>1</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq107.gif" /> a fraction is a broken number, and consequently the part of another number </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mn>2</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq108.gif" /> a fraction expresses the parts or part only of a unit </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mn>3</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq109.gif" /> when there be several fractions of one entire unit, they must be reduced to one denomination, and also into one fraction </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mn>4</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq110.gif" /> when an improper fraction is given, which is greater than a unit, it should be changed into a more appropriate form expressing the unit </p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi mathvariant="bold">Θ</mi></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq111.gif" /> Arithmetic </p></td></tr></tbody></table> </ephtml> </p> <p>In general, the distribution of task types ( <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub></math> </ephtml> to <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>8</mn></msub></math> </ephtml> ) across <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> , and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> , detailing both the frequency and percentage of each task type in each country. In <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> , the most common tasks are <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>2</mn></msub></math> </ephtml> , accounting for <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>21</mn><mo>%</mo><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><mn>15</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>20</mn><mo>%</mo><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><mn>14</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> , respectively, while <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>3</mn></msub></math> </ephtml> is the least frequent at <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>6</mn><mo>%</mo><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> . Interestingly, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>6</mn></msub></math> </ephtml> is entirely absent in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> . <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> 's most frequent task is <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>T</mi><mn>5</mn></mrow></math> </ephtml> , comprising <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>23</mn><mo>%</mo><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><mn>16</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> of the tasks, followed by <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>4</mn></msub></math> </ephtml> at <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>20</mn><mo>%</mo><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><mn>14</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>7</mn></msub></math> </ephtml> at <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>18</mn><mo>%</mo><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><mn>13</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> , while <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>2</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>3</mn></msub></math> </ephtml> are not observed. In <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub></math> </ephtml> is overwhelmingly dominant, representing <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>29</mn><mo>%</mo><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><mn>34</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> , with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>7</mn></msub></math> </ephtml> also notable at <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>24</mn><mo>%</mo><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><mn>28</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> . However, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>6</mn></msub></math> </ephtml> is extremely rare in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> , contributing only <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>0.8</mn><mo>%</mo><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> , the lowest value across all task types in the table. Overall, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> recorded the highest total number of tasks <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><mn>115</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> , followed by <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>M</mi><mi>Y</mi><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><mn>71</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>I</mi><mi>D</mi><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><mn>70</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> . The tasks highlight the variability in preferences and distributions among the three countries, with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub></math> </ephtml> being the most common in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>5</mn></msub></math> </ephtml> in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> , and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>2</mn></msub></math> </ephtml> leading in ID.</p> <p>Here, the function of reduction assumes various roles, where the tasks associated with the introduction of reductions are categorised into three types: 1) <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub></math> </ephtml> : for expressing or manipulating proper and improper fractions, 2) <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>2</mn></msub></math> </ephtml> : for comparing fractions, and 3) <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>3</mn></msub></math> </ephtml> : for ordering fractions. We found that all three institutions incorporate <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub></math> </ephtml> , whereas <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>2</mn></msub></math> </ephtml> are present only in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> , and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>3</mn></msub></math> </ephtml> is exclusively found in the <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> . Techniques for comparison and ordering, using <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>6</mn></msub></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>7</mn></msub></math> </ephtml> , and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>9</mn></msub></math> </ephtml> are justified by theories <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>θ</mi><mn>1</mn></msub><mo>-</mo><msub><mi>θ</mi><mn>4</mn></msub></mrow></math> </ephtml> , which are part of the overarching theory <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Θ</mi></math> </ephtml> , as described in Table 3.</p> <p>Subsequently, we examined sections of the textbooks that present <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub><mo>-</mo><msub><mi mathvariant="normal">T</mi><mn>3</mn></msub></mrow></math> </ephtml> before analysing their application in the arithmetic operations outlined in the tasks <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="normal">T</mi><mn>4</mn></msub><mo>-</mo><msub><mi mathvariant="normal">T</mi><mn>8</mn></msub></mrow></math> </ephtml> . We identified that textbooks from <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> , and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> each incorporate <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>5</mn></msub></math> </ephtml> , albeit presented differently. <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> explicitly labels its activity as 'reducing fractions,' introducing <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>5</mn></msub></math> </ephtml> to obtain a fraction with a smaller denominator—a process JP refers to as fraction reduction. This involves dividing both the numerator and denominator by the same number, similar to how Lortie-Forgues et al. ([<reflink idref="bib31" id="ref96">31</reflink>]) suggested where all fraction arithmetic procedures require whole number arithmetic calculations. Conversely, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> introduce fraction reduction under the term 'equivalent fractions,' also mentioned in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> , without explicitly naming the reduction process. In the same direction with Lortie-Forgues and Siegler ([<reflink idref="bib30" id="ref97">30</reflink>]) regarding qualitative understanding of fraction multiplication and division, these textbooks explain how fractions can be represented in different notations but hold the same value, starting with using fraction tapes and concluding with multiplication and division of the numerator and denominator by the same number to achieve the same denominator, identified as <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>1</mn></msub></math> </ephtml> .</p> <p>In exploring <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub></math> </ephtml> further, each textbook presents <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>3</mn></msub></math> </ephtml> or <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>4</mn></msub></math> </ephtml> —conversion between improper fractions and mixed numbers—in various ways. <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> prioritises <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>7</mn></msub></math> </ephtml> as a step, while <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> employs <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>2</mn></msub></math> </ephtml> . Additionally, both <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> present <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>2</mn></msub></math> </ephtml> using similar methods, incorporating both <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>2</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>7</mn></msub></math> </ephtml> . Moreover, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> holds exclusive rights to a form of reduction presentation in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>3</mn></msub></math> </ephtml> , through ordering fractions. Since ordering involves comparison, this task is justifiably based on its foundational rationale. However, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> differs in presenting reductions related to <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>3</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>4</mn></msub></math> </ephtml> without any justification for the techniques used in converting mixed numbers to improper fractions, merely prompting students to follow brief steps. Lortie-Forgues et al., [<reflink idref="bib31" id="ref98">31</reflink>] and Wu ([<reflink idref="bib55" id="ref99">55</reflink>]) just proved such approaches by <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> in how students memorize fraction arithmetic procedures without understanding them, which limits the understanding that they can convey. Furthermore, no fundamental reasons or concepts are provided to support the execution of this algorithm (refer to Table 4 and Fig. 4).</p> <p>Graph</p> <p>Graph: Fig. 4 The mixed number manipulation task in ID</p> <p>The subsequent task, which is a continuation of the preceding ones, focuses on placing and ordering fractions on a number line after the reduction process is completed. This involves placing the reduced fraction, whether an improper or proper fraction (if the expression is already in that form), on a number line and ordering them accordingly. Using a number line, suitable for integers as in basic arithmetic, may not directly apply to fractions due to potential issues with equivalent fractions. Meanwhile, word problems were easier than their symbolic counterparts (Mostert & Hickendorff, [<reflink idref="bib35" id="ref100">35</reflink>]). Also the "whole number mindset" is what the students currently have in number line prior to fractions, thus computation errors may become a whole number bias (Lortie-Forgues et al., [<reflink idref="bib31" id="ref101">31</reflink>]). In <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> , the narrative approach directly proceeds to order the fractions based on their positions. Hence, we assume that <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>3</mn></msub></math> </ephtml> effectively represents reduction, although its practical use in arithmetic operations might be limited. Meanwhile, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> , as identified, does not present <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>2</mn></msub></math> </ephtml> . Instead, it relies on equivalent fractions to instill the concept of the same denominators.</p> <p>In contrast, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> employs a distinct approach to using the number line for packaging <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>2</mn></msub></math> </ephtml> emphasising on the size of fractions. The task is performed through <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>2</mn></msub></math> </ephtml> similar to the one used by <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>8</mn></msub></math> </ephtml> . Previously, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> had used visuals to represent a unit and a fraction to contextualise mixed numbers. <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> adopts this but applies it within the framework of a number line. This strategy demonstrates to students how mixed numbers comprise both whole numbers and fraction, that according to Lortie-Forgues and Siegler ([<reflink idref="bib30" id="ref102">30</reflink>]) it seems a promising way of helping students understand that the mathematical operations have the same meaning in both cases. For task <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>2</mn></msub></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> consistently employs <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>6</mn></msub></math> </ephtml> , which, due to the comparative nature of the task, also incorporates <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>9</mn></msub></math> </ephtml> . The presentation of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>2</mn></msub></math> </ephtml> in both <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> precede discussions on finding a common denominator, with the manipulations detailed in Table 5. In <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> emphasises converting improper fractions to a combination of a whole number and a fraction, thereby creating a mixed number, where in the case of magnitude representation imposes an especially high processing burden due to their bipartite structure (DeWolf et al., [<reflink idref="bib18" id="ref103">18</reflink>]). Conversely, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> focuses on the positional value of numbers to explain the comparison between improper and proper fractions to emphasise their relative positions on the number line to aid in understanding fractional values and comparisons. Similarly, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> aims to transform improper fractions into mixed numbers which mirrors the Japanese approach. However, this method may also introduce unique educational strategies, possibly emphasising the procedural aspects of conversion, akin to <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> 's approach.</p> <p>Graph</p> <p>Reflecting on how technique <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>1</mn></msub></math> </ephtml> is employed by <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> in its <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> also conveys a similar procedure. However, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> predominantly explains this using <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>9</mn></msub></math> </ephtml> (using visuals of parts of a whole or unit). Moreover, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> provides the same algorithm but constructs its concept based on <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>1</mn></msub></math> </ephtml> instead of directly presenting the algorithm without any raison d'être, as seen in Fig. 4 and Table 3 for <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> , for comparison (see Fig. 5a, and Table 3 at <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> as comparison). Moreover, there is also <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> who brings the <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>5</mn></msub></math> </ephtml> to perform the manipulation (Table 4 and Fig. 5b).</p> <p>Graph: Fig. 5 The mixed number manipulation task of MY (a) and JP (b)</p> <p>Across the tasks in the three textbooks, each institution allows students to demonstrate how they perform reductions and introduces these concepts independently before proceeding to arithmetic operations. In this way, all the textbooks already follow the fractions arithmetic steps of Recorde ([<reflink idref="bib38" id="ref104">38</reflink>]) by placing reduction after numeration, and indeed align with our REM. Four out of six types of reductions are discussed in both institutions as described in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub><mo>-</mo><msub><mi mathvariant="normal">T</mi><mn>3</mn></msub></mrow></math> </ephtml> , employing techniques grounded in the technology of a theory. Moreover, the mathematical praxeologies differ regarding the variety of tasks, the techniques utilised and their justifications, and the relationships between tasks. Next, our investigation will extend to tasks <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="normal">T</mi><mn>4</mn></msub><mo>-</mo><msub><mi mathvariant="normal">T</mi><mn>8</mn></msub></mrow></math> </ephtml> to explore the role of reductions in arithmetic operations involving fractions. They highlight significant considerations of reduction especially on how it is transposed as seen through didactic transposition (Chevallard, [<reflink idref="bib15" id="ref105">15</reflink>]) to address the knowledge of scholarly version and their extension into arithmetic operations involving fractions as the knowledge to be taught.</p> <hd id="AN0189055851-15">Reduction in the Mathematical Praxeology of Arithmetic Operation</hd> <p>Following the reduction introduced previously, we now continue to examine the position and the function of reduction within the use of them in arithmetic operation. This discussion will now challenge how Wu ([<reflink idref="bib55" id="ref106">55</reflink>]) argued about the lack justification of the fractions arithmetic process in school literatures. Each of the three textbooks under study presents the same types of tasks, except that <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>6</mn></msub></math> </ephtml> is absent in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> . Within the task organisation, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>4</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>5</mn></msub></math> </ephtml> are typically categorised into operations with identical denominators and operations involving fractions with different denominators. Conversely, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> editions introduce an additional task type, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>6</mn></msub></math> </ephtml> , which encompasses operations that combine different types of fractions. Furthermore, regarding multiplication and division, both textbooks demonstrate approaches that reveal fundamental differences when analysed through our REM and task organisation. Drawing on the findings outlined in Table 6, we will explore the process of didactic transposition for each textbook.</p> <p>Table 6 Task mapping on how reductions are used in arithmetic operations</p> <p> <ephtml> <table rules="groups"><thead><tr><th align="left"><p>Type of task</p></th><th align="left"><p>Issued in</p></th><th align="left"><p>Technique</p></th></tr></thead><tbody><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi mathvariant="normal">T</mi><mn>4</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq192.gif" /> To add fractions </p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>J</mi><mi>P</mi><mo>,</mo><mi>I</mi><mi>D</mi><mo>,</mo><mi>M</mi><mi>Y</mi></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq32.gif" /></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>8</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq44.gif" /> Adding the numerators of the same form fractions <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>6</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq42.gif" /> Using visual of numbers line to picture the position of fractions; <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>9</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq196.gif" /> Using visual of part of a whole </p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi mathvariant="normal">T</mi><mn>5</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq197.gif" /> To subtract fractions </p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>J</mi><mi>P</mi><mo>,</mo><mi>I</mi><mi>D</mi><mo>,</mo><mi>M</mi><mi>Y</mi></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq32.gif" /></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>9</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq196.gif" /> Subtracting the numerators of the same form fractions; <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>9</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq196.gif" /> Using visual of part of a whole; <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>8</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq44.gif" /> Using visual numbers line to picture the position of fractions </p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi mathvariant="normal">T</mi><mn>6</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq202.gif" /> To add and subtract fractions </p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>J</mi><mi>P</mi><mo>,</mo><mi>M</mi><mi>Y</mi></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq203.gif" /></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>8</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq44.gif" /> Adding the numerators of the same form fractions; <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>9</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq196.gif" /> Subtracting the numerators of the same form fractions; <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>8</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq44.gif" /> Using visual numbers line to picture the position of fractions </p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi mathvariant="normal">T</mi><mn>7</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq207.gif" /> To multiply fractions </p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>J</mi><mi>P</mi><mo>,</mo><mi>I</mi><mi>D</mi><mo>,</mo><mi>M</mi><mi>Y</mi></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq32.gif" /></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>10</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq209.gif" /> Using visual of quantity; <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>9</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq196.gif" /> Using visual of part of a whole </p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi mathvariant="normal">T</mi><mn>8</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq211.gif" /> To divide fractions </p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>J</mi><mi>P</mi><mo>,</mo><mi>I</mi><mi>D</mi><mo>,</mo><mi>M</mi><mi>Y</mi></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq32.gif" /></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mrow><mi mathvariant="bold-italic">τ</mi></mrow><mn>9</mn></msub><mo>:</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq196.gif" /> Using visual of part of a whole </p></td></tr></tbody></table> </ephtml> </p> <p>Each textbook in the first place has already addressed the fraction introduction in a counter to the whole number bias described by Lortie-Forgues and Siegler ([<reflink idref="bib30" id="ref107">30</reflink>]). We identified that <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>1</mn></msub></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>2</mn></msub></math> </ephtml> , and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>3</mn></msub></math> </ephtml> all independently address fractions' expression and manipulation, meaning they are not tied to arithmetic operations between fractions. As per our REM, this section represents forms of reduction crucial in the learning steps of arithmetical arithmetic operations, constituting the second sequence in arithmetic learning. Moreover, although all these task types operate independently, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>2</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>3</mn></msub></math> </ephtml> are characterised differently from <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>1</mn></msub></math> </ephtml> . While <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>1</mn></msub></math> </ephtml> focuses on the manipulation of an individual fraction, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>2</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>3</mn></msub></math> </ephtml> involve comparing or ordering (repeatedly comparing) between two or more fractions. Therefore, the actions contained in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>2</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>3</mn></msub></math> </ephtml> necessitate the learned action in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>1</mn></msub></math> </ephtml> , where the reduction process is studied.</p> <p>We begin with the discussion on how addition—and, by extension, subtraction, which employs the same method—is approached, highlighting the role of reductions. For all tasks related to additions and subtractions, especially <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>4</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>5</mn></msub></math> </ephtml> , reductions play a nearly universal role when the fractions presented have different denominators (refer to Table 6). These praxeological components coincide with those problems made by (Mostert & Hickendorff, [<reflink idref="bib35" id="ref108">35</reflink>]) regarding fraction arithmetic and whole-number arithmetic problems. Additionally, the greater the number of representative fractions <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> encompasses, the more techniques are employed, such as <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>10</mn></msub></math> </ephtml> which belongs exclusively. <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> explicitly states that 'The calculation would be easy if the denominators were the same,' aligning with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>1</mn></msub></math> </ephtml> according to our REM and Recorde ([<reflink idref="bib38" id="ref109">38</reflink>]). Following this, the applied reduction is <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>1</mn></msub></math> </ephtml> in techniques <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>8</mn></msub></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>9</mn></msub></math> </ephtml> , or <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>8</mn></msub></math> </ephtml> . For example, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> introduces 'reducing fractions,' resembling <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>5</mn></msub></math> </ephtml> , and 'finding a common denominator,' or <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>1</mn></msub></math> </ephtml> , which is then directly applied in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>4</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>5</mn></msub></math> </ephtml> in the subsequent section. Thus, the role of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>1</mn></msub></math> </ephtml> is coherently connected with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>4</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>5</mn></msub></math> </ephtml> . However, similar to <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> encounters issues with the notation of the quantity of fractions as 'the number of equal parts,' such as <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mfrac><mn>1</mn><mn>5</mn></mfrac></mrow></math> </ephtml> , which interprets as two instances of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>5</mn></mfrac></math> </ephtml> s, but can be misconstrued due to its similarity to mixed numbers notation.</p> <p>Indeed, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> features a greater number of independent tasks, specifically <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>T</mi><mn>1</mn></msub><mo>-</mo><msub><mi>T</mi><mn>3</mn></msub></mrow></math> </ephtml> . However, we observed that eventually, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>2</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>3</mn></msub></math> </ephtml> serve a similar purpose in finding equivalent fractions or identifying the same denominator. The same denomination is necessary as according to Siegler and Pyke ([<reflink idref="bib42" id="ref110">42</reflink>]), students displayed higher accuracy on equal denominator addition and subtraction and unequal denominator multiplication than on all other types of problem, the same result was obtained by Siegler et al. ([<reflink idref="bib43" id="ref111">43</reflink>]). Despite the initial bypassing of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>2</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>3</mn></msub></math> </ephtml> in fraction arithmetic operations in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> context, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>4</mn></msub></math> </ephtml> requires the application of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>1</mn></msub></math> </ephtml> in its execution process, which is presented in several parts of the sequence in previous tasks. For instance, initially, students are given the opportunity to demonstrate their fraction manipulation skills using a number line visual or <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>8</mn></msub></math> </ephtml> . In this activity, students are first asked to 'make the fraction forms uniform' among the three proposed fractions, which involves converting mixed numbers into improper fractions. Although this task involves <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>1</mn></msub></math> </ephtml> , the <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>1</mn></msub></math> </ephtml> lacks justification as it relies solely on directly applying the 'proper-fraction algorithm.' Nevertheless, this indicates that <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>1</mn></msub></math> </ephtml> is still employed in executing the techniques identified in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>2</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>3</mn></msub></math> </ephtml> . Here, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> on this task lack its raisons d'être, align with the transcendentist approach as the reasons for being (Chevallard, [<reflink idref="bib15" id="ref112">15</reflink>]). In the end, both <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>4</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>5</mn></msub></math> </ephtml> generally utilise the least common multiple (LCM) as the quickest step in determining the same denominator.</p> <p>As in Table 7, in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> , the problem is structured around juice quantities to encourages students within a measurement framework. The guided solution prompts students to recognize that fractions must be expressed with a common denominator before addition can proceed, which align with our REM. The explicit formulation of the fraction conversion process, from <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn><mo stretchy="false">/</mo><mn>3</mn></mrow></math> </ephtml> to <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>3</mn><mo stretchy="false">/</mo><mn>6</mn></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mo stretchy="false">/</mo><mn>6</mn></mrow></math> </ephtml> , highlights the central role of reduction as an initial step. In contrast, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> presents a task with a less structured, narrative-based context involving Lukas and Komang sharing pieces of paper, or in another relevant term, the word problem (Mostert & Hickendorff, [<reflink idref="bib35" id="ref113">35</reflink>]). The guided solution includes two different methods, with the second explicitly directing students to find the LCM of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>3</mn></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>6</mn></mrow></math> </ephtml> before converting <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn><mo stretchy="false">/</mo><mn>3</mn></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn><mo stretchy="false">/</mo><mn>6</mn></mrow></math> </ephtml> into fractions with a denominator of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>6</mn></mrow></math> </ephtml> . This approach aligns with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> in that reduction is addressed early in the solution. However, the presence of two alternative solution pathways suggests that <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> offers more flexibility in approaching fraction addition, potentially leading to variability in how students engage with reduction as a conceptual step. The first solution appears to leave more room for students to infer the need for a common denominator rather than explicitly guiding them to compute the LCM, as Wu ([<reflink idref="bib55" id="ref114">55</reflink>]) asserted the contradictory demands of teaching relevance and the principles of mathematics.</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> , on the other hand, takes a markedly different approach by prioritizing mixed numbers in both the task formulation and the recommended solution process. Unlike <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> , where reduction is explicitly integrated into the first steps of fraction addition, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> separates the unit components from the proper fractions before performing operations. It suggests an alternative structure: the integer parts of mixed numbers are added separately from the fractional parts, which are subsequently reduced and combined. While this method may provide an intuitive path to results that remain in mixed number form, it introduces an additional step before reduction occurs. Unlike the direct denominator equalization seen in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> 's approach delays the Reduction process, treating it as a subsequent rather than initial operation.</p> <p>The epistemological trajectory of these differences is notable. <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> ensures that students internalise reduction as a fundamental prerequisite for fraction operations, supported with the most task on the fractions manipulation which then support the understanding of reduction. <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> introduces flexibility by offering multiple pathways, while <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> 's approach embeds additional operations before reduction takes place. This will also trigger students' strategies as, at minimum, they who understands an arithmetic operation should know the direction of effects on which the operation produces (Lortie-Forgues et al., [<reflink idref="bib31" id="ref115">31</reflink>]).</p> <p>In this section of the group of additions, the same steps and approaches are used in subtraction. Both institutions utilise similar tasks but with inverse steps. One aspect under the examination is the absence of arithmetic operations that produce or involve negative numbers. Students might question the possibility of numerically expressing fractions in negative integers—a topic we also address in our REM—or how they are visually represented. Thus, anticipation related to such cases might not have been fully considered.</p> <hd id="AN0189055851-16">Cases in Multiplication and Division</hd> <p>In analysing the multiplication and division tasks presented in the textbooks, a key focus remains on how reductions are applied within the arithmetic processes. Three specific reductions—r₁, r₂, and either r₃ or r₄—are identified as significant, alongside additional reductions where necessary, such as the transformation of mixed numbers into improper fractions. These types of reductions, in several cases (see for example, Lortie-Forgues & Siegler, [<reflink idref="bib30" id="ref116">30</reflink>]; Ni & Zhou, [<reflink idref="bib36" id="ref117">36</reflink>]), leads to errors such as <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>3</mn><mo stretchy="false">/</mo><mn>5</mn><mo>×</mo><mn>4</mn><mo stretchy="false">/</mo><mn>5</mn><mo>=</mo><mn>12</mn><mo stretchy="false">/</mo><mn>5</mn></mrow></math> </ephtml> due to involving inappropriately importing procedures from other fraction arithmetic operations—addition and subtraction.</p> <p>For the reduction of fractions of fractions or <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>2</mn></msub></math> </ephtml> , it is employed in tasks by <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> while not observed in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> . <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> considers it part of a quantity, leading to multiplication between fractions and a whole number. <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> views it as the multiplication of a divided unit in multiplied denominators. JP also transforms a unit into improper fractions, thus obtaining both forms of fractions for arithmetic operation as <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>2</mn></msub></math> </ephtml> . Unlike <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> , in the context of addition and subtraction, which focuses on mixed numbers, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> converts mixed numbers into improper fractions, using <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>4</mn></msub></math> </ephtml> or <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>6</mn></msub></math> </ephtml> . JP also introduces <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>5</mn></msub></math> </ephtml> in the steps for multiplying its fractions (see Table 8).</p> <p>Table 8 Reinterpretation of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> 's tasks involving reductions within multiplication</p> <p> <ephtml> <table rules="groups"><thead><tr><th align="left"><p>Reduction</p></th><th align="left" colspan="2"><p>Task</p></th></tr></thead><tbody><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>r</mi><mn>2</mn></msub></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq15.gif" /></p></td><td align="left" colspan="2"><p>Think about how to calculate <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mfrac><mn>4</mn><mn>5</mn></mfrac><mo>×</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq303.gif" />. (fraction of fraction [<math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>r</mi><mn>2</mn></msub></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq15.gif" />])</p><p><italic>Alternative guided solution provided:</italic></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mfrac><mn>4</mn><mn>5</mn></mfrac><mo>×</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq303.gif" /> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mfenced close=")" open="("><mn>4</mn><mo>×</mo><mn>2</mn></mfenced><mfrac><mn>1</mn><mrow><mn>5</mn><mo>×</mo><mn>3</mn></mrow></mfrac></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq306.gif" /> ▭<italic>s</italic>, so... </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mfrac><mn>4</mn><mn>5</mn></mfrac><mo>×</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>=</mo><mfrac><mrow><mo>□</mo><mo>×</mo><mo>□</mo></mrow><mrow><mo>□</mo><mo>×</mo><mo>□</mo></mrow></mfrac></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq307.gif" />=▭ </p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>r</mi><mn>4</mn></msub></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq17.gif" /> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>r</mi><mn>6</mn></msub></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq21.gif" /></p></td><td align="left" colspan="2"><p>Think about how to calculate <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mn>1</mn><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>×</mo><mn>2</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq310.gif" />. (A mixed numbers of units with fractions [<math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>r</mi><mn>4</mn></msub></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq17.gif" /> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>r</mi><mn>6</mn></msub></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq21.gif" />])</p><p><italic>Alternative guided solution provided:</italic></p><p>Calculate by converting the mixed numbers into improper fractions</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mn>1</mn><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>×</mo><mn>2</mn><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mfrac><mo>□</mo><mo>□</mo></mfrac><mo>×</mo><mfrac><mo>□</mo><mo>□</mo></mfrac></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq313.gif" /></p></td></tr><tr><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>r</mi><mn>5</mn></msub></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq18.gif" /></p></td><td align="left"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mfrac><mn>5</mn><mn>9</mn></mfrac><mo>×</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mo>=</mo><mfrac><mrow><mn>5</mn><mo>×</mo><mn>3</mn></mrow><mrow><mn>9</mn><mo>×</mo><mn>4</mn></mrow></mfrac></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq315.gif" /></p><p>(Fraction written in great numbers, which might be written in lesser numbers. [<math xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>r</mi><mn>5</mn></msub></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq18.gif" />])</p><p><italic>Alternative guided solution provided:</italic></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mfrac><mn>5</mn><mn>9</mn></mfrac><mo>×</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mo>=</mo><mfrac><mn>15</mn><mn>36</mn></mfrac></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq317.gif" /></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mfrac><mn>5</mn><mn>9</mn></mfrac><mo>×</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mo>=</mo><mfrac><mn>5</mn><mn>12</mn></mfrac></mrow></math><inline-graphic mime-subtype="GIF" href="10763_2025_10554_Article_IEq318.gif" /></p></td><td align="left"><p>'Normally, we would reduce the fraction during this calculation as shown on the right.'</p></td></tr></tbody></table> </ephtml> </p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> systematically incorporates reductions into its instructional approach, particularly in cases where multiplication involves fractions. The "fractions of fractions" ( <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>r</mi><mn>1</mn></msub><mrow><mo stretchy="false">)</mo></mrow></mrow></math> </ephtml> is explicitly demonstrated in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> where fractions are considered as units being subdivided further, according to Recorde ([<reflink idref="bib38" id="ref118">38</reflink>]). <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> employs <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>1</mn></msub></math> </ephtml> by treating a fraction as the multiplication of divided units before the multiplication of denominators. Moreover, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> emphasizes the transformation of mixed numbers into improper fractions to perform with a uniform fraction format before performing arithmetic operations. This transformation aligns with the application of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>3</mn></msub></math> </ephtml> or <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>4</mn></msub></math> </ephtml> in ensuring that mixed numbers do not disrupt the reduction process.</p> <p>For multiplication tasks, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> also performs <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>2</mn></msub></math> </ephtml> to guide students in cases where fractions can be rewritten with equivalent, simplified denominators. The guided solution pathways in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> 's materials encourage the application of reductions to obtain the most simplified expression before final computation. In contrast, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> adopts a fundamentally different strategy that does not align with the reduction-based framework seen in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> . Instead of employing explicit reduction strategies, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> separates the concepts of multiplication and division of fractions into discrete operations. For example, a problem such as <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mo>×</mo><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">/</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> is introduced through an additive interpretation, i.e., <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>3</mn><mo stretchy="false">/</mo><mn>4</mn><mo>+</mo><mn>3</mn><mo stretchy="false">/</mo><mn>4</mn><mo>=</mo><mn>6</mn><mo stretchy="false">/</mo><mn>4</mn></mrow></math> </ephtml> , rather than through direct fraction multiplication. Similarly, a problem like <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac><mo>×</mo><mi>c</mi></mrow></math> </ephtml> is rewritten as <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>a</mi><mo>×</mo><mi>c</mi></mrow><mi>b</mi></mfrac></math> </ephtml> to reinforce a procedural approach based on whole number multiplication followed by fraction notation. However, for students with initial knowledge of whole number (repeated addition or subtraction) in their conception of multiplication and division, there is chance that students might encounter whole number bias (Lortie-Forgues & Siegler, [<reflink idref="bib30" id="ref119">30</reflink>]; Ni & Zhou, [<reflink idref="bib36" id="ref120">36</reflink>]).</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> 's approach relies on conceptual explanations that frame multiplication of fractions as a process of determining "how many parts" exist within a given set or collection. However, this method does not establish direct connections to the reduction techniques found in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> 's framework, making it difficult to align <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> 's tasks <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>3</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>4</mn></msub></math> </ephtml> with the structured reduction strategies observed in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> . <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> , on the other hand, presents a distinct approach to reduction, particularly through the application of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>5</mn></msub></math> </ephtml> , which is framed as "fractions of a quantity." Nevertheless, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> 's approach has advantage in its presentation with narrative as Mostert and Hickendorff ([<reflink idref="bib35" id="ref121">35</reflink>]) suggested that in multiplication of a fraction with a whole number and in all types of fraction division, word problems were easier than their symbolic counterparts.</p> <p>Unfortunately, we did not find any connection with our REM in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> 's tasks <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>6</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>7</mn></msub></math> </ephtml> . On the other hand, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> presents <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mn>1</mn></msub></math> </ephtml> as 'fractions of a quantity'. However, the emphasis here is on consistency in using <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>2</mn></msub></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>τ</mi><mn>9</mn></msub></math> </ephtml> . This becomes a concern in how Malaysia uses a 'reliable' technique—applicable in all cases—and avoids the confusion of understanding varied approaches for students, agreeing Wu's ([<reflink idref="bib55" id="ref122">55</reflink>]) judgement on the presentation of fractions that has absence of reasoning. In conclusion, the use of reductions in tasks <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>4</mn></msub></math> </ephtml> to <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>8</mn></msub></math> </ephtml> is elaborated upon, with their role being most extensively performed by <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> .</p> <hd id="AN0189055851-17">Conclusion</hd> <p></p> <hd id="AN0189055851-18">Concluding Remarks</hd> <p>We recall our goal to examine three textbooks representing three different institutions that incorporate reduction in the process of fractions arithmetic to reveal the topographic praxis and the justifying logos of reduction in each textbook. The piece of knowledge that is the reduction takes different positions and function in the praxis-logos within various institutions represented by <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> , <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> , and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> mathematics school textbooks. In this way, we join the fractions arithmetic discussion while proposing inquiries regarding a topic of reduction in our research questions. The Chevallard's ATD framework ([<reflink idref="bib12" id="ref123">12</reflink>], [<reflink idref="bib13" id="ref124">13</reflink>], [<reflink idref="bib14" id="ref125">14</reflink>], [<reflink idref="bib15" id="ref126">15</reflink>]) demonstrates an effective and legitimate approach to identify eight types of mathematical tasks present in the textbooks, examined their praxeological organisation, and addressing the epistemology of the praxeological components that incorporate reduction. The term reduction in the topic of fraction as introduced by Recorde ([<reflink idref="bib38" id="ref127">38</reflink>]) together with the studies we have discussed throughout this paper, carried our proposed REM to be the primary basis of analysis.</p> <p>Here, in response to the first research question, among the tasks examined, expressing or manipulating fractions ( <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>1</mn></msub></math> </ephtml> ) is a common element across all three countries, highlighting its fundamental role in fraction arithmetic. However, some tasks were found to be unique to specific textbooks. For example, comparing fractions ( <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>2</mn></msub></math> </ephtml> ) is not explicitly included in MY, while ordering fractions ( <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>3</mn></msub></math> </ephtml> ) is a distinct task exclusive to ID. Furthermore, tasks <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>T</mi><mn>4</mn></msub><mo>-</mo><msub><mi>T</mi><mn>8</mn></msub></mrow></math> </ephtml> are generally consistent across the three countries, with the exception of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>6</mn></msub></math> </ephtml> , which is present only in <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> . These variations illustrate that reduction operates within different mathematical structures in each country, influenced by their respective pedagogical traditions and curriculum frameworks. Moreover, by analysing the praxeological components in depth of pinpoint <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>p</mi><mo>=</mo><mfenced close="]" open="["><mi>T</mi><mo stretchy="false">/</mo><mi>τ</mi><mo stretchy="false">/</mo><mi>θ</mi><mo stretchy="false">/</mo><mi mathvariant="normal">Θ</mi></mfenced></mrow></math> </ephtml> suggested by Chevallard ([<reflink idref="bib15" id="ref128">15</reflink>]), we found that each country exhibits specific differences in the sequencing of tasks, the range of techniques employed <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><msub><mi>τ</mi><mn>1</mn></msub><mo>-</mo><msub><mi>τ</mi><mn>10</mn></msub></mfenced></math> </ephtml> , the technologies used to justify mathematical procedures ( <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>θ</mi><mn>1</mn></msub><mo>-</mo><msub><mi>θ</mi><mn>4</mn></msub></mrow></math> </ephtml> ), and the overarching theoretical legitimacies ( <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Θ</mi></math> </ephtml> ). Additionally, we identified nine different forms of reduction ( <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>r</mi><mn>1</mn></msub><mo>-</mo><msub><mi>r</mi><mn>9</mn></msub></mrow></math> </ephtml> ), each of which plays a distinct role in the fraction arithmetic of <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>J</mi><mi>P</mi><mo>,</mo><mi>M</mi><mi>Y</mi><mo>,</mo></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> . The presence or absence of specific reductions reflects the underlying epistemological and pedagogical orientations of each educational system. This comparison between institutions runs a same direction with studies on textbooks comparison and analysis (e.g. Alajmi, [<reflink idref="bib1" id="ref129">1</reflink>]; González-Martín et al., [<reflink idref="bib24" id="ref130">24</reflink>]; Solis & Isoda, [<reflink idref="bib46" id="ref131">46</reflink>]; Takeuchi & Shinno, [<reflink idref="bib51" id="ref132">51</reflink>], see also at theoretical foundation), our institutional analysis provides the state-of-the-art in how a particular praxeological pinpoint focusing on the reduction is examined systematicaly and epistemologically both in the praxis level and logos level within ATD.</p> <p>Considering our second research question, we summarise how reductions are relatively transposed and how they take position and function in addition, subtraction, multiplication, and division as the consequences of the didactic transposition. Despite the different forms of praxis, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>J</mi><mi>P</mi><mo>,</mo><mi>I</mi><mi>D</mi><mo>,</mo></mrow></math> </ephtml> and <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> certainly have the reduction as part of their praxeological components. In <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">JP</mi></mrow></math> </ephtml> , reduction is explicitly addressed and closely integrated into mathematical justifications, with <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>r</mi><mn>1</mn></msub><mo>-</mo><msub><mi>r</mi><mn>6</mn></msub></mrow></math> </ephtml> playing a central role in supporting various techniques ( <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>τ</mi><mn>1</mn></msub><mo>-</mo><msub><mi>τ</mi><mn>9</mn></msub></mrow></math> </ephtml> ). The <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ID</mi></mrow></math> </ephtml> textbook, by contrast, introduces reduction in a more varied manner, with a unique emphasis on ordering fractions ( <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>3</mn></msub></math> </ephtml> ). Meanwhile, <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">MY</mi></mrow></math> </ephtml> adopts a highly visual approach, making extensive use of representations of part-whole relationships, which integrates reductions ( <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>r</mi><mn>1</mn></msub><mo>-</mo><msub><mi>r</mi><mn>6</mn></msub></mrow></math> </ephtml> ) into nearly all tasks. These differences illustrate that while the mathematical goals are aligned across the three educational systems—namely, to facilitate the correct execution of fraction arithmetic—the means by which this knowledge is structured and transposed vary considerably. Some of those findings also align with the additional findings in theoretical level about technological jump or the distant praxeological level from academic mathematics as asserted by Takeuchi and Shinno ([<reflink idref="bib51" id="ref133">51</reflink>]) and González-Martín et al. ([<reflink idref="bib24" id="ref134">24</reflink>]), respectively.</p> <hd id="AN0189055851-19">Implications and Limitations</hd> <p>In our analysis of how knowledge of fraction arithmetic is presented in textbooks and the potential consequences of these decisions, we suggest that this study may contribute to the ongoing discourse on didactic phenomena by offering an alternative to the current structure of knowledge within didactic design. Our direction aligns with our role in the Suryadi's DDR framework (Suryadi, [<reflink idref="bib47" id="ref135">47</reflink>], [<reflink idref="bib49" id="ref136">49</reflink>], [<reflink idref="bib50" id="ref137">50</reflink>]), for this analysis is prospectively prepared for both theoretical discussions and practical applications in the next phase in the framework for research.</p> <p>Our results, in line with existing reports, suggest that reduction is a primary cause of students' difficulties and errors in arithmetic operations. The institutional approach has revealed that the challenges identified by earlier research are not explicitly addressed in the textbooks we examined. These findings may be relevant to textbooks used in other educational settings, and we encourage further exploration of this hypothesis using our approach.</p> <p>Our analysis indicates that the praxeologies likely generated by textbooks are significantly transposed from the body of scholarly knowledge of mathematics, both mechanically and theoretically. However, this praxeological organisation used is limited on the systemic and epistemological aspect on the knowledge to be taught as what ATD believes in. Some of the organisations lack a coherent and operational procedure for arithmetical fractions—the praxis may concern on the implicit judgements, while actual implication in the didactical situations may differ, and the justification of properties is solely based on theoretical legitimacies. This issue, however, remains an open conjecture for further studies.</p> <hd id="AN0189055851-20">Acknowledgements</hd> <p>This study was made possible thanks to research students' collaborative works in Indonesia, Malaysia, and Japan. We would also like to acknowledge the funding contribution by Indonesian Endowment Fund for Education (LPDP).</p> <hd id="AN0189055851-21">Funding</hd> <p>The funding of this research is contributed by Indonesian Endowment Fund for Education (LPDP). Website: https://lpdp.kemenkeu.go.id/en/</p> <hd id="AN0189055851-22">Data Availability</hd> <p>Our research focused on the content of widely available school textbooks. 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  Label: Title
  Group: Ti
  Data: Institutional Relativity of Reduction within Arithmetical Fractions: An Analysis of Japanese, Indonesian, and Malaysian School Textbooks
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Sani+Sahara%22">Sani Sahara</searchLink> (ORCID <externalLink term="http://orcid.org/0000-0001-5426-4296">0000-0001-5426-4296</externalLink>)<br /><searchLink fieldCode="AR" term="%22Didi+Suryadi%22">Didi Suryadi</searchLink><br /><searchLink fieldCode="AR" term="%22Turmudi+Turmudi%22">Turmudi Turmudi</searchLink><br /><searchLink fieldCode="AR" term="%22Agus+Hendriyanto%22">Agus Hendriyanto</searchLink><br /><searchLink fieldCode="AR" term="%22Lukman+Hakim+Muhaimin%22">Lukman Hakim Muhaimin</searchLink>
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  Label: Source
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  Data: <searchLink fieldCode="SO" term="%22International+Journal+of+Science+and+Mathematics+Education%22"><i>International Journal of Science and Mathematics Education</i></searchLink>. 2025 23(7):2397-2425.
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  Data: Springer. Available from: Springer Nature. One New York Plaza, Suite 4600, New York, NY 10004. Tel: 800-777-4643; Tel: 212-460-1500; Fax: 212-460-1700; e-mail: customerservice@springernature.com; Web site: https://link.springer.com/
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  Label: Peer Reviewed
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  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 29
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2025
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  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Foreign+Countries%22">Foreign Countries</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Instruction%22">Mathematics Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Teaching+Methods%22">Teaching Methods</searchLink><br /><searchLink fieldCode="DE" term="%22Arithmetic%22">Arithmetic</searchLink><br /><searchLink fieldCode="DE" term="%22Fractions%22">Fractions</searchLink><br /><searchLink fieldCode="DE" term="%22Textbook+Evaluation%22">Textbook Evaluation</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+Concepts%22">Mathematical Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Epistemology%22">Epistemology</searchLink>
– Name: Subject
  Label: Geographic Terms
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Japan%22">Japan</searchLink><br /><searchLink fieldCode="DE" term="%22Indonesia%22">Indonesia</searchLink><br /><searchLink fieldCode="DE" term="%22Malaysia%22">Malaysia</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1007/s10763-025-10554-x
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 1571-0068<br />1573-1774
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Didactic transposition transpose knowledge relatively across various institutions, or the place where knowledges live within. In this study we analyse the position and the role of reduction of fraction arithmetic, specifically on how the piece of knowledge takes account, in the mathematics school textbooks of Japan, Indonesia, and Malaysia. We follow the institutional analysis by the Anthropological Theory of the Didactic (ATD) as framework for our examination. Moreover, we also seek to contribute to both theoretical discourse and didactical practice, as the study also takes a part in the prospective study in the Didactical Design Research (DDR). As our model of analysis, in this paper, we propose a Reference Epistemological Model (REM) as an objective basis for our interpretation to the knowledge to be taught. Our findings show the varieties of types, positions, and roles of reduction within the praxeological organisations of each textbook. The roles in which reductions perform are distinctively unique for each textbook, as they are positioned differently in terms of technique and technology. Additionally, the cases in the arithmetical fractions are also discussed.
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  Group: Date
  Data: 2026
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  Label: Accession Number
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  Data: EJ1492869
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        Value: 10.1007/s10763-025-10554-x
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      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 29
        StartPage: 2397
    Subjects:
      – SubjectFull: Foreign Countries
        Type: general
      – SubjectFull: Mathematics Instruction
        Type: general
      – SubjectFull: Teaching Methods
        Type: general
      – SubjectFull: Arithmetic
        Type: general
      – SubjectFull: Fractions
        Type: general
      – SubjectFull: Textbook Evaluation
        Type: general
      – SubjectFull: Mathematical Concepts
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      – SubjectFull: Epistemology
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      – SubjectFull: Japan
        Type: general
      – SubjectFull: Indonesia
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      – SubjectFull: Malaysia
        Type: general
    Titles:
      – TitleFull: Institutional Relativity of Reduction within Arithmetical Fractions: An Analysis of Japanese, Indonesian, and Malaysian School Textbooks
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