Learning to think and teach functorially: categorical covariation and effective noticing in Advanced Mathematics Education

This study examines how university algebraic topology courses can simultaneously develop advanced mathematical thinking and representational competence for teaching. Through an Inquiry-Based Learning intervention, we investigate the development of categorical representational competence, the ability to recognise and manipulate systematic relationships between algebraic and geometric representations. Using a multi-layered observation design, we analyse how students develop functorial thinking and categorical covariational reasoning and how observer groups transform these insights into pedagogical knowledge through embodied, relational noticing. Results demonstrate that explicit attention to functorial connections supports both mathematical maturity and the development of pedagogically-oriented awareness.