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FWIW I notice how remarkably bad I was taught math at school. When I started to go to university I really noticed how bad it was. At university the jump forwar

FWIW I notice how remarkably bad I was taught math at school. When I started to go to university I really noticed how bad it was. At university the jump forward was really noticeable. For example, at school they would show you a couple of simple explanations about derivative math or integrals, briefly and start with all the formulas. At university I used to have a teacher that started with: history of mathematics, why they were invented, made a point about its primarily practical origins. To explain things, he could most of the time come with real-life instances of application and there were much more often intuitive or geometric interpretations of the techniques used much more often even before starting the explanation itself to have an intuitive idea and visualization of what you were achieving. After that, I noticed that to learn math, the first thing is to develop an intuitive, non-mathy idea of what you are doing and later formalize it. At school and high school they just taught it as almost-memorize tables, apply formulas. Talking about Spain, btw.

robertlagrant 4y ago
When I came across the (obvious, in retrospect) visual explanation for why (a + b)(a + b) == a^2 + 2ab + b^2 I was blown away by how simple it was, and by why on earth I had to just memorise that formula in school.
- germandiago 4y ago
  This is exactly the kind of intuitions I am talking about. Much easier to explain like that.
- n4r9 4y ago
  By "visual explanation" do you mean imagining a square whose sides are length (a + b), and breaking it up into smaller squares and rectangles? Or do you mean mulitplying out the terms: (a + b)(a + b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2 ?
  - robertlagrant 4y ago
    The former, yes.
    - n4r9 4y ago
      It's interesting that some people find that way easier to remember. For me, multiplying out the terms seems faster, probably only because I've practised so many times in my life that I can picture the algebra in my head. It also seems more general, as it's a technique which you have to know anyway. I guess it comes down to being more geometrically minded vs algebraically minded. School should try to cater to both!
      - robertlagrant 4y ago
        I wouldn't use it to calculate, but I would use it to explain why the calculation is what it is.
rcarr 4y ago
This! Identical situation in the UK from my experience. Why maths is taught completely separated from its history and its purpose is baffling. Well actually it’s not: it’s because it serves the teachers. You hammer the formulas and patterns into the kids so they can pass the exams. Your school gets good grades and the principal is happy because he can now market his school as successful and the government inspectors are happy because the school is hitting its metrics. Meanwhile, the kids haven’t actually learned anything. As soon as the exam is passed they forget the formulas and go on with their lives. We should be explaining the story of maths and how it benefitted society. We should be asking kids about their interests and then showing them how mathematical tools can be used in those areas. We need to show kids how maths is tied to real life rather than just presenting them with a boring formulas to memorise.
- RugnirViking 4y ago
  > because it serves the teachers I don't think this is the case. I think it's because the teachers themselves don't have a good grasp on "the story of maths and how it benefitted society". There is a silly meme about asking high school maths teachers "how will we use this in life", and imo it's not because there isn't a good response, but rather that it requires a good understanding of the ways math is actually used. Few high school teachers have actually themselves used the math they teach for anything other than academic exercise. Someone trained in control theory or using physics equations can make things that appear almost magical using maths, and if they are talented, they can find a way to explain it to laypeople. However, people with that combination of talents are desired by just about everybody, from universities to companies, and high schools simply have no way to compete (not least because teaching high schoolers is a soul-crushing job for bureaucratic reasons)
  - rcarr 4y ago
    Yeah this is a very good point. The amount of people who basically go from school to university to school teacher is alarmingly high and bad for society in my opinion. Grown adults who have never known anything but the school system teaching the next generation. It’s ridiculous. I do still think the original point about hitting metrics is a major contributing factor though along with low pay and underfunding.

robertlagrant 4y ago

When I came across the (obvious, in retrospect) visual explanation for why (a + b)(a + b) == a^2 + 2ab + b^2 I was blown away by how simple it was, and by why on earth I had to just memorise that formula in school.

germandiago 4y ago
This is exactly the kind of intuitions I am talking about. Much easier to explain like that.
n4r9 4y ago
By "visual explanation" do you mean imagining a square whose sides are length (a + b), and breaking it up into smaller squares and rectangles? Or do you mean mulitplying out the terms: (a + b)(a + b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2 ?
- robertlagrant 4y ago
  The former, yes.
  - n4r9 4y ago
    It's interesting that some people find that way easier to remember. For me, multiplying out the terms seems faster, probably only because I've practised so many times in my life that I can picture the algebra in my head. It also seems more general, as it's a technique which you have to know anyway. I guess it comes down to being more geometrically minded vs algebraically minded. School should try to cater to both!
    - robertlagrant 4y ago
      I wouldn't use it to calculate, but I would use it to explain why the calculation is what it is.

rcarr 4y ago

This! Identical situation in the UK from my experience. Why maths is taught completely separated from its history and its purpose is baffling. Well actually it’s not: it’s because it serves the teachers. You hammer the formulas and patterns into the kids so they can pass the exams. Your school gets good grades and the principal is happy because he can now market his school as successful and the government inspectors are happy because the school is hitting its metrics. Meanwhile, the kids haven’t actually learned anything. As soon as the exam is passed they forget the formulas and go on with their lives. We should be explaining the story of maths and how it benefitted society. We should be asking kids about their interests and then showing them how mathematical tools can be used in those areas. We need to show kids how maths is tied to real life rather than just presenting them with a boring formulas to memorise.

RugnirViking 4y ago
> because it serves the teachers I don't think this is the case. I think it's because the teachers themselves don't have a good grasp on "the story of maths and how it benefitted society". There is a silly meme about asking high school maths teachers "how will we use this in life", and imo it's not because there isn't a good response, but rather that it requires a good understanding of the ways math is actually used. Few high school teachers have actually themselves used the math they teach for anything other than academic exercise. Someone trained in control theory or using physics equations can make things that appear almost magical using maths, and if they are talented, they can find a way to explain it to laypeople. However, people with that combination of talents are desired by just about everybody, from universities to companies, and high schools simply have no way to compete (not least because teaching high schoolers is a soul-crushing job for bureaucratic reasons)
- rcarr 4y ago
  Yeah this is a very good point. The amount of people who basically go from school to university to school teacher is alarmingly high and bad for society in my opinion. Grown adults who have never known anything but the school system teaching the next generation. It’s ridiculous. I do still think the original point about hitting metrics is a major contributing factor though along with low pay and underfunding.