I've never taught mathematics but I think it's much harder to teach the intuition, and I also think that more people are capable of learning the mechanics than the intuition. So from the point of view of someone designing a methodology and curriculum - you have to design for scale.

I have the same thing with learning languages: I despise focusing on the rules of conjugation and word order and rote memorizing rules, exceptions to them and exceptions to exceptions. I much prefer to bootstrap with some vocab and then acquire by immersion, even if that means I get the conjugations wrong in the beginning. But I also find myself in the minority by far, to the extent that pretty much all language teachers I ever had (except my high school French teacher - she was awesome letting me translate MC Solaar lyrics for grades) didn't even really seem to believe that there are other ways of learning other than studying rules.