> No amount of explanation and reasoning will help the students and instructors get around the fact that they just have to spend the time to develop fluency wit

> No amount of explanation and reasoning will help the students and instructors get around the fact that they just have to spend the time to develop fluency with the basics. Conversely, no amount of memorization and drilling will provide students with the intuition necessary to solve problems just beyond the grasp of naive symbolically-driven proof search algorithms. In a typical non-proof-based calculus sequence, tricky u substitution and integration by parts problems are the two places where you really see "I was good at math in high school" students start to struggle because they lack a good conceptual foundation in trig and logarithms. Up until that point the students with trig identities etched into their skulls by practice and drilling do just fine. But from that point onward, they really start to struggle because they need a bit of geometric intuition and some practice with more effective problem solving techniques than "blindly apply equations until something works". Really, both are needed -- drilling and practicing, as well as a solid (and preferably proof-backed) intuition about elementary properties of trig funcations and logarithms.