09.09.05
What is the difference between AP Calculus and Math 1a and 1b?
At Harvard we get a large and increasing number of students who have taken the Advanced Placement Calculus AB or Calculus BC exams. These courses have syllabi similar to those in our first-year calculus classes. The topics in Calculus AB roughly correspond to Math 1a (although Math 1a is a semester-long course and Calculus AB is a year-long course), and Calculus BC roughly corresponds to the Math 1a and Math 1b sequence.
But there are big differences between an AP course and our first-year undergraduate courses. Not all who have taken Calculus BC are necessarily ready to move on to the second-year courses such as 19, 20, 21a, 21b, 23, or 25. In our first-year undergraduate courses we go beyond the techniques of differentiation and integration, and spend a good deal of time on the conceptual aspects of calculus. For instance:
- What is a function?
- What do we mean by the derivative of a function?
- What do we mean by the integral of a function over an interval?
Armed with these flexible understanding of what functions, derivatives, and integrals are, we are able to apply it to many areas. So we develop many “modeling” techniques, such as
- Given a range of choices and an objective, find the optimal choice by setting up a function, differentiating it and finding its critical points.
- Given a scientific quantity determined by a product (work, flux, revenue), set up an integral to compute it when the dependent quantities are not constant.
- Given a quantity that changes over time and a principle which govern its rate of change, set up a and solve a differential equation to determine the quantity.
Finally, we spend more time in 1b on both differential equations and infinite series than is found in the BC class. Infinite series is a topic that seems from out of left of field, having nothing to do with derivatives or integrals. However, it turns out that functions can be expressed as power series, which are hugely important for mathematics beyond calculus. Not only that, they are used all the time: Calculators, cell phones, MP3 players, all digital devices depend on series to function.
For the reasons above, we often advise some students who have taken AP Calculus to take Math 1a or 1b. This shouldn’t be taken as a “repeat” of a course already taken. Instead, this is a chance to gain a more fundamental understanding of a subject very important in science. If your concentration requires a calculus course, it is because they need you to understand not only the techniques but the concepts which make it apply to your discipline.
Every year in Math 1a and 1b, we have students who have taken calculus before and those who have not taken it before. We feel that both groups will learn a lot. While there is no denying that those who have taken calculus before will find many aspects of our courses familiar, there will be parts which all students will find challenging.
All cases are unique, and we are happy to talk to any student about selecting a math course. During the opening days of each semester, please visit the math department’s advisors for information about your particular situation. A list of advisors (including me) and their availabilities can be found on the Math Department’s sectioning page. There is also a wealth of information on math for Harvard Undergraduates at the Math Department’s undergraduate page. Feel free to contact me as well!
Frank Linn said,
October 28, 2005 at 11:19 pm
How important is it for my child to strive harder and take BC Calculus than to take AB Calculus her senior year of “high expectation high school” if her goal is to prepare well for college? She is a great kid, smart, but so far no signs of math prodigy. Though she may be capable of either, I don’t want to assume that BC gives her some type of competitive advantage when perhaps it will push her too hard and turn her off from math. We are surropunded by parents steering the child towards BC Calculus. Help, Dad, Boston area
leingang said,
October 31, 2005 at 8:40 am
I’m not so much a fan of AP calculus; I think it and the race to get to it have become the de facto mathematics curriculum and they shouldn’t be. But I can’t advise your child to buck the trend based on my opinions.
I’ve never worked professionally advising high school students or prospective college students, so anything I say should be checked with actual guidance counselors. I would think that given the choice between BC and AB, the more advanced course would be the one that would most help your child’s application portfolio. So BC would best meet the goal of getting into college, which perhaps you had guessed.
As for which would prepare your child best for college, that depends on the college and the career (and the child). Here we advise just about everybody except the hard-core scientists and engineers that it’s completely possible to start college with Calculus I, finish all the math requirements for a degree, and still leave room for electives. No matter which course your child takes now, there is plenty of time for a great, interesting, challenging degree program in college.
Kareem said,
December 5, 2005 at 8:42 pm
I am currently in Pre-AP Calculus. (another name for Precalculus). Well I have a C fro the 1st semester. Would you advise me to take AP Calculus AB or AP Calculus BC? I know it’s a C, but I thought a C in Precalculus is actually pretty good. Please write back.
leingang said,
December 6, 2005 at 7:31 am
Let me first disclaim myself by saying I am a college math teacher, not a high school guidance counselor, and certainly not your high school guidance counselor. That’s who you should be listening to the most, along with your current math teachers, about which courses to take.
I don’t know much about Pre-AP Calculus–the only links I found were to a Virtual High School one-term course whose syllabus resembles AP Calculus AB “light.” If this is the course in which you got a C, taking AB might help you shore up what you missed this time around.
If your course is a true precalculus course, in which things like trigonometry, logarithms, functions and graphing are taught, then a C in that course may indicate you are not ready for calculus. If you go ahead with a calculus AP course, make sure you have enough time to catch up on the precalculus.
Precalculus (as I described in the paragraph above) is not just a hoop to be jumped through on the way to calculus. It consists of the fundamental tools with which we practice calculus. In some preliminary studies, among our freshmen, we found precalculus performance to correlate most strongly with success in any of our introductory math courses, more so than any standardized test or AP exam score.
grisham said,
July 2, 2007 at 3:59 pm
What is the difference between AP Calculus AB and AP Calculus BC?
leingang said,
July 5, 2007 at 6:42 am
That’s a good question, luckily a easy one! Calculus AB covers functions, limits, derivatives and their applications, the definite integral, the fundamental theorem of calculus, and integration by substitution. This is a one-year course.
Calculus BC does the same as AB in half a year. Then BC continues on to techniques and applications of integration, sequences, series, and power series.
The courses use the same document as their syllabus so you can easily see the difference between the two side-by-side.