Good points, Judy. I’d like to highlight your point about the comparison of examples. I think the feedback from comparing and contrasting examples generated by students can be a *very* effective teaching tool. If a student can answer *why* something is or isn’t a good example of a particular principle, then they have gone a long way towards mastering a concept.
I discovered only as an adult the powerful insight into original mathematical proofs. It’s one thing to memorize rote Pythagorean theorem; it’s another to see the proof Pythagoras offered of his hypothesis.
I learned the theorem as a secondary school student, but I never really grasped the why and how of it until I saw a youtube video showing how Pythagoras proved his theorem.