Meeting the Challenge of High School Calculus, IV: Recent History

David M. Bressoud, June, 2010

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The 2002 NRC Study

In 2002, the National Academy of Science published the report of its National Research Council on the advanced study of mathematics and science in high school. [1] They did a very good job of identifying both the strengths and weaknesses of the AP Calculus program. They commended the program for the changes of the past decade:

“The AP examination has improved under the new syllabus. The effort to promote conceptual understanding by asking non-standard questions and requiring verbal explanations is excellent. The fact that there is now a wider variety of applications of integration (and not from a prescribed list) encourages students to think about the meaning of an integral. The inclusion of graphing problems involving a parameter focuses attention on the behavior of a family of functions. The variety of representations of a function—by a graph and a table as well as by a formula—promotes a better understanding of the concept of function.” [1]

However, they also felt that much more could be done to probe conceptual understanding, and they deplored the fact that the course is so driven by the examination. They pointed out that the course pays slight attention to proof, that “the real world problems are limited to the most manageable, both conceptually and technically,” and that material that is not on the AP tests tends not to be taught either in calculus or in the prerequisite courses. [1]

The problems that the NRC study identified are real and integrally part of the nature of AP Calculus. At 195 minutes, the examination is both comprehensive and probing, but it is also exhausting for the students. It consists of 55 minutes for 28 multiple choice question for which calculators are not allowed, 50 minutes for 17 multiple choice questions for which calculators are allowed, 45 minutes for three free response questions (each with multiple subquestions) for which calculators are allowed, and 45 minutes for three free response questions (again with multiple subquestions) for which calculators are not allowed. While the test development committee makes a point of including unfamiliar questions that probe conceptual understanding, students do poorly on these under the pressures of such an extensive timed test.

Beyond its examination, the AP Program has very little control over how the course is taught. I have heard more than one anecdotal story of the teacher who spends the entire year drilling students on past tests. Students who report that they were subjected to such a course do fairly well when they take the AP exam, but develop a visceral hatred of calculus.

As the NRC report stated, material that is not tested generally is not taught. In particular, the AP Calculus exam is not a test a knowledge of precalculus. Students with significant gaps in their understanding of precalculus topics can do reasonably well on the AP examination. Certainly, a 3 is within their reach. In fact, Allison Ahlgren of the University of Illinois, Urbana-Champaign has found that over half of the students who earn a 3 on the AP Calculus AB exam fail to pass that university’s placement exam for entry into calculus. A satisfactory AP score does not preclude gaps in the student’s understanding of algebra or trigonometry that may impeded their further progress in mathematics. [2]

Rather than expanding the AP Calculus exam to make it also a test of precalculus, the proper response is a clear set of expectations prior to enrollment in AP calculus and controls to ensure that these students are adequately prepared. There is a real need for a placement test prior to admission to AP Calculus.

The AP Course Audit

In an ideal world, those who teach AP Calculus have a deep understanding of the concepts of calculus on which they can draw as they lead students through this course. In fact, there are many outstanding AP Calculus teachers, and the best calculus instruction taking place in this country is occurring in some of our high schools. However, far too many AP Calculus teachers lack the depth of knowledge and confidence in that knowledge that is needed to teach a course that is truly college-level.

The College Board does a reasonably good job of providing training and educational opportunities for high school calculus teachers. ExxonMobil and the Gates Foundation have put serious money into improving the quality of calculus teachers in urban and rural schools. But the pressures to adopt and expand AP Calculus programs have meant that the supply of qualified teachers does not meet the demand.

For many years, the College Board has been under pressure to install controls on who teaches the course and how it is taught. In a move that angered many as a feint that accomplished little of substance, the College Board introduced the AP Course Audit. This requirement began with the academic year 2007–08. Each teacher who wishes to list a course as AP Calculus must submit the proposed syllabus for auditing. Given the number of sample syllabi that are available both from the College Board and other venues and the lack of any follow-up on what actually is taught or how it is taught, the Course Audit is generally seen as a meaningless exercise.

Admittedly, it is very difficult to put in place any meaningful controls given the size of the AP Calculus Program. That brings us to the biggest change to the AP Calculus Program over the past two decades: the incredible expansion in the number of students who study AP Calculus.

NEXT: The Growth of the Program

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[1] National Research Council, Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools: Report of the Content Panel for Mathematics, 2002, National Academy Press, Washington, DC.

[2] Allison Ahlgren, private communication.

Access pdf files of the CUPM Curriculum Guide 2004 and the Curriculum Foundations Project: Voices of the Partner Disciplines.

Purchase a hard copy of the CUPM Curriculum Guide 2004 or the Curriculum Foundations Project: Voices of the Partner Disciplines.

Find links to course-specific software resources in the CUPM Illustrative Resources.

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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 This column does not reflect an official position of the MAA.



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