Here is a quote from CalTech’s description of its Freshman math course:
“…The typical high school courses, and the AP tests themselves, are woefully inadequate in explaning, or testing, why things work and how to justify one’s propositions…”
The link to the where I got this quote is: http://www.math.caltech.edu/~2011-12/1term/ma001a/#des
This is, of course, CalTech, but as I state in my comments on NPR, (see my post http://www.inside-higher-ed.com/ap-calculus-courses-discussion-on-npr/) it is my experience that students aren’t required to learn what is the essence of calculus, without which (a) it is not really calculus, and, more importantly, (b) the student’s depth of understanding doesn’t increase to the level required in today’s globally competitive world.
I believe that there is a fair amount of politics in play in today’s AP courses. Every school wants to “say” lots of its students “know” calculus. First, as just pointed out, that usually isn’t true. Second, students are rushed past learning important mathematics, some of it skipped, so that they can “learn” calcuus. On the way, they don’t learn important ideas, such as the Binomial Theorem (important as an application and important because of some of the ideas it contains), or even induction.