[wheelinsupial]

For what little it’s worth, the thing that finally made it click for me was a series of comments on HN that were discussing musical scales.

I don’t have any musical training, but I related it back to the practice and warm up sessions we had before we’d play an actual game in the sports I played as a kid.

Perhaps some explanation like that will get it to click with someone.

I also learned of the existence of soft question tags on Math Overflow and Math Stack Exchange that contained an incredible amount of guidance that I think was never possible in lectures. Sharing links to those websites in the syllabus may be helpful for the odd student that actually looks at the syllabus.

[kmill]

I'm teaching discrete math in January — I'll try the analogy, wish me luck!

As someone who's gone through the mathematical ringer, the analogy doesn't ring true to me, but it does sound pedagogically useful still (my students will be CS majors, so the math will be for training rather than an end). Even at the highest levels the definitions are of prime importance, though I suppose once you get to "stage 3" in Terry Tao's classification (see elsewhere in the thread) definitions can start to feel inevitable, since you know what the theory is about, and the definitions need to be what they are to support the theory.

Personal aside: In my own math research, something that's really slowed me down was feeling like I needed everything to feel inevitable. It always bugged me reading papers that gave definitions where I'm wondering "why this definition, why not something else", but the paper never really answers it. Now I'm wondering if my standards have just been too high, and incremental progress means being OK with unsatisfactory definitions... After all, it's what the authors managed to discover.

[wheelinsupial]

Yeah, sorry if I wasn't clear. I 100% agree with definitions, theorems, counterexamples, and proof techniques being incredibly important. Those are the "warm ups" or "scales" or things that need to be repeatedly drilled in my mind before trying to jump into the "game," which, to me, is solving problems.