My math teacher was not like the ones you describe, he really let me go off the reservation if I ever wanted to approach the problem in some wierd zany way. IMO

My math teacher was not like the ones you describe, he really let me go off the reservation if I ever wanted to approach the problem in some wierd zany way. IMO, that's how it should be. On the other hand, mastery of the tools and methods requires repeating it and spending a lot of time with them untill they "make sense" or rather, you get used to them. So that is useful as well. Most people can apply mechanical methods well, but the part where I see a disconnect is applying them to new problems and new contexts. It's like... Most people don't even realize you can do that. Would love to find a scalable way that teaching can approach that as well. It is a crucial mental tool.