Stillwell knows what he's talking about. In his fantastic book "Mathematics: A History" he makes the point that calculus is taught entirely backwards from its h

Stillwell knows what he's talking about. In his fantastic book "Mathematics: A History" he makes the point that calculus is taught entirely backwards from its historical development, and he says that approach ruins most undergrads in truly learning calculus. In line with Abel's dictum to "Study the masters, and not the pupils", it's obvious that sweeping the why and how of calcuclus under the rug, and instead presenting a polished and abstract end-product is a disaster as far as learning goes. As natural tinkerers and hackers, surely we can appreciate how much more you can learn by peaking behind the curtain, taking apart the machine and trying to put it back together again. Sure we can thank Cauchy and Weierstrauss for working out the logical kinks and putting centuries of ungrounded assumptions on a proof-based foundation, but the FTC is a hell of a lot easier to understand on an intuitive level if you started with series, derived integrals and then saw how derivatives came out of that, rather than the other way around. The real problem with mathematics education is that they are teaching to the least-common denominator on a mass-scale, adopting the attitude that 99% of students won't need to really know how it works under the hood. If students can just grind through the problem sets, it's good enough, so the entire curriculum, text books, and TA lectures are tailored to that. Unfortunately, this approach has infected even the pure math programs--I think it's silly to teach the group who will specialize in mathematics in the same way say biology and economics students are taught caluclus.