On the different ways that mathematicians use diagrams in proof construction

Abstract

The processes by which individuals can construct proofs based on visual arguments are poorly understood. We investigated this issue by presenting eight mathematicians with a task that invited the construction of a diagram, and examined how they used this diagram to produce a formal proof. The main findings were that participants varied in the extent of their diagram usage, it was not trivial for participants to translate an intuitive argument into a formal proof, and participants’ reasons for using diagrams included noticing mathematical properties, verifying logical deductions, representing ideas or assertions, and suggesting proof approaches.

Keywords:

• mathematicians
• diagrams
• proof construction

Acknowledgements

We are especially grateful to the mathematicians who participated in this study, and who so graciously allowed us to benefit from the depth of their mathematical experience – we found the thoughts of each mathematician fascinating and insightful. We appreciate their generosity in opening their work and reflections on the study tasks to analysis. We thank the editor and the anonymous reviewers for helpful comments. This work was supported by grants from the National Science Foundation (#EHR-1008317, #DUE-081736, and #DRL-0643734).