Joan Reinthaler

Sidwell Friends School

Washington, DC

AP CALCULUS AND THE LAW OF UNINTENDED CONSEQUENCES

In the beginning was a good idea and that idea was the Calculus AP exam. This exam would offer intellectually mature, well-prepared high school students a way to show university math departments how much calculus they understood, it would goad secondary schools into designing a curriculum that included calculus for their intellectually mature and able students and it would offer high-ability students something to strive for during their secondary school years,-- and it worked amazingly well ... at first. High school students who took the AP exam frequently had a better understanding of first-year calculus than did their college freshmen elders (not really surprising since, presumably, the most able and mature high school students were being compared to the average college freshmen). Schools did redesign curricula. Teachers went back to school to sharpen up their own calculus understanding and, in general, both high school and college educators seemed happy with what was going on.

... at first. But, as so often happens the law of unintended consequences has been invoked and "the bottom line", or the world of education's equivalent of the bottom line has seen a good thing and has run with it.

Secondary schools, or their supervising agencies began to get competitive about AP numbers. If their school has 10% more students enrolled in AP Calculus than our school does, maybe their school is 10% better than ours is. What to do? why get kids through algebra faster so that more of them can take calculus in our school. Never mind if they really understand algebra or if they are mature enough to really understand the basic ideas of calculus.

College admissions officers looked at AP Calculus enrollments (remember .. this is enrollments, not AP test scores. Decisions on admission are made well before kids ever take the exam) and saw a nifty flag to help them identify "the kids we want". So the message they communicate to high school college guidance counselors is "we are most interested in looking at kids who have taken AP Calculus" They may not say this in so many words but what is said, in so many words is “we are most interested in kids who are taking the most advanced courses your school offers”. It has taken about a nano-second for parents and school administrations to get this message and another nano-second for the pressure to mount to accelerate kids - ready or not, mature-enough or not, well-prepared enough or not - as quickly through algebra as humanly possible.

To make this happen, many public school systems are putting their “Gifted and Talented” kids into Algebra 1 in 7^th grade and geometry in 8^th grade. How many 7^th graders can really deal with the abstraction of a real Algebra course? At Sidwell this may be one or two a year, certainly nothing like the wholesale acceleration that is currently being advocated around the country. The fall-out from this is kids whose appropriate 7^th grade experiences - extensive work with geometry constructions, probability, pattern investigations, sequences, extended problems in the form of projects, and, perhaps, work with vectors and matrices, as well as continued work with the structure of the number system, the importance of 1 and zero, inverses and identity elements, in short, all the activities that feed into mathematical intuition - is short circuited - and kids who end up as sophmores or juniors in courses that require a maturity and ability to abstract and conceptualize that they simply are not ready for yet.

Meanwhile the message from university math faculties is that too many kids are arriving on their doorsteps with a year of highschool calculus under their belts who they find almost unteachable. These are kids whose algebra is weak, whose calculus is superficial and who think they already know everything. Want to know why students are arriving at the doorsteps of college and university mathematics departments so poorly prepared? Partly because of this almost irresistible pressure on them to rush through the material that they need to be prepared, and to study it before they are mature enough to understand it.

Sure there are kids for whom acceleration is an appropriate and joyful experience. These are the kids that the AP program was originally designed for. At Sidwell between 25% and 35% of our kids can reasonably expect to take calculus before graduating (our accelerated program begins in the 9^th grade). But, even for the most able kids, acceleration can mean a narrow focus on calculus as the ultimate mathematics nirvana. Increasingly, math-knowledgable people understand that calculus is not necessarily at the heart what should be taught to Mathematics concentrators in the 21^st century. At last year’s MAA convention, a colloquiem of emminent mathematicians agreed that Linear Algebra, Discrete mathematics and Statistics were going to be at least as important as Calculus and the student who arrives in college without some experience with these topics in highschool may very well be at a disadvantage.

At Sidwell, our main stream kids do not take calculus. They graduate with four years of algebra (during grades 8 - 12 - our regular program has two years of algebra after algebra 2) that includes rigorous work with proof, limits, vectors, matrices, data analysis and a variety of 3-D coordinate systems. They graduate with an intuition not only about traditional pre-calculus topics but about pre-linear algebra topics and discrete math topics as well because they have had the luxury of spending time looking at these topics from a variety of viewpoints and getting their hands dirty playing with them.

However it is getting harder and harder to resist the pressure from college admissions people to get the kids to calculus in their senior year and the overwhelming anxiety this arouses in some parents who want their kids to accelerate when they are not ready to.

If college and university math educators around the country really want to do something about the sorry state of the average freshman's math preparation, they ought to talk to their admissions officers and get them to stop using enrollment in AP Calculus as a convenient flag. They ought to convince them that having taken calculus in high school is not necessarily a virtue. They ought to convince them as forcefully as possible to communicate enthusiasm about schools that challenge their students with strong algebra programs. For too long admissions officers have spoken as the self-appointed mouthpieces of mathematics departments and their messages have not done the state of math education any favors.