When I pick up a book and start to learn something new (which is pretty much what I do in my free time), I usually do it with an IDE at hand. I try to implement as much of what I learn in code. If I can teach the computer how to do it, then I am sure I really understand it, and it often helps me find holes in my understanding.

I have significant trouble with doing this with mathematics, though. Programming languages (at least the ones I know) just don't seem to be good for expressing things like identities and invariants. Anyone have any suggestions on how to handle such things?