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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

#181

Post by j7ake »

physicsguy wrote: post #136
strken wrote: post #79

Earlier quoted context omitted.

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 tow…

The extreme case does not imply a binary scenario ie that there are those that can those that cannot.

Rather, learning ability is a continuum. people have varying degrees of ability to learn mathematics. Couple this with environmental factors and society generates a huge variability in mathematical ability that crosses income levels and other demographics.

This view is rejected by many because it is against the push for equality.

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

#182

Post by rafaelero »

wtetzner wrote: post #160
sdeframond wrote: post #137

Earlier quoted context omitted.

> 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.…

> 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. Being tall isn't a skill. I suspect you could be skillful enough at basketball to overcome the hight disadvantage. However, I think most people who might become that skillful see the high disadvantage (plus the general difficulty of becoming a pro basketba…

Intelligence is also not a skill, but the thing that makes you skillful in all cognitive tasks. Just like what height does to basketball players.

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

#183

Post by vundercind »

katzenversteher wrote: post #115
kristopolous wrote: post #56

there's thinking mathematically and then there's being able to fluently read math articles on wikipedia as if they're easier than ernest hemingway. I can do the former and the latter I will insist until my grave is impossible for me.

I have a lot of trouble reading math formulas, implemented as code I understand most stuff though. Is there a good math book or something similar that teaches things using code or helps translating formulas to code?

I realized some time in middle age that I have to convert formulas and equations to steps and things happening to something "passing through" each step—to algorithms. It's painful and slow and also the only way I stand a chance in hell of reading mathematical writing.

That's probably why math writing largely makes me feel dyslexic, while programming came naturally. And why I hate Haskell and find it painful to read even though I understand the "hard" concepts behind it just fine—it's the form of it I can't deal with, not the ideas.

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

#184

Post by throw4847285 »

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!

In college I took Formal Logic II as it fulfilled requirements in both my Comp Sci and Phil major. It turned out that PHIL 104 was cross listed as MATH 562, because the professor who taught Logic I was allowed to teach whatever he wanted for the followup class. I had technically taken the prereq, which was a basic CS logic course, but I was in way over my head. It was one of the most fun courses I took in college.

We were given the exact text of the final exam weeks in advance, and were allowed to do anything at all to prepare, including collaborating with the other students or asking other professors (who couldn't make heads or tails of it). The goal was to be able to answer 1 or 2 out of the 10 questions on the exam, and even if you couldn't you got a B+ at minimum.

I wish I had a better memory, but I believe one of the questions I successfully answered was to prove Post's Theorem using Turing machines? The problem is, I never used the knowledge from that class again, but to this day I still think about it. It would be amazing to go back and learn more about that fascinating intersection of philosophy and computer science.

What I loved the most was that it combined hard math with the kind of esoteric metaphysical questions about mathematics which many practitioners despise because they feel like it undermines their work. It turns out, when you go that deep it's impossible not to touch on the headier stuff.

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

#185

Post by ai4eva »

DiscourseFan wrote: post #7

This guy is unbelievably French (I mean in his intellectual character). Here I was expecting a kind of rehash of the 20th century movements of pure math and high modernism[0], but instead we get a frankly Hegelian concept of math or at least a Hegel filtered through 20th and 21st century French philosophy. [0] https://news.ycombinator.com/item?id=41962944

there is a debate between the intuitionists, formalists, and the symbolists nicely captured in the intro chapter of Heyting's Intuitionism.

constructive mathematics is close to computation and programming. and many including myself have a natural feel or intuition for it. A majority of euclids elements, and galois's original proof are constructive in nature.

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

#186

Post by bunderbunder »

penguin_booze wrote: post #139

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…

The ironic thing is, I swear that this must have been how math (at least more advanced math) was taught a century ago. Or at least, nowadays I've taken to relying on textbooks from the early-to-mid 20th century to learn new math. Maybe it's survivor bias and the only textbooks from back then that anyone remembers are the good ones.

I hate new textbooks because they're so built around instant gratification. They just come out and tell you how to solve the problem without building the solution up in any way. Maybe afterward they take a swipe at telling you how it works, but that's just completely the wrong way around IMO. It robs me of the chance to mull things over, try to anticipate how this will all come together in the end, and generally have my own "aha" moments along the way.

But, getting back to what you say, I think that it also engenders this tendency to reduce math to symbol manipulation. Because if they give you the formula in the first paragraph, then all subsequent explanation is going to end up being anchored to that formula. And IMO that's just completely wrong. Mathematical notation is at its best when it's a formalizing tool and mnemonic device for cementing concepts you already mostly understand. It's at its worst when it's being used as the primary communication channel.

(It's also an essential tool for actually performing any kind of symbolic reasoning such as algebraic manipulation, of course, but I'm mainly thinking about pedagogical uses here.)

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

#187

Post by Retric »

kdfjgbdfkjgb wrote: post #170
air7 wrote: post #92

Earlier quoted context omitted.

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.

> You're saying that the curiosity is "natural" hence one is either born with it or not. Why does curiosity being natural necessarily mean some people are born without it? It could also mean everyone (or every average human) is born with it, and overtime it gets pushed out of people.

Some infants explore vastly more than others.

So the minimum might not be zero, but it isn’t some fixed quantity.

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

#188

Post by nemo »

rafaelero wrote: post #182
wtetzner wrote: post #160

Earlier quoted context omitted.

> 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. Being tall isn't a skill. I suspect you could be skillful enough at basketball to overcome the hight disadvantage. However, I think most people who might become that skillful see the high disadvantage (plus the general difficulty of becoming a pro basketba…

Intelligence is also not a skill, but the thing that makes you skillful in all cognitive tasks. Just like what height does to basketball players.

>Intelligence is also not skill, but the thing that makes you skillful in all cognitive tasks.

Careful with that "all", even the most highly intelligent humans still have peaks and deficits in different domains.

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

#189

Post by davidbessis »

tgv wrote: post #161
chipdart wrote: post #147

Earlier quoted context omitted.

> cannot agree. It's just "feel-good thinking." Not really. There's nothing inherently special about people who dedicated enough time to learn a subject. > "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. What a bad comparison. So far in human history there were less than 200 people who ran 10…

> Not really. There's nothing inherently special about people who dedicated enough time to learn a subject. "You didn't work hard enough." People really blame you for that, not for lacking talent. > So far in human history there were less than 200 people who ran 100m in less than 10s. And many millions have tried. There may be 200 people who can run it under 10s, but there are thousands that can run it under 11s, and…

I'm the author of what you've just described as clickbait.

Interestingly, the 100m metaphor is extensively discussed in my book, where I explain why it should rather lead to the exact opposite of your conclusion.

The situation with math isn't that there's a bunch of people who run under 10s. It's more like the best people run in 1 nanosecond, while the majority of the population never gets to the finish line.

Highly-heritable polygenic traits like height follow a Gaussian distribution because this is what you get through linear expression of many random variations. There is no genetic pathway to Pareto-like distribution like what we see in math — they're always obtained through iterated stochastic draws where one capitalizes on past successes (Yule process).

When I claim everyone is capable of doing math, I'm not making a naive egalitarian claim.

As a pure mathematician who's been exposed to insane levels of math "genius" , I'm acutely aware of the breadth of the math talent gap. As explained in the interview, I don't think "normal people" can catch up with people like Grothendieck or Thurston, who started in early childhood. But I do think that the extreme talent of these "geniuses" is a testimonial to the gigantic margin of progression that lies in each of us.

In other words: you'll never run in a nanosecond, but you can become 1000x better at math than you thought was your limit.

There are actual techniques that career mathematicians know about. These techniques are hard to teach because they’re hard to communicate: it's all about adopting the right mental attitude, performing the right "unseen actions" in your head.

I know this sounds like clickbait, but it's not. My book is a serious attempt to document the secret "oral tradition" of top mathematicians, what they all know and discuss behind closed doors.

Feel free to dismiss my ideas with a shrug, but just be aware that they are fairly consensual among elite mathematicians.

A good number of Abel prize winners & Fields medallists have read my book and found it important and accurate. It's been blurbed by Steve Strogatz and Terry Tao.

In other words: the people who run the mathematical 100m in under a second don't think it's because of their genes. They may have a hard time putting words to it, but they all have a very clear memory of how they got there.

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

#190

Post by 1980phipsi »

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…

I find it is good to go back to things you struggled with in the past and come at them with a new and broader understanding.
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