Proof in the Context of Tertiary Mathematics: Undergraduate Inquiry-Based Learning in Abstract Algebra as a Precursor to Mathematical Proof

Abstract

This chapter examines how mathematical proof and its precursors are utilized and developed through an inquiry-based learning (IBL) abstract algebra class. The analysis draws on Weber’s (Proceedings of the Joint Meeting of PME 38 and PME-NA 36. PME, Vancouver, BC, 2014) notion of proof as a cluster concept, in which he suggested there are six aspects of proof. For each justification for a mathematical claim, we determined which aspects the justification might be understood as enacting. The analysis suggested that, while there were no claims supported by something that the professor or students called a proof, there were multiple opportunities during the class for students to make and support claims via observations around the Cayley tables they were discussing. Evidence found to support claims were categorized into (a) perceptual evidence, (b) computational evidence, and (c) evidence that did not meet any criteria for proof. We also draw on Czocher and Weber’s (J Res Math Educ 51:50–74, 2020) work to draw out ways that pedagogical considerations within IBL and lecture contexts might influence and be able to leverage the aspects of proof that were enacted, including questions about how to make the aspects mathematical reasoning and proof more transparent to students.KeywordsProofClaimCluster conceptTeachingAbstract algebraPerceptual evidenceComputational evidence