The role of proof validation in students’ mathematical learning

The study of proofs is a major obstacle in the transition from school mathematics to university mathematics. Given the importance of argumentation and proof in the spectrum of mathematical activities, the incoming students’ understanding, appreciation and knowledge of the nature and role of proof must be considered. I describe the results of an exploratory study of first year mathematics undergraduates’ criteria and learning process when validating mathematical arguments or proofs. The study is based on a series of written tasks and interviews conducted with first year honours mathematics students at NUI Galway. I presented the whole class with numerous proposed proofs of mathematical statements, and asked them to evaluate and criticize those. The first year students’ written comments on different and partly incorrect ‘proofs’ of mathematical statements revealed some information about their criteria when validating mathematical arguments. In recently held interviews with eight randomly chosen students I focussed on the learning experience during the process of proof validation. Considering the observed learning effect and its large potential extent during the process of proof validation I propose its practice in the teaching of mathematics.

In the first part of this article, I explain what is meant by proof validation and consider various aspects of this activity; including how and why mathematicians validate mathematical arguments. I will then describe our experiments and findings, and consider the undergraduates’ validation skills and practices in relation to previously characterized aspects of proof validation. I emphasize the learning effect during the process of proof validation and finally argue for explicit inclusion of its practice in the teaching of mathematics, as the development of validation skills not only improves the practice of validation itself, but also the ability to construct proofs, the understanding of mathematical context, the knowledge of proving strategies and the links between different areas of mathematics…