T.S. Eliot Teaches Evelyn Lamb Some Math

Well not exactly, but I do love the way Evelyn brings Eliot's words into a mathematical context for this wonderful meditation at her "Roots of Unity" blog:

http://tinyurl.com/lwo6cbz

Patrick Honner similarly notes in a tweet that "reinventing the wheel" is in a sense what a teacher needs to do in order "to learn to teach something well."

It is a common complaint that too often students learn mathematics procedures or methods, without really comprehending the deeper process behind those methods.
Evelyn's mention of calculus, in this regard, caused me to recall my own dismal experience with college calculus:

[Evelyn's words]: "Calculus is far from a basic topic in math, but it is one that is learned very early, before we understand much about just how intricate it is. We do a lot of exploring before we come back and truly understand calculus for the first time." 

My experience: most days our prof would spend 45 mins. with his back to us as he wrote on the blackboard some long drawn-out proof of whatever we were covering that day, and then with 3 minutes of time left on the clock would turn around and ask, "any questions?" Those of us who hadn't fallen asleep didn't have the nerve to explain that we were lost from step 4 on… and then the bell rang anyway. On the one hand it was a horrible way to teach, but in retrospect I also understand now that he believed to really grasp the 'intricacy' of calculus you needed to be shown how each piece of it was derived from the ground up.

I mention all this because most of my younger life I had little interest in (or patience for) proofs or explanations, but just wanted to learn the 'facts' of math and how to apply them. Only later in life did I come to appreciate that it is by fully understanding such proofs or processes that one acquires a deep grasp of mathematics and mathematical thinking. My concentration on doing the methods and working the numbers got me through high school in fine form, but probably contributed to my math downfall in college.
Moreover, in the logic, reasoning, and proofs of mathematics, once comprehended, lie more of the beauty that is so easily missed if you treat math as just a bag of manipulative symbolic tricks, as many may perceive it.
As Evelyn implies, math is more a constant "exploration" than a rote process or recitation, and that remains so whether you are a student… or teacher.