David M. Bressoud, April 2010
Last month, I introduced this series of columns on the problems created by the pressure to get ever more students into calculus while still in high school. In the interests of full disclosure, I need to begin this month’s column by declaring my own connection to the AP Calculus program. From 1999 until 2005, I was a member of the AP Calculus Development Committee, the committee that sets the syllabus and writes the exams for AP Calculus. For the last three years of that tenure, I chaired this committee. I am very aware of the program’s strengths, and also of its weaknesses.
I do believe that the AP Calculus program does a good job in the mission for which it was intended: providing an outlet for talented students who are ready for college work while still in high school and certifying those who are ready for advanced placement into the next mathematics course. As I explained in my column of this past June, AP Calculus: What We Know , there are multiple reliable studies that show that students who do well in the AP Calculus exam are at least as well prepared for Calculus II as students who have taken Calculus I at that institution. These studies were conducted at first-tier universities including Cornell University, University of Texas at Austin, University of Illinois Urbana/Champaign, and Stanford.
If readers question the level of understanding required by the AP Calculus exam, I challenge them to use the AP Calculus questions on their own students. The six free response questions given in each exam are publicly available and can be found at the College Board website . Students have 90 minutes to answer these questions. For three of them, they are permitted to make use of a graphing calculator. For the other three, calculators are not allowed. There also is a 90-minute multiple choice portion of the exam, again half with and half without a calculator. Given the limitations of administering an examination to over 300,000 students, I believe that the AP Calculus exams does a good job of probing student understanding of the principles and techniques of calculus.
My concern is not with the students who do well on the AP Calculus exam, but with those who do not earn or intend to take advantage of advanced placement. They are a large and growing component of our entering freshmen.
Graph 1 shows the growth in the AP Calculus program, from 25,000 exams in 1979 to 305,000 this past spring, plotted against fall term Calculus I enrollments at 2- and at 4-year undergraduate institutions, which have remained essentially unchanged at a combined total of about 250,000.
Graph 1: Data from CBMS and The College Board
From the US Department of Education longitudinal transcript studies, we know that the number of students who take the AP Calculus exam is around 50% of the high school students who enroll in a course entitled “calculus,” putting that number at around 600,000, or one-third of the traditional college-bound high school graduates each year (those who will enter college within a year). My best guess, based on limited studies by Karen Christman Morgan and David Lutzer (see last June’s launchings, ), is that between 150,000 and 200,000 of these students, between a quarter and a third of those who study calculus in high school, are entitled to and choose to take college credit for their calculus course. This includes not just AP credit but also credit earned for International Baccalaureate or dual enrollment programs. While a few tens of thousands of these students are earning and taking advantage of credit for more than a single semester of college calculus, the overwhelming majority of students in this group arrive entitled or intending to begin mathematics with Calculus II.
Given the dramatic growth in AP Calculus, one would expect an equally dramatic increase in the number of students enrolled in Calculus II in the fall term. Graph 2 shows that, if anything, fall Calculus II enrollments have fallen in the past quarter century. No one questions that the make-up of fall Calculus II classes has changed dramatically. They are now dominated by good first-year students who arrive with Calculus I credit rather than sophomores or juniors who have taken a slower route to Calculus II. What appears to have happened is that this latter group has essentially disappeared.
Graph 2: Data from CBMS
But if total college calculus enrollments have not increased, the shift in the make-up of fall Calculus II classes suggests that we are seeing more highly capable students in these classes, students who should be increasing our enrollments down the line. Unfortunately, for all mathematics at the level of calculus and above, enrollments have been trending slowly downward over the past quarter century (Graph 3). Despite the greater numbers of students arriving with advanced placement, we have seen no increase in the number of students continuing on to several variable calculus, linear algebra, differential equations, or advanced mathematics.
Graph 3: Data from CBMS, calculus level includes sophomore courses such as linear algebra and differential equations.
These data suggest that students who do not enroll and do well in high school calculus do not even entertain the thought of pursuing mathematics at the level of calculus and above when they get to college, and that those who do well in high school calculus are no better prepared or motivated than those students of a quarter century ago who began their study of calculus in college.
The situation is actually worse than this. Over this past quarter century, 4-year college enrollments have increased by close to 50%, and the population as a whole has gone up by 25%. The slow downward trend we have experienced in the absolute number of mathematics students constitutes a significant decrease in the percentage of college graduates who are prepared for mathematically intensive careers. This should be of concern to anyone who cares about our workforce preparation for this century.
Comparison of these trends does not constitute evidence of cause and effect, but I find it deeply disturbing that the AP Calculus program, which was created as a vehicle for facilitating the entry of more students into advanced mathematics, appears to be failing in this purpose.
Next month, I will look at the history of the AP Calculus program and the changes it has undergone with the intention of shedding some light on some of the mechanisms that may be behind the great calculus disconnect.
 David Bressoud, AP Calculus: What We Know. Launchings. MAA Online. June, 2009. www.maa.org/columns/launchings/launchings_06_09.html
 College Board, The AP Calculus AB Exam, http://apcentral.collegeboard.com/apc/members/exam/exam_questions/1997.html, The AP Calculus BC Exam, http://apcentral.collegeboard.com/apc/members/exam/exam_questions/8031.html
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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 email@example.com. This column does not reflect an official position of the MAA.