Generic examples in undergraduate topology: a case study

Abstract

Expert mathematicians use examples and visual representations as part of their informal mathematics reasoning when constructing proofs. In contrast, the ways undergraduates reason and work toward creating proofs is an open area of investigation, particularly in advanced mathematics. Furthermore, research on students’ reasoning in topology is largely unexplored. Building on findings from Gallagher [(2020). Identifying Structure in Introductory Topology: Diagrams, Examples, and Gestures. Doctoral Dissertation, West Virginia University, Morgantown, West Virginia. https://researchrepository.wvu.edu/etd/7599/ (MS #8610)], we present a case study of an undergraduate taking a first course in general topology and her use of generic examples in argumentation prior to producing formal proofs and counterexamples. We discuss patterns that emerged in her use of generic examples when proving true statements and disproving false claims. We compare and contrast this student’s generic examples with other generic examples in the literature from other content areas within mathematics, and we argue that generic examples are not one-size-fits-all but rather must be fit-for-purpose. Our results suggest that generic examples may be particularly useful for students writing proofs when they may not have access to specific examples.