David M. Bressoud, June, 2010
Contents of this article:
The Solow Committee and its Aftermath
The 2002 NRC Study
The AP Course Audit
The Growth of the Program
The AP Calculus program underwent significant changes in the 1990s. They were the result of the confluence of three powerful factors. The first was Calculus Reform and the recognition that it is not sufficient for students to master the manipulative skills of calculus, they also need conceptual understanding. This includes the ability to apply the tools of calculus to functions that are represented either graphically or with a table of values. Students also need to be able to explain what an answer means and to justify why it is correct.
The second development that was forcing change was the growing adoption of graphing calculators and the realization that this could be a useful tool for building conceptual understanding. As the result of studies conducted by Educational Testing Service (ETS) in 1990, it was decided to require scientific calculators for the 1993 and 1994 exams and begin testing with graphing calculators in 1995. In preparation for this transition, John Kenelly of Clemson University, John Harvey of the University of Wisconsin, and Wade Curry of the College Board organized the TICAP (Technology Intensive Calculus for Advanced Placement) workshops that were run at Clemson in the summers of 1993 and 1994. The leaders trained in these workshops then worked with local groups of teachers around the country. While the emphasis was on the use of technology, there was a good deal of discussion about the kinds of questions and explorations that technology enabled. The textbooks that were highlighted—textbooks that made effective use of graphing calculator technology—were all textbooks that had emerged from the Calculus Reform movement.
The third factor was the simple fact that there had been no systematic review of the AP Calculus syllabus since 1969 when the single curriculum and exam had been split into the separate AB and BC exams. Over the ensuing decades, the precalculus topics of the AB syllabus disappeared. In fact, beyond some material on differential equations and series, the syllabi for AB and BC Calculus were not substantially different. What was significantly different was the level of sophistication of the problems that students were asked to solve. This was reflected in the fact that in the early years after the split between AB and BC, almost one in three AP Calculus students chose to take the BC exam, but by 1990, this had dropped to one in six. Students would not take the BC exam unless they were supremely confident they would do well on it.
The Solow Committee and its Aftermath
In 1994, under the direction of Anita Solow of Grinnell College, a forum of College Board and ETS officers and 23 prominent mathematicians and mathematics educators convened to map out the future direction of AP Calculus. The new syllabi were adopted a year later and went into effect with the 1998 exam. The guiding principle in formulating the AB exam was that nothing should be included that is not normally covered or needed in the first semester of mainstream college calculus. Thus, for example, while l’Hospital’s Rule and integration by parts are important components of the BC syllabus, they were taken out of the AB syllabus. These syllabi have been subject to change since then. In 1999, it was decided to include slope fields in the AB syllabus because of their usefulness in introducing integration.They first appeared in the 2004 AB exam.
In addition to revising the syllabi, the Solow committee introduced an important shift in how BC Calculus was to be viewed. In their words, “Common topics [between the AB and BC syllabi] are covered at the same conceptual level, leading to a similar depth of understanding. Calculus BC is an extension of Calculus AB, rather than an enhancement.” This is actually a move that had begun with the 1992 exam for which two of the six free response questions were the same on both the AB and BC exams. In 1999, the AB and BC exams began to share three of the six free response questions. The explicit promise that the BC exam would not ask harder questions, just questions that covered more material, led to a dramatic increase in the percentage of students taking the BC exam, from 17% in 1997 to 19% in 1998. That trend has continued, surpassing 24% in 2009.
The other factor in 1998 that increased the percentage of students taking the BC exam was the AB subscore, allowing students who take the BC exam but do not do well enough to earn BC credit to still demonstrate that they have mastered the AB material well enough to earn a semester’s worth of credit. About 80% of students taking the BC exam earn a grade of 3 or higher. Half of those who fail to earn a grade of at least 3 on the BC portion manage to earn a 3 or higher for the AB subscore. These students who are “saved” by the AB subscore amount to only about 7,000 students out of the 300,000 that now take the AP Calculus exam each year, a small but significant population.
From 1995 through 1999, all free response questions allowed the use of graphing calculators. When computer algebra systems (CAS) on non-qwerty keyboard calculators first appeared, it was decided to allow them and to write free response questions that would not advantage students with access to CAS. This was not done because of a desire to encourage use of CAS, but rather a recognition of the practical difficulty of training exam proctors, many of whom know little mathematics and nothing about graphing calculators, to distinguish between those that do and those that do not possess CAS capabilities. But it soon was realized that it is difficult to write probing and truly CAS-neutral questions for BC topics. Since 1995, the multiple choice questions had always had a calculator portion and a non-calculator portion. In a practice that began in 2000, the last three free response questions also have to be answered without calculators.
But the greatest shift in the AP Calculus program has been in the broadening of the types of questions asked. Beginning in the early1990s, there has been a very deliberate effort to include questions that probe student understanding of calculus concepts not just through symbolic but also via graphical, tabular, and written descriptions of functions. There has been a much broader view of integration, including the need to probe student understanding of the Fundamental Theorem of Calculus. This means both using the definite integral as a means of creating an antiderivative and using antiderivatives to evaluate the limit implicit in the definite integral. The AP examinations now also require interpretations of answers and explanations of what is done and why.
NEXT: The 2002 NRC Study
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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.