Title: “What is a polygon?”: Exploring the Teaching of Mathematical Definitions

Abstract: Traditional approaches to teaching definitions tend to focus on memorization, often resulting in fragile understanding for students. How might we instead teach students to reason about mathematical definitions and author mathematical definitions? In this talk, I share my research focused on examining this question within the domain of geometry. I first problematize traditional approaches to the teaching of definitions, drawing upon research in education and philosophy of mathematics to show how traditional approaches limit opportunities for students to engage in ways of thinking that are important to the discipline of mathematics. Based on results from classroom studies with elementary school students and university prospective elementary teachers, I argue for an alternate approach to the teaching of mathematical definitions - one focused on definitional reasoning. When students engage in definitional reasoning, they make sense of definitions, their properties, and examples and non-examples. Through this research, my collaborators and I have developed teaching strategies that can be used within elementary, secondary, or university mathematics classrooms. Through these approaches, we aim to empower students to develop deeper understanding of concepts, greater mathematical agency, and a love of mathematics.