The Mathematical Practice of Learning from Lectures: Preliminary Hypotheses on How Students Learn to Understand Definitions

In this theoretical chapter, we argue that when mathematicians lecture, they not only cover mathematical content, but also model how student should learn mathematics. We analyze a corpus of eleven lectures in a variety of advanced mathematics courses to investigate ways in which mathematicians present the definitions of concepts and gain insight into how mathematicians may expect students to learn from lectures. We highlight how the instructors modeled what it means to study a concept and its definition and argue that students are expected to engage in independent study outside of class.