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.

KEYWORDS:

• Generic examples
• diagrams
• topology
• proof
• argumentation

Acknowledgements

The results presented in this manuscript are derived from an analysis of data collected by the lead author for their doctoral dissertation (Gallagher, 2020).

Disclosure statement

No potential conflict of interest was reported by the authors.

Human participants disclosure

Human participants took part in this research. This research was classified as minimal risk; no physical or mental risks were anticipated during the study. Ethics approval for this research was granted by the Institutional Review Board at West Virginia University under protocol/approval number 1711838593. All subjects were informed that they would be video and audio recorded solving mathematics problems, and any data they provided may be used for research purposes and published in scholarly manuscripts. All research subjects signed consent forms and were free to withdraw their consent and/or request to have their data removed from our data corpus at any time.

Notes

1 The facilitator neglected to include the condition that the sets $U$ and $V$ must be nonempty to form a separation of $X$. Neither Stacey nor the facilitator acknowledged this omission during Session 7.