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Probably figuring out math paths. I've always been bad at remembering formulas or theorems, but that's because I remember how to get to that formula or prove th

Probably figuring out math paths. I've always been bad at remembering formulas or theorems, but that's because I remember how to get to that formula or prove the theorem instead. E.g. in grade school, I never memorized the sum of cubes formula, but I knew a^3 + b^3 = a^3 - (-b)^3 and that the difference of cubes looks somewhat like the difference of squares, so I guessed a^3 + b^3 = (a - (-b))(<and figured out the rest.>) Having the whole path in your head makes it so you don't get stuck when you forget one step, and also makes it easier to make new connections.

_benj 2y ago
I like this idea. I'm guessing that it might be trying to learn math from proofs? or from first principles? I'm trying to figure out how could I find where to learn math in this way?
- programjames 2y ago
  Yes, essentially proofs. I didn't use that word because people usually only write proofs for each other, so they end up being much more formal. When it's just yourself, chicken scratch in the margins is good enough. How you learn it this way is by always asking yourself how something came to be. Why is the area of a triangle base * height / 2? Why are you allowed to subtract two equations (but not inequalities)? What is this curly thing in Green's theorem? How did someone come up with that alternating sign formula for the determinant? And so on.
- programjames 2y ago
  AoPS was a good resource when I was a kid: https://artofproblemsolving.com/community https://artofproblemsolving.com/community
kgeist 2y ago
Same, I never could remember formulas at school. What I did during exams was to "reconstruct" them visually: I'd sketch a few graphs at various random points and try to derive a formula from there. I also remember successfully solving problems by sketching geometrical shapes (triangles, lines etc.) and just trying to reason with basic logic + trial and error. I realized back then that a lot of math (at least school-level math) can be grokked if you visualize it geometrically/spatially, as manipulatable objects in space. I don't know why our teachers rarely explained it like that. For most students, it was like strange symbol manipulation rules that you must remember by heart and can't derive from scratch. Currently I'm fascinated with the way neural networks can be understood as a problem of trying to untangle tangled manifolds (to make them linearly separable) by "folding"/distorting space, using basic matrix manipulations... That way it's not magic anymore, it's something which appears so straightforward.

_benj 2y ago

I like this idea. I'm guessing that it might be trying to learn math from proofs? or from first principles? I'm trying to figure out how could I find where to learn math in this way?

programjames 2y ago
Yes, essentially proofs. I didn't use that word because people usually only write proofs for each other, so they end up being much more formal. When it's just yourself, chicken scratch in the margins is good enough. How you learn it this way is by always asking yourself how something came to be. Why is the area of a triangle base * height / 2? Why are you allowed to subtract two equations (but not inequalities)? What is this curly thing in Green's theorem? How did someone come up with that alternating sign formula for the determinant? And so on.
programjames 2y ago
AoPS was a good resource when I was a kid: https://artofproblemsolving.com/community https://artofproblemsolving.com/community

kgeist 2y ago

Same, I never could remember formulas at school. What I did during exams was to "reconstruct" them visually: I'd sketch a few graphs at various random points and try to derive a formula from there. I also remember successfully solving problems by sketching geometrical shapes (triangles, lines etc.) and just trying to reason with basic logic + trial and error. I realized back then that a lot of math (at least school-level math) can be grokked if you visualize it geometrically/spatially, as manipulatable objects in space. I don't know why our teachers rarely explained it like that. For most students, it was like strange symbol manipulation rules that you must remember by heart and can't derive from scratch. Currently I'm fascinated with the way neural networks can be understood as a problem of trying to untangle tangled manifolds (to make them linearly separable) by "folding"/distorting space, using basic matrix manipulations... That way it's not magic anymore, it's something which appears so straightforward.