Mathematicians’ Assessments of the Explanatory Value of Proofs

Mejía-Ramos, Juan Pablo; Evans, Tanya; Rittberg, Colin Jakob; Inglis, Matthew

doi:10.1007/s10516-021-09545-8

Cited by 8 publications

(9 citation statements)

References 38 publications

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“…Reference [5] is one of the most interesting and enjoyable mathematics books I have read in many years. Some of the proofs from [5] were used by [6] recently as research materials to investigate the nature of mathematical explanations. I wanted to undertake research on rigour and insight in mathematics education but was dissatisfied with the choice of theorem and, to a lesser extent, the selection of proofs used.…”

Section: Letmentioning

confidence: 99%

Sums of the first n odd integers

Sangwin¹

2023

Math. Gaz.

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There is a long tradition of mathematicians valuing collections of proofs, e.g. [1] is a famous recent collection. In the past, professional mathematicians have also written many different proofs of single important theorems, e.g. [2]. Proofs of the Pythagorean Theorem were collected by [3], and [4] discusses mathematical style through proofs of the irrationality of . The recent work of [5] is a discussion of mathematical style via a comparison of 99 different ‘proofs’ the following ‘theorem’.

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Section: Letmentioning

confidence: 99%

Sums of the first n odd integers

Sangwin¹

2023

Math. Gaz.

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“…In this paper, we build on the approach of Mejía-Ramos et al (2021), by using comparative judgement to gather undergraduate mathematics students' opinions on the rigour and insight of a set of proofs of the same theorem. Our main contribution is to extend the approach to investigate potential differences between two constructs-rigour and insight.…”

Section: Research Focusmentioning

confidence: 99%

“…CJ is also becoming a well-established as a research tool (e.g., Bisson et al, 2016;Jones et al, 2019). As noted above, Mejía-Ramos et al (2021) used CJ to determine mathematicians' views of the explanatory value of a collection of proofs. Other recent work has used CJ to investigate students' comprehension of proofs by judging the quality of their written summaries of proofs (Davies et al, 2020), and to investigate mathematicians' and undergraduate students written conceptions of mathematical proof (Davies et al, 2021).…”

Section: Comparative Judgementmentioning

confidence: 99%

“…In more recent work, Mejía-Ramos et al (2021) investigated mathematicians' views of the explanatory value of nine proofs, taken from a collection of different proofs of the same result (Ording, 2019). They used a comparative judgement approach, where the mathematicians made a series of paired comparisons between proofs, each time choosing the one they thought "best explains why the proposition holds".…”

Section: Introductionmentioning

confidence: 99%

“…The number of judges is different in each case, so we also checked the split-halves reliability when choosing an equal number of judges in each case (similar toMejía-Ramos et al, 2021). This gave similar results: 0.890 for rigour and 0.665 for insight.…”

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confidence: 98%

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Investigating insight and rigour as separate constructs in mathematical proof

Sangwin¹,

Kinnear²

2021

Preprint

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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.

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