PROCEEDINGS OF THE 18TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION
2015
Pittsburgh, Pennsylvania
Developing abstract knowledge in advanced mathematics: Continuous functions and the transition totopology
Page: 415
Despite intuitive foundations, the nature of the transition to abstract topology often results in students’ reliance on dissociated collections of definitions and theorems, without any integrated cognitive structure. In recent decades, there have been numerous analyses of proof, symbols, and the encapsulation of processes as factors in student comprehension, as well as content-specific studies examining which mental constructions support the development of coherent schemata for particular topics. I will expand this research by categorizing students’ understanding in the domain of topology. In a year-long, mixed-methods study, I will analyze the components involved in the development of an axiomatic schema for continuous functions in topological contexts. I will compare this model with actual student constructions in an introductory topology course, collected through task-based interviews and a path-analysis on the coded data. The goal is to confirm the theoretical model, or to provide support for altering the model to increase its validity.