Oh it can be so much worse than that—actively pushing students away from that kind of understanding. I’ve had math teachers mark answers wrong because I (correctly) derived a rule I’d forgotten instead of phrasing it the way they taught it, or because they couldn’t follow the derivation. Before college, I can think of maybe two of my teachers who actually seemed to understand high school math in any deeper way.
I think this post is great for here on LW, but if someone wanted to actually start teaching students to understand math more deeply, calling it common sense probably comes off as condescending, because it doesn’t feel that way until you get comfortable with it. There’s a lot to unlearn and for a lot of people it is very intimidating.
Personally I wish we treated math class at least some of the time as a form of play. We make sure to teach kids about jokes and wordplay and do fun science-y demonstrations, but math is all dry and technical. We assign kids books to read like A Wrinkle in Time and The Phantom Tollbooth. But, I don’t think my elementary school teachers had any clue what a tesseract was, or what the Mathemagician and Dodecahedron are all about, and so that whole aspect of these books was just a lost opportunity for all but maybe 3 kids in my grade.
More specifically, the correctness of the proof (at least in the triangles case) is common sense, coming up with the proof is not.
The integrals idea gets sketchy. Try it with e^(1/x). It’s just a composition of functions so reverse the chain rule then deal with any extra terms that come up. Of course, it’s not integrable. There’s not really any utility in overextending common sense to include things that might or might not work. And you’re very close to implying “it’s common sense” is a proof for things that sound obvious but aren’t.
Sure. And I’m of the opinion that it is only common sense after you’ve done quite a lot of the work of developing a level of intuition for mathematical objects that most people, including a significant proportion of high school math teachers, never got.
Oh it can be so much worse than that—actively pushing students away from that kind of understanding. I’ve had math teachers mark answers wrong because I (correctly) derived a rule I’d forgotten instead of phrasing it the way they taught it, or because they couldn’t follow the derivation. Before college, I can think of maybe two of my teachers who actually seemed to understand high school math in any deeper way.
Wanted to add:
I think this post is great for here on LW, but if someone wanted to actually start teaching students to understand math more deeply, calling it common sense probably comes off as condescending, because it doesn’t feel that way until you get comfortable with it. There’s a lot to unlearn and for a lot of people it is very intimidating.
Personally I wish we treated math class at least some of the time as a form of play. We make sure to teach kids about jokes and wordplay and do fun science-y demonstrations, but math is all dry and technical. We assign kids books to read like A Wrinkle in Time and The Phantom Tollbooth. But, I don’t think my elementary school teachers had any clue what a tesseract was, or what the Mathemagician and Dodecahedron are all about, and so that whole aspect of these books was just a lost opportunity for all but maybe 3 kids in my grade.
More specifically, the correctness of the proof (at least in the triangles case) is common sense, coming up with the proof is not.
The integrals idea gets sketchy. Try it with e^(1/x). It’s just a composition of functions so reverse the chain rule then deal with any extra terms that come up. Of course, it’s not integrable. There’s not really any utility in overextending common sense to include things that might or might not work. And you’re very close to implying “it’s common sense” is a proof for things that sound obvious but aren’t.
Sure. And I’m of the opinion that it is only common sense after you’ve done quite a lot of the work of developing a level of intuition for mathematical objects that most people, including a significant proportion of high school math teachers, never got.