So I am frantically going through and brutally editing this talk I have to give for real tomorrow (!) and it's actually kind of exciting and fun. I haven't worked so much against as many tight deadlines as I have had to this week in quite a while.
One principle that I am only slowly learning by being faced directly with trying to live by it is that, in talks at least, it is possible and indeed a good idea to put examples before definitions. I catch myself, over and over again, writing fragments of the talk as "ok, give some rules that classify valid foo-things, explain explain explain what they rules are supposed to mean, so that the audience can in principle determine what valid things are if they have missed nothing in the preceding stream of mumbling, finally, 20 slides later, give an example of a valid foo-thing."
But no! People find it easier to have an example, even if they can't fully understand it, to have something to hang onto when slogging through the definitions.
In doing the math itself, you have to (in some sense) put the definitions at the beginning. The definitions have to be utterly, perfectly clear before anything can really be done with them, not that I'm discounting the pragmatic value of fuzzy intuitions before that point. And, you know, in writing a paper, it's the same deal as giving a talk; giving examples early is good for humans. It's only in the pure content of doing the math that everything has to be rigorously sorted in definitional-dependency order, and explaining math to human beings is not at all the same thing as doing the math. Big surprise, eh? In fact it's sort of directly opposed, which I think explains why people who love doing math like me wind up being suboptimal explainers by default, until they learn better. I like to think I am in the process of learning better...
Okay, it's about midnight and I think I have the revised slides done modulo drawing a diagram or two. They are significantly more example- and motivation-driven than before, and I feel good about that. I think I've got about an hour of useful work-time tonight before my brain becomes useless, and a couple hours in the morning if I get up early enough.
The fact that the weather is warm at night again affects my general mood terrifically well. Everything seems a million times more okay and relaxed and manageable when I don't feel that on-edge just-a-little-bit-too-cold feeling all the time.