## Layers of Abstraction: Theory and Design for the Instruction of Limit Concepts

### Abstract

Imagine asking a first-semester calculus student to explain the definition of thederivative using the epsilon-delta definition of a limit. Given the difficulty of each of these concepts for students in such a course, you might not be surprised at the array of confused responses generated by a question requiring understanding of both. Since the central ideas in calculus are defined in terms of limits, research on students' understanding of limits and the ways in which they can develop more powerful ways of reasoning about them has significant implications for instructional design. Throughout this paper we will focus on calculus courses intended as an appropriate introduction for

students who have never seen limits or derivatives and that is not intended to be a

rigorous treatment of analysis.