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Everyone is capable of, and can benefit from, mathematical thinking

quantamagazine.org

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Re: Everyone is capable of, and can benefit from, mathematical thinking

#41

Post by elbear »

shrubhub wrote: post #38
junto wrote: post #23

Earlier quoted context omitted.

> I agree with the sentiment of this. I think our obsession with innate ~~mathematical~~ skill and genius is so detrimental to the growth mindset that you need to have in order to learn things. I strongly believe that the average human being can be exceptional in any niche topic given enough time, dedication and focus. The author of the book has picked out mathematics because that was what he was interested in. The r…

So you're saying success at maths isn't an inbuilt ability. Instead, it depends on an (inbuilt) ability to hyper focus... Which you are just born with?

Not even that. It depends on the learned ability to stop pushing yourself when your focus is wavering. That's how you develop aversion towards the topic. Let your natural curiosity draw you to particular topics (that's why you might have a winding road through the subject).

Re: Everyone is capable of, and can benefit from, mathematical thinking

#42

Post by chipdart »

gsabo wrote: post #15

I agree with the sentiment of this. I think our obsession with innate mathematical skill and genius is so detrimental to the growth mindset that you need to have in order to learn things. I've been working a lot on my math skills lately (as an adult). A mindset I've had in the past is that "if it's hard, then that means you've hit your ceiling and you're wasting your time." But really, the opposite is true. If it's e…

> I agree with the sentiment of this. I think our obsession with innate mathematical skill and genius is so detrimental to the growth mindset that you need to have in order to learn things.

I would argue something different. The "skill" angle is just thinly veiled ladder-pulling.

Sure, math is hard work, and there's a degree of prerequisites that need to be met to have things click, but to the mindset embodied by the cliche "X is left as an exercise for the reader" is just people rejoicing on the idea they can needlessly make life hard for the reader for no reason at all.

Everyone is familiar with the "Ivory tower" cliche, but what is not immediately obvious is how the tower aspect originates as a self-promotion and self-defense mechanism to sell the idea their particular role is critical and everyone who wishes to know something is obligated to go through them to reach their goals. This mindset trickles down from the top towards lower levels. And that's what ultimately makes math hard.

Case in point: linear algebra. The bulk of the material on the topic has been around for many decades, and the bulk of the course material,l used to teach that stuff, from beginner to advanced levels, is extraordinarily cryptic and mostly indecipherable. But then machine learning field started to take off and suddenly we started to see content addressing even advanced topics like dimensionality reduction using all kinds of subspace decomposition methods as someting clear and trivial. What changed? Only the type of people covering the topic.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#43

Post by w10-1 »

Not sure this article captured it for me.

Plato's Meno has Socrates showing that even a slave can reason mathematically.

It's not really math alone but modeling more generally that activates people's reasoning. Math and logic are just those models that are continuous+topological and discrete+logic-operation variants, both based in dimension/orthogonality. But all modeling is over attribution - facts, opinions, etc., and there's a lot of modeling with a healthy dose of salience - heuristics, emotions, practice, etc. Math by design is salience-free (though it incorporates goals and weights), so it's the perspective and practice that liberates people from bias and assumptions. In that respect it can be beautiful, and makes other more conditioned reasoning seem tainted (but it has to work harder to be relevant).

However, experts can project mathematical models onto reality. Hogwash about quantum observer effects and effervescent quantum fields stem from projecting the assumptions required to do the math (or adopt the simplifying forms). Yes, the model is great at predictions. No, it doesn't say what else is possible, or even what we're seeing (throwing baseballs at the barn, horses run out, so barns are made of horses...). Something similar happens with AI math: it can generate neat output, so it must be intelligent. The impulse is so strong that adherents declare that non-symbolic thinking is not thinking at all, and discount anything unquantifiable (in discourse at least). Assuming what you're trying to prove is rarely helpful, but very easy to do accidentally when tracking structured thinking.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#44

Post by quus »

I’m actually interested in the “can benefit from” claim in this title. I don’t particularly doubt that most people could become reasonably good at math, but I wonder how much of the juice is worth the squeeze, and how juicy it is on the scale from basic arithmetic up to the point where you’re reading papers by June Huh or Terry Tao.

As anti-intellectual as it sounds, you could imagine someone asking, is it worth devoting years of your life to study this subject which becomes increasingly esoteric and not obviously of specific benefit the further you go, at least prima facie? Many people wind up advocating for mathematics via aesthetics, saying: well it’s very beautiful out there in the weeds, you just have to spend dozens of years studying to see the view. That marketing pitch has never been the most persuasive for me.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#45

Post by globalnode »

gsabo wrote: post #15

I agree with the sentiment of this. I think our obsession with innate mathematical skill and genius is so detrimental to the growth mindset that you need to have in order to learn things. I've been working a lot on my math skills lately (as an adult). A mindset I've had in the past is that "if it's hard, then that means you've hit your ceiling and you're wasting your time." But really, the opposite is true. If it's e…

As a kid I was also terrible at maths, then later became obsessed with it as an adult because I didn't understand it, just like OP. It was the (second) best thing I've ever done! The world becomes a lot more interesting.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#46

Post by lynx23 »

faangguyindia wrote: post #35
gsabo wrote: post #15

I agree with the sentiment of this. I think our obsession with innate mathematical skill and genius is so detrimental to the growth mindset that you need to have in order to learn things. I've been working a lot on my math skills lately (as an adult). A mindset I've had in the past is that "if it's hard, then that means you've hit your ceiling and you're wasting your time." But really, the opposite is true. If it's e…

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Re: Everyone is capable of, and can benefit from, mathematical thinking

#47

Post by guerrilla »

quus wrote: post #44

I’m actually interested in the “can benefit from” claim in this title. I don’t particularly doubt that most people could become reasonably good at math, but I wonder how much of the juice is worth the squeeze, and how juicy it is on the scale from basic arithmetic up to the point where you’re reading papers by June Huh or Terry Tao. As anti-intellectual as it sounds, you could imagine someone asking, is it worth devo…

Is it worth it to be able to think better, have a growth mindset and learn how to learn? Yes. Everyone can benefit from that. Pushing on into higher math? No, very few people can benefit from that.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#48

Post by diffeomorphism »

junto wrote: post #23
gsabo wrote: post #15

I agree with the sentiment of this. I think our obsession with innate mathematical skill and genius is so detrimental to the growth mindset that you need to have in order to learn things. I've been working a lot on my math skills lately (as an adult). A mindset I've had in the past is that "if it's hard, then that means you've hit your ceiling and you're wasting your time." But really, the opposite is true. If it's e…

> I agree with the sentiment of this. I think our obsession with innate ~~mathematical~~ skill and genius is so detrimental to the growth mindset that you need to have in order to learn things. I strongly believe that the average human being can be exceptional in any niche topic given enough time, dedication and focus. The author of the book has picked out mathematics because that was what he was interested in. The r…

Caveat here is that "talent" and "dedication" is linked to speed at least in the beginning. For instance, any student can learn calculus given enough time and advice even starting from scratch. However, the syllabus wants all this to happen in one semester.

This gives you vicious and virtuous cycles: Students' learning speed increases with time and past success. So "talented" students learn quickly and have extra time to further explore and improve, leading to further success. Students who struggle with the time constraint are forced to take shortcuts like memorizing "magic formulas" without having time to really understand. Trying to close that gap is very hard work.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#49

Post by quus »

guerrilla wrote: post #47
quus wrote: post #44

I’m actually interested in the “can benefit from” claim in this title. I don’t particularly doubt that most people could become reasonably good at math, but I wonder how much of the juice is worth the squeeze, and how juicy it is on the scale from basic arithmetic up to the point where you’re reading papers by June Huh or Terry Tao. As anti-intellectual as it sounds, you could imagine someone asking, is it worth devo…

Is it worth it to be able to think better, have a growth mindset and learn how to learn? Yes. Everyone can benefit from that. Pushing on into higher math? No, very few people can benefit from that.

Math doesn’t seem to me the only source of thinking clearly, or learning how to learn, etc. And if I’m searching for an aesthetic high, there are definitely better places to look — and ones that don’t require such a long runway.

Re: Everyone is capable of, and can benefit from, mathematical thinking

#50

Post by defrost »

quus wrote: post #44

I’m actually interested in the “can benefit from” claim in this title. I don’t particularly doubt that most people could become reasonably good at math, but I wonder how much of the juice is worth the squeeze, and how juicy it is on the scale from basic arithmetic up to the point where you’re reading papers by June Huh or Terry Tao. As anti-intellectual as it sounds, you could imagine someone asking, is it worth devo…

I'll second guerrilla - you can absolutely benefit from mathematical thinking without pushing into territory higher than undergaduate studies.

You can even benefit from the thinking taught in good high school coursework (or studying online).

At an arithmetic, bookkeeping level you can better appreciate handling finances and the seductive pitfalls surrounding wagers (gambling, betting, risk taking).

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