David M. Bressoud, July, 2010
In my column of June 2009 , I looked at what we do know about those who study calculus while in high school. I will not repeat that information here, but I will summarize what is known:
- We have a pretty good idea of how many students study calculus of some form in high school each year, about 600,000, roughly twice the number who take the AP Calculus exam. This constitutes 20% of all high school graduates.
- We have much less precise knowledge of what happens to them. We know the number of students who earn 3, 4, or 5 on the AP exam, just over 200,000 in 2009, but not how many of each category are entitled to use their credit, nor how many of these actually do use it. We do not know what happens to those who do not earn or use credit for calculus taken in high school. Very little is known about what happens to students who have gone through a dual enrollment program in calculus.
- Of the students who earn and choose to use college credit in calculus in order to place into a more advanced course, we do know that they generally do well, though even the most recent of the published studies now relies on data that are over ten years old. Recent studies undertaken at the University of California, San Diego and the University of Illinois at Urbana-Champaign  suggest that a 3 on the AB exam is problematic for advanced placement, but that students with a 4 or higher on the AB exam or a 3 or higher on the BC exam are at least as well prepared for mainstream Calculus II as the students who have taken Calculus I at that university.
Two studies that are currently underway should help us understand the effects of the movement of calculus into the high schools. Both studies are sponsored by the REESE program of the NSF Division of Research on Learning in Formal and Informal Settings.
- The Factors Influencing College Success in Mathematics (FICS-Math) study out of Harvard will identify factors in the high school preparation, including the taking of calculus while in high school, that influence success in Calculus I. This study should also help to identify the relative benefits of AP Calculus and other forms of calculus instruction.This study is currently analyzing data collected in fall, 2009.
- The MAA’s study of Characteristics of Successful Programs in College Calculus will provide national data on the number of students enrolled in Calculus I who have passed through each of the different types of high school calculus, as well as further information on the effect of each type of program on success in Calculus I. This study will be collecting its data via a national survey to be conducted in fall, 2010.
But that still leaves many unanswered questions. In particular:
- What do high school students really need to know before they begin calculus in high school, not just to be able to pass high school calculus but to enter college with the foundation needed for success in mathematics beyond calculus?
- What is the common high school experience of calculus and how well does this prepare students for college-level work in mathematics? I know that many of the very best calculus classes taught in the United States today are taught in some of our high schools. I also know that some of the very worst are taught there. AP Calculus AB has a far broader syllabus than any single semester of college calculus that I know of. On the other hand, it is taught at a much slower pace than any college class because it is spread over an entire academic year. It is presented in small bites, often of less than 45 minutes, that are frequently interrupted by announcements and school activities. And the course is dominated by the AP exam. I have talked with many high school teachers who have expressed their frustration that their time in the calculus classroom needs to be spent on preparing students to answer certain standard types of questions quickly and accurately, and they feel that they do not have the time to develop understanding. What is the effect on student aspirations of their experience of high school calculus?
- What happens to students who take calculus in high school but are not prepared to start calculus when they get to college? The most recent national data we have is from the US Department of Education’s National Education Longitudinal Study (NELS:88)  that looked at students in the high school class of 1992. Of those students who completed calculus while in high school, 31% enrolled in precalculus when they got to college. What is the current percentage of students who go through high school calculus but arrive in college with serious deficiencies in their knowledge of precalculus, and what happens to them when they get to college? How does this affect motivation to pursue a career in science or engineering, and what strategies are most effective in addressing student deficiencies while continuing to move them toward careers in science or engineering?
- What fraction of the students who earn a 3 or higher on an AP Calculus exam use it to exempt out of any mathematics at the college level? From 1979 to 2009, the number of students entering college each year who received one term of college credit for calculus studied while in high school (AP, IB, dual enrollment, or enrollment in a local college) grew from less than 15,000 to over 150,000. Yet over this same period, the number of students enrolling in Calculus II in the fall in 4-year undergraduate programs actually fell from 100,000 to 85,000. It is not unreasonable to expect many students to use a 3 or higher on an AP Calculus exam to avoid taking mathematics in college, but I have heard anecdotal stories that the movement of calculus into the high school has increased the fear of college-level mathematics among entering students. Is there any validity to this explanation? And what has happened to the students who need to begin with a semester of precalculus before entering Calculus I? There seem to be very few of them taking Calculus II in the fall. Are they still making it into STEM careers?
- What is the success rate in calculus for those students who see it for the first time when they enter college? How many of these students make it successfully through a full year of calculus? There appears to be an assumption among entering college students that calculus is impossible to master unless one has studied it before in high school. Has this assumption become self-fulfilling?
- And, finally, there is the issue of the alternatives to calculus in high school. In particular, what is the effect of statistics as a senior high school course, and does it help or hinder student preparedness for college calculus?
 D. M. Bressoud, AP Calculus: What We Know, Launchings, MAA Online, June, 2009. www.maa.org/columns/launchings/launchings_06_09.html
 Private communications from Alison Ahlgren at UIUC and Bruce Arnold at UCSD.
 National Center for Education Statistics, National Education Longitudinal Study of 1988, nces.ed.gov/surveys/nels88/
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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 firstname.lastname@example.org. This column does not reflect an official position of the MAA.