I think we are teaching mathematics the wrong way around. Instead of "this is how we solve a quadratic function; this is how we graph a function; coincidentally, the graph of a quadratic function is a parabola", start from the parabola; the quadratic function is its specification
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The relationship between arithmetic, linear algebra and geometry comes across as a surprising coincidence, which is like telling a joke backwards while forgetting to mention its punchline.
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It's as if we taught music by starting with finger exercises, then teaching the manipulation of lots of complicated notation, and mentioning in passing that it could theoretically help with all that singing and instrumental playing that we are doing intuitively every day.
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End of conversation
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(cue quotes of Einstein + Feynman describing working using their built up intuition/models)
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