Features and purposes of mathematical proofs in the view of novice students: observations from proof validation and evaluation performances

Abstract

This thesis describes a comprehensive exploratory study of the approaches taken by novice students to the validation and evaluation of mathematical proofs. A theoretical framework based on sociocultural learning theories was considered suitable as a basis to develop a new terminology and schema for observations and interpretations of proof validations and evaluations. In this theoretical framework learning is seen as accessing and participating in the practice of an expert community. The theoretical considerations of this thesis build on Hemmi's conception of proofs as artifacts in the community of practice. Philosophical theories about the evaluation of artifacts are specialized to the case of mathematical proof. A result of these considerations is a schema that extends Hemmi's model of proofs as artifacts, and provides both a theoretical basis and an analytic tool for consideration of the practice of proof evaluation and for interpretation of specific instances of proof evaluation. The study is based on a series of tests and interviews with first year honours mathematics students at the National University of Ireland, Galway. The students were asked to evaluate and criticize numerous proposed (correct and incorrect) proofs of mathematical statements. The participants' written comments and the interview discussions on different and partly incorrect proofs give insights into their criteria for valuing a mathematical proof, their habits when performing proof validations and evaluations and their knowledge about features and purposes of mathematical proofs. This thesis describes the theoretical framework as well as the design, observations and findings of the written and oral exercises. It also includes a discussion about advantages and shortcomings of the schema that has been developed and used for the interpretation of evaluations of mathematical proofs and a discussion about possibilities for further research.