Theoretical Model for Understanding Homotopies Through Mathematical Argumentation in Undergraduate Mathematics Students in Peru

This qualitative case study formulates a theoretical model for understanding homotopies through mathematical argumentation among undergraduate mathematics students at a Peruvian university. The study addresses a documented gap in the literature: although mathematical argumentation and homotopy theory have been studied as separate fields, no prior research integrates them pedagogically at the university level. Data were collected through an open-question questionnaire and semi-structured interviews with five purposively selected participants and analyzed via thematic analysis (TA) using Atlas.ti v9, generating seventy-eight codes organized into four categories. The results reveal three argumentation components—initial data (hypotheses), axiomatic substrate, and conclusion (thesis)—and three epistemologically grounded thinking modes: synthetic-geometric (SG), analytic-topological (ATp), and analytic-algebraic (AAg), connected by conceptual articulators. A fourth emergent category—comprehension obstacles—documents specific epistemological errors (EPs) and conceptual difficulties in topological contexts. The proposed model was validated through expert judgment, obtaining Aiken’s V = 1.00 across all evaluated dimensions. The findings advance the understanding of mathematical argumentation in formal proof contexts and offer a theoretically grounded framework applicable to the teaching of advanced topology in higher education.