I think the key, in part, is to resist that mathematicians’ tendency to abstract away individual problems into general solutions or categories of solutions or entire subfields, and spend some time with the specific problems that mathematicians are or have been interested in. But it also helps a lot if, in that specific problem, you get that mathematical move of discarding whatever doesn’t matter to the structure of the problem. After all, that’s a big part of what you’re trying to teach: how to think like a mathematician. You just to have to unlearn what a mathematician already assumes first.