How France, Japan and Indonesia Teach Linear Equations: Three Textbooks, Three Philosophies

Every year, millions of lower secondary school students around the world meet their first real algebra: the linear equation in one variable. It looks deceptively simple — find the value of x that makes a statement true — but it marks one of the most consequential cognitive thresholds in school mathematics, the passage from arithmetic, where numbers and answers dominate, to algebra, where letters stand for quantities and relationships must be manipulated as objects in their own right. A new comparative study published in SN Social Sciences by Dilham Fardian, Didi Suryadi, Sufyani Prabawanto and Al Jupri of Universitas Pendidikan Indonesia, together with Hamid Chaachoua of Université Grenoble Alpes and Tatsuya Mizoguchi of Tottori University, asks a question that sounds narrow but turns out to be profound: when three very different countries — France, Japan and Indonesia — put linear equations into their official lower secondary textbooks, what exactly are they asking students to do, and what hidden mathematical and institutional choices shape those pages?

The study is grounded in the Anthropological Theory of the Didactic, or ATD, a framework developed by the French didactician Yves Chevallard that treats mathematical knowledge not as a free-floating abstraction but as something produced, transformed and institutionalized within human institutions. In ATD terms, knowing a piece of mathematics means mastering a praxeology — a tightly linked pair of praxis and logos. The praxis block consists of types of tasks (what students are asked to do, such as solving 3x + 5 = 20) and techniques (the procedures they use to do it, such as performing the same operation on both sides). The logos block consists of technologies — the discursive justifications that explain why a technique works — and theories, the broader bodies of knowledge in which those justifications are embedded. By dissecting textbooks into these four components, the researchers could compare not merely which equations appear in each country’s books, but the entire epistemological architecture supporting them.

The choice of textbook as the object of study is itself significant. Textbooks occupy a privileged position in the ATD’s model of didactic transposition, the chain of transformations that carries scholarly knowledge — the mathematics of research mathematicians — into knowledge to be taught, then into knowledge actually taught in classrooms, and finally into knowledge learned by students. Textbooks are the most concrete, inspectable embodiment of the knowledge to be taught, and decades of international assessment research, including the TIMSS textbook studies of Valverde and colleagues, have shown that they strongly shape the opportunities students get to learn. The authors situate their comparison across the nine levels of didactic co-determination in the ATD framework, which run from the specificity of a single classroom moment up through the school, the national curriculum, and the wider society. This allows them to connect a worked example on a textbook page to the curricular and cultural conditions that produced it.

The three countries were chosen because they represent genuinely distinct institutional traditions. France has a centralized curriculum and a long, internationally influential didactic research tradition; the analyzed textbook, Mission Indigo for cycle 4, is a mainstream commercial title from Hachette Éducation. Japan operates a national Course of Study with a rigorous textbook screening system, and its textbooks are widely admired for coherence and careful problem sequencing; the Japanese volume analyzed comes from the official 2024 textbook database. Indonesia, meanwhile, has undergone repeated curricular reform and relies on textbooks distributed through a national system, with the analyzed book carrying reference number 9786022445142. All three systems target linear equations in one variable at lower secondary level, so the comparison holds the mathematical content roughly constant while letting the institutional variables vary.

The analysis reveals that although the three countries cover similar content, they organize it in strikingly different ways. The differences appear first at the level of types of tasks: the distribution and progression of equation-solving tasks, from simple one-step equations to equations requiring distribution, collecting like terms, or handling variables on both sides, varies across the three books. They appear again at the level of techniques: the canonical solving procedures presented to students differ, reflecting different answers to a classic didactic question — should equations be solved through a balance-like manipulation that preserves the equality, or through inverse operations applied systematically to isolate the unknown? Prior experimental work, including cross-cultural studies comparing the balance method with the inverse method in Australia and Malaysia, has shown that this choice is not pedagogically neutral, since different techniques load differently on students’ conceptual and procedural knowledge.

The deepest differences emerge in the logos block. The textbooks differ in the technologies they offer — the explicit or implicit explanations of why the solving techniques are legitimate — and in the theories on which those explanations rest. These differences matter because the transition from arithmetic to algebra is precisely where students must reconstruct their understanding of the equals sign. A large body of cognitive research, cited in the study, shows that many children interpret the equals sign as a command to compute an answer rather than as a symbol of relational equivalence, and that this misconception acts as a measurable barrier to early algebra learning. A textbook’s technology — whether it presents equation solving as undoing operations, as maintaining balance, or as reasoning about equivalent equations — shapes which conception of equality students are positioned to build. The three countries’ books, the analysis suggests, embody different institutional answers to this epistemological problem.

The researchers interpret these contrasts as reflections of distinct institutional choices concerning curricular design, textbook development and pedagogical orientation, rather than as mere national idiosyncrasies. In the ATD’s terms, the praxeologies of linear equations in each book are ecologically adapted to their institutional habitat: a centralized curriculum with strong didactic research input in France, a tightly screened national Course of Study in Japan, and a reform-driven, centrally distributed system in Indonesia. What looks like a page of routine exercises is, on this reading, the visible tip of a deep structure of decisions about what algebra is for, how it should be justified, and how students should be moved from computing answers to reasoning about relationships.

The study is careful about its own limits. The authors restrict their claims to the knowledge to be taught as represented in the textbooks; they do not claim to describe what teachers actually do in French, Japanese or Indonesian classrooms, nor what students ultimately learn. Within the ATD framework, the natural next steps are to extend the analysis to scholarly knowledge, taught knowledge and learned knowledge, completing the full circuit of the didactic transposition. The authors also note that the approach builds on their own earlier work analyzing Indonesian textbooks and on a recent five-country comparison of quadratic equation curricula involving Brazil, France, Japan, Spain and Vietnam, situating the present study within a growing international research program that uses praxeological tools to make curricula comparable in a rigorous, non-superficial way.

For a general audience, the significance of the work lies in what it says about educational reform. International comparisons of mathematics achievement often stop at scores, leaving policymakers to guess at the causes of national differences. This study demonstrates a method for tracing those differences back to concrete, analyzable design decisions — which tasks are posed, which techniques are taught, which justifications are given — in the very materials students hold in their hands. It suggests that improving algebra learning may depend less on adding content than on restructuring the praxeologies students encounter, particularly around the fragile moment when arithmetic gives way to algebra. The data supporting the study are openly available through the publishers’ databases for the French, Japanese and Indonesian textbooks, and the full article, published on 13 October 2026 in volume 6 of SN Social Sciences as article number 534, was funded by Indonesia’s Ministry of Education, Culture, Research, and Technology through the PMDSU scheme.

Subject of Research: Comparative didactic analysis of linear equation instruction in lower secondary mathematics textbooks from France, Japan, and Indonesia

Article Title: Comparing lower secondary mathematics textbooks from France, Japan, and Indonesia: a praxeological analysis of linear equations

Article References: Comparing lower secondary mathematics textbooks from France, Japan, and Indonesia: a praxeological analysis of linear equations. (n.d.). https://doi.org/10.1007/s43545-026-01675-3

Image Credits: AI Generated

DOI: 10.1007/s43545-026-01675-3

Keywords: mathematics education, linear equations, textbook analysis, Anthropological Theory of the Didactic, praxeology, algebra, didactic transposition, comparative education, France, Japan, Indonesia, curriculum