Launchings from the CUPM Curriculum Guide:

The Crisis of Calculus

David M. Bressoud April, 2007

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In 2004, I wrote two articles for Focus on the phenomenal growth of high school calculus and its implications for what and how we teach in college: “The Changing Face of Calculus: First-Semester Calculus as High School Course”[1] and “The Changing Face of Calculus: First- and Second-Semester Calculus as College Courses”[2]. The most dramatic illustration of what is happening was the accompanying graph that compared the number of students taking the Advanced Placement© Calculus exams with the number taking first-semester mainstream calculus[3] in the fall term. With results from the latest CBMS survey of college and university mathematics and statistics departments, it is now possible to update that graph.



Enrollments at 2- and 4-year colleges have remained flat, while the number of AP Calculus tests has continued to grow at about 6% per year.[4]

Of course, these are only partial numbers. First-semester calculus is taught in other terms. But at most colleges, well over half of the enrollments in the first calculus class are in the fall term. It is safe to say that total college enrollment in mainstream first-term calculus is under 500,000, probably closer to 400,000. On the other hand, not all or even most students who study calculus in high school take the AP exam. According to the latest NAEP study, 16% of 12th-graders in 2004–05 were taking or had taken a course called Calculus.[5] This implies that about 500,000 high school students studied calculus in 2004–05.[6]

The problem is that relatively few of these students earn recognition for college-level work in calculus. Most colleges give credit beginning at either a 3 or a 4 on the AP Calculus exams. In 2006, 173,000 of the test takers earned 3 or higher, 129,000 earned 4 or higher.[7] Students do earn college credit for calculus in other ways including the International Baccalaureate program, enrollment in local colleges, and dual-enrollment programs (about which I will say much more in a future column), but these numbers are relatively small, a few tens of thousands. The total number of students who entered college in 2006 and actually received college credit for calculus taken while in high school was almost certainly below 200,000, roughly a third of those who had studied calculus.

What is worse, many of the students who arrive in college with a calculus course on their high school transcript are not even considered ready for calculus. College-administered placement exams often direct them to pre-calculus or college algebra. I am unaware of any estimate of the number of such students, but anecdotally it appears to be high.
This constitutes a crisis. Students who choose to study calculus in high school are a reasonable approximation of the top quintile of high school graduates. They are some of our best candidates for demanding careers in science, technology, engineering, and the mathematical sciences. I do not believe that, on the whole, we who teach in the colleges serve any of them very well. Those who have received college credit for the calculus they studied often find themselves in courses that articulate poorly with their high school experience. Those who start college by repeating first-term calculus find themselves re-treading familiar territory, but at a much faster pace, in larger classes, and with an instructor who is unable to give them the individual attention that they experienced when they struggled with these ideas the previous year. Those who are directed back to pre-calculus or college algebra find themselves in a frustrating and humiliating position with regard to mathematics.

The MAA and NCTM are very much aware of this problem. Bernie Madison, as chair of the MAA Committee on Articulation and Placement, has made it one of his top priorities. The recently constituted MAA-NCTM Committee on Mutual Concerns, chaired by Ann Watkins, has chosen as its first item of focus the problems of articulating calculus instruction in the high schools with that in colleges. There are three urgent tasks that we face at this time.

First, we need a much better grasp of the extent and nature of the problem. Of the students who receive college credit for calculus studied in high school, how many never go on to the next mathematics class? How many choose to retake the class they are entitled to skip? How many continue but encounter insurmountable difficulties making the transition to college classes? What are the most successful strategies for these students and how do these choices affect their perception of mathematics and ability to complete the mathematical training needed for their chosen careers? Of the students who retake the course they studied in high school, how many of them now succeed? How many of them, in retrospect, feel that they have made good use of their time? What are the most successful strategies for these students and how do these choices affect their perception of mathematics and ability to complete the mathematical training needed for their chosen careers? How many students take calculus in high school but are deemed inadequately prepared to study calculus when they get to college? What are the most successful strategies for these students and how do these choices affect their perception of mathematics and ability to complete the mathematical training needed for their chosen careers?

Second, we need to communicate our findings to high school students, their parents, their teachers and counselors. We need a clear set of expectations of the preparation needed before beginning the study of calculus. We need collaborative ventures within the mathematical community, embracing both high school and college teachers, that articulate and disseminate these expectations and assist K-12 mathematics teachers in preparing students to meet these expectations. Several national organizations have been working on such sets of expectations, especially NCTM[8], The College Board[9], and Achieve [10] (an organization that works with the National Governor’s Association). The MAA’s PMET (Preparing Mathematicians to Educate Teachers) has begun this work, and the MAA special interest group SIGMAA TAHSM (Teaching Advanced High School Mathematics) is dedicated to fostering the dialogue between college and high school teachers, but there is much left to do.

Third, we need to use these findings to re-evaluate the way we teach calculus in college. Most colleges still teach a calculus sequence predicated on the assumption that students taking the first course in the sequence have never seen calculus before, and students in subsequent courses have come through the previous courses at that college. The norm today is that most students taking first-term calculus have studied calculus in high school, and most of them have no intention of continuing the study of mathematics beyond that course. The norm in most colleges is that a significant number, often a majority, of the students taking the second course in calculus has earned credit for calculus learned in high school. This is their first mathematics course in college. Once we know what is working and what is not, where we are failing to meet the needs of our students, then we can revise our curricula to meet these needs.

The landscape has changed dramatically in just the past few years. There is a juggernaut pushing calculus into the high schools, a massive and inexorable force driven by competition to get into college, to get into the best colleges, to get into the best colleges with generous financial aid. Whether or not we believe that calculus should be studied in high school, we cannot afford to ignore the fact that it has moved into the high school curriculum and is increasingly seen as a prerequisite for college. When and how we teach calculus carries implications all the way back to pre-Kindergarten and all the way forward into graduate school. We need an accurate assessment of the strains that the movement of calculus into the high school curriculum is putting on our educational system and an identification of the points of fracture. Armed with such an assessment, we will be prepared for the intensive and sustained effort that will be required as we address the articulation of high school to college mathematics and its repercussions throughout the entire mathematics curriculum.

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[1] Bressoud, David M., The Changing Face of Calculus: First-Semester Calculus as High School Course, Focus, vol. 24 (2004), issue 6, pp. 6–8.

[2] Bressoud, David M., The Changing Face of Calculus: First- and Second-Semester Calculus as College Courses, Focus, vol. 24 (2004), issue 8, pp. 14–16.

[3] A calculus course is mainstream if it leads to upper division courses in the mathematical sciences. In Fall, 2005 enrollment in non-mainstream Calculus I was 108,000.

[4] College enrollments from Lutzer, David J., James W. Maxwell, and Stephen B. Rodi, Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States, Fall 2000 CBMS Survey, American Mathematical Society and CBMS Survey, Fall 2005, preliminary table A.1 at www.math.wm.edu/~lutzer/cbms2005/Appendix1PdfSept5/05Appendix1Table%20A.1page2.pdf. AP enrollments from AP Program Summary Report, latest at apcentral.collegeboard.com/apc/public/repository/ap06_prog_summary_rpt.pdf

[5] National Assessment of Educational Progress, 12-th Grade Reading and Mathematics 2005, National Center for Education Statistics, Department of Education, NCES 2007-468.

[6] Based on 16% of 3,089,000 high school graduates in 2005 (NCES data for number of high school graduates).

[7] AP Central, 2006 Calculus AB & BC Grade Distributions at apcentral.collegeboard.com/apc/members/exam/exam_questions/151231.html

[8] While not a set of standards, the recent NCTM Focal Points for Prekindergarten through Grade 8 Mathematics does articulate basic expectations for mathematics education for pre-K through grade 8. See www.nctm.org/focalpoints/

[9] The College Board, Standards for College Success: Mathematics and Statistics, www.collegeboard.com/prod_downloads/about/association/academic/mathematics-statistics_cbscs.pdf

[10] Achieve, Inc., Secondary Mathematics Expectations, to appear.

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David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota, he was one of the writers for the Curriculum Guide, and he currently serves as Chair of the CUPM. He wrote this column with help from his colleagues in CUPM, but it does not reflect an official position of the committee. You can reach him at bressoud@macalester.edu.