Thoughts on Advanced Placement Precalculus
By: David Bressoud @dbressoud
The College Board is now actively exploring the possibility of an AP Precalculus course. I have agreed to be part of the advisory board for this exploration and am sharing my thoughts and concerns in this article. The College Board welcomes “academic conversations” about this possibility. Those who would like to participate in this conversation should contact Jason Van Billiard jvanbilliard@collegeboard.org.
David Bressoud is DeWitt Wallace Professor Emeritus at Macalester College and Director of the Conference Board of the Mathematical Sciences
My initial reaction to an AP Precalculus course was negative. Precalculus is a high school course. The fact that it is taught for credit at many colleges and universities is an indictment of inadequate high school preparation. Nevertheless, I do see two significant advantages to offering Precalculus as an Advanced Placement course: It will take some of the pressure off getting into AP Calculus in high school, especially from students who now skip precalculus in high school so that they can get AP Calculus onto their transcript. And it should help to improve the quality of precalculus as taught in high school since it would bring with it a clear set of expectations for what students will learn as well as the College Board’s efforts to prepare and support teachers for the implementation of this course.
There are two significant considerations that must be faced before introducing such a course. One is practical, the other gets at deeper issues of future directions for mathematics education.
The practical consideration is that AP Precalculus will not receive credit at many colleges and universities. Macalester is not alone in not offering either College Algebra or Precalculus (courses for which the distinction is not always clear). Even where they are taught, many universities do not award credit toward graduation for College Algebra or Precalculus.
Nevertheless, both College Algebra and Precalculus are heavily taught as dual enrollment courses, thus conferring college credit. Over the spring and fall 2015 terms, College Algebra was taught as dual enrollment to 136,000 students (46,000 with 4-year colleges, 90,000 with 2-year colleges) and Precalculus was taught as dual enrollment to 63,000 students (31,000 with 4-year colleges, 32,000 with 2-year colleges). These numbers are from six years ago and are almost certainly much higher today. Dual enrollment suffers from the fact that in many or most cases there is no nationally recognized control over the quality of instruction (see my column from July 2007: “The Dangers of Dual Enrollment”). This can make it difficult or impossible to use this credit when attending an out-of-state college or university. AP credit for Precalculus would be more widely accepted.
Furthermore, even when college credit is not available, AP courses on a transcript do improve the prospect of admission to the college of one’s choice. There should be a market for AP Precalculus.
The deeper issue involves the intent of high school mathematics. I have long railed against the fact that preparation for calculus dominates and drives the high school curriculum, even and especially for those students who have no desire to ever study it. There is so much more to mathematics. As the college pathways programs have shown, many students are much better served by statistics or quantitative reasoning than by college algebra. This suggests that high school students also would be better served by options that explore mathematical modeling or begin to develop their ability to use statistics or data science. Several states are actively building such programs. The states of Washington, Georgia, and Texas are working with the Charles A. Dana Center at the University of Texas, Austin on the Launch Years project. Virginia recently announced the opening steps toward the Virginia Mathematics Pathways Initiative. All of these are intended to give students options that remain rigorous and build mathematical ability while enabling them to pursue the mathematics that is most relevant to them. The introduction of an AP Precalculus course seems a step away from this vision of high school mathematics.
But none of the high school pathways programs intend to eliminate preparation for calculus, including the opportunity to take AP Calculus, as an option. Moreover, for under-resourced schools that today do not offer AP Calculus or Statistics, introduction of AP Precalculus would be a much easier lift than trying to build alternate pathways.
Photograph of a moving SUV, taken at 1/1000th of a second, from Project DIRACC (Developing and Investigating a Rigorous Approach to Conceptual Calculus), developed by Pat Thompson, Mark Ashbrook, and Fabio Milner at Arizona State University. This and next image that zooms in on a window of the car are used to illustrate the relationship between instantaneous and average rate of change, one of the important foundations for calculus.
A Little History
The idea that precalculus is a high school subject and that, ideally, science, math, and engineering majors should begin college mathematics with calculus is a decision forged in the 1950s and early ‘60s. When an AP Mathematics course was first proposed in the early 1950s, there was considerable debate whether or not it should include calculus. At that time, there was still a general expectation that calculus was best left until the sophomore or even junior year of college. I have been told that it was largely the physics community that convinced mathematicians that their students needed to see calculus in the first year. Even so, the early years of AP Mathematics focused largely on sophisticated treatment of precalculus topics.
In the early 1960s, the Committee on the Undergraduate Program in Mathematics (CUPM) of the Mathematical Association of America (MAA) created a curriculum of 13 courses, Math 1 through Math 13, that they encouraged every mathematics department to offer. Math 1 and Math 2 were the first two semesters of single variable calculus. Recognizing that not everyone headed into the sciences or engineering would be prepared to start with calculus, they added a Math 0 with precalculus topics.
Through 1968 there was only one AP Calculus exam, covering the entire year of single variable calculus. Knowing that there were many students who might be able to complete a single semester of college calculus while in high school, but not both semesters, the AB and BC exams were created. Many of the people who created this split had also been on the CUPM committee. The original vision was that one exam would cover the material from Math 0 and Math 1, while the other would cover Math 1 and Math 2. Given that 0, 1, and 2 look like low AP exam scores, 0, 1, and 2 became A, B, and C. The original understanding of Calculus AB was that it would include a component of purely precalculus topics.
Taken at a shutter speed of 1/1000th of a second, it appears that the car is frozen in time. But if you zoom in on the trim around the window (see Section 4.3 of Project DIRACC for a video of the zoom), the streaks reveal that the car was moving. One can even estimate the length of the streaks to approximate the velocity of the car. See my Launchings column from June 2017, “Re-imagining the Calculus Curriculum, II,” for more on Project DIRACC.
While knowledge of and facility with the mathematics of precalculus is presumed for Calculus AB, it has been a very long time since there were any questions on the exam that specifically tested knowledge of precalculus. Major revisions were made to the AP Calculus syllabi and exams in the mid-1990s, and there was a serious effort to rename the exams since the A in Calculus AB referred to precalculus material. However, even then few teachers were aware of what those letters signified, and none of the alternatives met with approval. We are stuck with Calculus AB and Calculus BC. Like the SAT, they are now letters without actual meaning.
In some sense, an AP Precalculus is simply a return to earlier understandings of what the Advanced Placement program might test.
It is worth noting that over the past fifteen years there have been increasing calls for an AP Multivariable Calculus course. Last year almost 50,000 students took the BC exam in their junior year or earlier. Many high schools now offer multivariable calculus as an option for these students, and many of the large universities that attract high performing students struggle to determine how to place these students. The greatest problem with the creation of such a course is that it would further encourage acceleration, especially in those schools with the most advantaged students. If AP Multivariable Calculus were an option, parents of above average socio-economic status would want to ensure that their children had the opportunity to include it on their high school transcript.
What AP Precalculus Should Not Be
There may be a temptation to look at the syllabi for Precalculus as taught for credit in colleges and universities and mimic that. I believe that would be a serious mistake. Access to calculus in college is usually controlled via placement exams. These assess knowledge of and facility with precalculus topics, in line with the often heard refrain that no one fails calculus in college because they do not know calculus, but because they do not know the precalculus.
While a distortion of the true state of things, there is some truth to this. As taught in most colleges and universities, calculus presumes fluency in the use of the basic constructs of precalculus from which calculus is built: ratios, algebra, function notation, and especially transcendental functions, with some reference to the nature of limits. Students who are not fluent in the use of these tools will flounder in the fast-paced and demanding environment of university calculus. Precalculus is there to bring them up to speed.
Generally speaking, it is a terrible course: a collection of facts and procedures, most of which students have seen before but not mastered, now coming at them much faster. There is little cohesiveness to the course. A study quite a few years ago at Arizona State University revealed that a large percentage of the students who successfully navigated this course were so put off by the experience that they failed to enroll in calculus, even though that was the ostensible reason for taking Precalculus (see my column from January 2010, “The Problem of Persistence”).
We now know that students are much better enabled to succeed in calculus if the precalculus topics are presented as needed in parallel with the calculus course, either by adding additional hours to the calculus course, by stretching Calculus I over two terms, or by offering a paired contemporaneous course in precalculus that is closely tied to what students are learning in calculus. College precalculus is not a course worth emulating.
There is another reason why AP Precalculus should not mimic university precalculus. It will be taught in high schools where it might well be the last mathematics course these students take. Precalculus at university level should serve no purpose other than preparing students to succeed in calculus. Precalculus taught in high school certainly should serve this purpose, but it also has a responsibility to provide a capstone experience, leaving students with some sense of the power and potential of mathematics. That is a tall order, but it is the one that this development committee must face if this course is to succeed.
A Vision for AP Precalculus
AP Precalculus needs a theme, a trajectory, a story line that ties the different elements together and provides motivation for the students. The most natural is modeling dynamical systems. While that might seem to more appropriately apply to calculus, the tools of precalculus need to be understood in the context of modeling change. Linear functions express constant rates of change. Quadratic functions have linear rates of change. Exponential functions are about models of change where the rate of change is proportional to the amount that has accumulated. Trigonometric functions model oscillating behavior. Rates of change, and especially average rates of change with some indication of the connection to instantaneous rates of change are important. Accumulation problems can be studied—using software—in ways that prefigure integration.
Precalculus needs to build fluency with algebraic tools as well as transcendental functions. It must develop the ability to use function notation with the sophisticated understanding that calculus will require. But it also must provide an opportunity to model the phenomena of the world around us, challenging students to broaden their understanding of the importance of mathematics.
The members of the AP Precalculus advisory board are in agreement that this course, should it come to fruition, faces a difficult challenge. On the one hand, it needs to be deep enough that it enables students to develop their ability to tackle unfamiliar problems and build mathematical understanding. On the other hand, it must be broad enough to cover all of the topics that colleges will look for in Precalculus. We believe that this is not an impossible task, and we on the advisory board look forward to assisting as it unfolds.
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