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I'm at UPenn, probably doing differential geometry and mathphys. Although it turns out that I'm pretty bad at algebraic geometry, so mathphys might not be an option for me for much longer. Sheaves are scary.

Witten gave a talk here yesterday; I understood a tiny little bit because of my interest in mathphys and having done quite a bit of symplectic stuff, but I think only two people in the entire room (basically all of the math and physics departments) really had any idea what he was talking about.
Also, he's not that good a lecturer or that good at explaining things.

Perhaps. In my experience, it's because as you progress in maths and theophys, you internalize a lot of crap. And once you've internalized it, it becomes "obvious" and therefore it is beneath your abilities to explain.
And this process never, ever stops.
Because a lot of higher mathematics is a series of epiphanies; once you've seen something, a bunch of stuff comes together at once and it's all clear in hindsight. Never mind the hours upon hours of staring at the wall trying to piece these little disparate bits together; once you've seen the big picture it's trivial where each piece fits and you wonder why you didn't see it earlier.
And thus, when it comes time to explain it to someone else, you only remember the end and if you're lucky the beginning, but the miles and miles in between have vanished.
So you can't explain how you got to where you are and you can't understand why the student isn't already there with you.

I'm currently doing a jigsaw puzzle, so I'm getting this feeling every few minutes. I've sunk like at least 20 hours total so far into this thing and every piece I place it's like "oh, duh. Why didn't I see that before?" but before that it's like "this piece looks exactly like every other piece, and I can't find the piece I actually want."
Why does the Earth have so much ******** water on it?

And yeah, a lot of theophys ends up talking about moduli spaces and invariants and while these sometimes have nice differential structures, usually they're not smooth manifolds (or indeed manifolds at all) so you need to use algebraic methods to deal with singular parts and all that jazz. Plus it's a lot easier to calculate and manipulate in algebraic terms than in analytic terms.

Sure, there are textbooks. And a lot of those textbooks are horrible because the authors don't know how to explain either.
It didn't sink in for me until recently, but unlike primary and high-school teachers, the main criterion for becoming a college or university professor is knowing a sufficient amount of stuff, as opposed to being able to teach.

That seems like such an obvious conclusion. If the professor can't explain it while he's teaching, it's probably not going to get better in book format. It might become more thorough or more organized, but often the primary missing points are still missing. This is especially true since most college textbooks, I feel, come out of lecture notes that were already prepared for a class.

MIT is doing good work in trying to up the teaching ante. Every year they host workshops for grad students and professors on various aspects of public speaking, writing good science papers, designing problem set and test questions, organizing lectures, etc. Some of it's mandatory before you become a TA, but most of it...well, there's not as much enthusiasm as I would like.

And to Mecill's point, "trivial" should be eliminated from the mathematical lexicon. "Obvious" and "trivial" concepts are ways for arrogant mathematicians (theory people in general) to either avoid having to explain themselves, avoid remembering earlier proofs, or generally feel better than everyone else.

Sorry, forgot I wanted to say something about this part, too. Nonlinear is almost never a good way to explain a concept, but it's a great way to get a class to explore a concept. As long as you lay out the foundation and always keep clear 1) the assumptions or previous material involved (the beginning) and 2) the end goal/concept, explicable in simple everyday terms (the simple end), getting a class to explore different possible paths of reasoning from A to B can help a lot more people grasp the concept than you just showing them one path.

This speaks to a more general phenomenon of asking inverse questions. Say you've gone over what eigenvectors are, how to find the eigenvectors of a matrix, etc. You think most of the class has the idea, but not the practice, and some of the class is still struggling on the concept. If you can get them to work as a class or at least in small groups on an inverse problem like:
"Find as many 3x3 matrices with these (three) eigenvectors as you can"
you engage a lot more students in thinking about eigenvectors in several different ways.

The other thing that most people try to do when being "nonlinear" in their explanations - which actually works if done right - is bring in outside examples. Rather than just giving a strict, formal, abstract definition of a term/concept, give an example of its use in your field. In math, when you define an eigenvector you'd give an example of an eigenvector of a relatively simple matrix. Then, give another example from a slightly different field of math: the eigenvector/eigenfunction of a differrential operator. Finally, bring in something that looks completely unrelated at first, like a coupled spring or spinning top, and explain - with direct and clear references to the definition of the concept - how this new example fits in. Getting students to analyze the similarities and differences between these examples can really solidify the concept in their heads with a method of reasoning of their choosing. Then you can ask conceptual questions about the definition and applications to see if they're on the right track.

One of my classes requires me to draw geometric figures a lot. I do them in crayon. I'd forgotten how bad the resolution of crayon is.

It turns out that of the two explanations for why bosonic string theory has 26 dimensions, one of them is really unsatisfying.
It boils down to needing to deal with a factor of Σn, where the sum ranges from n = 1 to infinity. Yes, the sum is of all the positive integers.
We need to set (D-2)/2 times the sum to equal -1.
Obviously, this is nonsense. Not so obviously, we replace Σn with ζ(-1), where ζ is the Riemann-Zeta function ζ(s) = Σn^{-s}; or more accurately, the meromorphic continuation of the Riemann-Zeta function. It still blows up at 1 and at -2k for positive integer k, but it is finite at -1, and equal to -1/12.
So we get that -(D-2)/24 = -1, and thus that D = 26.
FIND THE FISHY STEP.

There is also a rather nice explanation via algebraic geometry which does not involve summing all the positive integers and then playing coy with function representations.