# Introductory Calculus: Through the Lenses of Covariation and Approximation

## Files

Appendix A (619 KB)

Spring 5-10-2020

Portfolio

## Degree

Master of Arts (MA)

Mathematics

## School or Department

Mathematical Sciences

## Abstract

Over the course of a year, I investigated reformative approaches to the teaching of calculus. My research revealed the substantial findings of two educators, Michael Oehrtman and Pat Thompson, and inspired me to design a course based upon two key ideas, covariation and approximation metaphors. Over a period of six weeks, I taught a course tailored around these ideas and documented student responses to both classroom activities and quizzes. Responses were organized intonarratives, covariation, rates of change, limits, and delta notation. Covariation with respect to rates of change was found to be incredibly complex, and students would often see it as a series of steps rather than a simultaneous occurrence. With regards to rates of change, students went from seeing the average rate of change as some mean of variation to a change in y divided by the change in x within some acceptable error bound. Limits were a new concept to students, and they ended the course with an understanding of limits as finding an approximation for some value within an acceptable bound. Similar to limits, delta notation was also new to the students. Although it helped students better articulate their thoughts, the context in which students used it to describe change was oftentimes not mathematically rigorous. Besides these four narratives, evidence was also shown that students may gain deeper insights from problems based outside of the traditional physics context, such as velocity. These findings resulted in a list of suggestions of how the course might be implemented in the future so as to better ensure that students have a deeper conceptual understanding of derivatives.

## Keywords

Rates of Change, Covariation, Calculus, math ed

## Subject Categories

Educational Methods | Other Mathematics

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