I remember a supervision (tutorial) back at university where a supervisor got really exasperated us for pointing out that something in an equation was actually a 1x1 matrix and not a scalar and that this meant that the operations being done were technically invalid.
I’ve discussed this specifically with several people who teach linear algebra professionally, and they all said that most students learn better bottom-up: examples first, then algorithms, then eventually maybe concepts (but for a lot of them the concepts are just confusing).
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I can’t learn that way; I have to get the concepts first, then the algorithms, then work examples. My assumption that everyone would learn math better if only the explanations came first is apparently an instance of “typical mind fallacy.”
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My ideal approach to teaching maths is a kind of concept sandwich: * examples of problems * hand wavy idea about what a solution might look like * work through what the precise version of that must look like * examples of solutions with a bunch of cross linking and checking.
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