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

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”.

easy_things -> comfort_zone

This perspective has discouraged so many people from exploring their potential

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.

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.

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?

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.

But also, some people are stupid, right?

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).

> 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.

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.

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.

> 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.

> 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.

The inborn part is how quickly you get results (good or bad). Stupidity is the results.

If we spent 50% of time thinking productively - inborn thinking speed would matter. But in my estimate even 5% is generous.

So it matters far more what kind of feedback you have to filter out the wrong results, and how much time you spend thinking - than how quickly you can do it.

Also practice helps with speed.

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.

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).

> 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.

"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.

> 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.

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 thungs.

> 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.

>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.

Intellect is like a gas, it will expand to fill its container. The container, in humans, is epigenetic and social — genetics only determines how hot or cold your gas is, ie how fast and how fluidly it expands, but you’re taught your limits — it’s best to see stupid as not how limited you are relative to other but what limits you have now and may abandon in the future.

That said, some people received a smaller starting container, and might need some help cracking it. That’s the work of those who think they’ve found a bigger one.