Friday, November 1, 2013

Limits to Growth

Priscilla Bremser—Middlebury College

For your next mathematical modeling project, download “AP Program Size and Increments” from Using the number of Advanced Placement exams given annually from 1989 (463,644 exams) to 2013 (3,938,100), develop a model that describes the growth of the program. In your analysis, discuss possible adverse consequences of such growth. I can suggest one or two.

The College Board tells students that AP courses will help them “stand out in college admissions.” Guidance counselors, along with college admissions officers, advise students to take the most challenging courses at their schools. Dutifully heeding this advice, high school students rush through the mathematics sequence to get to calculus, often taking as many as six other AP courses before they graduate.

At the end of this frenzy, a number of bright, hardworking students have weak algebra skills, effectively neutralizing any advantage they might have earned. They may have placed out of Calculus I, but they are only marginally prepared for Calculus II.

Over time the AP program has shifted from being a way to meet the needs of a few students who are ready for a challenge to a de facto admissions requirement for many who may not be. Having used their AP credit to get into Middlebury, a number of our students try to take calculus again, saying “I know I got a 5 on the exam, but I didn’t really understand it.” If placement into advanced college classes is truly the main objective, then something is amiss.

Breadth over Depth

Mathematics majors have told me that they didn’t see an ε or δ until junior year of college. Their AP Calculus courses did not include the precise definition of a limit, upon which calculus stands. The College Board’s course description calls only for “an intuitive understanding of the limiting process,” followed by a list of topics so exhaustive that I’ve never seen a single college course cover them all. Apparently “rigor” and “challenge” lie in breadth, not depth.

Students in high school AP programs who love mathematics may end up with a weak conceptual understanding of their favorite subject. Meanwhile, students better suited to a different math course feel compelled to take AP Calculus to enhance their transcripts. Once they all get to college, their math professors have some students who earned 5s on the exam as well as others who scored 3 or lower (58 percent of those who took the AB exam in 2013). At a conference I heard one mathematician say to another, “We’re trying to figure out how to deal with students who have taken the AP.” Join the club.

Figure the Expenses

Why has this happened? At $89 per exam, some grumble that it’s all about money. Defenders of the program would point out that the College Board is “a not-for-profit membership organization.” Still, nonprofits exist to perpetuate themselves and seem to be taking a grow-or-die approach. The majority of students who took the Human Geography AP exam in 2013— 67,070 of them—were in ninth grade. Are that many 14-yearolds truly mature enough to take a college-level course?

There’s no going back to the time when the AP program was simply a way for well-prepared students to get advanced placement. Indeed, at my own institution, the faculty voted down a proposal to do away with giving course credit for high AP scores, choosing instead to limit each student to five such credits. Meanwhile, the College Board advertises the program as a way to “save on college expenses.” College may be too expensive, but this purported remedy blithely disregards the significant differences between high school and college.

For extra credit on the modeling assignment, use demographic data to estimate the carrying capacity of this system. What will growth rates look like in the coming years? At what costs?

Priscilla Bremser is a professor of mathematics at Middlebury College. Her interests include number theory, mathematics education at all levels, and appreciating the Vermont landscape on foot, bicycle, and skis.

This article was published in the November 2013 issue of Math Horizons.