Meeting the Challenge of High School Calculus, VI: The Need for Guidelines

David M. Bressoud, August, 2010

In the past year, the number of students taking the AP Calculus exam jumped by over 7%, exceeding 325,000 this past spring. The rush to enroll ever more students into AP Calculus is undiminished, and the need to ensure that the students who take it really are prepared is greater now than ever before. We desperately need a re-affirmation of the 1986 MAA-NCTM statement on high school calculus. But this is only a start. We also require enforced guidelines on what it means to be ready for calculus in high school so that students will not be pushed into a course for which they are not prepared, making it a dead end to their mathematical aspirations.

It is surprising that success in AP Calculus, measured as a 3 or higher on the AB or BC exam, has held up despite the rapidly increasing numbers (see Graphs 1 and 2).

Graph 1: Percentage of AP Calculus Exams earning a 3 or higher. [1]

Graph 2: Number of AP Calculus Exams earning a 3 or higher. [1]

Does this mean that standards have dropped or that high schools are doing a better job of recruiting and preparing students?

There are real and effective measures in place to ensure the continued integrity and validity of the exam. The AP Program’s reputation rests upon them. Having been involved in AP Calculus test construction, I know that many of the multiple choice questions are judiciously chosen from an exam given several years earlier. Performance on these questions is used in setting the cut scores for a 2, 3, 4, or 5 so that a given score on the current exam means what it would have on the earlier exam. This does not preclude a gradual diminishment of expectations over a long period of time since other factors do play a role in determining cut scores. But, from my own experience, I know that some of those factors— including the influence of the AP Calculus Development Committee that writes the exam and the periodic comparability studies that directly compare performance of high school and college students—usually tighten the requirements for passing scores.

The Fordham Foundation published a very informative study in 2009 [2] that asked AP teachers about their perceptions of the program. The study involved all AP subjects. The responses from AP Calculus teachers might be different. Nevertheless, the responses are suggestive. Regarding the statement: “The standards for grading AP exams have not been watered down—for example, a score of 3 means the same thing today as it did five years ago,” 69% agreed, 14% disagreed, and 16% were unsure [2, p. 27].

But even if the exam scores mean the same that they did ten or twenty years ago, there is a pernicious effect from the sheer magnitude of the number of students who fail to earn a passing score. Only about half the students who study calculus in high school take the AP Calculus exam. It is roughly a third of all high school calculus students who earn college credit. I worry about the other two-thirds. In 1989, about 120,00 students studied calculus in high school. The 80,000 who did not earn college credit constituted a manageable population. Even if the standards today are the same as they were in 1989, today we have over 600,000 studying calculus in high school. The 400,000 who do not earn college credit, many of whom have short-changed their pre-calculus preparation to get into high school calculus, are having a serious impact on the science and engineering pipeline when they arrive in college.

In next month’s column, I will look at what colleges and universities are and should be doing about this. But this month I want to focus on what happens in the high schools. The Fordham study supplies a great deal of additional detail.

Teachers attributed the growth in AP programs primarily to student desire to make their college applications look better (90% agreed that this was an important factor), but not far behind was pressure from the school to improve its ranking and reputation (76% agreed that this was important). The only other reason credited by over half of the teachers for the growth in the AP program was student desire to earn AP credits to save money or graduate faster (58% considered this an important factor). Only 14% considered better preparation of their students to be an important factor in the growth of the AP program [2, p.25].

In fact, 39% of the teachers felt that the quality of their AP students had declined, measured in terms of aptitude and capacity to do the work. By comparison, 43% felt it had stayed about the same and 16% believed it had improved [2., p.23]. Sixty percent agreed with the statement that “Many parents push their children into AP classes when they really don’t belong there,” and 56% agreed that “Too many students overestimate their abilities and are in over their heads when they take AP classes” [2, p. 26]

Most significantly, when offered a variety of options for improving the AP program including grouping students by ability, changing the curriculum so that it is less broad but deeper, or limiting class size, by far the most popular was “Conduct more screening of students to ensure that they are ready to do AP-level work before they get in those classrooms.” Sixty-three percent of teachers felt that this would improve the program [2, p. 30]

There is a significant and vocal minority who feel that AP is beneficial even for students who are not prepared for the AP course. Though a majority disagreed, 38% of the surveyed teachers agreed with the statement “The more students taking AP courses the better—even when they do poorly in the course, they benefit from the challenge and experience.” There may be AP subjects for which this is true. It is signally not true for calculus. Because of the strong hierarchical structure of mathematics, students who aspire to technical or scientific careers but who fail to master critical stages in their mathematical development will almost certainly be forced to back up when they get to college and retake courses they thought they had already passed. This can be deeply discouraging and is a poor use of the time and resources of both students and universities.

How might we ensure that students who enroll in AP Calculus are ready for this course? One option is explicitly to restore Mathematics A (precalculus) material to the AP exam. In an exam that already covers such an extensive syllabus, I worry about what that would do to the coverage of the calculus material. It also seems to get at the students at the wrong end of the course.

A better solution is to require students to take a placement exam to qualify for enrollment in the course. Today, this is the norm for access to calculus at most colleges and universities and a few, including the University of Illinois, require it even of their students who have completed their own precalculus course. There are some good placement exams. The MAA, in collaboration with Maple, has produced an online exam, the Maplesoft-MAA Placement Test Suite, that tests both skills and conceptual understanding and that allows repeated tries so that it measures not just what is immediately accessible but gives the student time to bring back more deeply seated knowledge.

Another option is the ALEKS test of precalculus, which the College Board could shape to conform to its needs. This not only allows for multiple attempts but provides online tutoring to bring students up to speed when there are only a few specific topics in which they are deficient. Alison Ahlgren, who runs the ALEKS placement program at the University of Illinois, told me that, if students need it, they can take the entire summer to work on the placement exam. It is sufficiently rigorous that those who eventually pass really are ready for calculus. Those who don’t pass have been given sufficient opportunity to come up to speed that they themselves accept that they are not ready.

In and of itself, there is nothing wrong with the dramatic growth of the AP Calculus program. Getting more students into an experience of college-level mathematics, and especially making this opportunity available to students from underrepresented groups, should be encouraged. But these students must be ready for this experience.

[1] These are taken from the Archived AP Data available at

[2] Ann Duffett and Steve Farkas. 2009. Growing Pains in the Advanced Placement Program: Do Tough Trade-offs Lie Ahead? Thomas B. Fordham Foundation.

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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.