Didactic transposition of trigonometric ratios: A comparative study of school and university textbooks

Abstrak

Trigonometric ratios constitute a fundamental topic in mathematics that bridges school-level and university-level mathematics education. Nevertheless, many students struggle to grasp the relationships among sine, cosine, and tangent, in part due to inconsistencies in conceptual emphasis and sequencing across different educational stages. This study aims to compare the praxeological organization of trigonometric ratios in an Indonesian Grade X senior high school mathematics textbook and in Gelfand’s Trigonometry, drawing on the Anthropological Theory of the Didactic (ATD), particularly the notion of didactic transposition. A qualitative document analysis approach is adopted to examine the four components of praxeology—types of tasks, techniques, technology, and theory—in each textbook. ​ The analysis reveals substantial epistemological differences between the two texts. The university-level textbook develops trigonometric ratios through a deductive and coherent sequence in which sine and cosine are established as primary ratios, and tangent is subsequently introduced as a derived concept grounded in these foundations. By contrast, the school textbook is predominantly application-oriented, foregrounding the tangent ratio at an early stage and providing only limited theoretical justification for its introduction and use. This reorganization reflects a didactic transposition that shifts the focus from theoretical coherence toward a more procedural orientation in school mathematics. ​ In response to these findings, an alternative praxeology is proposed to reconstruct the epistemological relationships among techniques, technological discourse, and theoretical underpinnings for trigonometric ratios. The proposed instructional sequence begins with the identification of the sides of a right triangle, followed by defining sine and cosine as fundamental ratios, then deriving tangent from these primary ratios, and finally engaging students in contextual problem situations. This reconstructed praxeology is designed to foster a more balanced integration of procedural fluency and conceptual understanding, while maintaining epistemological coherence and accommodating classroom constraints. Moreover, the study offers concrete guidance for textbook authors and instructional designers on organizing content sequences and providing conceptual justifications that are consistent with the ATD framework and its praxeological perspective.