Investigating insight and rigour as separate constructs in mathematical proof

Abstract

In this paper we investigate undergraduate mathematics students' conceptions of rigour and insight. We conducted comparative judgement experiments in which students were asked to judge different proofs of the same theorem with five separate criteria: rigour, insight, understanding, simplicity and assessment marks. We predicted, and our experiment found, that rigour is a reliable construct. We predicted that insight is also a reliable construct but asking students to judge on the basis of ``which proof gives you more insight into why a theorem is true'' did not result in a reliable judging scale. Our analysis suggests two distinct dimensions: rigour and marks contribute to one factor whereas simplicity and personal understanding relate to a second factor. We suggest three reasons why insight was related almost equally to both factors. First, while comparative judgement was suitable for assessing some aesthetic criteria it may not be suited to investigating students conceptions of insight. Second, students may not have developed an aesthetic sense in which they appreciate insight in ways which are regularly discussed by mathematics educators. Lastly, insight may not be a coherent and well-defined construct after all.

What makes a good educational research summary? A comparative judgement study of mathematics teachers' and mathematics education researchers' views

“…The use of CJ continues to grow in more recent research, for example with students in product design to support their understanding of quality (Ivo et al, 2021) and with mathematicians to investigate their conceptions of the notion of proof (B. Davies et al, 2021). Recently, Sangwin and Kinnear (2021) also used CJ to judge several different aspects of mathematical proof, evaluating whether they were reliable constructs by post‐hoc analysis. On the whole, CJ is considered effective in evaluation of complex, creative and composite work involving skills such as making connections between learned ideas, applying understanding to novel contexts, constructing arguments and demonstrating chains of reasoning (Marshall et al, 2020) which we suggest includes knowledge brokering work such as the research summaries evaluated in this study.…”