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

The study, published in Discover Education, combined a close theoretical reading of the official textbooks with empirical testing of real students. The researchers analyzed both the teacher and student editions of the mandated textbooks using the Anthropological Theory of the Didactic, a framework developed by the French didactician Yves Chevallard that treats mathematical knowledge as a set of organized practices. Central to this framework is the concept of praxeology, the pairing of tasks and techniques with the explanations and theories that justify them. A well-designed textbook, in this view, does not merely show students how to graph an inequality; it explains why the shaded region represents an infinite set of solutions and why two overlapping regions must both be satisfied simultaneously. The Indonesian textbooks, the analysis found, repeatedly present the how while omitting the why.

That imbalance has a technical name in the literature: a praxeological imbalance, in which the practical block of tasks and techniques crowds out the logos block of justification and theory. The researchers documented concrete examples. In one opening problem, students weigh apples and oranges on a scale and are asked to build a system of inequalities through trial and error. The task is engaging, but it never explicitly instructs students to convert kilograms to ounces, a hidden technical demand that derails learners who have not mastered unit conversion. The completed mathematical model is then printed directly beside the problem, inviting imitation rather than independent reasoning. Elsewhere, the book leaps from single inequalities to full systems without scaffolding the crucial idea that a system’s solution is the intersection of two half-planes, not simply the area below each line.

To test whether these theoretical weaknesses translate into real difficulties, the team administered a diagnostic assessment of eight open-ended tasks to 80 students at a public vocational school in Banten, Indonesia, and then conducted in-depth interviews with 20 of them, selected purposively for showing distinctive error patterns. The tasks were deliberately mapped to the obstacle categories predicted by the textbook analysis: four targeted epistemological obstacles involving inequality symbols, solution sets and half-planes, three targeted didactical obstacles arising from instructional design, and one addressed ontogenic obstacles in applying the concepts to real-world contexts. The design meant that each error a student made could be traced back to a specific feature of how the material had been presented.

The results were remarkably consistent with the predictions. Half of the students, 50 percent, could draw boundary lines but could not interpret what the inequality symbols meant graphically, leaving them unable to decide which side of the line to shade. One student told the researchers, I have drawn the lines, but I don’t know whether to shade the upper or lower part. Twenty percent failed to identify the intersection of two regions as the solution set, instead shading everything below both lines; one admitted, I didn’t know I had to look for the intersection, I thought it was enough to shade below the lines. Another 30 percent chose test points randomly from the graph without substituting them into the inequalities, revealing a disconnect between graphical and symbolic representations that echoes Raymond Duval’s classic work on the cognitive difficulty of converting between mathematical registers.

The researchers also catalogued ontogenic obstacles, those rooted in students’ own developmental readiness. Conceptual gaps were the most common, affecting 55 percent of participants, who lacked command of prerequisites such as plotting lines and finding axis intercepts. Instrumental obstacles, meaning weak technical skills like substitution and arithmetic, appeared in 30 percent; one student wrote that 3 plus 3 equals 7 while checking a point against the inequality x plus y is at most 7. Psychological obstacles, including math anxiety and fear of failure, affected 15 percent, with students reporting they did not dare to try because they were afraid of being wrong. These affective barriers, the authors note, consume working memory resources and compound the conceptual difficulties, a finding consistent with the broader literature on cognitive load and mathematics anxiety.

Perhaps the most consequential category was the didactical obstacle, the kind produced by instruction itself. Forty-five percent of students shaded regions by pure imitation of textbook patterns, with one explaining, I shade the area below the line because that’s how it is in the book, but I’m not sure if it’s correct. Thirty-five percent had never been taught to seek the intersection of two constraints, and 20 percent showed no habit of algebraic validation at all, selecting points visually because, as one put it, the teacher usually never asks to check points. The researchers interpret these behaviors as products of an implicit didactical contract in which students see mathematics as a sequence of authoritative steps to be copied rather than a logical system requiring justification. When the textbook provides worked templates but no validation prompts, students learn that verification is not part of the job.

The team argues that these findings establish more than a correlation. By triangulating the textbook task layouts, the diagnostic error patterns and the interview testimonies, they built a mapping matrix linking specific textbook features to specific student errors. The high rate of shading mistakes corresponds to the books’ emphasis on mechanical table completion and the placement of finished models next to problems, which lowers cognitive demand and invites copying. The intersection and validation errors correspond to the textbooks’ silence on those very procedures. In the researchers’ framing, the obstacles are structurally embedded in the instructional material, not merely individual deficits, which means they can be anticipated and designed away.

The proposed remedies are concrete. The authors recommend a three-phase scaffolding model in which students first explore a contextual problem with guiding questions, then compare their reasoning with a partially completed model, and only finally see a fully worked example accompanied by explicit justification of each inequality. They call for standardized validation prompts after every graphing task, prerequisite activation boxes, and what-if prompts that ask how changing a strict inequality to an inclusive one alters the boundary. Comparative context strengthens the case: studies of Chinese textbooks by Liping Fan and Yan Zhu found more systematic scaffolding and explicit linking of models to tasks, suggesting the Indonesian materials lag behind international best practice. The researchers also urge a formal praxeological review of all government-published textbooks, so that every technique is paired with its conceptual rationale, and recommend that teachers supplement the books with think-aloud protocols and activities that make students justify their shading choices aloud.

The study’s limitations are acknowledged: it examined a single vocational school population and did not test an alternative textbook design longitudinally, so generalization requires caution. Yet its central message travels well beyond one Indonesian classroom. Textbooks are the primary, and often the only, reference in many mathematics classrooms worldwide, and any systematic weakness in their design propagates quietly through millions of learners. Systems of linear inequalities matter because they model budget constraints in economics and resource limits in engineering, and students who master them only mechanically tend to falter in linear programming, optimization and calculus. If the obstacles are built into the pages, the researchers conclude, then the fix belongs there too, in a deliberate redesign that treats conceptual understanding, solution validation and reflection not as extras but as the core of what a mathematics textbook is for.

Subject of Research: The impact of mathematics textbook design on learning obstacles and student performance in systems of linear inequalities in two variables

Article Title: Assessing the impact of mathematics textbooks on learning obstacles and student performance in systems of linear inequalities in two variables

Article References: Nasrullah, A., Suryadi, D., Prabawanto, S., Hendriyanto, A., Muhaimin, L. H., & Dewi, D. A. (2026). Assessing the impact of mathematics textbooks on learning obstacles and student performance in systems of linear inequalities in two variables. Discover Education, 5(1), Article 1036. https://doi.org/10.1007/s44217-026-02064-x

Image Credits: AI Generated

DOI: 10.1007/s44217-026-02064-x

Keywords: mathematics education, textbook analysis, linear inequalities, learning obstacles, didactic transposition, praxeology, Merdeka Curriculum, algebra learning, cognitive load, math anxiety, scaffolding, Indonesia