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

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

#131

Post by card_zero »

air7 wrote: post #92
elbear wrote: post #41

Earlier quoted context omitted.

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).

parent comment was a bit tounge-in-cheek but I'll continue the sentiment: You're saying that the curiosity is "natural" hence one is either born with it or not. I think that there is no way around the fact that it will be hard and uncomfortable to mimic the progress of someone that has an innate inclination towards a subject (be it talent or focus or curiosity) artificially.

Hey, that doesn't have to be what "natural curiosity" means. Besides which it makes no sense to say people are born with complex interests. I mean, OK, your genes might incline you a certain way, but that's not the same thing.

Being interested in a subject is massively helpful to learning it. But interest arises circumstantially, it's an emotion. The grim reality that it would be really useful to you to learn a certain subject does not necessarily make you interested in the subject, unfortunately. (Perhaps "financially interested", but that's something else.)

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

#132

Post by nestes »

tgv wrote: post #121
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 cannot agree. It's just "feel-good thinking." "Everybody can do everything." Well, that's simply not true. I'm fairly sure you (yes, you in particular) can't run the 100m in less than 10s, no matter how hard you trained. And the biological underpinning of our capabilities doesn't magically stop at the brain-blood barrier. We all do have different brains. I've taught math to psychology students, and they just don't…

Not the original poster, but I want to push back on one thing -- being capable of something and being one of the best in the world at something are hugely different. Forgive me if I'm putting words in your math -- you mentioned "placing the bar for mathematical skill pretty" low but also mentioned running a sub-10s 100m. If, correspondingly, your notion of mathematical success is being Terence Tao, then I envy your ambition.

I do broadly agree with your position that some people are going to excel where others fail. We know there trivially exist people with significant disabilities that will never excel in certain activities. What the variance is on "other people" (a crude distinction) I hesitate to say. And whatever the solution is, if there is even a solution, I'd at least like for the null hypothesis to be "this is possible, we just may need to change our approach or put more time in".

On a slightly more philosophical note, I firmly believe that it is important to believe some things that are not necessarily true -- let's call this "feel-good thinking". If someone is truly putting significant dedicated effort in and not getting results, that is a tragedy. I would, however, greatly prefer that scenario to the one in which people are regularly told, "well, you could just be stupid." That is a self-fulfilling prophecy.

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

#133

Post by setopt »

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…

> 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 easy, then it means you already know this material, and you're wasting your time.

It’s a well-established effect in pedagogics that learning vs. difficulty has a non-monotonic relationship, where you don’t learn efficiently if the material is either too hard or too easy compared to your current level. There is an optimum learning point somewhere in-between where the material is “challenging” – but neither “trivial” nor “insurmountable” – to put it that way.

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

#134

Post by nestes »

sureglymop wrote: post #114

I think for most people the issue is that they never even get to the fun stuff. I remember not really liking math right until university where we had set theory in the first semester, defined the number sets from scratch went on to monoids, groups, rings etc. That "starting from scratch" and defining everything was extremely satisfying!

Interesting, I somewhat of an opposite reaction, although I am certainly not a mathematician. Once everything became definitions, my eyes glazed over - in most cases the rationale for the definitions was not clear and the definitions appeared over-complicated.

It took me some time, but now it's a lot better -- like a little game I somewhat know the rules of. I now accept that mathematicians are often worrying about maximal abstraction or addressing odd pathological corner cases. This allows me to wade through the complexity without getting overwhelmed like I used to.

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

#135

Post by jvanderbot »

diffeomorphism wrote: post #48
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…

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 tim…

Indeed, speed is often read as "smarts" whereas I would maintain it's much more often "preparation". We can't on one hand believe in the plasticity and retrainability of the mind, while simultaneously believing that speed is something only a few are born with. On the nature/nurture scale, I think it's 20/80 or so - but prodigies and geniuses have an interest that keeps them thinking and learning 10x or 100x more than other kids, and a little bump that lets them get started easier and therefore much earlier.

This sets them up for fantastic success very quickly. [1] shows a great example of this.

I'm fond of saying "You can do anything you want, but wanting is the hard part", because to truly be a grandmaster, genius-level mathematician, olympic athlete, etc, requires a dedication and amount of preparation that almost nobody can manage. Starting late, with emotional baggage, kids, and having to spend 5 years relearning how to learn? Forget it.

1. https://danielkarim.com/how-to-become-a-genius-the-polgar-ex...

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

#136

Post by physicsguy »

strken wrote: post #79
physicsguy wrote: post #21

Earlier quoted context omitted.

That certain countries both now and in the past have had significantly higher mathematical ability among the general population and much higher proportions going on to further study suggests that ability isn’t innate but that people don’t choose it. In the Soviet Union more time was spent teaching mathematics and a whole culture developed around mathematics being fun.

Why would ability not be innate just because some people with the ability don't use it? Or more specifically, two of my friends teach special needs children in the 50 to 70 IQ band. Who are we going to blame for them not becoming mathematicians? The teachers, for not unlocking their hidden potential? The kids, for not trying hard enough? Claiming that the only thing holding them back is choice seems as cruel as it is…

Well, I guess what I mean is that most people have some level of general intelligence that when applied correctly can generally give good results in most subjects. In general the people who do well in school do well in everything, even if they have a preference, and as such could do well in most of those subjects if they went on to further study. The evidence tends to be that in lower income countries people push towards subjects more likely to bring financial stability than those they prefer which bears this out somewhat.

There are some extreme cases of course but I’m not sure the general public needs to worry too much about those, most of us aren’t an Einstein nor do we have learning disabilities.

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

#137

Post by sdeframond »

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…

> I strongly believe that the average human being can be exceptional in any niche topic given enough time, dedication and focus.

I respectfully, but strongly, disagree. There's a reason most NBA players are over 2 meters tall, and one does not become taller with time, dedication nor focus.

It might be different for intellectual skills but I am not that sure.

Almost anyone can become decent at almost anything though. Which is good already!

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

#138

Post by mr_mitm »

benreesman wrote: post #123
mr_mitm wrote: post #113

Earlier quoted context omitted.

Wouldn't it then follow that all students of the same teachers end up with the same skill level in math? Not sure that's the case.

Cantor gave his life to the Continuum Hypothesis, Hilbert gave much of his life to similar goals. You’re making an argument somewhat along those lines, but given that I didn’t stipulate a convergence condition your conclusions can be dismissed by me. If it were a valid argument then we’d need Gödel.

Did you mean to reply to another post? I don't follow at all.

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

#139

Post by penguin_booze »

To my mind, the premature formalization of the math is the principal contributor to gas lighting and alienation of people from maths. The reduction of concepts to symbols and manipulation thereof, is an afterthought. It's misguided for them to be introduced to people right at the outset.

People need to speak in plain English [0]. To some mathematicians' assertion that English is not precise enough, I say, take a hike. One need to walk before they can run.

Motivating examples need to precede mathematical methods; formulae and proofs ought to be reserved for the appendix, not page 1.

[0] I mean natural language

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

#140

Post by aeonik »

nestes wrote: post #134
sureglymop wrote: post #114

I think for most people the issue is that they never even get to the fun stuff. I remember not really liking math right until university where we had set theory in the first semester, defined the number sets from scratch went on to monoids, groups, rings etc. That "starting from scratch" and defining everything was extremely satisfying!

Interesting, I somewhat of an opposite reaction, although I am certainly not a mathematician. Once everything became definitions, my eyes glazed over - in most cases the rationale for the definitions was not clear and the definitions appeared over-complicated. It took me some time, but now it's a lot better -- like a little game I somewhat know the rules of. I now accept that mathematicians are often worrying about m…

My dad always told me growing up today math was like a game and a puzzle, and I hated that. I also hated math at the time. It felt more like torture than a game.

I didn't fall in love with math until Statistics, Discrete Math, Set Theory and Logic.

It was the realization that math is a language that can be used to describe all the patterns of real world, and help cut through bullshit and reckon real truths about the world.

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