I think a course like that would need to be called something like “Mathematics Appreciation,” in analogy to “Music Appreciation.” I would never hire someone whose only music education was the latter to actually play in a band or write a movie score, and I would never hire someone who didn’t know how to do differentiation or elementary integration by hand to do any work that required an understanding or use of calculus.
Now, there is such a thing as overkill in assigning calculus homework; integration problems can be made arbitrarily difficult, and past a certain point I don’t think there’s much educational payoff.
But learning advanced math is hard and requires hard work. Sorry about that. Math students often start off in elementary school finding everything easy and intuitive, but then at some point everyone hits the wall. Same thing can happen when learning a language or developing a physical skill. At that point there is no alternative to working hard in order to make progress past that wall.
This analogy doesn't make any sense. Music is not like math and the music industry is not like industries that are related to calculus.
Music theory is NOT required to play in a band or write a movie score. Check Gustavo Santaolalla's for an example of a great composer with ZERO musical bg.
No, music is not like math in that sense. Music is like math in the sense that carpentry is like music or math, or sports are like music or math. You can't be a spectator, you have to grind thousands upon thousands of exercises to be good in it.
Hehe, honestly I just assume everyone here has a copy of GEB (or has infused it into their being)
But yeah, I read TC's post as being less "you can't play music without formal theory" and more "taking a music appreciation class won't (solely) imbue you with the ability to play violin"
...nor would a course which provides a cursory pass over advanced concepts give you applicable (let alone employable) skills without at least a little of the manual labour needed to really feel the intuitions (?)
There probably _are_ some undiscovered clever ways to teach foundations such that you form useful or higher level synaptic links faster - but does seem you still need to then add some brute force to gain adequate fluency..
PS. I don't think that excludes the possibility of Ramanujans or Dizzy Gillespies existing - whether your hard work is unstructured or structured, top-down or bottom up, the key point is - beyond a superficial level there are eventually "no shortcuts"
Now, there is such a thing as overkill in assigning calculus homework; integration problems can be made arbitrarily difficult, and past a certain point I don’t think there’s much educational payoff.
But learning advanced math is hard and requires hard work. Sorry about that. Math students often start off in elementary school finding everything easy and intuitive, but then at some point everyone hits the wall. Same thing can happen when learning a language or developing a physical skill. At that point there is no alternative to working hard in order to make progress past that wall.