Meeting the Challenge of High School Calculus, III: Foundations

*David M.
Bressoud, *May, 2010

Mathematics 0,1,2 becomes Calculus A,B,CThe CUPM vision for the calculus sequence greatly influenced the subsequent structure of AP Calculus. CUPM described Mathematics 1 as a general overview of both differential and integral calculus, focusing on the problems of determining areas and tangents, and including volumes of solids of revolution as an application of the definite integral.

The second semester, Mathematics 2, was to return to the topics of Mathematics 1, developing them in greater depth. Here is where the students would see a proof of the general chain rule, treatment of implicit and inverse functions, and more sophisticated techniques of integration. It would include the calculus of curves (tangents, arc length, curvature), and separable differential equations. The third semester would introduce multivariable calculus.

They also laid out an alternate track for the second and third semesters: a second semester covering multivariable calculus, followed by a third semester that was an introduction to real analysis including rigorous treatment of limits and continuity as well as sequences and series.

When the AP Calculus Development Committee considered how to split the existing course, they largely followed the CUPM model. The basic course would include the precalculus material from Mathematics 0 and assume that the students had mastered the general overview of calculus of Mathematics 1. The more advanced course would cover both Mathematics 1 and Mathematics 2. They may have briefly entertained the notion of calling these courses Calculus 0-1 and Calculus 1-2, but recognizing that the numbers might get confused with AP scores, they instead settled on Calculus AB and Calculus BC.Echoes of the CUPM structure remain in the AP Calculus curriculum. While most college calculus spends the first semester on differential calculus with only a brief introduction to integration, the AP Calculus AB curriculum includes much more integral calculus, especially in terms of applications such as volumes of solids of revolution. The retention of this broad overview of both differential and integral calculus has been intentional. The common current college curriculum teaches the first semester under the assumption that students will continue to the second. The AP program makes no assumption that Calculus AB students will ever see calculus again, and thus seeks to give a more comprehensive view of what calculus entails.

The Calculus AB curriculum also suffers from the fact that, if the course is to be widely accepted by colleges, it must include every topic that is commonly covered in the first semester of college calculus. In practice, the College Board periodically surveys the 300 departments that receive the most AP Calculus scores to determine the topics they cover in each semester of calculus. Any topic that is considered important by a substantial percentage of the departments is one that should be in the AB syllabus. The result is a very extensive syllabus for AP Calculus AB.

The result is tension of depth versus breadth. For well-prepared students with teachers who know how to teach AP Calculus effectively, this is not a problem. These students enter college with the depth of understanding required for college work and a breadth of knowledge of calculus that goes well beyond what their peers who studied Calculus I at that college have learned. But there are many high school classes where the focus turns entirely toward mastering enough techniques that the students can perform adequately on the AP Calculus exam. These students enter college knowing both more and less than their peers.

The fact is that many college students survive their Calculus I experience by memorizing technique rather than achieving understanding. No student I have ever encountered really wants to do it this way, but it is a common survival technique when the pace of the course becomes overwhelming. But neither the college nor the high school students who have earned Calculus I credit by relying on memorization of technique are truly prepared to build on that knowledge to succeed beyond Calculus I.

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David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota, and President of the MAA. You can reach him at bressoud@macalester.edu. This column does not reflect an official position of the MAA.

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