TY - JOUR

T1 - Investigating insight and rigour as separate constructs in mathematical proof

AU - Sangwin, Chris

AU - Kinnear, George

PY - 2024/7/5

Y1 - 2024/7/5

N2 - 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.

AB - 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.

UR - https://doi.org/10.35542/osf.io/egks4

U2 - 10.35542/osf.io/egks4

DO - 10.35542/osf.io/egks4

M3 - Article

SN - 1479-4802

JO - Research in Mathematics Education

JF - Research in Mathematics Education

ER -