> 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”.

> 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.

> 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.

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.

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

I think that's also a trap. Even professional athletes spend a little bit of their time doing simple drills: shooting free throws, fielding fly balls, hitting easy groundstrokes.

Sometimes your daily work keeps up the "easy" skills, but if you haven't used a skill in a while, it's not a bad idea to do some easy reps before you try to combine it with other skills in difficult ways.

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.

> 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.

I am trying to stress pushing through these barriers with my kid right now. The second her brain encounters something beyond its current sphere she just shuts down.

I have heard it is the ego protecting itself by rejecting something outright rather than admitting you can't do it. It still happens to me all the time. My favorite technique was one I heard from a college professor. He starts reading while filling a notepad with sloppy notes, once a page is filled he just throws it away. He claimed it was the fastest way to "condition his brain to the problem space". More than the exercise I like the idea that your brain cannot even function in that space until it has been conditioned.

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 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 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.

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

I'd like to improve my mathematical knowledge (which has degraded to basically zero).

Can you talk about the path or resources you are using to improve here?

This perspective has discouraged so many people from exploring their potential

How have you been working on it? Asking for a friend ;)

easy_things -> comfort_zone

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.

Sounds really cool.

It reminds me of programming, that moment when new code starts to really sync up and code goes from being a bunch of text to more intuitive structures. When really in the zone it feels like each function has its own shape and vibe. Like a clean little box function or a big ugly angry urchin function or a useless little circle that doesn't do anything and you make a note to get rid of. I can kinda see the whole graph connected by the data that flows through them.

> rotated shapes, slot machines, or origami

Or gears (like Seymour Papert), or abacus beads, or nomograms, or slide rules, etc etc. Anyone have any more, throw them out!

Is mathacademy good? I have been thinking of giving it a month of a try. You say "stressful", which I'm not sure is a mis-type or not.

I ordered Mathematica at my local library by the way, and can now forget about it until I get an SMS one day informing me of its arrival. Thank you for confirming that it's worth it!

I really want to try MathAcademy.com. How quickly do you think someone doing light study could move from a Calc 1 -> advanced stuff using that site? In my case I could put in at least 30 minutes to an hour a day.

>>It's like an imagination sport.

Honestly speaking I think this is a wrong way to teach people to think about Math. Math is just one of those things which feels hard because people struggle to hold long trials of manipulations in their head. Especially if they are manipulations to something very large, evolved slowly over hundreds of steps. People are not coming short, its just how the human mind works.

IMO, the right way to teach Math is to teach people that its just base axioms, manipulation rules. And after that its how you evolve the base axiom using rules. People need to be taught how to make one valid change at a time. Of course this means tons of paper work and patience. But that is what Math actually is. Its taking truth and rules, to make new ones.

Im teaching this to my kid, and she often goes like this is it?? its really just laborious paper work??

Im using this method and LLM help at times these days to learn Algorithms and Data Structures. When you start working things from base conditions and build from there. A lot of Algos that otherwise seem like the domain of novel inventions just seem to follow from the manual steps you just worked, and then translated into a program.

When you remove all the fluff, Patience and Paper work is all there is to Math.

Would you say the book ventures more into the practical side of learning this stuff or is it closer to the tone of this article? I found this article hard to gain anything from. A lot of just motivational cliche statements and nothing really groundbreaking or mind altering. If the book is better at that, I'd love to read it. If it's stories and a lot of fluff, I'd rather skip. So I'm curious what you are getting from it and how practical and applicable it feels to you?

I actually heard about this book very recently, and it's coming up soon on my (never-ending) reading list.

Happy to hear you're enjoying it, makes me even more confident that I should read it :)

+1 for Math Academy. I’ve been using it daily for over a year now (started October 2023). I summarized my experiences after 100 days here in case it helps anyone: https://gmays.com/math

Thanks for sharing this. I was debating buying the book.

This sounds like a book I needed for one of my early comp sci classes in college. It was called something like Think Like a Programmer: An Introduction to Creative Problem Solving. Maybe it was this, maybe it was something like this.

I mean to say, just applied scientific thinking is important. Even if you never get into pure math or computer programming, applying concepts like "variables", "functions" or "proofs" is universally useful.

I'm an adult who's been programming computers professionally for 20 years, and went to school for it, and I've lost most mathematical skills past what I'd learned by 6th grade or so, from lack of use.

People who aren't even working in a field that's STEM-adjacent have even less use for stuff past simple algebra and geometry (the latter mainly useful just for crafting hobbies and home-improvement projects) and a handful of finance-related concepts and formulas.

I expect to go to my grave never having found a reason to integrate something, at this rate.

The result is that any time I try to get back into math (because I feel like I should, I guess?) it's not really motivated by an actual need. The only things that don't bore me to tears for sheer lack of application ends up being recreational math problems, and even that... I mean, I'd rather just read a book or do almost anything else.

I feel the opposite.

Before high school, math is just a grind of memorization and unmotivated manipulation of numbers.

Many students (ok, me, but I expect the same was true of others), get turned on to math for the first time when they encounter proofs in high school geometry and also actual applications in high school physics.

It's a revelation to students that math can be a way to go from one truth to another and thus find new truths. It's a way of thinking and that can be very exciting.

Tragically, many students disengage before this can happen because of sheer boredom and the tedium of endless math drills. Once they develop a gap in their knowledge it becomes difficult to progress unless those gaps are addressed. For lots of students, it all ends with fractions. You'd be surprised how many adults don't really understand fractions. For others it ends with algebra, and for the college bound it ends with calculus.

Only math majors and a minority of engineering/science/CS folks get past the "standard sequence" of math courses in college and gain an appreciation for the really interesting stuff that comes AFTER all that.

Mathematics is the conversion of a large number of object languages in to a single meta language that lets us talk about all of them.

The sin of modern mathematics is that it's meta language is so ill define that you need towers of software to manipulate it without contradiction. Rewriting all of it into s-expressions with a term rewriting system for proofs under a sequent calculus is an excellent first step to making it accessible.

We do not need to go back to the 16th century when men were men, an numbers positive. If people want to look at what math talks about instead of how it talks about let them pick up stamp collecting.

What does premature formalization mean and when does it occur? Do you mean formal in the sense of using formulaic, rote manipulations or formal in the sense of proofs and rigor?

As someone who went on to study mathematics at the graduate level, I was bored out of my mind in high school math and most subjects. What's missing from a lot of primary and secondary school education is context, and that's what makes it boring. Math wasn't easy because I was particularly good at it. It was easy because it was just blindly following formulas and basic logic.

Something is very wrong with our educational system because almost all math at the primary and secondary levels is basic logic. So when people with this maximum level of mathematics education say they're "bad at math" or "don't get math", it means that they lack extremely basic logic and reasoning skills.

In my mind, we need to teach mathematics in a contextual way (note that I don't necessarily mean applications) in a way to enrich the reasoning and exploring of it. This should include applications, yes, but not be fully concentrated on applications. Sometimes one needs to just learn and think without being tied to some arbitrary standard of it being applied.

This sentiment comes up all the time here. Mathematics uses formalism because it's easier.

It's easier to read "a(b+c+d) = ab+ac+ad" than

> If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments.

It's well known that good notation is exactly the one that elevates good intuition. For example, the Legendre symbol has the property that (a/p)*(b/p)=(a*b/p), an important visual cue that you wouldn't get from writing down (in way too many words) what the Legendre symbol actually means.

Also, most actually good mathematical textbooks aren't just dumps of formulae and proofs and they do contain motivation, examples, pictures, etc. You're attacking a strawman. But you can't just relegate the formalism and proofs to the appendices, that's crazy.

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.)

I believe you're right, even though I don't have any evidence except for my own experience.

This issue becomes very clear when you see how many ways there are to express a simple concept like linear regression. I've had the chance to see that for myself in university when I pursued a bunch of classes from different domains.

The fact that introductory statistics (y = a + bx), econometrics (Y = beta_0 + beta_1 * X) and machine learning (theta = epsilon * x, incl. matrix notation) talk about the same formula with quite different notation can definitely be confusing. All of them have their historical or logical reasons for formulating it that way, but I believe it's an unnecessary source of friction.

If we go back to basic maths, I believe it's the same issue. Early in my elementary school, the pedagogical approach was this: 0. only work with numbers until some level 1. introduce the first few letters of the alphabet as variables (a, b, c) - despite no one ever explaining why "variable" and "constant" are nouns all of a sudden 2. abruptly switch to the last letters of the alphabet (x, y, z), two of which don't exist in my native language 3. reintroduce (a,b,c) as sometimes free variables, and sometimes very specific things (e.g., discriminant of a quadratic equation) 4. and so on for greek letters, etc.

It's not something that's too difficult to grasp after some time, but I think it's a waste to introduce this friction to kids when they're also dealing with completely unrelated courses, social problems, biological differences, etc. If you're confused by "why" variables are useful, why does the notation change all the time, and why it sometimes doesn't - and who gets to decide - this never gets resolved.

Not to mention how arbitrarily things are presented, no explanation of how things came to be or why we learn them, and every other problem that schools haven't tackled since my grandparents were kids.

I have never disagreed more with a comment. You can fully decide that you're not interested in mathematics, after having taken all of the math classes that you could possibly be offered before university, without ever encountering a proof, or even a mathematical definition.

I do agree that explaining why mathematical concepts are useful is something that's often lacking in mathematical curricula, but not that the problem is premature abstraction. Like another commenter said, the opposite is true. The way children are first introduced to (and therefore soured to) something that adults call "math" is by performing pointless computation that has as much to do with actual math as lensmaking has to do with astronomy.

The same goes for language skills, by the way. In the US, 21% of adults are illiterate, and 54% of adults have literacy below sixth-grade level.[1] This is higher than in other developed countries. For example, in Germany, 10% are illiterate, and 32% have literacy below fifth-grade level.[2]

General intelligence also seems to have been trending downward since the 1970s (the reverse Flynn Effect)[3]. It has been measured in the US and Europe.

So, while it is true that the education system and other factors have an influence, the idea that "everybody is capable of X" is wrong and harmful. It's the equivalent of "nobody needs a wheelchair" or "everybody can see perfectly." People are different. A lot of nerds only hang out with other nerds, which screws up their perception of society.

[1]: https://www.thenationalliteracyinstitute.com/post/literacy-s... [2]: https://leo.blogs.uni-hamburg.de [3]: https://www.sciencedirect.com/science/article/pii/S016028962...

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.

>A nice sentiment but clearly a large % of people never do learn even basic mathematical thinking and seem very confused by it

Any healthy/able individual could learn to deadlift twice their bodyweight with sufficient training, but the vast majority of people never reach this basic fitness milestone, because they don't put any time into achieving it. There's a very large gap between what people are capable of theoretically and what they achieve in practice.

> So is there some scientific study backing up the claim that all these people could easily learn it [emphasis added]

Who said it would be easy?

I do not think that Bessis's argument is entirely "made up"

We are not really taught (thought) to think, we are taught to memorize. Until one actually tries to think, you really can't tell if they're able to do it.

One size doesn't fit all is what I believe Common Core math is attempting. The part that it misses is that a student should probably be fine demonstrating one modality instead of having to demonstrate them all

I also have long felt that anyone who has the ability to read and write at a high-school level in their primary language also has the mental capacity to learn to do math at a high school level. As a pedagogical challenge, I think the main stumbling block for most people with math is not the complexity of it but rather the dryness of how its taught. The rules of language are at least as complex, but many more people learn language at a high-school level than math. There are lots of reasons for that, most obviously language is used more in people's every day lives.

Effort, combined with the right motivation, can overcome most perceived barriers

Lots of people seem to get permanently lost right around when operations on fractions are introduced. Other places, too, but that seems like the earliest one where a lot of people get lost and never really find their way back.

Factoring was another that lost a lot of folks in my class. Lots of frustration around it seeming both totally pointless and the process involving lots of guessing, several classmates were like "well, fuck math forever I guess" at that point, like if they'd been asked to dig a ditch with a spoon and then fill it back in.

I don't know how widespread this phenomenon is but in the book Do Not Erase [0] it seems that there are quite a few prominent mathematicians who do use visual representations in their work.

[0] https://en.wikipedia.org/wiki/Do_Not_Erase:_Mathematicians_a...

Sadly, when I was a postdoc, an eminent mathematician I was working under once shared a story that he found amusing that one of his colleagues was once asked a question in the form: "This might be a stupid question, but..." and the response was "There are no stupid questions, only stupid people."

Run into too many people like that, who I daresay are common in the field, and it's easy to see how people become dispirited and give up.

How much of math aversion stems from a chain reaction of ineffective instruction

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.

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.

Yes, I agree! And also that a lot of the fun stuff is hidden behind historically opaque terminology. Although I'm also sympathetic to the fact that writing accessible explanations is a separate and hard to master skill. Once you understand something it can be really hard to step back into the mindset of not understanding it and figuring out an explanation that would make the idea "click".

I think a lot of maths is secretly a lot easier than it appears, but just missing an explanation that makes it easy to get the core idea to build upon.

For example, I've been meaning to write an explorable[0] for explaining positional notation in any integer base (so binary, hexadecimal, etc) in a way that any child who can read clocks should be able to follow. Possibly teaching multiplication along the way.

Conceptually it's quite simple: imagine a counter that looks like an analog clock, but with the digits 0 to 9 and a +1 and -1 button. We can use it to count between zero and nine, but if we add one to nine, we step back to zero. Oh no! Ok, but we can solve this by adding a second counter. Whenever the first counter does a full circle, we increase it by one. A full circle on the first counter is ten steps, so each step on the second counter represents ten steps. But what if the second counter wants to count ten steps? No problem, just add a third! And so on.

So then the natural question is... what if we have fewer digits than 0 to 9? Like 0 to 7? Oh, we get octal numbers. 0 and 1 is binary. Adding more digits using letters from the alphabet?

The core approach is just a very physical representation of base-10 positional, which hopefully it makes it easy to do the counting and follow what is happening. No "advanced" concepts like "base" or "exponentiation" needed, but those are abstractions that are easy to put on top when they get older.

I've asked around with friends who have kids - most of them learn to read clocks somewhere between four and six, and by the time they're eight they can all count to 100. So I would expect that in theory this approach would make the idea of binary and hexadecimal numbers understandable at that age already.

EDIT: funny enough the article also mentions that precisely thanks to positional notation, almost every adult can immediately answer the question "what is one billion minus one".

[0] https://explorabl.es/

If you’re interested in computer science, have you ever looked at the Software Foundations course by UPenn? It follows a similar approach of having you build all sorts of fascinating math principles and constructions from the ground up. But then it keeps going, all the way up to formal methods of software analysis and verification.

https://softwarefoundations.cis.upenn.edu/

totally agree! in high school, lots of things were vaguely defined. I remember, I didn't fully understand what "f o g" was until I was given the definition of a monoid. Also limits and derivations, once you get the proper definition, you can pretty easily derive all the formulas and theorems you use in high school. Also in high school, we mostly did calculations and simple deductions, but at university we were proving everything. Nice change of perspective.

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.

Calculus isn't on the standard SAT or even the Math Subject II tests [0], at least not in the early 2000s.

They taught us calculus because it's a prereq for engineering and physics and that's been important since the space race.

[0] https://en.wikipedia.org/wiki/SAT_Subject_Tests

Eh, I mean it's not on the SATs, but why isn't it? In Canada, we had a similar calculus based curriculum up to the first year of university. A little bit of linear algebra thrown into the mix. Why is that? Well you need calculus to do any form of engineering, physics, certain domains of chem/bio, stats, certain domains of economics, etc etc. Math in society is first and foremost a tool. I say this as a person who majored in Pure Math and focused on algebra and number theory. For the vast majority of students, it truly is about the practicality. Math just has the layer of abstraction that makes it hard to enjoy without deliberate framing unlike the sciences or humanities.

If someone still on X could send them the hn@ycombinator.com support address, that'd be useful.

I haven't read his Human Understanding, but his Second Treatise is really weak in ways that can't really be blamed on lack of mathematical training (unless we're going with "all rigorous thinking is math") so there may be more to it in his case than just "he didn't study math enough".

One thing I can agree with is that if one is stressed out and have poor psychological habits you will suffer and be miserable regardless.

I would say focusing on mindfullness(like vipasaana) can go a long way in this. But mindfulness is not an intellectual exercise, one has to live it. Do multiple hours of meditation a day and it gets you somewhere good in a few months.

Didn’t we have that experiment during Covid? A bunch of people got paid to stay at home, sometimes for 2 years. How many Grammy nominations since then have gotten to musicians that came out of that? A new face at the Oscars? MOMA exhibiting an artist that was a barista before the pandemic?

At least anecdotally, many people around me now have more children, on the other hand.

I must disagree. I consider art same as entertainment to the most part. I would want to be good at math and I also disagree that it has anything to do with mental capacity. It's not a competition, I don't need to be better at math than others but my pursuit of other things like cryptography, better algorithms, and understanding physics is limited by my primitive understanding of mathematics.

If I was a multi-millionaire, learning lots of math on my free time would be one of the things I would pursue while chilling at my beach house.

UBI scoped for self-actualization would be nice, imagine all the mighty works people would make if not for The Markets making us organize around badware.

Come on, get real. If people had all of their needs covered a lot more of them would sit around getting high and playing video games than perfecting their art.

Yeah, but we are all in software so for us economy is not a concern. All of us should learn maths.

This topic is probably the worst possible topic for Quanta.

The book, as I understand it, is about the life changing power of mathematical thinking. Quanta's mission is to make deep mathematics and mathematicians a "sexy" "human interest" topic, by making it as non-mathematical as possible while keeping a superficial veneer of mathematics.

Too often people think of learning as accumulating knowledge and believe blockers are about not enough knowledge stored.

That would be like strength training by carrying stuff home and believing that the point is to have a lot of stuff at home.

Intelligence is about being able to frame and analyze things on the fly and that ability comes from framing and analyzing lots of different things, not from memorizing the results of past (or common forms of) analysis.

Pure math is probably not worth the squeeze. I think more important to everyday life is systems thinking and a bit of probability/stats, mainly bayesian updates. "Superforecasting" was an eye-opening book to me, I could see how most people would benefit massively by it.

Similar to systems thinking, just the ability to play out scenarios in your head given a set of rules is a very useful skill, one which programmers tend to either be good at because of genetics or because we do it every day (i.e. simulate code in our head). You can tell when someone lacks this ability when discussing something like evolutionary psychology. Someone with a systems thinking mindset and an ability to simulate evolution tend to understand it as obvious how evolutionary pressures tend to, and really must, create certain behavior patterns (on average), while people without this skill tend to think humans are a blank slate because it's easier to think about, and also is congruent with modern sensibilities.

This skill applies in everyday life, especially when you need to understand economics (even basic things like supply and demand seems elusive to many), politics etc.

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.

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).

i’ve always found that mathematical thinkers do better, no matter the role. it’s difficult to express how, it’s kind of intangible, but it’s just a mental process that helps break down tasks and come up with unique solutions.

that being said, my marketing pitch for math was always, nothing else is even close to as intellectually stimulating as math.

I also studied it and got several degrees, but I don't think that it actually benefited me. I think high school math is incredibly important to be able to think clearly in a quantitative way, and one university-level statistics course, but all the other university math... I dont think it helped me at all. I am disappointed by it because I feel that I was misled to believe that it would be useful and helpful.

Have you ever ascribed numbers to real life personal problems? I find that managing to frame something bothersome into a converging limit somehow, really dissolves stress.. A few times at least.

The different notations in calculus are unfortunately a historical accident and you're going to see all of them if you do anything with it (it's certainly not true that everyone in the field uses the same notation). I agree that it's frustrating but so are irregular verbs in Spanish/French/other languages - you can't really change it.

> One is that math nerds at school insisted on intimidating for the win and I just hated it.

Only an adult can look and see what that was - immature, insecure little boys, desperately trying to show off as bigger/more mature or kick down anybody showing any weakness or mistake. Often issues from home manifesting hard. Its trivial to look back without emotions, but going through it... not so much.

If my kids ever go through something similar (for any reasons, math nerds are just one instance of bigger issue) I'll try reasoning above, not sure if it will help though.

> One is that math nerds at school insisted on intimidating for the win and I just hated it.

For me the worst part was the teachers that encouraged that behavior and did the same.

This was a very tiring blog post for me. And I have a quip about posts that open with questions but close without obvious definite answer, no matter how simple it is.

Try this remarkable book:

Who Is Fourier?: A Mathematical Adventure

https://www.amazon.com/Who-Fourier-Mathematical-Transnationa...

It started off as a bunch of non-math literate folks teaching themselves math from scratch, including trigonometry, calculus etc, and ending in Fourier series. It is a very approachable and fun book.

I was the same In high school.

2 weeks ago I hired a professor to help me learn math again so I can attend University computer science.

I can tell you, you can and should.

I'm totally addicted to math, I work as a programmer once I finish my work for the day I spend all my free time learning math again.

I'm still going over the very basics like 9 th grade stuff but I can see already it's going to go fine! I'm enjoying it so much!

Math Academy and (optionally for a deeper study) the Art of Problem Solving books.

Check out Susan Rigetti's guide: https://www.susanrigetti.com/math

Possibly making a small, focused game!

This is how I got back into learning maths... through necessity and immediate application.

List of good books, sorted by difficulty:

- Maths: A Student's Survival Guide (ISBN-13 978-0521017077)

- Review Text in Preliminary Mathematics - Dressler (ISBN-13 978-0877202035)

- Fearon's Pre-Algebra (ISBN-13 978-0835934534)

- Introductory Algebra for College Students - Blitzer (ISBN-13 978-0134178059)

- Geometry - Jacobs ( 2nd ed, ISBN-13 978-0716717454)

- Intermediate Algebra for College Students - Blitzer (ISBN-13 978-0134178943 )

- College Algebra - Blitzer (ISBN-13 978-0321782281)

- Precalculus - Blitzer (ISBN-13 978-0321559845)

- Precalculus - Stewart (ISBN-13 978-1305071759)

- Thomas' Calculus: Early Transcendentals (ISBN-13 978-0134439020)

- Calculus - Stewart (ISBN-13 978-1285740621)

The main goal of learning is to understand the ideas and concepts at hand as “deeply” as possible. Understanding is a mental process we go through to see how a new idea is related to previous ideas and knowledge. By “deeply” we mean to grasp as much of the ideas and relations between them as possible. A good metaphor for this is picturing knowledge as a web of ideas where everything is somehow related to everything else, and the more dense the web is, the stronger it becomes. This means that there might be no “perfect” state of understanding, and otherwise it is an on-going process. You could learn a subject and think you understand it completely, then after learning other subjects, you come back to the first subject to observe that now you understand it deeper. Here we can use a famous quote from the mathematician John V. Neumann: “Young man, in mathematics you don't understand things. You just get used to them”, which I think really means that getting “used to” some subject in Mathematics might be the first step in the journey of its understanding! Understanding is the journey itself and not the final destination.

Solve as many exercises as you can to challenge your understanding and problem-solving skills. Exercises can sometimes reveal weaknesses in your understanding. Unfortunately, there is no mathematical instruction manual for problem-solving, it is rather an essential skill that requires practice and develops over time. However, it could be greatly impacted by your level of understanding of the subject. The processes of learning and problem-solving are interrelated and no one of them is dispensable in the favor of the other. There are also general techniques that could be helpful in most cases which are found in some books on problem-solving (which are included in the roadmap).

Teach what you have learned to someone else or at least imagine that you are explaining what you learned to someone in the best possible way (which is also known as the Feynman Technique). This forces you to elaborately rethink what you have learned which could help you discover any weaknesses in your understanding.

Learning how and when to take notes is not easy. You don't want to waste your time copying the entire book. Most modern books have nice ways to display important information such as definitions and theorems, so it's a waste of time to write these down since you can always return to them quickly. What you should do is take notes of how you understood a difficult concept (that took you a relatively long time to understand) or anything that you would like to keep for yourself which is not included in the book, or to rewrite something in the book with your own words. Notes are subjective and they should be a backup memory that extends your own memory.

Read critically. Books are written by people and they are not perfect. Don't take everything for granted. Think for yourself, and always ask yourself how would you write whatever you are reading. If you found out a better way to explain a concept, then write it down and keep it as a note.

Cross-reference. Don't read linearly. Instead, have multiple textbooks, and “dig deep” into concepts. If you learn about something new (say, linear combinations) — look them up in two textbooks. Watch a video about them. Read the Wikipedia page. Then write down in your notes what a linear combination is.

Learning is a social activity, so maybe enroll in a community college course or find a local study group. I find it's especially important to have someone to discuss things with when learning math. I also recommend finding good public spaces to work in—libraries and coffee shops are timeless math spaces.

Pay graduate students at your local university to tutor you.

Take walks, they're essential for learning math.

Khan Academy is not enough. It has broad enough coverage, I think, but not enough diversity of exercises. College Algebra basically is a combination of Algebra 1, Algebra 2, relevant Geometry, and a touch of Pre-Calculus. College Algebra, however, is more difficult than High School Algebra 1 and 2. I would tend to agree that you should start with either Introductory Algebra for College Students by Blitzer or, if your foundations are solid enough (meaning something like at or above High School Algebra 2 level), Intermediate Algebra by Blitzer. Basically, Introductory Algebra by Blitzer is like Pre-Algebra, Algebra 1, and Algebra 2 all rolled into one. It's meant for people that don't have a good foundation from High School. I would just add, if it is still too hard (which I doubt it will be for you, based on your comment), then I would go back and do Fearon's Pre-Algebra (maybe the best non-rigorous Math textbook I've ever seen). Intermediate Algebra is like College Algebra but more simple. College Algebra is basically like High School Algebra 1 and 2 on steroids plus some Pre-Calculus. The things that are really special about Blitzer is that he keeps math fun, he writes in a more engaging way than most, he gives super clear—and numerous—examples, his books have tons of exercises, and there are answers to tons of the exercises in the back of the book (I can't remember if it's all the odds, or what). By the time you go through Introductory, Intermediate, and College Algebra, you will have a more solid foundation in Algebra than many, if not most, students. If you plan to move on to Calculus, you'll need it. There's a saying that Calculus class is where students go to fail Algebra, because it's easy to pass Algebra classes without a solid foundation in it, but that foundation is necessary for Calculus. Blitzer has a Pre-Calculus book, too, if you want to proceed to Calculus. It's basically like College Algebra on steroids with relevant Trigonometry. Don't get the ones that say “Essentials”, though. Those are basically the same as the standard version but with the more advanced stuff cut out.

> IMO everyone has a ceiling when it comes to math.

Yes, but it's higher than you think: https://www.justinmath.com/your-mathematical-potential-has-a...

God that's one of the most pessimistic views of humanity I've seen on the web, and I'm pretty pessimistic. Developed nations have a 99.2% literacy rate per wikipedia. 69% of OECD students meet "level 2" mathematical proficiency (https://www.oecd.org/en/topics/sub-issues/mathematics-litera...), which indicates an understanding above the basic mathematical operators like multiplication and division.

FWIW I did not go to any fancy school, I was educated in the state system.

Not sure about schools for gifted, but I think lots of people here a bit live in bubbles.

Maybe industrialized countries, a bit better off neighborhoods, STEM classes in high school, and then university? (Not everyone but ... Most here?)

Frankly, its overrated. Now you can adjust your priors.

I don't have dyscalculia but behind the numbers I have my own intuitive system(s) that I jump to sometimes when doing arithmetic. I think we all do since the early days arithmetic in school or what not, perhaps the dyscalculia folk missed making some connection at some point. I feel that arithmetic with numbers without that intuitive system is rote memory..

> There are actual techniques that career mathematicians know about.

Your best example is the decimal system in contrast to roman numerals. I think that explains the point well. The zero is one of those tricks, and most people know it now, but that wasn't true until very recently.

I think you've simply redefined genius. Many years ago I read an article on youth football, if I remember correctly, and in it there was a bit about the writers visit to the Ajax Youth Academy. In it he writes of a moment during practice when a plane flies over and all the 7(?) year olds on the pitch look up to see it except for one kid who keeps his eyes on the ball. That kid (of course) grows up to be a very good midfielder for Real (I'm forgetting the exact details, I think its Wesley Sniejder?). My point is: whatever that motive energy is that manifests as the single minded pre-occupation with math at an age when everybody else's attention is all over the place is that inherent thing that people call genius. I have read many of Thurston's non-mathematical writings about himself and in it this sort of singular pre-occupation is also clear -- which is why he developed his preternatural geometric vision.

> I see that many people are confused [...] by my take that math talent isn't primarily a matter of genes

Speaking only for myself, I'm not confused at all. Rather I vigorously disagree with this statement, and think that stumping for this counterfactual premise leads to cruel behavior towards children (in particular) who plainly do not have what it takes to learn, for example and in particular, algebra.

> In other words: the people who run the mathematical 100m in under a second don't think it's because of their genes.

This is not their subject of expertise, and they are simply wrong. Why? Simpson's Paradox, ironically.

Ironically, the ones who don't do well at school (inc maths) are the ones who then become trades like builders/plumbers etc or run small businesses like a shop. And so are regularly working with numbers via estimates and billing.

Waste of time. Just talk in terms of what they want to hear. They are just interested in the payoffs (not in the details).

As info explodes and specialists dive deeper into their niches, info asymmetry between ppl increases. There are thousands of specialists running in different directions at different speeds. Leaders can't keep up.

Their job is to try to get all these "vectors" aligned toward common goals, prevent fragmentation and division.

And while most specialists think this "sync" process happens through "education" and getting everyone to understand a complex ever changing universe, the truth is large diverse groups are kept in sync via status signalling, carrot/stick etc. This is why leaders will pay attention when you talk in terms of what increases clout/status/wealth/security/followers etc. Cause thats their biggest tool to prevent schisms and collapse.

I used to judge myself for not understanding everything in math articles on Wikipedia, but as time has gone on I've realized that their purpose isn't really to be an introduction, but a reference. Especially as the topics become more esoteric. So they're not really there for you to learn things from scratch, but for people who already understand them to look things up. Which is why you'll sometimes see random obscure & difficult factoids in articles about common mathematical concepts.

(don't have any examples on-hand atm, this is just my general perception after years of occasionally looking things up there)

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?

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.

I was actually thinking Jean Paul Satre when I read his answers

I guess I assumed it was a reference to Alfred Whitehead's and Bertrand Russell's book Principia Mathematica, which predates Wolfram himself by several decades.

???

"Mathematician Tai-Danae Bradley and physicist Gabe Perez-Giz offer ambitious content ... Previous host Kelsey Houston-Edwards "

I just published the comment above by mistake and couldn't find a way to delete it.