The Calculus Trap (2005)

I would say that calculus is the first "real math" that students study which is so complicated that they can't really learn the first principles without years of effort. That's why classes teach the most immediately useful parts instead of covering the details. Someone who only studies those basics might even come out thinking that calculus is just a bunch of techniques, without grasping how gorgeous the underlying concepts and abstractions are.

Take geometry as something I think you regard as "real math." How come geometry classes never prove that it's impossible to trisect the angle? Or double the cube? It turns out that compass and straightedge can only produce quadratic constructions, and Wantzel showed that these problems require cubics. This shows just how limited geometry really is, which is partially why students can feel that they have a handle on the entire topic. (Also, few if any secondary schools cover non-Euclidean geometries, or Euclid's book X as it explores incommensurable magnitudes.)

I would also say that trigonometry is the first applied math course that students learn, not calculus.

Let's go to calculus. Take for example the chain rule, D(g∘f)(c) = Dg(f(c))∘Df(c). This is one of those rules everyone learns in Calculus I. Can you prove it? I once could. Here's the start of the proof from my text book:

"The hypothesis implies the c is an interior point of the domain of h = g∘f. (Why?) Let e>0 and d(e, f) be as in Definition 39.2. It follows from Lemma 39.5 that there exists strictly positive numbers g, J such that if |x-c|And so on for another 10 lines of the book.

Do you really want to subject all calculus students to that level of detail? I don't. Who other than a mathematician needs to learn about Lebesgue integration and measure theory, which are even more advanced topics in calculus?

BTW, my undergrad had multivariable and vector calculus as a single Calculus III class, so there were only 3 semesters of calculus before getting into the foundations of calculus. Also, calculus books do have some proofs. For example, every chapter I looked at from http://ocw.mit.edu/resources/res-18-001-calculus-online-text... contains a few proofs.

Your example has a 15 year old going to community college or university in order to go to a calculus course. You named it the "calculus trap", which includes as a negative the social problem of having a 15 year old in a class full of 19 year olds. This implies that it's unlikely that the high school offers calculus.

My observation is that more and more high schools offer calculus in the high school, so the 15 year old you described, who was taking college courses to learn calculus, is now more likely to be a 15 year old at high school taking courses with 16 year olds (like your second son).

In that case, the severity of the trap is lessened, no? If only because the age gap is so much less.

I'm not saying that you are wrong about how math knowledge should be developed. I point out only that the arguments from your hypothetical case feel a bit out of date.

Than again, suppose the 15 year old does take calculus at high school, then takes a tertiary education class at age 16. What class might that be? That's about the time the standard college curriculum branches away from calculus, to include algebra, differential equations, or discrete math.

In that case, it's not really a "calculus" trap, no? :)

Also, I read the reviews of KTEM. Cross-cultural observational comparisons are often enticing, but it's hard to draw firm conclusions from them. Had you read, say, a comparison with the Finnish model then perhaps you might have drawn different conclusions?

My hypothesis, btw, is that the US is entirely too car dependent. Extracurricular activities like a city math club would be much easier if teens had ready access to transportation independent of their parents.

Many people that are very advanced in math get bored. Topics like number theory can reintroduce the fun and wonder in math that has been beaten out by the system. Bringing "Wow, how can that possibly be?" back into mathematics should be a goal in and of itself. (This isn't meant to degrade the "master the basics to master the complex" argument either)