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there is a constant tension in any field between fox/hedgehog, breadth-first/depth-first competence where in their own contexts they are absolutely correct abou

there is a constant tension in any field between fox/hedgehog, breadth-first/depth-first competence where in their own contexts they are absolutely correct about the supriority of their approaches, but quite wrong outside of it. we could frame these in a general category of Endian conflicts, where depth-firsts think the breadth-firsts are handwavey bullshitters, and breadth-firsts have a finite amount of patience for depth-first's concrete thinking and denial of abstractions. I often state I don't know anything about math as if there's a python library and a wikipedia page that's usually enough for my purposes, and then use a kind of profane math to do stuff instead of the sacred math that seems mostly to be about arguing and telling people what is impossible. Learn math for real, it's admirable and useful, and maybe someone will hire you to turn their handwavings into something someone wants.

mlyle 2y ago
Yes, but since an effort to educate students in math is universal, it's worth thinking about the different outcomes we want: A. To create top flight mathematicians who can push the frontiers of the field forward. Arguably not a whole lot of what we do in K-12 and the first couple of years of college isn't really aimed at this for the most part, since there is such a strong applied math push and the proofs stuff we teach in K-12 is broken. B. To create people competent enough in math to be engineers and scientists. Most math systems are pretty squarely aimed at this. C. To create people competent enough to live a life which tangentially touches mathematics (even if they are in a field like most of finance or accounting or whatever, the amount of mathematics they will do is limited). Here, I think we go pretty far off: getting a person just barely through Algebra 2 or trig doesn't serve them well; you'd be better off teaching them first and foremost not to be scared of mathematical reasoning, about general problem solving ("look, you can just hold up the shape and rotate it!" "we can figure out the length of the board with a compass!"), and strengthening their general arithmetic and lower math skills. I think we need to diversify out from path "B" to do both "A" and "C" better.
- Kim_Bruning 2y ago
  * How about showing kids the intrinsic beauty that can be found in parts of it? * Or the way you can make endless fun puzzles you can solve on a rainy day?
  - mlyle 2y ago
    That's a separate question-- how (especially earlier) math classes should grab student engagement and encourage all students to progress as much as possible. Re: your point: I've said often that I think most math classes should be roughly equal amounts of * Exploration/play/intuition-building * Rote practice of problems, facts, specific problem solving or symbolic manipulation techniques. * Rigor / careful explanation (The exact balance may shift a bit, from younger students doing the first two a bit more and older students doing the latter a lot more). I think if you do this better, more of the populace will happily make it past Algebra 2. But still, even so, others will still get stuck in real understanding somewhere between Algebra I and Geometry, and I think we need to decide what to do then so as to best serve those students.

mlyle 2y ago

Yes, but since an effort to educate students in math is universal, it's worth thinking about the different outcomes we want: A. To create top flight mathematicians who can push the frontiers of the field forward. Arguably not a whole lot of what we do in K-12 and the first couple of years of college isn't really aimed at this for the most part, since there is such a strong applied math push and the proofs stuff we teach in K-12 is broken. B. To create people competent enough in math to be engineers and scientists. Most math systems are pretty squarely aimed at this. C. To create people competent enough to live a life which tangentially touches mathematics (even if they are in a field like most of finance or accounting or whatever, the amount of mathematics they will do is limited). Here, I think we go pretty far off: getting a person just barely through Algebra 2 or trig doesn't serve them well; you'd be better off teaching them first and foremost not to be scared of mathematical reasoning, about general problem solving ("look, you can just hold up the shape and rotate it!" "we can figure out the length of the board with a compass!"), and strengthening their general arithmetic and lower math skills. I think we need to diversify out from path "B" to do both "A" and "C" better.

Kim_Bruning 2y ago
* How about showing kids the intrinsic beauty that can be found in parts of it? * Or the way you can make endless fun puzzles you can solve on a rainy day?
- mlyle 2y ago
  That's a separate question-- how (especially earlier) math classes should grab student engagement and encourage all students to progress as much as possible. Re: your point: I've said often that I think most math classes should be roughly equal amounts of * Exploration/play/intuition-building * Rote practice of problems, facts, specific problem solving or symbolic manipulation techniques. * Rigor / careful explanation (The exact balance may shift a bit, from younger students doing the first two a bit more and older students doing the latter a lot more). I think if you do this better, more of the populace will happily make it past Algebra 2. But still, even so, others will still get stuck in real understanding somewhere between Algebra I and Geometry, and I think we need to decide what to do then so as to best serve those students.