How do you learn how a math equation, proof, or algorithm works? What's the fastest way? These questions have persisted in my mind as I go through school. The final answer that I arrived at was simply: work out an example of the algorithm in your head.
The learn-by-example algorithm is completely obvious in hindsight. That is perhaps the best and only way to learn concepts. In his Art of Computer Programming books, the "father of computer science" Donald Knuth said that the only way to understand how an algorithm worked was to give it an example input and see how it worked. When renowned physicist Feynman struggled to understand physics during his stay in Japan, he found that the simplest way to fully understand a physics concept was to construct some physical object that matched what his colleagues were describing, and then analyze the physics of that physical object, no matter how simple or ridiculous. To derive the new theorems of relativity, Einstein carried out specific thought experiments, not abstract mathematics. Every famous man of science has learned by thinking through specific instances! The concept was staring at me right in the face.
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