Mathematicians’ perspectives on their pedagogical practice with respect to proof

Abstract

In this article, nine mathematicians were interviewed about their why and how they presented proofs in their advanced mathematics courses. Key findings include that: (1) the participants in this study presented proofs not to convince students that theorems were true but for reasons such as conveying understanding and illustrating methods, (2) participants believed students did not appreciate the complex processes involved in reading a proof but often did not teach these processes to students, (3) the participants used superficial methods to assess students’ understanding of a proof and (4) some participants questioned whether proof was the best way to convey mathematics to all of their students.

Keywords:

• mathematicians’ pedagogy
• proof
• proof presentation
• undergraduate mathematics education

Acknowledgements

This work was supported by a grant from the National Science Foundation (DRL-0643734). The views expressed in this paper are not necessarily those of the National Science Foundation. I would like to thank Eric Knuth for useful comments about the design of this study as well as Tim Fukawa-Connelly and Evan Fuller for useful feedback on earlier versions of this manuscript.

Notes

Notes

1. An individual holds an axiomatic proof scheme ‘when a person understands that at least in principle that a mathematical justification must have started with undefined terms or axioms’ 20 (p. 273).

2. Generic proofs usually would not qualify as formal mathematical proofs.

3. These comments were made in relation to the proof that bounded monotonic sequences converge or when speaking about proof presentation in general. The proof that √2 is irrational does not lend itself to example presentation.

4. There is an important difference between these mathematicians’ comments and Rowland's recommendations. These mathematicians would use generic proofs to accompany a formal demonstration, whereas Rowland would precede the formal presentation with a generic proof, or use the generic proof in lieu of a formal demonstration.

5. M1 did note that with hard work, he had developed the proving capabilities of ‘tone deaf’ students in the past, but not to the point that these students were fluent with reading or writing proofs.