[Raidion]

I feel like teaching is mostly a one size fits all endeavor, but people learn and think in different ways, just like a processor is optimized for some operations but not others.

Take simple arithmetic like 12x17. Some people do the long form multiplication (carry the one..), some people say it's 12x10+12x7. Some remember 12x12 from times tables and go 12x12+12x5. Some people make it 24x8+12 => 48x4+12 => 50x4-8+12 etc. Some do it on the abacus in their heads.

All valid, though some are slightly more optimal than others. Good teachers empower alternative solutions and try to help people connect what they already know to what they already understand.

[jasomill]

As someone who only learned multivariable calculus successfully via independent study of exterior differential forms as a result of off-handed comments from the differential geometry professor who taught my first-year calc course, I wholeheartedly agree.

Oh, and 12×17 = 10×17 + 2×17 = 10×17 + 2×10 + 2×7 = 170 + 20 + 14 = 204.