(I numbered my questions- it ended up being a long post!) (1) I'm also wondering if anyone has any good metaphors for difference between proof and "drills" or "techniques". Maybe learning "techniques" is sort of like getting good at scales, whereas proof is actually playing songs? I feel that math through high school calculus is pretty much mostly learning different manipulation techniques, with the exception maybe of geometry. But those geometry proofs, often the only proof in any HS math curriculum, end up seeming kind of like an aberrance in their methodology- open-ended, no one single procedure to memorize like in say Algebra II or in finding the derivative; a lot of gruntwork for, perhaps, relatively simple payoffs. I have to admit, I myself often don't find some of the geometry proofs that fun- and yet they can be daunting because we aren't used to thinking that way in math, no series of steps to follow and memorize. Maybe that's partly because I'm approaching proof wrong- it can be easier if you build your way from the conclusion back on simple proofs so you don't miss overlooked steps, for example. But, in general, (2) how can I make proof less daunting/more interesting to others? (3) Is it possible to "like math" without liking proofs? (4) Why do you think it is that people who say they like math also say they "hate proofs"? Is it just because our brain's reward centers that reward us for generating correctness haven't been trained to handle those sorts of challenges throughout our primary/secondary math education? Finally, (5) I'm wondering how/why the US got so stuck on "math = techniques" in the first place. (6) Is it like that in other countries?