communication

I have to give a short talk tommorow, and I've been so wrapped up in details, its hard to draw back and think about the big picture.
Previously I've been far into equations and then my talks are just one equation after another. If the equations are abstract enough, then the person who listens thinks, "Ahh, yes, its very simple at the heart of it all, its just about linear algebra, invariant subspaces, etc. I don't have a great grasp of those concepts, and I don't know exactly how they relate to the problem at hand, but math is pretty solid stuff."
Now I've been doing computational work for awhile and my tendency is to give plot after plot without clearly defining things. "Trust me," these plots say. "This code works and is smart."

In both these cases, the audience doesn't learn very much. At the moment, I'm considering showing one equation, and that seems like a dramatic gesture.

I used to wonder how people could be bad teachers but still know things. If they can't explain it to me, why should I think they understand it? Now I realize that you can learn things in either a narrow way or a broad way. In a narrow way, each thing you learn is connected to one solid thing. You follow a thread. In the end, you may have gone quite far and found something quite interesting, but all you can do is tell other people how to follow the same thread you did. If there are independent ways of checking that something is right... the pie is delicious, or parachute does indeed open, then that may be good enough. To learn something in a broad way is to carve out a path to the result in which each step along the way is connected to something else. This means that you can enter the process at any point and feel relatively comfortable. This is what you need to do in order to teach something. Then, if the person you are teaching has different knowledge, assumptions, understanding than you, then they can still find a way to the answer that is compatible with what they know.

I suppose the same goes for personal communications, which is in some ways about teaching people about yourself. I always feel uncomfortable arriving anywhere in the narrow sense I just described. I then feel isolated and that no one can help me from there since it was by means of such a narrow path that I got there. If there are enough rewards, people push themselves through narrow situations to get to the big prize at the end. But if the journey was so narrow that it becomes untellable, then a big piece of personal history may be lost, which may or may not be compensated by the prize.