Mathematical proofs are something that I'm still struggling with in general (both constructing and understanding them). Let's take the relatively simple proof that sqrt(2) is irrational. The presentation is fairly typical: it's terse, no motivation for any step is provided, and the whole setup is confusing. Even worse, I don't even see how I can apply the idea of example cases to this proof. It's not a general property, so I can't look at other cases.

It's a particular property, which you can apply to other cases by substituting some other particularity -- for example, replace 2 by any other number throughout and see whether the proof still goes through. Doing this sort of thing will tell you how the proof works and why it works.

As an aspiring polyglot, I also picked up the French Michel Thomas course and will test it this week.

I'm interested in knowing how this goes. I've never got very far when learning other languages, although I have to say I'm not impressed by the Michel Thomas web site or the Amazon reviews. I suspect that his method would drive me up the wall.