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

https://www.quantamagazine.org/mathematical-thinking-isnt-what-you-think-it-is-20241118/

by sonabinu • 2 years ago

669 382 2 years ago

benreesman5• 2 years ago

I’m far from being any kind of serious mathematician, but I’ve learned more in the last couple years of taking that seriously as an ambition than in decades of relegating myself to inferiority on it.

One of the highly generous mentors who dragged me kicking and screaming into the world of even making an attempt told me: “There are no bad math students. There are only bad math teachers who themselves had bad math teachers.”

thefaux• 2 years ago

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.

benreesman5• 2 years ago

I think we can recognize Pauli for his identification of one of the few magic gadgets we accept around spin statistics without accepting his educational philosophy: “Das ist nicht einmal falsch.”

He was right on the nature of the universe, he was wrong on making a better world. I for one forgive him on the basis of time served.

Jensson2• 2 years ago

Isn't that a positive statement, that you can ask questions without worry since they aren't stupid?

ckw• 2 years ago

That’s a south park quote: https://youtu.be/wWfacULP1o0?si=ddBWXFuQMoxxY-Yu

cchi_co2• 2 years ago

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

benreesman5• 2 years ago

According to an excellent mentor: all of it minus epsilon.

mr_mitm3• 2 years ago

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.

benreesman5• 2 years ago

Cantor gave his life to the Continuum Hypothesis, Hilbert gave much of his life to similar goals.

You’re making an argument somewhat along those lines, but given that I didn’t stipulate a convergence condition your conclusions can be dismissed by me.

If it were a valid argument then we’d need Gödel.

diffeomorphism5• 2 years ago

Doesn't follow. Bell curve in, shifted bell curve out. Ideally this also tweaks the variance a bit.

In other words: Some students flourish despite their teachers, some flourish because of them.

jyscao• 2 years ago

everyone can, and should, try to improve their mathematical thinking — not necessarily to solve math problems, but as a general self-help technique

Agreed with the above. Almost everyone can probably expand their mathematical thinking abilities with deliberate practice.

But I do not think this is innate, even though it often manifests in early childhood. Genius is not an essence. It’s a state. It’s a state that you build by doing a certain job.

Though his opinion on mathematical geniuses above, I somewhat disagree with. IMO everyone has a ceiling when it comes to math.

felideon• 2 years ago

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...

limit499karma2• 2 years ago

the provocative claim

Leibniz made that claim centuries ago in his critical remarks on John Locke's Essay on Human Understanding. Leibniz specifically said that Locke's lack of mathematical knowledge led him to (per Leibniz) his philosophical errors regarding the nature of 'substance'.

https://www.earlymoderntexts.com/assets/pdfs/leibniz1705book...

vundercind7• 2 years ago

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".

limit499karma2• 2 years ago

Leibniz wasn't saying that "rigorous thinking" is only available to mathematically trained or that Locke's reasoning was not "rigorous".

His critique of Locke was that one can not have a correct model of human understanding (or world model) based on purely philosophical means, and that the lack of exposure to certain aspects of modern mathematics (that was emerging at that time) was the basis of Locke's misunderstandings.

tkgally2• 2 years ago

I studied math hard for several years in college and graduate school—purely out of interest and enjoyment, not for any practical purpose. That was more than forty years ago, but Bessis's description of the role of intuition in learning and doing math matches my recollection of my subjective experience of it.

Whether that youthful immersion in math in fact benefitted me in later life and whether that kind of thinking is actually desirable for everyone as he seems to suggest—I don't know. But it is a thought-provoking interview.

Frummy2• 2 years ago

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.

tkgally2• 2 years ago

That’s an interesting approach. I don’t think I’ve done that myself, but I can see how it could be helpful.

One positive effect of having studied pure mathematics when young might have been that I became comfortable with thinking in multiple layers of abstraction. In topology and analysis, for example, you have points, then you have sets of points, then you have properties of those sets of points (openness, compactness, discreteness, etc.), then you have functions defining the relations among those sets of points and their properties, then you have sets of functions and the properties of those sets, etc.

I never used mathematical abstraction hierarchies directly in my later life, but having thought in those terms when young might have helped me get my head around multilayered issues in other fields, like the humanities and social sciences.

But a possible negative effect of spending too much time thinking about mathematics when young was overexposure to issues with a limited set of truth values. In mainstream mathematics, if my understanding is correct, every well-formed statement is either true or false (or undecided or undecidable). Spending too much time focusing on true/false dichotomies in my youth might have made it harder for me to get used to the fuzziness of other human endeavors later. I think I eventually did, though.

plsbenice34• 2 years ago

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.

agtech_andy• 2 years ago

I used to get very frustrated that others could not intuit information the way I could. I have a lot of experience trying to express quantities to leaders and policymakers.

At the very minimum, I ask people to always think of the distribution of whatever figure they are given.

Just that is far more than so many are willing to do.

cen4• 2 years ago

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.

namaria2• 2 years ago

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

This is overthinking it. People with power tend to be interested in outcomes. They can't evaluate all the reasoning of all their reports. It comes down to building credibility with a track record and articulating outcomes, when you want to advise decision makers.

DiscourseFan2• 2 years ago

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

[0]https://news.ycombinator.com/item?id=41962944

sonabinu2• 2 years ago

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

ai4eva2• 2 years ago

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.

tracerbulletx• 2 years ago

A nice sentiment but clearly a large % of people never do learn even basic mathematical thinking and seem very confused by it. So is there some scientific study backing up the claim that all these people could easily learn it or are we just making it up because its a nice egalitarian thesis for a math popularization book?

logicchains• 2 years ago

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.

Jtsummers• 2 years ago

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?

Jensson2• 2 years ago

It is easy to learn for some.

physicsguy2• 2 years ago

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.

strken• 2 years ago

Why would ability not be innate just because some people with the ability don't use it?

Or more specifically, two of my friends teach special needs children in the 50 to 70 IQ band. Who are we going to blame for them not becoming mathematicians? The teachers, for not unlocking their hidden potential? The kids, for not trying hard enough? Claiming that the only thing holding them back is choice seems as cruel as it is wrong, to me.

Yeah, we're probably not cultivating anywhere near the potential that we could, but I personally guarantee you I am not Ramanujan or Terence Tao.

sublimefire• 2 years ago

have had significantly higher mathematical ability among the general population

This is not really true is it? There were not that many standardized testing globally to measure such claims. Many people were in poverty and did not get tested, did not go to schools, or finished schools very early (5, 9 years). Many more kids go to school these days.

In the Soviet Union more time was spent teaching mathematics and a whole culture developed around mathematics being fun

It is just wrong. It was the same as now, except it was critical for people to show results because otherwise you had grim perspectives in the life, there was little "fun". People wanted to get into universities to get better jobs and to get better apartments, to be able to leave their parents. You could not just buy places, but a good position in some public body would guarantee you a nice place. FYI engineers could earn more in comparison to other jobs, not to mention if you could get into defense industry.

barrenko• 2 years ago

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.

brodo• 2 years ago

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...

j2kun• 2 years ago

What a weird comment. Are you trying to argue by analogy that a decent fraction of the population are not capable of literacy? It seems self-evident that low literacy rates have nothing to do with innate ability. I see no evidence to suggest that math is any different.

crazygringo• 2 years ago

The idea that 21% of US adults are "illiterate" doesn't mean they can't read, by the way:

English literacy test results from 2014 suggest that 21% of U.S. adults ages 16 to 65 score at or below PIAAC literacy level 1, meaning they have difficulty "[completing] tasks that require comparing and contrasting information, paraphrasing, or making low-level inferences." Included in that 21% is the 4.2% of respondents who were unable to be assessed due to language barriers, cognitive disability, or physical disability.

https://en.wikipedia.org/wiki/Literacy_in_the_United_States

A lot of people aren't aware that the academic definition of "literacy" was changed around 1950 to no longer refer to "alphabetical literacy", which is still what most people think literacy means (in lay usage).

https://en.wikipedia.org/wiki/Literacy

And this particular survey ignored people who didn't speak English but were entirely literate in their native language. Which obviously has nothing to do with the educational system or people's intelligence.

agnishom• 2 years ago

Gentle Reminder that the author of this article used to have a wonderful math channel: https://www.youtube.com/c/pbsinfiniteseries

lupire2• 2 years ago

???

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

gsabo• 2 years 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.

junto• 2 years 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 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”.

ponderings• 2 years ago

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.

shrubhub2• 2 years ago

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?

diffeomorphism5• 2 years ago

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.

graemep• 2 years 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.

LoganDark6• 2 years ago

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.

Malidir3• 2 years ago

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.

dennis_jeeves2• 2 years ago

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.

InDubioProRubio• 2 years ago

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.

sdeframond3• 2 years ago

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!

hilbert424• 2 years ago

"What we should be teaching children is HOW to learn."

Absolutely correct. And that begins with getting their interest, thus their attention; and that's a whole subject in and of itself.

Gimpei• 2 years ago

I don’t know if I agree. Grad school was profoundly humbling to me because it really showed me that there are a LOT of people out there that are just much much better than me at math. There are different levels of innate talent.

red75prime2• 2 years ago

The belief that some people have an innate skill that they are born with is deeply unhelpful.

In practice your views result in stinting access of non-existent (in your opinion) talented children to a faster education track. They don't exist therefore they don't need different treatment (finer points get lost when the idea disseminates). Quite a hot theme in American education two (or so) years ago.

scubbo• 2 years ago

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 see you're familiar with Ryan North.

myworkinisgood• 2 years ago

Math is a stepping stone to critical thinking skills. And while one can probably learn those skills in any way, math forces you to learn those skills by learning the method of writing proofs. No other field forces you to push your critical thinking skills to the limit that math does.

solarized• 2 years ago

easy_things -> comfort_zone

chipdart3• 2 years 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 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.

hehehheh• 2 years ago

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.

theclansman2• 2 years ago

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.

globalnode• 2 years ago

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.

doublerabbit2• 2 years ago

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.

khafra• 2 years ago

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.

llm_trw11• 2 years 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.

sn98• 2 years ago

LLMs hallucinate and are completely the wrong tool for this. People need to learn about the history of AI.

Expert-systems using techniques from the 80s are doing this right now [0].

[0] https://www.mathacademy.com/how-our-ai-works

bamboozled• 2 years ago

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

sn98• 2 years ago

I'd strongly recommend Math Academy for at least 30 minutes per day.

bdjsiqoocwk• 2 years ago

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.

wslh• 2 years ago

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

tgv7• 2 years ago

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.

nestes2• 2 years ago

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.

chipdart3• 2 years ago

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.

llm_trw11• 2 years ago

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.

theclansman2• 2 years ago

Anybody can do everything if we restrict everything to things everyone can do.

bloqs• 2 years ago

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.

Barrin92• 2 years ago

You don't need to be able to run 100m in less than 10 seconds. But almost everyone probably could run a marathon in three and a half hours. How many people do you think have actualized their physical potential, or how far is the average person removed from it?

If someone's smart enough to get into a psychology class they are smart enough to be thought basic undergrad math. It wasn't your teaching failure necessarily, but it was someone's teaching failure at some point.

Not everyone can play Rachmaninov like Lugansky or do math like Terence Tao, but there is absolutely no doubt that almost all people are magnitudes away from their latent potential in almost all domains. I'm fairly certain you could teach any average person how to play Rachmaninov decently. You could bring any person to a reasonably high mathematical level. You can get any person to lift a few hundred pounds.

Most people today read at a 7th grade level, can't do basic math, and are out of air after 3 flights of stairs. "Everyone can do everything" is maybe not literally right but directionally right given how utterly far removed we are from developing practically anyone's potential.

antegamisou2• 2 years ago

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.

Bad example, it's much more likely to create a musical prodigy by providing early and appropriate guidance. Of course this is not easy as it assumes already ideal teaching methods and adequate motivation to the youngling, but even those with some learning difficulties have the potential to excel. The subtypes of intellect required to play complex music and absord advanced abstract math subjects are quite different, former requiring strong short-term memory (sightreading) the latter fluid intelligence -I think almost everyone is familiar with these terms by now and knows that one can score high/low on certain subtypes of an IQ test affecting the total score-.

BTW IDK if the Rachmaninoff choice was deliberate to imply that even the most capable who lack the hand size won't be able to perform his works well yeah, but there are like 1000s of others composers accessible that the audiences appreciate even more. Attempting to equate music with sports in such manner is heavily Americanized and therefore completely absurd. Tons of great pianists who didn't have the hand size to interpret his most majestic works and of others. Tons of others who could but never bothered. There have been winners of large competitions who barely played any of his works during all stages of audition or generally music requiring immense bodily advantage. Besides, it's almost 100% not a hand size issue when there are 5 year old kids playing La Campanella with remarkable fluidity.

And even in this case this isn't even the point. Most conservatory alumni today are 100x skilled than the pianists of previous generations... yet they all sound the exact same, their playing lacks character/variability, deepness, elegance to the point where the composers ideas end up distorted. And those can be very skilled but just have poor understanding of the art, which is what music is, not the fast trills/runs, clean arpeggios, very strict metronomic pulse.

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.

Well the vast majority of people in the Soviet Union were very math literate, regardless of what they ended up working as (although indeed most became engineers) and in quite advanced subjects. This is obviously a product of the extensive focus of primary and secondary education on the sciences back then.

So the point isn't to make everyone have PhD level math background and I heavily dislike the dork undertones/culture that everyone should love doing abstract math on their freetime or have to have some mathematical temperament' . But let's not go the other way and claim that those not coming close to achieving the knowledge those in the top % of the fields possess, they are chumps.

dghughes• 2 years ago

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.

setopt• 2 years ago

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.

willtemperley• 2 years ago

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.

User23• 2 years ago

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.

dkarl• 2 years ago

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.

ericmcer2• 2 years 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.

beezlewax• 2 years ago

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?

sn98• 2 years ago

If you know how to multiply and divide, Math Academy will take you to college level math pretty quickly.

te_chris• 2 years ago

Agree. I’ve been trying to learn ML and data for a few years now and, around 2021 I guess, realised Maths was the real block.

I’ve tried a bunch of courses (MIT linalg, Coursera ICL Maths for ML, Khan etc etc) but what I eventually realised is my foundations were so, so weak being mid 30s and having essentially stopped learning in HS (apart from a business stats paper at Uni).

Enter a post on reddit about Mathacademy (https://www.mathacademy.com/). It’s truly incredible. I’m doing around 60-90 minutes a day and properly understanding and developing an intuition for things. They’ve got 3 pre-uni courses and I’ve now nearly finished the first one. It’s truly a revelation to be able to intuit and solve even simple problems and, having skipped ahead so far in my previous study, see fuzzy links to what’s coming.

Cannot recommend it enough. I’m serious about enrolling in a Dip Grad once I’ve finished the Uni level stuff. Maybe even into an MA eventually.

namaria2• 2 years ago

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.

magicalhippo3• 2 years ago

I'm not a math teacher, but I do enjoy math, and I have helped several family members and friends with math courses.

I've long thought that almost all have the capability to learn roughly high school level math, though it will take more effort for some than for others. And a key factor to keep up a sustained effort is motivation. A lot of people who end up hating math or think they're terrible at it just haven't had the right motivation. Once they do, and they feel things start to make sense and they're able to solve problems, things get a lot easier.

Personally I also feel that learning math, especially a bit higher-level stuff where you go into derivations and low-level proofs, has helped me a lot in many non-math areas. It changed the way I thought about other stuff, to the better.

Though, helping my family members and friends taught me that different people might need quite different approaches to start to understand new material. Some have an easier time approaching things from a geometrical or graph perspective, others really thrive on digging into the formulas early on etc. One size does not fit all.

cchi_co2• 2 years ago

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

magicalhippo3• 2 years ago

It sounds like trivial insight, but at least in my experience many adults and even teachers have this "it's hard so it's ok to not want to do it" attitude towards math. And I think that is very detrimental.

sethammons5• 2 years ago

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

vundercind7• 2 years ago

The part that it misses is that a student should probably be fine demonstrating one modality instead of having to demonstrate them all

I cannot overstate enough how consistently and extremely this has turned my kids off from math. 3-for-3 on absolutely hating this. Having to solve the same thing five different ways just pisses them off, and, like... yeah, of course it does. They want to finish the work and go play and it feels like you're just fucking with them and disrespecting their time by making them solve the same problem several times, even if that's not the intent.

o_nate• 2 years ago

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.

block_dagger• 2 years ago

Statistical (Bayesian) thinking is an extremely underrated way of thinking of almost everything.

DiscourseFan2• 2 years ago

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

ai4eva2• 2 years ago

lol yea.

bayesian thinking doesnt come to me naturally.. i have no intuition for it. seems forced. believe me - i have tried. but there are those who are swearing by it.

vundercind7• 2 years ago

I'm pretty down on it just by association. So many people who are super-into it seem to be doing a lot of that rigorously-wrong engineer-brain thing.

w10-1• 2 years ago

Not sure this article captured it for me.

Plato's Meno has Socrates showing that even a slave can reason mathematically.

It's not really math alone but modeling more generally that activates people's reasoning. Math and logic are just those models that are continuous+topological and discrete+logic-operation variants, both based in dimension/orthogonality. But all modeling is over attribution - facts, opinions, etc., and there's a lot of modeling with a healthy dose of salience - heuristics, emotions, practice, etc. Math by design is salience-free (though it incorporates goals and weights), so it's the perspective and practice that liberates people from bias and assumptions. In that respect it can be beautiful, and makes other more conditioned reasoning seem tainted (but it has to work harder to be relevant).

However, experts can project mathematical models onto reality. Hogwash about quantum observer effects and effervescent quantum fields stem from projecting the assumptions required to do the math (or adopt the simplifying forms). Yes, the model is great at predictions. No, it doesn't say what else is possible, or even what we're seeing (throwing baseballs at the barn, horses run out, so barns are made of horses...). Something similar happens with AI math: it can generate neat output, so it must be intelligent. The impulse is so strong that adherents declare that non-symbolic thinking is not thinking at all, and discount anything unquantifiable (in discourse at least). Assuming what you're trying to prove is rarely helpful, but very easy to do accidentally when tracking structured thinking.

quus6• 2 years ago

I’m actually interested in the “can benefit from” claim in this title. I don’t particularly doubt that most people could become reasonably good at math, but I wonder how much of the juice is worth the squeeze, and how juicy it is on the scale from basic arithmetic up to the point where you’re reading papers by June Huh or Terry Tao.

As anti-intellectual as it sounds, you could imagine someone asking, is it worth devoting years of your life to study this subject which becomes increasingly esoteric and not obviously of specific benefit the further you go, at least prima facie? Many people wind up advocating for mathematics via aesthetics, saying: well it’s very beautiful out there in the weeds, you just have to spend dozens of years studying to see the view. That marketing pitch has never been the most persuasive for me.

guerrilla2• 2 years ago

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.

quus6• 2 years ago

Math doesn’t seem to me the only source of thinking clearly, or learning how to learn, etc. And if I’m searching for an aesthetic high, there are definitely better places to look — and ones that don’t require such a long runway.

defrost4• 2 years ago

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).

quus6• 2 years ago

My claim isn’t really that there’s no benefit or utility to math — that’s obviously false — but that maybe its benefits to regular people are more modest than the cheerleaders want to admit.

purplethinking• 2 years ago

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.

kian• 2 years ago

Abstract Algebra, Combinatorics, and Discrete Mathematics are all definitely worth the squeeze; and incidentally something that could easily be taught to middle- and high-schoolers with the right examples.

pokerface_863• 2 years ago

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.

quus6• 2 years ago

yeah, I wish we could be more specific here. Like I appreciate the things you’re saying, and I don’t doubt the truth or sincerity of that opinion — but I feel like I’ve encountered some math PhD students who were surprisingly inflexible in the way they thought, and very parochial in their intellectual interests. I guess I just haven’t (anecdotally) found the transference of math training to other domains as much as others seem to.

dboreham• 2 years ago

Careful there. They'll start voting logically..

kristopolous3• 2 years ago

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

katzenversteher2• 2 years ago

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?

vundercind7• 2 years ago

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

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

atribecalledqst2• 2 years ago

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)

kristopolous3• 2 years ago

I've heard that and I think it's silly. They handwave away why nothing should ever be explained. Wikipedia doesn't work like that for any other topic.

You'll see something like a mathematical proof with no explanation and it's end of article. The edit history will have explanations aggressively removed.

The equivalent would be the article for say, splay tree, to have no diagrams and just a block of code - feeling no obligation to explain what it is or if you looked up a chemical and it would just give you some chemical equation, some properties and feel no obligation to tell you its use, whether it's hazardous or where you might find it... Or imagine a European aristocrat and all that is allowed is their heraldry and genealogy. Explanations of what the person did or why they're important are forbidden because, it's just a reference after all.

Nope, these math people are a special kind of bird and I'm not one of them.

openrisk2• 2 years ago

There is this element of abstract mathematical thinking that many young people get exposed to at some point in the educational system but just never "get it" and they disconnect. This is where it goes awry as the gap only widens later on and its a pity.

Working with symbols, equations etc. feels like it should be more widely accessible. Its almost a game-like pursuit, it should not be alienating.

It might be a failure of educators recognizing what are the pathways to get the brain to adopt these more abstract modes of representing and operating.

NB: mathematicians are not particularly interested in solving this, many seem to derive a silly pleasure of making math as exclusive as possible. Typical example is to refuse to use visual representation, which is imprecise but helps build intuition.

agentultra2• 2 years ago

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...

openrisk2• 2 years ago

There is this long-running (and quite fascinating I think) debate about the different "types" of mathematical thinking. Logical vs Intuitive, Geometric vs Algebraic etc. Can't recall where exactly but I remember reading about a 19th century mathematician that crowed they had not a single figure in their masterpiece.

Visualization is probably not a silver bullet but a lot of people are visual thinkers so maybe it would help a few more to reach a higher level in mathematical thinking.

vundercind7• 2 years ago

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.

fifteen1506• 2 years ago

But, is that profitable? I'm both being sarcastic and real with this question.

If I can earn an extra 1 million being 'dumb' and thus ensure quality healthcare, education, housing, is it smart to try to be smart?

This is the true tragedy of the commons (or the reverse tragedy, to be precise).

ggm• 2 years ago

I want to say yes, but I have two counters. One is that math nerds at school insisted on intimidating for the win and I just hated it.

The second is notation. I had a snob teacher who insisted on using Newton not Leibniz and at school in the 1970s this is just fucked. One term of weirdness contradicting what everyone else in the field did. Likewise failure to explain notation, it's hazing behaviour.

So yes, everyone benefits from maths. But no, it's not a level playing field. Some maths people, are just toxic.

jajko• 2 years ago

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.

bmitc2• 2 years ago

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.

I'm not so sure that adults always get it or rise about this. This happens in the workplace all the time.

dennis_moore• 2 years ago

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.

Tainnor11• 2 years ago

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.

KevinMS• 2 years ago

But what is the difference between math talent and plug-n-chug math talent? That seems to be the most significant filter.

LoganDark6• 2 years ago

I have an autistic friend with dyscalculia. They see numeric digits as individual characters (as in a story), each with their own personalities. Each digit has its own color, its own feelings. But they are not quantities; they don't make up quantities. Numbers are very nearly opaque to them. I wonder how this theory would apply to them. Do they still perform mathematical thinking? They're still capable of nearly all the same logic that I am, and even some that I'm not (their synesthesia gives them some color/pattern/vibes logic that I don't have)... just not math.

tartoran• 2 years ago

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..

LoganDark6• 2 years ago

Indeed, I do also have an intuitive system for this; that's what I mean by "make up a quantity". When something makes up a quantity for me, then I can perform math on it. I have to wonder if my friend with dyscalculia doesn't have a "quantity" system like I do. They can't really perform multiplication or division but can technically perform basic addition and subtraction, though it takes them a while. I think it may be the rote memory you mention doing that for them.

jonplackett• 2 years ago

Has anyone here self-taught themselves math in later life?

I studied up to A level (aged 19) but honestly started hating math aged 16 after previously loving it.

It’s a big regret of mine that I fell out of love with it.

I self taught myself coding and Spanish and much enjoy self study if I can find the right material.

Any suggestions?

sriram_malhar• 2 years ago

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.

AntoniusBlock• 2 years ago

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

webdev1234568• 2 years ago

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!

rnewme2• 2 years ago

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 inâ€”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.

hackable_sand• 2 years ago

Possibly making a small, focused game!

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

sn98• 2 years ago

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

ChaitanyaSai2• 2 years ago

"mathematics is a game of back-and-forth between intuition and logic" I teach/guide Math at our school (we run a small school and currently have kids under age 10) and this is so so true.

I just wrote about this. In fact, you can even see this at play in the video of the kids talking https://blog.comini.in/p/what-happens-in-math-class

rnewme2• 2 years ago

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.

ChaitanyaSai2• 2 years ago

Tiring because the answer wasn't revealed? That was the whole point :) It's the path, not the summit.

GrantMoyer• 2 years ago

Does the quip have to do with someone asking what the quip is?

jojobas• 2 years ago

- Hey teach, will I really need all these logarithms, derivatives and vectors in my adult life?

- No, but the smarter kids might.

practal• 2 years ago

This interplay between intuition and logic is exactly what makes the magic happen. You need intuition to feel your way forward, and then logic to solidify your progress so far, and also for ideas maybe not directly accessible via intuition only. I've experienced that myself, and it is even well-documented, because I wrote technical reports and such at each stage. My discovery of Abstraction Logic went through various stages:

1) First, I had a vague vision of how I want to do mathematics on a computer, based on my experience in interactive theorem proving, and what I didn't like about the current state of affairs: https://doi.org/10.47757/practal.1

2) Then, I had a big breakthrough. It was still quite confused, but what I called back then "first-order abstract syntax" already contained the basic idea: https://obua.com/publications/practical-types/1/

3) I tried to make sense of this then by developing abstraction logic: https://doi.org/10.47757/abstraction.logic.1 . After a while I realized that this version only allowed universes consisting of two elements, because I didn't distinguish between equality and logical equality, which then led to a revised version: https://doi.org/10.47757/abstraction.logic.2

4) My work so far was dominated by intuition based on syntax, and I slowly understood the semantic structures behind this: the mathematical universe consisting of values, and operations and operators on top of that: https://obua.com/publications/philosophy-of-abstraction-logi...

5) I started to play around with this version of abstraction logic by experimenting with automating it, giving a talk about it at a conference, (unsuccessfully) trying to publish a paper about it, and implementing a VSCode plugin for it. As a result of using that plugin I realized that my understanding until now of what axioms are was too narrow: https://practal.com/press/aair/1/

6) As a consequence of my new understanding, I realized that besides terms, templates are also essential: https://arxiv.org/abs/2304.00358

7) I decided to consolidate my understanding through a book. By taking templates seriously from the start when writing, I realized their true importance, which led to a better syntax for terms as well, and to a clearer presentation of Abstraction Algebra. It also opened up my thinking of how Abstraction Algebra is turned into Abstraction Logic: https://practal.com/abstractionlogic/

8) Still lots of stuff to do ...

I would not be surprised if that is exactly the way forward for AIs as well. They clearly have cracked (some sort of) intuition now, and we now need to add that interplay between logic and intuition to the mix.

sureglymop• 2 years ago

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

yodsanklai• 2 years ago

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.

vanderZwan• 2 years ago

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/

nestes2• 2 years ago

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.

aeonik• 2 years ago

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

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

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

e79• 2 years ago

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/

throw4847285• 2 years ago

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.

atribecalledqst2• 2 years ago

Last year I read How to Solve It and the first half of one of Polya's other books - Mathematics and Plausible Reasoning. I certainly didn't commit them to memory, and I never systematically tried to apply them during self-study, but they do sometimes help give me a pointer in the right direction (i.e. trying to think of auxiliary problems to solve, trying to find a way to make the known & unknown closer together... etc.).

Auxiliary problems are something that always screwed me in college, when we were doing Baby Rudin, if a proof required a lemma or something first I usually couldn't figure out the lemma. Or in general, if I didn't quickly find the 'insight' needed to prove something, I often got frustrated and gave up.

This material seems like it would be good to actually teach in school, just like a general 'how to think and approach mathematical problems'. Feels kinda weird that I had to seek out the material as an adult...

One other thing I got out of the Polya books, was I realized how little I remember about geometry. So many of their examples are geometrical and that made them harder for me to grok. That's something I wish I could revisit.

trialAccount3• 2 years ago

trial

trialAccount3• 2 years ago

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

revskill• 2 years ago

I conducted an interview with leetcode 2 years ago while not doing any leetcoding before. Surprisingly, by just applying some math tricks i finished them and got into later rounds. So yes math tricks are helpful.

penguin_booze• 2 years ago

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

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

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

[0] I mean natural language

bmitc2• 2 years ago

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.

llm_trw11• 2 years ago

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.

Tainnor11• 2 years ago

Weirdly enough, mathematicians have been manipulating expressions and writing proofs for centuries without (routinely) stumbling into contradictions all without the need of formal proof calculi or s-expressions.

I have nothing but admiration for projects like Lean and Coq and working in them can be a lot of fun (coupled with a lot of frustration when "obvious" things sometimes take an inordinate amount of time to prove), but Wiles' proof of FLT (the corrected version) was published in 1995. We're almost 30 years later and people are just now working on a formalisation which could take many years (https://leanprover-community.github.io/blog/posts/FLT-announ...). Mathematicians can't afford to be waiting for proof systems to catch up, at least not right now.

vundercind7• 2 years ago

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.

11101010001100• 2 years ago

Do you play any games that require mathematical reasoning? You might realize you are using integration without calling it integration i.e. calculating expected values.

jiggawatts• 2 years ago

For nostalgia I keep a copy of Mathematica on my laptop, so I can pretend to be an an actual computer scientist and not just an overpaid button-presser.

Ten years ago I used it to fit a nonlinear model to some performance metrics to predict the behaviour of a disk array past the maximum load level I was allowed to use for testing.

That’s the last time I did something that Excel couldn’t handle.

In university I learned how matrix exponentiation can be used to calculate the maximum throughput of a mesh network. In real life everyone just buys 10x the bandwidth they need.

It’s depressing.

crispyambulance2• 2 years ago

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.

red75prime2• 2 years ago

Tragically, many students disengage before this can happen because of sheer boredom and the tedium of endless math drills

It's tragic and all, but were there attempts to teach non-preselected children/adolescents interesting stuff first? What are the results?

bunderbunder• 2 years ago

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.)

noqc• 2 years ago

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.

fluoridation4• 2 years ago

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.

TomasBM• 2 years ago

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.

Tainnor11• 2 years ago

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.

fHr• 2 years ago

For example finance is such an important aspect of our lifes and you just need some understanding of math principles to understand how to make good financial choices.

Malidir3• 2 years ago

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.

lugu• 2 years ago

As a young child your brain is much more suitable to learn languages. You can make kids learn 4 languages effortlessly in the right context. When you grow up, slowly shift the focus to abstract thinking. And that shift can rely on building intuition using visualisation and experience.

Relic0935• 2 years ago

That's a very important thougt and I belive the world would be better, if more people would connect with their mathematical side.

musgravepeter• 2 years ago

https://www.smbc-comics.com/comic/grind

agentultra2• 2 years ago

And yet it is flame-bait to suggest that programmers benefit from mathematical thinking. I've not met a more passionate and divided crowd on the issue. Most traditional engineers wouldn't disagree that they use and benefit from mathematical thinking. Programmers though?

I don't think there's a single answer as to why many dislike it so much. Some folks view it as a way to gate-keep programming. Others view it as useless ("I've been a successful programmer all my life and I've never used math").

On the other side of the coin there are many who view our craft as a branch of applied mathematics -- informatics if you will.

trialAccount3• 2 years ago

The author: Would love to participate but account creation seems to be broken on Hacker News https://x.com/davidbessis/status/1859561768915173466

macintux• 2 years ago

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

mbbbackus• 2 years ago

I've been reading the author's book, Mathematica, and it's awesome. The title of this post doesn't do it justice.

He shows that math skill is almost more like a sports talent than it is knowledge talent. He claims this based on the way people have to learn how to manipulate different math objects in their heads, whether treating them as rotated shapes, slot machines, or origami. It's like an imagination sport.

Also, he inspired me to relearn a lot of fundamental math on MathAcademy.com which has been super fun and stressful. I feel like I have the tetris effect but with polynomials now.

sourcepluck2• 2 years ago

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!

Shosty1232• 2 years ago

I've had a MathAcademy subscription for some time and it's quite good. I'd say it's best at generating problems and using spaced repetition to reinforce learning, but I think it falls short in explaining why something is useful or applicable. I don't know, most math education seems to be "here's an equation and this is how you solve it" and MathAcademy is undoubtedly the best at that, but I wish there were resources that were more like "here's how we discovered this, what we used to do before, why it's useful, and here's some scenarios where you'd use it."

gravypod• 2 years ago

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.

Rendello• 2 years ago

I can't speak to the advanced stuff but here's my stats on Fundamentals I:

Total time on site (gathered from a web extension): 40h 30m Total days since start: 32

Total XP earned: 1881

Since "1 XP is roughly equivalent to 1 minute of focused work", I "should have" only spent 31 hours. I did the placement test and started at ~30%, and now I'm at 76%. I'd say 75% is stuff I learned in HS but never had a great handle on, 25% I never knew before.

Overall, I'm quite happy with the course. I'm learning a lot every day and feel like I have stronger fundamentals than I did when I was in school. The spaced review is good but I do worry I'll lose it again, so I'm thinking of ways I can integrate this sort of math into my development projects. It's no Duolingo, you really do have to put in effort and aim for a certain number of Xp per day (I try for 60 XP rather than time).

sn98• 2 years ago

Hard to say but this should give you an idea [0].

At that rate, less than a year is reasonable.

[0] https://www.justinmath.com/what-is-the-highest-sustainable-d...

ericmcer2• 2 years ago

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.

hosh2• 2 years ago

There's a lot of interesting discrete math that can supercharge programming at different levels of scale. What's pretty cool is that it reveals a layer of understanding when I watch my toddlers learn math from counting.

One of the interesting things is being able to exactly describe how something is an anti-pattern, because you have a precise language for describing constraints.

chankstein38• 2 years ago

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?

jolt42• 2 years ago

Agree. The article turned me off as well. No specific example, felt like an ad.

dfxm12• 2 years ago

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.

edanm• 2 years ago

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 :)

gmays• 2 years ago

+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

sonabinu2• 2 years ago

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

kamaal3• 2 years ago

>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.

heisenzombie2• 2 years ago

The author (and Grothendieck, liberally quoted in the book) disagree with you.

I think the reason you disagree is that it sounds like you’re teaching your child to be good at math class (a perfectly valid and good thing to do). Being good at math class requires being good at rational/logical thinking and computation. It also has only glancing similarities to anything that the author would recognise as mathematics.

delta_p_delta_x• 2 years ago

This is pretty interesting. I did reasonably well in maths up to the A-levels, and then absolutely collapsed in university. I never got a grade better than B- in any maths-adjacent class. Discrete maths was my worst topic, I barely scraped a pass. And the irony was that I majored in CS and physics.

I should probably find a time machine and re-do everything.

itissid5• 2 years ago

"It's the economy stupid" is what I would say. Mental capacity is capacity. Most of us don't study math not because we don't want to but because we can't.

I bet if you asked in a survey of people that if you were given a UBI that covered all your expenses and needs what would you do? It would be perfection of the self or art. Both of these are what is practicing and learning math.

itissid5• 2 years ago

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.

itissid5• 2 years ago

I can actually tell you how to do this right now. Take a 10 day vipassana course and practice the recommended two hours a day. If you use only therapy, seriously follow the therapist, but do some meditation too.

Once you do this, you will soon develop Attention(Concentration) and Equanimity(Inner Calmness in the face of tremendous external circumstances).

Soon you start realizing some inescapable facts:

1. Your current moment is ever changing.

2. One attaches more of the self(the ego, I me and mine) with the past and tries to predict his state in the future and ends up miserable. Don't think with the I, you are bound to suffer. The right action is timeless and free of the I. This is the reduction of the ego.

3. There is tremendous joy in focusing on being present in the moment. If you are into running and all you are doing is taking joy in your feet, breath and posture all the time for 5,10 miles you know what I mean.

This is the key to everything. No amount of book reading on self improvement can get this to you

notepad0x905• 2 years ago

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.

itissid5• 2 years ago

Thanks.

art same as entertainment

Could you volunteer me how much time you spend on it? And how is your day job?

my pursuit of other things like cryptography, better algorithms, and understanding physics is limited by my primitive understanding of mathematics.

Could you volunteer why you would want to learn these subjects? Is it your day job or is it something you would like to pursue in the future.

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.

I said UBI

nradov4• 2 years ago

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.

itissid5• 2 years ago

And how far would that get one doing that? 1, 5, 10, 40 years?

ahoka• 2 years ago

Perhaps some, but why is that a problem? There are already people who do this.

intelVISA2• 2 years ago

Is that not better than the current alternative where they're forced to become grifters?

manvillej• 2 years ago

I would perfect my video gaming art. Teenagers in their basement would fear me after school.

intelVISA2• 2 years ago

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.

ilrwbwrkhv• 2 years ago

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

lifter3101• 2 years ago

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.

XajniN• 2 years ago

Reading most of the answers here, I can only conclude most of you were home schooled or went to some fancy schools for gifted children.

An average human is unable to even write properly. Even basic mathematical operations like multiplication and division are too complicated from their perspective.

cutemonster14• 2 years ago

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?)

denkmoon• 2 years ago

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.

hilbert424• 2 years ago

"High school students are often unhappy with math, because they think it requires some innate things that they don’t have,” Bessis said. “But that’s not true; really it relies on the same type of intuition we use every day."

Agreed, but from my observation mathematics is often taught with a rigor that's more suited to students with a highly mathematical and or scientific aptitude (and with the assumption that students will progress to university-level mathematics), thus this approach often alienates those who've a more practical outlook towards the subject.

Mathematics syllabuses are set by those with high mathematical knowledge and it seems they often lose sight of the fact that they are trying to teach students who may not have the aptitude or skills in the subject to the degree that they have.

From, say, mid highschool onwards students are confronted with a plethora of mathematical expressions that seemingly have no connection their daily lives or their existence per se. For example students are expected to remember the many dozens of trigonometrical identities that litter textbooks (or they did when I was at school), and for some that's difficult and or very tedious. I know, I recall forgetting a few important identities at crucial moments such as in the middle of an exam.

Perhaps a better approach—at least for those who are seemingly disinterested in (or with a phobia about) mathematics—would be to spend more time on both the historical and practical side of mathematics.

Providing students with instances of why earlier mathematicians (earlier because the examples are simpler) struggled with mathematical problems and why many mathematical ideas and concepts not only preceded but were later found to be essential for engineering, physics and the sciences generally to advance would, I reckon, go a long way towards easing the furtive more gently into world of mathematics and of mathematical thinking.

Dozens of names come to mind, Euclid, Descartes, Fermat, Lagrange, Galois, Hamilton and so on. And I'd wager that telling students the story of how the young head-strong Évariste Galois met his unfortunate end—unfortunate for both him and mathematics—would never be forgotten by students even if they weren't familiar with his mathematics—which of course they wouldn't be. That said, the moment Galois' name was mentioned in university maths they'd sit up and take instant note.

Yes, I know, teachers will be snorting that there just isn't time in the syllabus for all that stuff, my counterargument is that it makes no sense if you alienate students and turn them off mathematics altogether. Clearly, a balance has to be struck, tailoring the subject matter to suit students would seem the way with the more mathematically inclined being taught deeper theoretical/more advanced material.

I always liked mathematics especially calculus as it immediately made sense to me and I always understood why it was important for a comprehensive understanding of the sciences. Nevertheless, I can't claim that I was a 'natural' mathematician in the more usual context of those words. I struggled with some concepts and some I didn't find interesting such as parts of linear algebra.

Had some teacher taken the time to explain its crucial importance in say physics with some examples then I'm certain my interest would have been piqued and that I'd have showed more interest in learning the subject.

generationP• 2 years ago

If the article is in any way representative of the book, then I'm not sure what there is to be learned from the book. That mathematical skills can be honed through practice? That it happens at an intuitive, pre-rigorous level before it is ready to be written down on paper? How surprising. And I doubt he can disprove the genetical component of intelligence, only show that there are other components to mathematical productivity as well.

At least I know that David Bessis's mathematical work is not as shallow as this. His twitter thread on the process https://x.com/davidbessis/status/1849442592519286899 is actually quite insightful. I would guess this also made it into the book in some longform version, but I don't know whether I would buy the book just for that.

lupire2• 2 years ago

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.

davidbessis8• 2 years ago

Great to see so many reactions to my interview, thanks!

I see that many people are confused by the interview's title, and also by my take that math talent isn't primarily a matter of genes. It may sound like naive egalitarianism, but it's not. It's a statement about the nature of math as a cognitive activity.

For the sake of clarity, let me repost my reply to someone who had objected that my take was "clickbait".

This person's comment began with a nice metaphor: '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.'

Here was my reply (copy-pasted from my post buried somewhere deep in the discussion):

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

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

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

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

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

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

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

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

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

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

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

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

samatman3• 2 years ago

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.

davidbessis8• 2 years ago

I think you really are confused. You are mistakenly equating "non-primarily genetic" with "easily teachable".

The story is much more complex than "if it's not genetic then everybody should get it". It's quite cruel to assume that if you don't get math today you'll never get it, and there are tons of documented counter-examples of kids who didn't get it at all who end up becoming way above average.

If you think that Descartes, Newton, Einstein, Feynman, Grothendieck (to just cite a few) are all equally misled on their own account because of Simpson's Paradox, which statistical result will to bring to the table to justify that YOU are right?

By the way, Stanislas Dehaene, one of the leading researchers on the neuroscience of mathematical cognition, is also on my side.

alganet• 2 years ago

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.

zyklu5• 2 years ago

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.

davidbessis8• 2 years ago

Indeed, it does involve redefining genius as a "state", or "flow", or "trajectory".

When I say it's not primarily genetic, many people wrongly assume there's an entirely explainable and replicable way of accessing this state. There isn't.

The 20,000 hours rule is a bit misleading, because who gets to invest 20,000 hours into something? How do you create this drive, this trajectory? You must have a good hope that it'll yield something worth the effort.

This is why the injunction to "work harder" so often misses the mark.

However, even if only a tiny fraction of the population will end up becoming a "genius", it's very important to debunk the myth, because the real story has valuable lessons for everyone: it gives concrete and pragmatic indications on what one should be on lookout for.

It's not fully teachable up to genius level, but the directionality is teachable and extremely valuable.

what9001• 2 years ago

What an incredibly neuro-typical thing to claim

johnp314• 2 years ago

It's a shame the title begins with 'Mathematica', makes one think the book is about the Wolfram software. That's the first thing I thought of when I saw the title. Hopefully Wolfram doesn't sue him for copyright violation or some such infringement.

tombert• 2 years ago

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.

exprofmaddy• 2 years ago

Some very nice related works that dispel widespread math myths: (1) What Is Mathematics, Really? Hersh, 1997. https://books.google.com/books/about/What_is_Mathematics_Rea... (2) Where Mathematics Comes From. Lakoff and Nunez, 2000. https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From

sethpurcell• 2 years ago

Given the sentiment in these comments, I figure this crowd might be interested in the book "Measurement" by Paul Lockhart (the guy who wrote "A Mathematician's Lament")

He's of the opinion that math should be taught not as jumping through hoops for "reasons", but as an art, enjoyable for its own sake, and that this would actually produce more confident and capable thinkers than the current approach. (I think the argument applies to almost all education but his focus is just on math.)

aleden2• 2 years ago

In my high school we were basically only instructed to get good at applied math. Calculus. Which more often than not was simply "plugging it in". Most of that work is trivially automatible through Mathematica. When I reached a university, I took number theory and abstract algebra and it blew my mind that math was actually so beautiful in a way that defied explanation. When I took real analysis I finally saw the side of calculus that didn't seem like a waste of time.

One day, I went back to my high school and spoke to my computer science mentor back then [1]. I passionately asked him why we were never exposed to group theory. The answer, he said, was the SAT. None of that stuff is on the SAT, so it can't be justified teaching.

[1] The great Andrew Merrill

BigGreenJorts• 2 years ago

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.

aleden2• 2 years ago

Many people dismiss mathematics because they aren't interested in it. I definitely wasn't interested in doing dozens and dozens of error-prone problem sets that mainly boiled down to performing arithmetic. I don't need to do that.

My number theory professor was a brilliant mind, someone who had spent lots of time at the Institute for Advanced Study, and he absolutely sucked at adding/subtracting/multiplying numbers. And that was something he freely admitted. It wasn't important to his work, and it isn't important to mine.

sn98• 2 years ago

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

javier_e06• 2 years ago

What is 13% of 91? I don't know. Do you? But I now 10% of 91 is 9.1 I got somewhere eh? Hey also I know that 1% of 91 is 0.91 Duh! lets triple that. 0.91 x 3 = 0.9x3 + 0.01x3 = 2.7 + 0.03 = 2.73 Now lets add 9.1.. 11.83 Weee! (Now my date is rolling her eyes and the waiter is stone faced)

plasticeagle• 2 years ago

Two thoughts

1) It's tragic that being "bad at math" is often positioned as some kind of badge of honour.

2) It's definitely not the case that everyone is capable of mathematical thinking. Having spent a certain amount of time trying to teach one of my kids some semblance of mathematical thinking, I can report confidently that his ability in this area is almost non-existent. His undeniable skills lie in music and writing, but definitely not in maths.

Yes, music and maths have some things in common. But musical thinking is not mathematical thinking.

mncharity• 2 years ago

I've lately been struck by people having a life difficulty due to "missing a clue" absent some experience. The person poorly conceptualizing and executing physical therapy, for lack of athletic experiences. The person poorly handling cognitive decline, for lack of a grasp of work processes. The person variously failing from having physical discomfort as an abort criteria, for lack of experiences normalizing its deferral.

"I have this clue at hand" can have broad impacts. Software development's emphasis on clarity, naming, and communication protocols, helped me a lot with infant conversation. Math done well, can be a rich source of clues, especially around thinking clearly.

There's an idea that education should provide more life skills (like personal finance). And another, that education should have a punch list (as in construction), of "everyone at least leaves with these". Now AIish personalized instruction will perhaps permit delivering a massive implicit curriculum, far larger than we usually think of as a reasonable set of learning objectives. Just as a story can teach far more than the obstensible moral/punchline of the story, so too might each description, example, question and problem, dynamically tuned in concert. So perhaps it's time to start exploring how to use that? In the past, we worked by indirection - "do literary criticism, and probabilisticly obtain various skills". And here, from math. Perhaps there's a near-term opportunity to be more explicit, and thorough, about the cluefulnesses we'd like to provide?

metalman• 2 years ago

To use an absolutist statement about human well bieng tied to the authors pet ideals,is a classic example of magical thinking, and a lack of understanding in general,tied to a character assination of the many many excellent non numerative humans living and thriving everywhere. Not nice at all.

etherealG• 2 years ago

Shouldn’t that headline read “or can benefit from”. Not a shining example of good mathematical thinking. Oh the irony.

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

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