Comment by gsabo
10 hours ago
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 easy, then it means you already know this material, and you're wasting your time.
> 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 reality is that this rule applies to everything.
The belief that some people have an innate skill that they are born with is deeply unhelpful. Whilst some people (mostly spectrum) do seem have an innate talent, I would argue that it is more an inbuilt ability to hyper focus on a topic, whether that topic be mathematics, Star Trek, dinosaurs or legacy console games from the 1980’s.
I think we do our children a disservice by convincing them that some of their peers are just “born with it”, because it discourages them from continuing to try.
What we should be teaching children is HOW to learn. At the moment it’s a by-product of learning about some topic. If we look at the old adage “feed a man a fish”, the same is true of learning.
“Teach someone mathematics and they will learn mathematics. Teach someone to learn and they will learn anything”.
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.
Thank you for the insight that academic (in a very broad sense) bulk-fixed-time approach does in fact produce both of the cycles, and the gap indeed only widens with time (speaking from personal experience, especially from my life as an undergrad student).
Reminds me of my personal peeve that "studying" should not be "being taught", studying is pursuit of understanding, "being taught" is what happens in primary school (and I'm aware I'm simplifying here).
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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...
> 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!
> 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 basketball player) and take a different path through life. It's also possible that the amount of time that would be needed to grow your skill past the height disadvantage is too long, so it's not feasible to do it to gain a position in the NBA.
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>I strongly believe that the average human being can be exceptional in any niche topic given enough time, dedication and focus.
And this also gives the proponent (you in this case) an excuse to blame a person for not focusing hard enough or not being dedicated enough if they don't grasp the basics, let alone excel.
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).
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The boostrap skill is the ability to obsess over something. To focus and self-reward on anything is a heaven sent. Good thing we do not medicate that if we are unable to get that energy on the road, that base skill.
>The belief that some people have an innate skill that they are born with is deeply unhelpful. Whilst some people (mostly spectrum) do seem have an innate talent, I would argue that it is more an inbuilt ability to hyper focus on a topic, whether that topic be mathematics, Star Trek, dinosaurs or legacy console games from the 1980’s.
Nonsense!
The brain you are born with materially dictates the ceiling of your talent. A person with average ability can with dedication and focus over many years become reasonably good, but a genius can do the same in 1 year and at a young age.
We have an education system which gives an A Grade if you pass the course, but 1 person may put on 5 hours a week and the other works day and night.
What makes you think that "genius" is nature and not nurture? I'd love to see the evidence for this; i'm deeply skeptical.
Edit: I don't mean to argue that there aren't genetics involved in determining aptitude on certain tasks, of course, but the assumption that genius is born and never made feels like a very shallow understanding of the capacity of man.
I've had some success converting people by telling them others had convinced them they were stupid. They usually have one or two things they are actually good at, like a domain they flee to. I simply point out how everything else is exactly like [say] playing the guitar. Eventually you will be good enough to sing at the same time. Clearly you already are a genius. I cant even remember the most basic cords or lyrics because I've never bothered with it.
I met the guitar guy a few years later outside his house. He always had just one guitar but now owned something like 20, something like a hundred books about music. Quite the composer. It looked and sounded highly sophisticated. The dumb guy didn't exist anymore.
But also, some people are stupid, right?
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> The author of the book has picked out mathematics because that was what he was interested in. The reality is that this rule applies to everything.
My first thought when the article got to the dialog between logic and intuition bit was that the same is true for school level physics.
> Whilst some people (mostly spectrum) do seem have an innate talent
I think the only thing in autism that I'd call an innate talent is detail-oriented thinking by default. It'd be the same type of "innate talent" as, say, synesthesia, or schizophrenia: a side effect of experiencing the world differently.
> a side effect of experiencing the world differently
A side effect for which there is a substantial, lifelong, and most importantly wide cost, even if it occasionally confers usually small, usually fleeting, and most importantly narrow advantage.
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> 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.
> 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.
I saw a lot of this when I went to college for engineering, some professors had this ability (or willingness) to make hard things simple, and others did the opposite, it was the same with the books, I dreaded the "exercise for the reader" shit, I don't think I ever did any of those, so those were all proofs I never got.
I think the ML people want to get (a narrow band) of stuff done and ivory towered people want to understand a prove things. ML is applied mathematic. Both are needed.
> I think the ML people want to get (a narrow band) of stuff done and ivory towered people want to understand a prove things. ML is applied mathematic. Both are needed.
I don't agree. First of all, ladder-pulling in math is observed at all levels, not only cutting-edge stuff. Secondly, it's in applied mathematics where pure math takes a queue onto where to focus effort. See how physics drives research into pure math.
> 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.
Absolutely. There's also a pernicious idea that only young people can master complex maths or music. This is a self-fulfilling prophecy - why bother try if you're going to fail due to being old? Or perhaps it's an elitist psy-op, giving the children of wealthy parents further advantage because of course no-one can catch up.
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 get it. I remember the frustration of the section's head when a student asked "what's a square root?" We all know how many of our fellow pupils struggled with maths. I'm not saying they all lacked the capability to learn it, but it can't be the case they all were capable but "it was the teacher's fault". Even then, how do you explain the difference between those who struggled and those who breezed through the material?
Or let's try other topics, e.g. music. Conservatory students study quite hard, but some are better than others, and a select few really shine. "Everyone is capable of playing Rachmaninov"? I don't think so.
So no, unless you've placed the bar for "mathetical skill" pretty low, or can show me proper evidence, I'm not going to believe it. "Everyone is capable of..." reeks of bullshit.
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.
There's a difference between being able to memorize what a square root is and being able to do math - which to mathematicians means being able to organize a proof.
I've found that the people who most believe in math being a genetic ability are the ones who do not work in the symbolic world of modern math, but in the semantic world of whatever the field the math describes is.
The two are rather different.
Strangely enough, you'd be hard pressed to find a mathematician who doesn't know what a square root is.
And I didn't mention genetics. Nature is complicated.
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> 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 100m in less than 10s.
I think you're just reflecting an inflated sense of self worth.
> 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 hundreds of thousands that can run it under 12s. Unless you've got clear evidence that those people can actually run 100m in less than 10s if they simply try harder, I think the experience of practically every athlete is that they hit a performance wall. And it isn't different for maths, languages, music, sculpting (did you ever try that?), etc. Where there are geniuses, there also absolute blockheads.
That's not to say that people won't perform better when they work harder, but the limits are just not the same for everyone. So 'capable of mathematical reasoning' either is some common denominator barely worth mentioning, or the statement simply isn't true. Clickbait, if you will.
This is mostly correct. Working memory plays a huge component in grokking more complicated mathematical components, and IQ itself is separated into performance and verbal IQ (which together constitute your IQ score) and its demonstrably robust. Some people find this easier than others and that is OK.
I dont disagree with the premise that mathematical thinking can benefit anybody, but its absurd notion that everything abstract is teachable and learnable to all is a fantasy of a distinctly left-wing variety, who would have you believe that everything is just social conditioning and human beings dont differ from one-another.
I think most people can become fairly skilled in useful fields if educated properly, and the people who can't are a small group that can be cared for. I agree that even in a better education system, people aren't all going to be equally skilled in the same fields, just that most people can contribute something of value.
Anybody can do everything if we restrict everything to things everyone can do.
> If it's easy, then it means you already know this material, and you're wasting your time.
One thing I'm anticipating from LLM-based tutoring is an adaptive test that locates someone's frontier of knowledge, and plots an efficient route toward any capability goal through the required intermediate skills.
Trying to find the places where math starts getting difficult by skimming through textbooks takes too long; especially for those of us who were last in school decades ago.
>and plots an efficient route toward any capability goal through the required intermediate skills.
LLMs currently can't find efficient paths longer than 5 hops when given a simple itinerary. Expecting them to do anything but a tactical explanation of issues they have seen in training is extremely naive with something as high dimensional as math.
I grow increasingly convinced that the difference in “verbal” and “mathematical” intelligence is in many ways a matter of presentation.
While it’s indisputable that terse symbolic formalisms have great utility, one can capture all the same information verbally.
This is perhaps most evident in formal logic. It’s not hard to imagine a restricted formalized subset of natural language that is amenable to mechanical manipulation that is isomorphic to say modal logic.
And finally, for logic at least, there is something of a third way. Diagrammatic logical systems such as Existential Graphs capture the full power of propositional, predicate, and modal logic in a way that is neither verbal nor conventionally symbolic.
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.
I haven't been able to grasp maths as a kid nor as an adult.
I've tried night classes, tutors, activities. Nothing sticks.
Even the standard 12x tables I struggle at. I want to understand it but my brain just can't understand the non-practicality side of things.
My best friend was like that. Couldn't see the practicality until he got bit by a geology and water science bug. He went from calling me to get help figuring out percentages to doing chemistry equations in his head because he "got" the applicability.
My brother's mom tutors math. One of her insights with a former student was that they were in need of forming some number sense. She started by walking them both out to the street: "how many tires are there on this street of parked cars?" The student, already flummoxed, started panic guessing. So she started with counting.
For times tables, have you developed any intuition around it? For me, times tables are rectangles composed of unit squares and that helps with my intuition. Modern Common Core standards in the US focuses a lot on exposing different mental models to students. And after seeing the same 4x6 enough times your brain will automatically associate that with its solution. Instead of calculating, it is memorized.
My brain doesn't require car tires, geology, or other practical needs: it likes puzzles. I struggle with medical stuff and I can feel my brain switching to meh-mode and hardly anything sticks. I don't know how many times I have been told about the different kinds of sugar and how your body uses that energy and I would still have to look it up.
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This perspective has discouraged so many people from exploring their potential
When I was a young adult, i spent a lot of time on math and physics.
I was initially celebrated for the mathematical talent.
But as life progressed, I my family started seeing me as an academic loser.
Basically, no girls would be interested in me because "mathemetical talent" doesn't help you with that.
And i seen handsome men had more respect from society than spending countless time on math.
So, i later gave up because my family kept pressuring me to attain real success, girls, money and car and i became a programmer.
Funny enough, I was still a loser in societal view doesn't matter I started clearly half a million a year.
So most people don't try hard at math because math is not rewarding, for most people.
It's much better to build physique, music talent, comedic talent, this helps you get girls and respect from peers.
Most people don't try hard at the gym. Most people don't try hard at music. Most people aren't comedians.
This reads like the foreword to the incel handbook.
And that's obviously bad because incels are not real people with feelings.
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"A loser in societal view"... What does that objectively mean? That only reads like you had or have a low sense of self worth. It must've been your perceived definition of what society is because how could you have come to such a conclusion? I think I'd actually subconsciously tend more to viewing someone as "a loser" if they made such a statement because it comes off as self victimization (without an apparent explanation to an outside observer).
And what's the shtick about girls? What are and were you looking for, love and a genuine relationship or attention to compensate for something? Personally I think your values and personality are what matter most and personality is usually what people fall in love with. Though charisma can help a lot to get the ball rolling. Most of what it takes is to treat people normally and nicely and you will have as much of a chance to find love as most people.
Though respect from peers and attention from women ideally shouldn't be your driving force. I think curiosity and passion are much better driving forces that don't involve such external factors and possibilities for insecurities.
Your post reads as if it expresses a frustration and a sense of entitlement. You may not be intrinsically entitled to the things you think you are. Think about that for a bit and try to be rational.
> ...my family kept pressuring me to attain real success, girls, money and car and i became a programmer.
As a child of the 80s and 90s, "getting girls as a programmer" made me snort. Nerds do seem to have it a bit better now; the money/financial security of software development helps. But as a whole, we developers are still less socially capable than our sales/hr/marketing counterparts
You will stay a loser as long as you care about what some fictional mystical society thinks of you.
Do the stuff you're good at, provide for your family, earn the respect of your peers and forget about the rest.
I assume OP is an Indian. And from what I've observed, Indian society is highly paternalistic and status-seeking in nature. Parents demand marriage and grand-children as soon as their offspring hit a certain age and success.
This just demonstrates that you dont understand how sexual selection works. For men, yes, aesthetic appearance is a considerable (main?) component in initial attraction, which is further tempered by compatible personality after that initial connection. For women, social value is the principle signifier, which is then tempered by facial symmetry, not demonstrating socially unacceptable habits and having sone degree of physical security, but the latter is the most variable across cultures.
Handsome men are largely irrelevant.
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I took an online electronics tech course 15 years ago and what got me was my math skills were atrocious. Not shocking since like learning a new language or music use it or lose it is the obvious answer to why I sucked. I spent half my time re-learning math just so I could complete the course.
It's funny because I've had the opposite heuristic most of my line: the things I want to do most are whatever is hardest. This worked great for building my maths and physics skills and knowledge.
But when I started focusing on making money I've come to believe it's a bad heuristic for that purpose.
How have you been working on it? Asking for a friend ;)
easy_things -> comfort_zone
Amazingly, I believe that today, with the myriad of tools available, anyone can advance in sciences like mathematics at their own pace by combining black-box and white-box approaches. Computers, in this context, could serve as your personal “Batcomputer” [1]. That said, I would always recommend engaging in social sciences with others, not working alone.
Who knows? You might also contribute meaningfully to these fields as you embrace your own unique path.
[1] https://dc.fandom.com/wiki/Batcomputer