Toward a Practical, Intellectually Honest, and Humanizing Conceptualization of Proof

Abstract

Harmful and inaccurate narratives about the nature of mathematics, and about the mathematical incapability of historically marginalized groups, are pervasive and longstanding. The persistence of these narratives, and their documented negative impacts in school, the discipline, and beyond, have led some scholars to conclude that there is something about mathematics itself, and the way that it has been enacted in schools, that upholds and reproduces these narratives – even suggesting that ideologies of oppression, such as whiteness or antiblackness, are intertwined with mathematics. Despite the strength of this criticism, the idea that proof might be implicated in this reproduction of harm has remained largely unexplored. Through an analysis of my own practice in a course for pre-service secondary teachers on mathematical proof, this dissertation investigates and unpacks proof’s role in the reification of these harmful narratives and asks whether it is possible for proof (as it exists in teaching and learning in schools) to be different. At the same time, it also considers what it might take for teachers to disrupt this reification. Specifically, this conceptual-analytic first-person research project considers data drawn from the design and enactment of proof-focused tasks to investigate the following questions: What might a conceptualization of proof that is simultaneously grounded in the discipline, useful for teaching, and aimed at disrupting longstanding and pervasive harms look like – what might be its character, and how might such a conceptualization shape the mathematical interactions between teacher and students? What might teaching that is oriented to conceptualizing proof in this way entail? My analysis thus tests and refines possible alternative conceptualizations of proof, grapples with the integrity and coherence of such conceptualizations, and identifies challenges for teaching with them in mind. In addition to demonstrating how a conceptualization of proof might carry and uphold potentially harmful ideological commitments, I argue that alternative conceptualizations of proof can make possible more humanizing ways of interacting mathematically, while still being useful for teaching, and maintaining disciplinary integrity. In particular, I identify affordances and constraints of a conceptualization of proof that is principally concerned with the sharing of insight and building consensus – ultimately concluding that such a conceptualization seemed to support more informal and idiosyncratic contributions from students as well as encourage more asset-oriented interactions where the collective seemed to value and invest in student insights on their own terms.