4 ms·

As always, this is written by someone who admits to never having learned to reason abstractly about algebraic objects, because it is hard. Quoting from the OP:

by cs702 4y ago

As always, this is written by someone who admits to never having learned to reason abstractly about algebraic objects, because it is hard. Quoting from the OP: > The traditional arithmetic I learned in the early grades made sense to me because I could relate it back to real-world things. Later, when I was learning how to design, finding the area of a circle proved essential to practical tasks such as sizing hydraulic and pneumatic cylinders. I could do trigonometry by visualizing, for example, the cables on a suspension bridge. But algebra: no. For a different perspective, I recommend reading instead Susan Rigetti (nee Fowler)'s posts about overcoming challenges to learn Math and Physics. She says she had to solve what felt like "millions of exercises" to become familiar with many abstract concepts, until they finally clicked for her and she was able to reason about them. In her words: "Solving problems is the only way to understand mathematics. There's no way around it." https://www.susanrigetti.com/math https://www.susanrigetti.com/math https://www.susanrigetti.com/physics https://www.susanrigetti.com/physics https://www.susanjfowler.com/blog/2016/8/26/from-the-fledgling-physicist-archives-if-susan-can-learn-physics-so-can-you https://www.susanjfowler.com/blog/2016/8/26/from-the-fledgli...

PartiallyTyped 4y ago
I’d argue that people need to be taught basic group theory and abstract algebra first, preferably in the form of puzzles to get familiar with more abstract thinking, and to see things in terms of composition of objects instead a of mixing numeracy in the mix. I think that numbers create all sorts of problems because people focus on them as they are familiar and miss the bigger picture.
- qsort 4y ago
  I don't think it really works, at least not for most people. Abstract algebra (i.e. actual algebra, as opposed to doing useless stuff to poor polynomials) requires one to have developed the mathematical maturity to iron out informal thinking. Good luck explaining "prime" and "irreducible" are not the same. I think a path that would work better is through problem solving, like olympiad problems. But even that definitely isn't for everyone.
- yellowcake0 4y ago
  This reminds me a bit of the New Math pedagogy, popular in the US during the 1950s-1970s. Without the proper context, group theory and abstract algebra are deeply uninteresting subjects. It would be like trying to read Bertrand Russell before you learned your ABCs, ridiculous. It's important to start with numbers because they are familiar.
btilly 4y ago
It is also written by an autistic person. Cognitive barriers are easier to overcome for some people than others.
pj_mukh 4y ago
It really does feel like all of this hand-wringing around the state micro-managing syllabi (esp in Math), is sort of skirting around the core issue of having better math teachers. This involves paying teachers paying more and treating them with respect? Nah, lets have endless conversations about how whether we should not learn math at all!
- nostromo 4y ago
  We pay our teachers well in the US compared to our better-performing peers. And who doesn't respect teachers? More money isn't the solution.
  - hollandheese 4y ago
    >We pay our teachers well in the US compared to our better-performing peers. And who doesn't respect teachers? Have you been to the US? Because most people here don't respect teachers. And we pay teachers almost nothing.
    - Turing_Machine 4y ago
      > And we pay teachers almost nothing. That is largely not true, though certainly true in some cases. Teacher salaries are set by the local school district and range from "reasonable" to "not great". Overall, teachers make about $51,000/year while bachelor's degree holders as a whole make about $59,600/year. Against that, you have to recognize that the teacher pay is for nine months, while the bachelor's degree number is for twelve months. If the typical bachelor's degree holder worked only nine months, one would expect their salary to be about $44,700, considerably less than the typical teacher. Sources: https://mint.intuit.com/salary/teacher https://mint.intuit.com/salary/teacher https://nces.ed.gov/programs/coe/indicator/cba https://nces.ed.gov/programs/coe/indicator/cba
      - threatofrain 4y ago
        Many teachers have a masters, although places like Florida are looking to change that.
cgriswald 4y ago
> "Solving problems is the only way to understand mathematics. There's no way around it." I'll read the linked documents at some point, but taking it per se, it should be noted that doing problems as a method of understanding may be necessary, but is not sufficient. I had a professor who taught how to solve problems and then assigned way way too many problems. I could manipulate the symbols, but I didn't grok what I was doing until I picked up a different text and started learning on my own. How many problems are needed definitely varies from person to person, and I'm not entirely convinced there aren't other ways to understand mathematics; but I suspect problems are the fastest and easiest.
- jalk 4y ago
  Yes - building intuition around a subject is key (the aha moment perhaps), which is exactly what Grandin struggles with in the abstract disciplines.
2devnull 4y ago
This is something I’ve always wondered about. It seems individual differences in innate capacity explains a lot when it comes to math ability, but don’t young math prodigies also need to do the brute force time consuming stuff too? Does it take them as many practice problems to learn to differentiate polynomials or are they able to skip over much of the drudgery somehow? If so, how? There are many concepts which require exposure and practice in order to master, so how can one speed up that part as some children seem able to do. In music I’ve heard the answer is that no, you can’t speed it up, you have to do the practice. But then how to explain the Mozarts of the world who somehow seem to have, innately, the knowledge that for others requires training and time?
- Jensson 4y ago
  > Does it take them as many practice problems to learn to differentiate polynomials or are they able to skip over much of the drudgery somehow? If so, how? Why would it take everyone the same amount? It isn't like there is a natural law that you need to drill the same problem 20 times to understand it, why wouldn't someone's brain be able to process it and understand it after doing it the first time or even just looking at someone else doing it?
- zozbot234 4y ago
  > But then how to explain the Mozarts of the world who somehow seem to have, innately, the knowledge that for others requires training and time? Those kids didn't really start out with "innate" knowledge, or anything like that. They get intensive training starting at a very early age, that's how they do better than others. Also, practice is not necessarily boring. When you're actively learning/mastering a skill at close to optimum effectiveness, practice is quite enjoyable and it's easy to rack up a lot of it.
  - 2devnull 4y ago
    I suspect that’s true, it sort of has to be, but then it seems to contradict various anecdotes. There’s a notion of the prodigy as a different type of person, a qualitative difference in kind rather than a difference in degree (of effort, iq, focus etc).
xtiansimon 4y ago
I believe most people who underperform in math had bored and uninspired teachers in their education. I found a joy with math only by accidentally reading biography of Newton, Godel Escher Bach, a book about Cantor, and a compendium of math papers for educators. Then I could see all my reasons for avoiding math were excuses for poor insight to the beauty and interrelatedness of mathematics. You do the work for intimate understanding for each move. If I could learn mathematics like learning sports (simple ideas worked into muscle memory) I would appreciate the result, and you can keep the abstract problems. I’d rather solve real problems.

by cs702 4y ago

PartiallyTyped 4y ago

I’d argue that people need to be taught basic group theory and abstract algebra first, preferably in the form of puzzles to get familiar with more abstract thinking, and to see things in terms of composition of objects instead a of mixing numeracy in the mix. I think that numbers create all sorts of problems because people focus on them as they are familiar and miss the bigger picture.

qsort 4y ago
I don't think it really works, at least not for most people. Abstract algebra (i.e. actual algebra, as opposed to doing useless stuff to poor polynomials) requires one to have developed the mathematical maturity to iron out informal thinking. Good luck explaining "prime" and "irreducible" are not the same. I think a path that would work better is through problem solving, like olympiad problems. But even that definitely isn't for everyone.
yellowcake0 4y ago
This reminds me a bit of the New Math pedagogy, popular in the US during the 1950s-1970s. Without the proper context, group theory and abstract algebra are deeply uninteresting subjects. It would be like trying to read Bertrand Russell before you learned your ABCs, ridiculous. It's important to start with numbers because they are familiar.

btilly 4y ago

It is also written by an autistic person. Cognitive barriers are easier to overcome for some people than others.

pj_mukh 4y ago

It really does feel like all of this hand-wringing around the state micro-managing syllabi (esp in Math), is sort of skirting around the core issue of having better math teachers. This involves paying teachers paying more and treating them with respect? Nah, lets have endless conversations about how whether we should not learn math at all!

nostromo 4y ago
We pay our teachers well in the US compared to our better-performing peers. And who doesn't respect teachers? More money isn't the solution.
- hollandheese 4y ago
  >We pay our teachers well in the US compared to our better-performing peers. And who doesn't respect teachers? Have you been to the US? Because most people here don't respect teachers. And we pay teachers almost nothing.
  - Turing_Machine 4y ago
    > And we pay teachers almost nothing. That is largely not true, though certainly true in some cases. Teacher salaries are set by the local school district and range from "reasonable" to "not great". Overall, teachers make about $51,000/year while bachelor's degree holders as a whole make about $59,600/year. Against that, you have to recognize that the teacher pay is for nine months, while the bachelor's degree number is for twelve months. If the typical bachelor's degree holder worked only nine months, one would expect their salary to be about $44,700, considerably less than the typical teacher. Sources: https://mint.intuit.com/salary/teacher https://mint.intuit.com/salary/teacher https://nces.ed.gov/programs/coe/indicator/cba https://nces.ed.gov/programs/coe/indicator/cba
    - threatofrain 4y ago
      Many teachers have a masters, although places like Florida are looking to change that.

cgriswald 4y ago

> "Solving problems is the only way to understand mathematics. There's no way around it." I'll read the linked documents at some point, but taking it per se, it should be noted that doing problems as a method of understanding may be necessary, but is not sufficient. I had a professor who taught how to solve problems and then assigned way way too many problems. I could manipulate the symbols, but I didn't grok what I was doing until I picked up a different text and started learning on my own. How many problems are needed definitely varies from person to person, and I'm not entirely convinced there aren't other ways to understand mathematics; but I suspect problems are the fastest and easiest.

jalk 4y ago
Yes - building intuition around a subject is key (the aha moment perhaps), which is exactly what Grandin struggles with in the abstract disciplines.

2devnull 4y ago

This is something I’ve always wondered about. It seems individual differences in innate capacity explains a lot when it comes to math ability, but don’t young math prodigies also need to do the brute force time consuming stuff too? Does it take them as many practice problems to learn to differentiate polynomials or are they able to skip over much of the drudgery somehow? If so, how? There are many concepts which require exposure and practice in order to master, so how can one speed up that part as some children seem able to do. In music I’ve heard the answer is that no, you can’t speed it up, you have to do the practice. But then how to explain the Mozarts of the world who somehow seem to have, innately, the knowledge that for others requires training and time?

Jensson 4y ago
> Does it take them as many practice problems to learn to differentiate polynomials or are they able to skip over much of the drudgery somehow? If so, how? Why would it take everyone the same amount? It isn't like there is a natural law that you need to drill the same problem 20 times to understand it, why wouldn't someone's brain be able to process it and understand it after doing it the first time or even just looking at someone else doing it?
zozbot234 4y ago
> But then how to explain the Mozarts of the world who somehow seem to have, innately, the knowledge that for others requires training and time? Those kids didn't really start out with "innate" knowledge, or anything like that. They get intensive training starting at a very early age, that's how they do better than others. Also, practice is not necessarily boring. When you're actively learning/mastering a skill at close to optimum effectiveness, practice is quite enjoyable and it's easy to rack up a lot of it.
- 2devnull 4y ago
  I suspect that’s true, it sort of has to be, but then it seems to contradict various anecdotes. There’s a notion of the prodigy as a different type of person, a qualitative difference in kind rather than a difference in degree (of effort, iq, focus etc).

xtiansimon 4y ago

I believe most people who underperform in math had bored and uninspired teachers in their education. I found a joy with math only by accidentally reading biography of Newton, Godel Escher Bach, a book about Cantor, and a compendium of math papers for educators. Then I could see all my reasons for avoiding math were excuses for poor insight to the beauty and interrelatedness of mathematics. You do the work for intimate understanding for each move. If I could learn mathematics like learning sports (simple ideas worked into muscle memory) I would appreciate the result, and you can keep the abstract problems. I’d rather solve real problems.