Apprenticeship
Yesterday, I also got to watch my tutor figure out how to prove something that he had, perhaps, been shown long ago, but had long since forgotten. Proofs have often puzzled me: given a statement like "Prove that x is a Y if z(x) is prime" (I'm making that up but it's the right flavor) I would have no idea where to start. What I saw yesterday is you really don't have to have any idea how the proof itself will start: you start by just writing down everything you know about the problem, and then begin "messing around" with the equations you have. Sooner or later, you (hopefully) will see the "finish line," as my tutor put it, and you will have your proof. Then you can clean things up, and get the concise, knee proves one finds published. No book has ever indicated to me that proofs are done that way! And this is a very Oakeshottian point (or Polanyian, if you like your case for apprenticeship from a scientist).