When I was a Computer Science grad student at the University of Pennsylvania, my most challenging courses in Mathematics were taught by Professor of Philosophy Scott Weinstein. Scott taught mathematically intense courses in recursion theory, model theory and computational learning. In every one of those courses Scott would walk in carrying nothing but a small tin of Altoids mints, and dive into complex mathematical proofs using chalk and blackboard. Not once did I see him bring notes or reference texts. This was also the case for the course on the theory of finite models that Scott taught by way of learning the subject for his own research.
In my teens, around the time my own mathematical education was pivoting from computation to proof, I observed my college friend Krishna Gopinathan figure things out from first principles. While the teacher would teach us the epsilon-delta formulation of sequence limits as a mantra or a procedure, Krishna was able to discern the intent and recreate the ideas from first principles. He used to call it internalization.
There is the related idea of the Feynman method. Feynman was reported to have said that if you can't reformulate and explain something to a young child, you probably have not understood it. Feynman gave a set of lectures on his formulation of quantum electrodynamics with the aim of making the key ideas accessible to a non-technical audience.
When my children were in school, I emphasized learning things along these lines. If you understand what is going on with that sleight of hand in algebra called completion of squares, the quadratic formula need not be a black box to you. After all, excepting in rare cases of a flash of inexplicable insight, most concepts and arguments were conceived in some logical fashion by another human being like us. What was their logical trail to the idea?
This is a lifelong process. There are many classical, fundamental ideas I still haven't grasped in that way: I would love to be able to understand the core building blocks of Galois' argument in showing the unsolvability of quintic equations using radicals.
The idea of internalizing concepts and ideas in Mathematics and Physics doesn't have to wait till one gets to college. The goal of Project Shiksha is to help students in India, particularly in underserved communities. Specifically, we plan to create material that helps these students see mathematics as inquiry and inference.
In my teens, around the time my own mathematical education was pivoting from computation to proof, I observed my college friend Krishna Gopinathan figure things out from first principles. While the teacher would teach us the epsilon-delta formulation of sequence limits as a mantra or a procedure, Krishna was able to discern the intent and recreate the ideas from first principles. He used to call it internalization.
There is the related idea of the Feynman method. Feynman was reported to have said that if you can't reformulate and explain something to a young child, you probably have not understood it. Feynman gave a set of lectures on his formulation of quantum electrodynamics with the aim of making the key ideas accessible to a non-technical audience.
When my children were in school, I emphasized learning things along these lines. If you understand what is going on with that sleight of hand in algebra called completion of squares, the quadratic formula need not be a black box to you. After all, excepting in rare cases of a flash of inexplicable insight, most concepts and arguments were conceived in some logical fashion by another human being like us. What was their logical trail to the idea?
This is a lifelong process. There are many classical, fundamental ideas I still haven't grasped in that way: I would love to be able to understand the core building blocks of Galois' argument in showing the unsolvability of quintic equations using radicals.
The idea of internalizing concepts and ideas in Mathematics and Physics doesn't have to wait till one gets to college. The goal of Project Shiksha is to help students in India, particularly in underserved communities. Specifically, we plan to create material that helps these students see mathematics as inquiry and inference.
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