When Topology Deceives: Structural Covariational Surprise in Understanding Metric-Dependent Homeomorphisms through Digital Collaboration

We investigate how undergraduate mathematics students develop understanding of homeomorphism through theoretical covariational thinking and metaphorical reasoning. Thirty second-year geometry students at a southern Italian university explored stereographic projections using Euclidean and Taxicab metrics, mediated by GeoGebra. Students discovered that while Euclidean stereographic projection constitutes a homeomorphism, the Taxicab counterpart does not, despite both metrics inducing identical topologies. Through collaborative “Geometric Thinking Groups”, students experienced ‘structural covariational surprise’, a productive cognitive conflict arising from unexpected patterns in how metric and topological structures covary. GeoGebra functioned as a cognitive partner enabling dynamic mathematical experimentations. Data analysis reveals how digital tools facilitate coordination of visual-geometric and formal-analytical thinking, leading to understanding of the interplay between metric structure, topological structure, and homeomorphic behaviour. Findings show how theoretical covariation, supported by conceptual metaphors and digital mediation, develops advanced mathematical concepts while revealing distinctions between topological equivalence and metric-dependent homeomorphic behaviours.