25 - Preparing for Calculus and Beyond: Some Curriculum Design Issues

Summary

This paper outlines an alternative to the topic-driven design principle that is the basis for most precalculus courses, arguing that the real power of mathematics lies in the methods used to produce results as much as in the results themselves. It describes a fourth-year high school course that adopts this design, with examples and student work.

Introduction

Curriculum design in US precollege mathematics is largely topic driven; a course is defined by the topics it treats. The major criteria for including a topic in any particular course include:

• does it review and deepen important ideas from previous courses?

• is it a prerequisite for likely subsequent courses?

• did it fall through the cracks in earlier grades?

• does it appear on high stakes tests?

As one moves up the grades, the effects of this design principle compound. By the time one reaches the fourth year of high school, we end up with 18-chapter, 800-page compendia of topics that range from trigonometry to data analysis to complex numbers. These monster texts all go under the name “precalculus,” which is therefore defined as everything from trigonometry to data analysis to complex numbers.

Of course, there's much more in these texts than what one needs as preparation for any of the current calculus offerings. Indeed, it's a well-known fact among high school teachers that one can only finish slightly more than half of these chapters in a given year, and yet many students who go on to calculus from such experiences have what they need to get respectable grades.