AP Calculus

Stanislaw Ulam
Stanislaw Ulam
The Polish - American mathematician Ulam. Ulam discovered the key insight that made the development of the Hydrogen Bomb possible. He also taught university calculus. Coincidence?
My son took the AP calculus BC exam last year, and while the experience is still relatively fresh, I thought I should offer my thoughts on the exam.
There are in fact two AP calculus exams: AP calculus AB, and AP calculus BC. The BC has a few more topics than the AB, and the collegeboard claims it is equivalent to a full year of university calculus.

Is AP Calculus BC equivalent to one year of university calculus?

No. In addition to the calculus of a single variable, first year calculus should include an introduction to multivariable calculus. (Some double integration and partial differentiation perhaps, depending on the university) The BC exam only covers single variable calculus.
Also, in the topic of series, the root test is, strangely enough, NOT covered. This is a rather egregious omission.
Having said that, BC's coverage of single variable calculus is almost university level. Had BC calculus included the epsilon-delta concepts for limits, and the root test, it would be pretty much the same.

How to prepare.

In my opinion, preparation can be broken down into three parts:
  1. Learn the required material
  2. FRQ practice
  3. Speed exercises
The Material:
With regard to the required material, there are no mysteries. The collegeboard spells out precisely what is covered in the exam on its website. There are excellent preparatory books that cover all topics on the exam, and previous years exams are readily available.

FRQ, or free response questions:
In these problems, students are required to show their work. A correct answer with no supporting work will be awarded zero points. Most high school students are not skilled at showing their work. Additionally, some students who have the habit of doing steps in their head, will have to learn to write it down, at least for the FRQ. It is most important that the previous years AP Calculus exams, and the collegeboard's guidelines for graders of these problems, are downloaded and studied. The graders have no leeway in awarding partial credit. What will and will not be awarded a point is clearly spelled out. The reason for this is so that the grading of the FRQ questions is as uniform as possible, regardless of where the exams are graded.

The exam is timed. Therefore, a student could have successfully mastered the material, but by taking an extra minute or two for each problem, do very poorly. Stated differently, the exam does not just measure calculus knowledge. If one student solves a problem in 3 minutes, and another student solves the same problem in 5 minutes, there is no objective reason to conclude the faster student has a better knowledge of calculus, yet the AP exam is likely to reach that conclusion.
There is a story about the world famous mathematician Grigori Perelman and speed in mathematics. Perelman became immortalized in mathematics when he solved the Poincaré Conjecture, which is one of the Millennium Prize Problems. One of Perelman's teachers,Yuri Burago, had this to say about him:
"There are a lot of students of high ability who speak before thinking. Grisha was different. He thought deeply. His answers were always correct. He always checked very, very carefully. He was not fast. Speed means nothing. Math doesn't depend on speed. It is about deep ."
Yuri Burago
It bears repeating. The AP Calculus exam does not just measure your calculus knowledge. You could in fact have complete mastery of all the topics on the test, and yet do poorly if you worked at a slightly leisurely pace. Therefore, the topics where the student is working slowly have to be identified, and exercises devised to increase that speed.

The advantages of taking the exam

  • Admissions: Getting a 5 (which is the top score) on the exam can help with university admissions.
  • Financial: Many universities give credit for a good exam result. You could therefore save on tuition.
  • Practice: Nobody becomes an expert overnight.The AP exams give you a chance to practice, and test, your skills. (This was in fact the only reason my son took the exam*)
  • Focus: Many high school students lack focus. Having a roadmap explicitly laid out of what one needs to learn allows the student to focus, thereby saving time and energy.

The disadvantages of taking the exam

Preparing for the exam, even if one knows the material, is time consuming, and takes time away from learning new material.

Paradoxically, in my view, the main disadvantage of the exam is one risks lowering academic ambition and retarding a students progress. (aka, the poverty of low expectations) Many academically minded high school students could easily learn far more than what is covered in the AP calculus BC exam.(In fact, most go on to do precisely that at university) Their ambition should be higher - to learn more advanced material than is covered on the AP exam. The goal of a 5 on the AP can be specious.

Stanislaw Ulam

"Without the invention of the infinitesimal calculus, most of our technology would have been impossible. Should we therefore say that calculus is bad?"
The Polish - American mathematician Ulam was responsible for the ingenious insight that made the development of the Hydrogen Bomb possible. Edward Teller had been stumped for years on the H-bomb problem. And while Teller is called "The father of the H-bomb", truthfully, without Ulam (and calculus!) it would have never happended. The above quote is taken from his chapter on the Hydrogen Bomb (or "Super", as it was called) in his book "The Adventures of a Mathematician" (Now a movie, by the way...).
The above, albeit strange, quote is interesting as the development of the calculus is one of the greatest intellectual achievements of mankind. There is no question of its excellence. There is no question that high school students who learn it are fortunate. Yet if in the above quote, one substitutes Ulam for infinitesimal calculus, and the H-bomb for technology, one sees that Ulam was actually wrestingly with the moral dilemma posed by his decisive role in the development of the H-bomb.

*I have been going on about the AP calculus exam and how to prepare for it. One may reasonably ask then how my son did. Being relatively young and inexperienced in such test-taking (he was in 7th grade), he did not know that guessing was actually allowed on the multiple choice part of the exam. (Unlike some exams, in the AP exam, there is no penalty for guessing wrong.) He therefore left five questions unanswered. Yet, he still got a five, which is the highest score possible. Being familiar with my son's calculus knowedge, and his working style, I have to assume it is very likely he got the answered questions correct. (By the way, with regard to the unanswered questions, typically, one can eliminate a couple of the choices in a multiple choice question, as they are often obviously bad. So with some luck, educated guesses in 5 questions should yield maybe two or three correct answers.)