Comparative judgement and proof

Abstract

Proof is a central concept in mathematics, pivotal both to the practice of mathematicians and to students’ education in the discipline. The research community, however, has failed to reach a consensus on how proof should be conceptualised. Moreover, we know little of what mathematicians and students think about proof, and are limited in the tools we use to assess students’ understanding. This thesis introduces comparative judgment to the proof literature via two tasks evaluated by judges performing a series of pairwise comparisons. The Conceptions Task asks for a written explanation of what mathematicians mean by proof. The Summary Task asks for a summary of a given proof, available to respondents as they complete the task. Having established robust evidence supporting the reliability and validity of both tasks, I then use these tasks to develop an understanding of the conceptions of proof held by mathematicians and students. I also generate insights for assessment, leading to an argument for the unidimensionality of proof comprehension in early undergraduate mathematics. In conducting this research I adopt a mixed methods approach based on the philosophy of pragmatism. By using a range of methodological approaches, from statistical modelling to thematic analysis of interviews with judges, I develop a multi-faceted understanding of both the validity of the tasks, and the behaviours and priorities of the participants involved. The Conceptions Task outcomes establish that mathematicians primarily think of proof in terms of argumentation, while students emphasise the arguably more philosophically naive notion of certainty. The Summary Task outcomes establish that references to the method of proof and key mathematical objects are most valued by mathematician judges. Further, from correlational analyses of various quantitative measures, I learn that the Summary Task scores are meaningfully reflective of local proof comprehension but are not related to more general measures of mathematical performance. Several open questions are identified. In particular, there is still much to learn about judges’ decision-making processes in comparative judgment settings, the dimensionality of proof comprehension, and the range of proofs for which the Summary Task is applicable. Future work on these questions is outlined in the final chapter, alongside the practical applications and theoretical implications of this work.

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ORCID for Ben Davies: orcid.org/0000-0003-1134-7205

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