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### The sky is not falling (anymore? yet?)

Thursday, March 19, 2009 - 12:25am

A few months ago David Bressoud (incoming president of the MAA) wrote an article for Notices of the AMS called "Is the Sky Still Falling?" It refers to a 1995 Notices article (titled "The Sky is Falling") which worried about a rapid decline in the number of students enrolling in mathematics courses.

Bressoud's article contains good news and bad news. The good news is that enrollment in college math courses is up. One bit of bad news is that enrollment in all college courses is up, too, and the percentage of enrollments which are math courses continues to drop.

What has grown at an even quicker rate is the enrollment in Advanced Placement calculus. Bressoud writes:

In the spring of 1985, 46,000 students took the Advanced Placement Calculus exam. In spring 2008, the number was 292,000. By 2009, it will be well over 300,000. In fact, the number of AP Calculus exams given each year has grown steadily over the past decade at an average rate of over 7% per year with no sign yet that it is approaching its inflection point....

Based on the NELS study from spring 2004, the NAEP transcript study from 2005 , and the growth of AP Calculus since then, it is safe to conclude that we have reached the point where each year over half a million high school students study calculus.

I don't want to bash AP Calculus, or the concept of teaching calculus in high school. I took calculus in high school and I turned out fine. :-) But I do think that colleges—especially selective colleges—need to adapt to an incoming student body that is very differently prepared for college mathematics than ten or 20 years ago.

On the other hand, I'm not willing to cede all of Calculus to the College Board. I think that calculus has more to offer than the computing of derivatives and integrals and solving optimization problems where the answer is always the unique critical point. I think that professional mathematicians have unique insights on calculus, including what it means to "know" it. And, speaking existentially, I guess, I think that taking a college level course and teaching it in high school makes it different from a college course.

Prof. Bressoud goes on to suggest three actions college mathematics educators should undertake to deal with the issue of enrollment (quoting the italicized sentence):

1. We need to understand what happens in college to students who study calculus in high school. The College Board says that students who take AP courses do better (not just place farther, but do better in sequent courses) in college than those who do not. It does not seem they go further in mathematics, though, otherwise enrollments in college math courses would be growing as rapidly as AP enrollments are.
2. We need to know more about the preparation of the students who take calculus in college and what they need in order to succeed once they get to our classes In the mad dash for high school students to reach calculus by their senior or junior year, Euclidean geometry is nearly gone. Analytic geometry, trigonometry, and other precalculus topics seem to have been given short shrift as well. Today more students know how to find the derivative of a quadratic function, but fewer students can draw the graph of a quadratic function. I don't mean to grouse, but prerequisites are there because it's hard to communicate a difficult concept without a firm ground below one's feet.
3. Mainstream calculus should not be the only entry to good college-level mathematics. This is an intriguing idea. Bressoud cites the explosion in biological sciences and thinks of the entry-level mathematics courses that could be designed for them. I think a dialogue between departments can get a good symbiosis between science and mathematics courses.

I still think that calculus is an important entry-level mathematics course, maybe not the only one, but an important one, and that entry-level mathematics courses should be designed and taught by mathematicians. But college math educators need to be aware of the changing levels of preparation in our students—what they know now that they didn't used to know, what they used to know that they don't know now, and also what they want to know that we can teach them.

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