All of these things are a genuinely useful notation that really does help you solve problems, but people teach them without highlighting the ambiguity implicit in them as if you can just learn the formal (and informal!) rules of manipulating the notation and it'll be fine.
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On the notation being poorly explained... a thing I've always struggled with is I get rows and columns mixed up in matrices until I realised (in the last year!) that matrices follow the way you would write it out in simultaneous equation.
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Matrices are "just" simultaneous equations, and you certainly get taught how to solve simultaneous equations with matrices, but the way that matrices are the natural compact notation for simultaneous equations is really not spelled out enough in intro classes.
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Two big mathematical skills are finding the: 1) right concepts to express the problem clearly. 2) right notation to express arguments about those concepts compactly. Different areas express each of these to different degrees, and people can be good at one but not the other.
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A big failure of mathematics education is that we expect people to figure the thought process out on their own by just showing them examples of it. This only works sometimes. Linear algebra is a bit where it failed for me, because I'm much better at concepts than I am at notation
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Replying to @DRMacIver
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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Replying to @Meaningness @DRMacIver
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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Replying to @Meaningness
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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Replying to @DRMacIver @Meaningness
I don't think people can understand concepts without examples, and I don't think people can remember examples without concepts - everyone will do better if you can properly shore each up with the other, but different people will do differently well depending on which you fail at.
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Replying to @DRMacIver @Meaningness
I also think people do much better at concepts if you hold their hand for developing the intuition around them. A lot of the time when students are confused about a concept it's not because the concept is confusing it's because we're shit at teaching the intuition.
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