This discussion was created from comments split from: Is AP Calculus BC (and AB) too easy?.
Sorry to be opening such an old thread back up again, but I think this is an incredibly interesting conversation and would like to chime in 
. I’ve been taking Calculus BC this year and essentially agree with what others have said. However, I think the issue with the US math education system starts way before Calculus BC. I’ve always found it so strange that we get essentially no proof before geometry, get a year of some fairly intuitive proofs, and then get no proof after that (until eventually we enter Analysis courses in college). Now, I do recognize that some proofs are advanced and require prerequisites (for instance, you can’t prove the Fundamental Theorem of Algebra in Algebra II as it used results from Complex Analysis); however, many are possible and don’t have this issue.
I think essentially what we lose in the education system is that math is ultimately built on axioms. You essentially have to take an approach of axioms, leading to lemmas, which lead to theorems which progress into more complex theorems. However, instead of that we simply get a barrage of theorems (the end product) without getting any of the theory behind them and in doing so lose the true elegance of mathematics which has been developing for thousands of years.
I’m not suggesting replacing calculus with an analysis course, but I do think proofs of theorems should not just be reserved for geometry but rather put throughout the mathematics curriculum (for example, why don’t we teach the proof of the binomial theorem by mathematical induction? It may be challenging but also is an incredibly important result for mathematics). Speaking of induction, I think we should put basic proof techniques into the syllabus. Mathematics comes down to spotting patterns and using both inductive & deductive reasoning, not just memorizing a bunch of rules/formulas. If mathematical induction and other such skills were taught, some of those skills would be much better obtained (you have to reason through a proof).
I think there’s room for more challenging, thought provoking questions even if they aren’t technically proofs. I’ve been preparing for the Oxford MAT and Cambridge STEP (entrance exams for math degrees at Oxford and Cambridge universities), and they ask some incredibly intriguing questions. For instance, one MAT question asks you to find the integral from 0 to n of the greatest integer of 2^x. That question actually gets you to think of the meaning of an integral. We don’t have some magic formula for integrating greatest integer functions, so you have to break the greatest integer function up into rectangles and figure out width/height of each, add some up and spot/deduce a pattern. That’s far more instructive than just applying an integration formula. Or another example- a Cambridge question asks you to compare the integral of x^2 versus (the integral of x)^2 (ie. integral of x^2 inside the integrand vs. the integral of just x and squaring the whole thing). It’s a common student error to incorrectly assume that they’re always the same, so the exam asks you what bounds of the integrand you have to apply to ensure that both are the same.
Neither of those two questions (or others like it) are found in the typical AP calculus course or exam. And I think that’s a major shame because it really obscures the actual beauty of mathematics. I think we need to insert not just proofs but even more importantly problems that make students think critically and resort to seeing patterns in lieu of just applying formulas.
Out of curiousity, why are you preparing for both the MAT & STEP? trust you know that you can only apply to one or the other…
Hi @collegemom3717 ; thanks for that information. I did know that. It’s mostly because I’m applying to Imperial College London also which requires the MAT. Thus, I know regardless of whether or not I apply to Oxford I’ll have to take the MAT anyway. Thus, most of my prep. so far has been for the MAT (although I’ve looked at one or two STEP questions in case I apply to Cambridge) as it’s first and counts for two universities in the UK.
Good to know. You sound like the kind of mathmo that thrives in the Oxbridge environment. If you haven’t already, read the course pages really carefully- that’s where you see the differences between the unis.
@collegemom3717 , thanks so much for that advice; I’ve begun looking at the course pages (and will be visiting next summer which should help to give more insight). I’m probably veering more towards Oxford right now, just because I’m very interested in the math and physics combination, and Oxford is just beginning to offer a MMathPhys 4th year, where you get a dual masters in both.
I have found Calculus BC too be a little bit hard due to the rigors of my teacher and the class, however on the exam i think it is way too easy to get a 5. After 1st semester, I took a practice exam and I still managed to clutch a 5 without knowing any Calculus C like Taylor and Maclaurin series, and area under polar coordinates. I mean, you literally need a D to get a 5 on the exam lol.
@cghar191 - yes I agree with you; I’m glad your teacher went beyond the scope of the AP though! I think the number of Calc. BC students obtaining 5’s (almost 50%) is too high, and they ought to scale it like other AP exams such that approximately 20-25% of students get 5’s. I don’t think they should do that by just making the curve more difficult but rather though putting more challenging questions on the exam. I think they do a much better job with the AP Physics C courses for instance where the questions actually challenge the test taker/make them think (and as a result, there are fewer 5’s).
What does everybody else think about whether it’s time to re-engineer calc. AB/BC to increase the amount of rigor in those courses?