College Math Teaching

August 14, 2012

A quick thought on the “Interdisciplinary” focus on the undergraduate level

Filed under: academia, calculus, editorial, integrals, mathematics education, pedagogy, quantum mechanics — collegemathteaching @ 9:15 pm

A couple of weeks ago, I attended “Mathfest” in Madison, WI. It was time well spent. The main speaker talked about the connections between algebraic geometry and applied mathematics; there were also good talks about surface approximations and about the applications of topology (even the abstract stuff from algebraic topology).

I just got back from a university conference; there the idea of “interdisciplinary education” came up.

This can be somewhat problematic in mathematics; here is why: I found that one of the toughest things about teaching upper division mathematics to undergraduates is to reshape their intuitions. Here is a quick example: suppose you were told that \int^{\infty}_1 f(x) dx is finite and that, say, f is everywhere non-negative and is continuous. Then lim_{x \rightarrow \infty} f(x) = ? Answer: either zero or it might not exist; in fact, there is no guarantee that f is even bounded!

This, of course, violates the intuition developed in calculus, and it is certainly at odds with the intuition developed in science and engineering courses. Example: just look at the “proofs” that the derivative (or second derivative) operator is Hermitian provided f is square integrable that you find in many quantum mechanics textbooks.

Developing the proper “mathematics attitude” takes time and it doesn’t help if the mathematics student is too immersed in other disciplines…at least it doesn’t help if an intellectually immature math student is getting bad intuition reinforced from other disciplines.

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