Amid the many signs of underperformance in American education, the growth and geographic spread of Advanced Placement courses in high schools is a widely admired success story. For several decades, AP calculus has been a standard-bearer for this movement, representing a singular mark of academic accomplishment for the ablest students, their teachers, and their schools.
Unfortunately, for the majority of those students, AP calculus has become not a steppingstone but a stumbling block. Each year hundreds of thousands of our best first-year college students are precluded from careers in science, technology, engineering, and mathematics because they fail to advance beyond AP calculus. We urgently need to fix the problems with how AP calculus fits into the college curriculum.
In 1965, the Mathematical Association of America’s Committee on the Undergraduate Program in Mathematics published “A General Curriculum in Mathematics for Colleges,” establishing what would become the standard undergraduate curriculum. The committee noted how well its curriculum fit with the AP curriculum. It observed, however, that students who took calculus in high school were still a tiny minority. Nine thousand students took the AP calculus exam that spring, of the 1.4 million who would start college that fall.
Of the 2.7 million students who matriculated in the fall term of 2008, 300,000 had taken the AP calculus exam, and at least 200,000 more either took the course but chose not to take the exam or participated in another high-school calculus program. Those students—almost one in five of those entering college—constitute the elephant that too many colleges and universities continue to ignore. While some issues must be dealt with in high schools, we in higher education have a responsibility to meet the needs of those students.
The clearest evidence that we have a serious problem comes from the number of students taking Calculus II, the second term of college calculus, in the fall. In 1990, about 50,000 students earned scores of 3 or higher on the AP calculus exam, enabling most of them to begin college mathematics with Calculus II or higher. That fall, according to the Conference Board of the Mathematical Sciences, 110,000 students enrolled in Calculus II. In 2005, 160,000 students earned scores of 3 or higher on the AP calculus exam. That fall, total enrollment in Calculus II was down to 104,000. That drop is even worse than it appears; tens of thousands of students who did not take AP calculus instead earned college credit for calculus via the International Baccalaureate program or programs that allow high-school students to also enroll in college courses. Those students, too, could have taken Calculus II but did not.
The decline in enrollment in Calculus II is also more significant than the numbers suggest. The Conference Board of the Mathematical Sciences breaks down its data by the type of institution. Between 1995 and 2005, fall-term Calculus II enrollments dropped by 20 percent at two-year colleges, undergraduate colleges, and comprehensive universities. At research universities, a 29-percent increase kept the overall total from falling too far. But fall-term enrollments in Calculus II at research universities are almost perfectly correlated with the number of incoming students who intend to major in engineering, which has increased by 31 percent from 1995 to 2005.
We have very little evidence of what has caused this decline or the general decline in enrollments in mathematics at the level of calculus and above. Professors point to what they consider to be poor mathematics instruction in the high schools. High-school teachers point to outdated pedagogical practices in many colleges, especially professors’ reliance on lectures and failure to engage students in active learning.
There is some indication that the pressure to offer calculus in high school is pushing underprepared students prematurely into the subject. The National Education Longitudinal Study of the high-school class of 1992 reported that of every 10 students who studied calculus in high school, three had to take precalculus when they got to college. Three others did not continue with any calculus in college. No more-recent data are available, but the phenomenal growth of students taking calculus in high school suggests that the problem has not abated.
An even bigger problem is the imperfect alignment of AP-calculus course content with the college curriculum. The Committee on the Undergraduate Program in Mathematics originally envisioned Calculus I as an overview, covering most of the common techniques and applications of both differentiation and integration. Calculus II would revisit those ideas in greater depth. Calculus AB, the AP-calculus program designed to correspond to Calculus I, still provides such an overview. But the way calculus is now taught at most colleges, most of the techniques and applications of integration are postponed until the second semester. Calculus I is now a more theoretical and sophisticated treatment of differential calculus. For the best students in the best schools—perhaps a third of those who study calculus in high school—that is not a problem. They are able to follow the expanded syllabus at the level required to be ready, even overprepared, for Calculus II.
But the majority of high-school calculus students earn a satisfactory grade by focusing on algorithms and procedures rather than understanding. Given the breadth of the material to be mastered, that is a common coping mechanism. Many students who retake Calculus I in college think they already know the material, but then get slammed midsemester when the level of sophistication required turns out to be higher than expected. Few of those students recover to complete the course or continue studies in mathematics.
Three things need to happen if we are to meet the challenges created by the explosive growth of high-school calculus:
1. Get more information about what happens to students who study calculus in high school. How many of them are deemed unready for calculus when they get to college? How many retake Calculus I, and how successful are they? How many never take another calculus course? What are the factors that affect their decisions? How important is it to future mathematical success to study calculus while in high school?
2. Establish and enforce guidelines for high-school programs offering calculus. There is nothing inherently wrong with the growth of AP or other high-school calculus courses. In fact, now that those programs have become the norm, students at schools that do not offer calculus may be disadvantaged. But calculus offered in high school must be designed to facilitate success in college mathematics for all students who take it, rather than creating obstacles for all but the very best.
3. Re-examine first-year college mathematics. Colleges and universities must ensure that there is an appropriate next course for students who studied calculus in high school but are not yet ready for college-level calculus—a course that acknowledges and builds on what they have learned while preparing them for further mathematics.
The time has come for colleges to stop assuming that Calculus I and II constitute two halves of a single course. We need to return to the original vision of Calculus I as a general overview of the themes and tools of calculus. Most of the students who take Calculus I never get to Calculus II. The traditional college Calculus I course serves those students very poorly, giving little indication of why calculus is important or what one might do with it outside the classroom. Moreover, if we try to explain calculus at a high level of sophistication before students understand its fundamental concepts and what it can accomplish, then we open the door to misleading perceptions and unnecessary frustration.
Some students take calculus in high school with the intention of avoiding mathematics in college. Departments of mathematics need courses that entice, engage, and encourage those students. At most colleges, students are expected to pass calculus before moving on to higher mathematics. But there is no reason why statistics, linear algebra, geometry, or discrete mathematics cannot be used instead of calculus as a bridge to higher-level mathematics. Eventually, all mathematics majors will need to continue with calculus and the courses that build directly upon it. But because it has become a stumbling block, we need more options that allow students to pursue challenging mathematics while postponing calculus.
If we want to facilitate a smooth transition from high-school calculus into college mathematics, then we must get more information about the true difficulties students face, ensure that students are ready before they begin calculus, and rethink the college mathematics curriculum. We need engaging, intellectually satisfying courses that will inspire students to continue their study of mathematics.
