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I think what you call “parsing” is largely indistinguishable from mastery in a lot of fields, particularly in abstract mathematics

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Knowing the rules of chess doesn’t make you a chess master. Knowing the syntax and semantics of a programming language doesn’t make you a master software architect.

It depends what you include in your definition of "parsing". For example, the chess master looks at the board and "parses" it in a very different way to the amateur player: he includes his knowledge of thousands of games played, analysis of potential future outcomes, and so on.

I disagree. "Parsing" is the first level of understanding. If you are not moving past the parsing level, you have not achieved any kind of mastery.

My experience is that mastery means more like "you have a mental model which gives you 'intuitive' reasons to accurately classify things as true/false and provides some motivation for the reasoning".

An example: you see someone has solved a degree 4 equation by repeatedly applying the quadratic equation, getting 8 solutions. "No way."

Another example: watch a famous baking show and you see somebody put a bunch of different sized pieces of bread in the oven at the same time. Right away: "aren't they going to cook at different rates?" Sure enough, some burned, some raw.

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