Learning Obstacles in Constructing Proofs of Geometric Transformation: A Didactic Transposition Analysis

Abstract

This study aims to identify and analyze the learning obstacles experienced by preservice mathematics teachers in constructing proofs of geometric transformations through a didactic transposition analysis from scholarly knowledge to learnt knowledge. A descriptive qualitative approach was employed, involving textbook analysis, classroom observations, diagnostic tests, and interviews with 19 preservice teachers. The findings indicate that learning obstacles are systemic and layered, emerging through the transformation of knowledge across transposition stages. These obstacles comprise epistemological, didactical, and ontogenetic dimensions that are interrelated. Epistemologically, obstacles arise from a reduction in deductive structure and conceptual formality, reflected in students’ difficulties with quantifiers, proof logic, and the relationships among formal definitions of functions, injective functions, and surjective functions. Proof is often reduced to empirical generalization and visual verification, while formal representations such as two-variable functions and the domain ℝ² remain conceptually underdeveloped. Didactically, obstacles are reinforced by instructional practices characterized by dominant visual representations, limited proof strategies, and minimal opportunities for students to construct deductive arguments. Ontogenetically, students exhibit insufficient readiness for formal abstraction, shaped by predominantly procedural and illustrative learning experiences. These findings demonstrate that learning obstacles are not merely individual difficulties but are produced through mismatches across levels of knowledge in the didactic transposition process, highlighting the need for instructional designs that preserve epistemological coherence.