In college, I took a discrete math course with the world's most unprepared, distractible professor. It was incredible. He would come in with nothing planned in particular, we could ask about concepts from the textbook and he would invent a problem on the spot. Then he'd run through various problem-solving strategies until one worked. I learned so much about how a mathematician thinks from this class.
This was in sharp contrast to my calculus classes where the results were basically thrown at you fully-formed. If you're lucky, you might get to walk through a proof with the professor, but you're never going to see how they mentally navigate the search space.
