Difficult dialogs about degenerate cases: A proof script study

Abstract

Contradictions play an important role in mathematics. Yet, much remains unknown about the ways undergraduate mathematics students’ attend to contradictions, especially outside of proofs by contradiction. To address this gap in the literature, 20 proof scripts were analyzed to explore students’ noticed proof problematics for a proof involving a logically degenerate case. Findings indicate that while students generally attended to the encountered contradiction as a problematic point, many struggled to address its role in the proof and held conceptions of proofs by cases that enabled them to dismiss the logically degenerate case rather than recognize an inconsistency in the emerging mathematical theory. Findings from the study not only support calls for further research on students’ reasoning about proofs by cases and the role of contradictions but also provide evidence of the viability of the proof script methodology as a mechanism for identifying difficulties observed by students but unseen by experts.