Undergraduate students’ theoretical covariational reasoning to understand the concept of homeomorphism with GeoGebra

Research in mathematics education claims that encouraging students’ covariational reasoning is beneficial for the development of mathematical concepts (Bagossi, 2024; Thompson & Carlson, 2017). Moreover, changing theory, exploring the meaning of a concept within each mathematical theory, makes richer the semiotic representations set and stronger its understanding. Just think about the use of Taxicab geometry to deeply understand a Euclidean geometry concept (Kemp & Vidakovic, 2023). The simultaneous variation of two representations (quantities, relationships, etc.) within distinct theories is said to be theoretical covariation (Miranda & Saliceto, 2025). At the university level, theoretical covariational reasoning processes can help in understanding and manipulating more advanced mathematical concepts. In the experiment I’m going to discuss, undergraduate mathematics students were required to explore the definition of homeomorphism in the domain of topology. Each group was required to reason with respect to two different metrics supported by GeoGebra. The digital environment helped them to enhance reasoning about spatial figures, verifying if the graphical representation of the object corresponds to what is visualised in the mind. But what happens from a mathematical point of view? They discover that a homeomorphism with respect to a given metric is not necessarily one with respect to another metric, even in the case where the metrics are topologically equivalent. In particular, the study focuses on stereographic projection, which is a widely used application in the STEAM fields. The theoretical covariation between the TaxicabEuclidean stereographic projections helps them visualise and reflect on the reasons for which the Taxicab-stereographic projection is not a homeomorphism. The data are analysed through the lens of the IK-MWS theoretical model (Miranda et al., 2025). The results are promising: the use of the digital tool in synergy with the change in geometry refines the visualisation and construction processes and thereby the discursive processes within the theories.