How Do Mathematicians Make Sense of Definitions?

The purpose of this paper is to share preliminary results from a pilot study on mathematical definitions. Interviews with university mathematicians were designed to gain insight into mathematicians\u27 processes for developing understanding of new definitions. We asked the participants to talk about what helps them understand a new definition and how they support students’ understanding of definitions. We also observed them while they engaged in a definition task. Analysis revealed a noticeable difference in the emphasis on examples between what the participants described that they do and what they actually did while working on the definition task. We hypothesize that mathematicians’ processes for making sense of a definition necessarily involve considering the definition’s usefulness within a particular mathematical setting. Furthermore, these data indicate that mathematicians see examples as a multi-faceted, but not comprehensive, tool for understanding definitions