Earlier quoted context omitted.
> In the countries that excel over the United States in mathematics instruction, proof is part of every mathematics course, not just part of geometry, for example. Can you expand on that? What do students do in other mathematical areas if they don't do proofs? I did all of my studies in France and I don't how much of a difference there is (though all my friends who have studied both in the US and in France found the…
US math courses for high schoolers often goes along the following path: Algebra, Geometry, Algebra II (basic linear algebra, but often not discussed as such, and other concepts), Trigonometry, Precalculus (1/2 semester each of the last two) and then Calculus. Of course, that adds up to 5 years so unless a student took Algebra I in middle school (year 8) they won't complete the full sequence, leaving calculus for coll…
Wow! I can't even imagine how that works. When I went to school (and that's in Europe, not Asia), the notions of theorem and proof were taught in the fifth year of school (out of 12 pre-college years -- 8 in elementary, 4 in high school). That was, indeed, as part of geometry. Starting from high school, however, pretty much no formula is given without prior demonstration. Subjects where they can't do that (like mechanics -- that's taught in Physics prior to pre-calculus and calculus) are a royal fuckup. You aren't usually expected to be able to reproduce the proofs for those formulae, but a) you're strongly encouraged to study them, because they help you with the exercises and b) it's usually implied that, if you can't remember the formula, but you can prove it, it's OK. For example, if you can't finish a problem in a test because you don't know the formula, but you obviously try to find it, the professor will sometimes give you full credit, depending on circumstances and following a short talk.
I didn't even like math back then, save for geometry, but I reckon this was the most important thing I learned there. Unless you're dumber than a toaster, you're typically able to apply what you learned to everyday phenomenae after ten years of school; the bonus you get is also being able to reason rigorously and correctly, which is an important takeaway no matter what path you choose to follow.