>"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."
While I do agree, we have to remember why most math classes actually exist: to teach calculus to physicists and engineers, and, as my stepfather's undergraduate advisor once said, "to keep the children from running in the halls".
(For the mathematician's extremely self-centered view of "children" as "anyone who has yet to ace two semesters of real analysis".)
I've been starting into real analysis myself via Pugh's textbook[1] after not taking a serious math class since multivariable calculus, and found that, once I get past the applied stuff, I really like the approach of building up calculus from its foundations in real numbers (taken as Dedekind cuts), limits (Cauchy-convergent sequences), the set-theoretic construction of functions, and the construction of topological and metric spaces "from scratch". But I can tell that I like it because, deep down, I have the mind of a theoretical computer scientist (which is what I like to be when I'm not writing firmware), which is a kind of mathematician. I appreciate that someone has to teach the applied classes to the people who aren't going to kvetch about "how can I trust that works!?" and who demand to just get their math over with as quickly as possible.
While I do agree, we have to remember why most math classes actually exist: to teach calculus to physicists and engineers, and, as my stepfather's undergraduate advisor once said, "to keep the children from running in the halls".
(For the mathematician's extremely self-centered view of "children" as "anyone who has yet to ace two semesters of real analysis".)
I've been starting into real analysis myself via Pugh's textbook[1] after not taking a serious math class since multivariable calculus, and found that, once I get past the applied stuff, I really like the approach of building up calculus from its foundations in real numbers (taken as Dedekind cuts), limits (Cauchy-convergent sequences), the set-theoretic construction of functions, and the construction of topological and metric spaces "from scratch". But I can tell that I like it because, deep down, I have the mind of a theoretical computer scientist (which is what I like to be when I'm not writing firmware), which is a kind of mathematician. I appreciate that someone has to teach the applied classes to the people who aren't going to kvetch about "how can I trust that works!?" and who demand to just get their math over with as quickly as possible.
[1] -- http://www.amazon.com/Mathematical-Analysis-Undergraduate-Te...