Mathematicians’ Perspectives on Features of a Good Pedagogical Proof

Abstract

In this article, we report two studies investigating what mathematicians value in a pedagogical proof. Study 1 is a qualitative study of how eight mathematicians revised two proofs that would be presented in a course for mathematics majors. These mathematicians thought that introductory and concluding sentences should be included in the proofs, main ideas should be formatted to emphasize their importance, and extraneous or redundant information should be removed to avoid distracting or confusing the reader. Study 2 is a quantitative study assessing the extent to which a larger group of mathematicians (N = 110) agreed or disagreed with the eight mathematicians interviewed in Study 1. This quantitative study confirmed the findings of Study 1 by demonstrating a high degree of agreement among mathematicians regarding how they would revise proofs for pedagogical purposes.

ACKNOWLEDGMENTS

We are grateful to the editor and the anonymous reviewers for helpful comments, to Aron Samkoff for transcribing interviews in the first study, and to the mathematicians who participated in the two studies. This work was supported by grants from the National Science Foundation (#EHR-1008317, #DUE-081736, and #DRL-0643734) and by an Investigating Student Learning Grant from the Center for Research on Learning and Teaching at the University of Michigan.

Notes

For purposes of brevity, we did not discuss “introducing a variable for the slope of a line,” “adding a word for emphasis,” or “adding a picture” for added revisions or “renaming a variable” for altered revisions. Mathematicians’ use of pictures in Study 1 is discussed in Samkoff, Lai, and Weber (2012).

This situation differs from elementary mathematics in which there are frameworks that researchers use to analyze the quality of instruction (e.g., Hill et al., 2008).