The Strange Role of Calculus in the United States
By: David Bressoud @dbressoud
David Bressoud is DeWitt Wallace Professor Emeritus at Macalester College and Director of the Conference Board of the Mathematical Sciences
This month’s column is a collection of the main points from a paper I recently published in ZDM Mathematics Education, “The strange role of calculus in the United States,” available here. I have often written about calculus instruction, but the point of this article, written for an international audience, is to clarify how truly dysfunctional we are in the way we treat calculus in the United States. In a nutshell, we consider it to be a college-level course, but teach it to a fifth of all high school students. This is accomplished in a hit-or-miss fashion that privileges those of higher economic status. We then act as if no one taking Calculus I has studied it before they get to college, but present it in a way that disadvantages those who really have not.
After presenting what I consider to be the damning case against our approach to calculus in the United States, I will conclude with some thoughts on where we might go from here.
How we differ
In other countries, it is common to teach at least some calculus to at least some students in high school. In many countries, secondary school preparation for university involves one more year of study than in the U.S. Whether or not there is an extra year, calculus forms part of the curriculum for students on a track for STEM studies in university. The huge difference is that in other countries where calculus is taught as a secondary subject, there is a national curriculum that sets expectations for what is taught and the level of mastery that is expected. Those who teach this curriculum are given the preparation they need to do so. This also means that for those who teach calculus in university, there is a relatively uniform expectation for what students have seen and can do.
In the United States, there usually is no explicit pre-service preparation for teaching calculus. What in-service support is available is largely hit-or-miss, highly dependent on the resources of the individual school district. Individual school resources also play a huge role in who takes calculus while in high school. For schools in wealthy districts where the expectation is that virtually all students will continue on to post-secondary education, algebra by 8th grade and calculus by 12th is the norm. But barely over half of all schools in the United States offer calculus. Many students without access to calculus in their high school can enroll in a local college or university or take the course online, but this requires high motivation and often additional costs. Even then, the National Survey of Science & Mathematics Education published in 2018 found that 17% of high school students do not have access to any means of studying calculus.
Why they take it
One hint into the dysfunctionality of our system is revealed in the study by Rosenstein and Ahluwahlia at Rutgers into student reasons for taking calculus while in high school. They found that students who took an AP Calculus exam and scored 4 or 5 had reasons that were in line with the initial impetus for the AP program: they really liked mathematics, they want to learn more, and they hoped to get advanced placement when they got to university, moving directly into more advanced courses. This represents about a quarter of those who enroll in calculus in high school.
For the others, three reasons dominated:
They intended to restart calculus in college and wanted to improve their chances of a good grade.
They knew that calculus would look good on their high school transcript when they applied to colleges.
They had taken Algebra I at or before eight grade and calculus was what the rest of their cohort was doing.
For most of these students, the rationale is some combination of the three. Dan Kennedy has reminded me of something that Dan Teague at the North Carolina School of Science and Mathematics has said, "Most students don't want to take AP Calculus; they just want to have taken AP Calculus."
Only just over half of the students who study calculus while in high school continue with mainstream calculus when they get to college. It is not at all clear that calculus in their last year of high school is the mathematics course that serves them best.
One syllabus for all
Figure 1:Distribution of intended majors of all male students in mainstream Calculus I.
N=3816. Source: Bressoud, 2017
The experience of those who want to repeat calculus when they get to college points to the dysfunction on the post-secondary side. Unlike most other countries, in the United States almost all of the calculus, regardless of the students’ intended major, is taught by the mathematics department. Although sometimes carrying multiple labels, there is a remarkably uniform syllabus and set of expectations for what is taught as mainstream calculus across the U.S. Teaching the same course to future mathematicians, engineers, and biologists makes sense in our system that values a flexible approach to the choice of major. The heterogeneity of student interests and level of commitment to mathematics is reflected in the variety of intended majors revealed in the first MAA study of college calculus and the finding that men and women, taking the same calculus class, are typically heading toward very different majors: engineering and the physical sciences for men, biology and education for women (Figures 1 and 2).
Figure 2: Distribution of intended majors of all female students in mainstream Calculus I. N=3137.
Source: Bressoud, 2017
But the greatest disparities occur in levels of preparation. The same calculus course frequently contains students who have already demonstrated mastery of the full year of single variable calculus and those who have never seen any of its topics and are struggling to understand its most basic concepts. Ten years ago, it was common for two-thirds of the students in mainstream calculus to have enrolled in this course in high school. Today that must be significantly higher. Even twenty-five years ago, in a survey I conducted at Penn State, there was a general perception that one could not succeed in Calculus I unless it had already been studied in high school. Given the prevalence of grading on a curve and implicit or, sometimes, explicit expectations for the percentage of students who would receive each grade, students are given the impression their grade is less dependent on what they have learned than on how they compare to their peers. Given the vast differences in where students start, it is not surprising that students from under-resourced schools become deeply discouraged and abandon hope of a STEM career.
Our colleges and universities have been very slow in acknowledging and dealing with this problem. Most still utilize a curriculum for Calculus I that assumes that no one has seen calculus before, but they move at a pace and test at a level that implicitly presumes this is all familiar territory. For those who have studied calculus, there is a terrible waste by not building on what they do know. For those who have not, too often the supports that they need are simply not there.
Problems of inertia
The reaction of many university faculty is to rail against any calculus being taught in high school, under the assumption that only they can teach it well. My article presents the evidence that it can be taught well, and, when taught as intended as a course designed to prepare students for advanced placement, it is every bit as effective as the course offered at universities. Moreover, the reasons that I cited for taking this course when advanced placement is not desired are so strong that a few grouchy professors are not going to change anything. For almost forty years MAA and NCTM have promoted the policy that calculus in high school should be limited to those who are fully prepared to study it and are doing so with the intention of seeking advanced placement. The College Board has supported these positions. At best, this may have slowed the expansion of high school calculus to all who can afford it.
This gets to my final point. What has happened to calculus in the United States has exacerbated the inequities of our educational system. In 2019, 12.7% of all White graduating seniors had taken an AP Calculus exam, and two-thirds of them had earned a three or higher. That same year, only 3.7% of Black seniors had taken an AP Calculus exam, and only 40% of them earned a three or higher.
Challenge for the future
Our current system grew out of extensive work on the post-secondary mathematics curriculum that was led by prominent mathematicians and mathematics educators from the early 1950s through the early 1960s. They established uniform expectations for the first years of college mathematics. In the process, this shaped the last years of high school mathematics. Now, sixty years on, we are overdue for a fresh re-evaluation of mathematics instruction across this critical bridge. There is wide agreement that computational and data science need to be part of the mathematical preparation of students facing the demands of the 21st century workplace. But the singular focus on calculus sucks the oxygen out of these efforts. Even when offered, few high school students who are capable of studying calculus will invest the time and energy needed to begin a serious investment into learning computational and data science.
Calculus will always be important. But we are now at an opportune moment to reconsider the mathematical trajectories through grades 11 to 14. This calls for a calculus path that, when it circles back, does so intentionally and in a manner that neither privileges the privileged nor disadvantages the disadvantaged. At the same time, there is an opening to develop a truly co-equal path that develops computational and data skills.
The original envisioning took a full decade. We should expect such a re-envisioning to also be a decade-long undertaking.
References
Bressoud, D. (2017). Introduction, pp 1–8 in D. Bressoud (Ed.), The Role of Calculus in the Transition from High School to College Mathematics. Washington, DC: MAA Press. Retrieved December 26, 2019 from https://www.maa.org/sites/default/files/RoleOfCalc_rev.pdf
Rosenstein, J. G. & Ahluwalia, A. (2017). Putting the brakes on the rush to AP Calculus. pp 27–40 in D. Bressoud (Ed.), The Role of Calculus in the Transition from High School to College Mathematics. Washington, DC: MAA Press. Retrieved December 26, 2019 from https://www.maa.org/sites/default/files/RoleOfCalc_rev.pdf
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