Science thinking, different teachers

I had a professor who told us that the first time he studied thermodynamics, as an undergrad, he didn't understand it; the second time he studied it, as a grad student, he thought he understood it; then later, when he had to teach it, he realized that he didn't understand it as well as he thought he did.
Oh yes I can believe that. To tell you the truth, there are things I thought I understood about QM and statistical TD that I only really understood when I had to explain them simply on internet forums, even though I can't recall all the maths now and need to look it up. That's one of the thing that - occasionally -makes taking part in these discussions rewarding. ;)
Of the math teachers I've seen close up, the most impressive was a PhD student (concentration: "Category Theory") who regarded neglect of his teaching responsibilities as a moral failing - so he talked his overseers on the relevant committees to support what they regarded as a waste of his time taking classes in teaching. How to teach math. This was at the U of Minnesota - some faculty there have studied the teaching of math, have theories and data and so forth, have worked out approaches for the various "learning styles" of young people, etc.

Sounds kind of dubious, to anyone who has run into the various fads and "New Math" approaches that claim to derive from such academic research. But it worked for him - his students were noticeably engaged, focused, and high performing. And they seemed to panic, flip out, throw in the towel and declare themselves hopeless, in much lower percentages.