> When my students encounter a math problem they can’t answer, I have them put it in the error log with an explanation of how they did and how they knew how to do it.
I'm gonna assume a step where they learned how to do it?
TFA's method is for incremental discovery expertise. Feynman talks about an inverse, where he maintained a list of interesting problems, and when he learnt a new technique, tried it on each one.
But Feynman's actual breakthroughs came from playfully looking at phenomena.
I think the incremental skills are basics like reading, writing and arithmetic - it's harder to really get to grips with something you've noticed without them.
I mean, Einstein famously didn't have adequate math for special relativity and sought help. He was however the one to notice something.
A library of techniques is a poor substitute for actual thought.
> Einstein famously didn't have adequate math for special relativity and sought help.
General relativity.
Special Relativity is an extremely subtle insight into relatively simple mathematics, general relativity is basically a chasm of rich detail that requires advanced mathematics to express and use.
Beyond examining the mathematics for yourself, you can see evidence for this in that there are very little texts considering "mathematical" (i.e. for mathematicians) on special relativity but many on general relativity. (For what it's worth, I find some of these mathematics-first one's to be remarkably poor pedagogically and far enough away from any useful physics that I sometimes question why some of them exist, although I am a very long way from being a mathematician).
I'm gonna assume a step where they learned how to do it?
TFA's method is for incremental discovery expertise. Feynman talks about an inverse, where he maintained a list of interesting problems, and when he learnt a new technique, tried it on each one.
But Feynman's actual breakthroughs came from playfully looking at phenomena.
I think the incremental skills are basics like reading, writing and arithmetic - it's harder to really get to grips with something you've noticed without them.
I mean, Einstein famously didn't have adequate math for special relativity and sought help. He was however the one to notice something.
A library of techniques is a poor substitute for actual thought.