[HN Gopher] Everyone is capable of, and can benefit from, mathem...
___________________________________________________________________
Everyone is capable of, and can benefit from, mathematical thinking
Author : sonabinu
Score : 512 points
Date : 2024-11-21 01:45 UTC (21 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| benreesman wrote:
| 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 wrote:
| 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.
| benreesman wrote:
| 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.
| Jensson wrote:
| Isn't that a positive statement, that you can ask questions
| without worry since they aren't stupid?
| cutemonster wrote:
| Maybe, I guess it's easily misinterpreted though O.O
| ckw wrote:
| That's a south park quote:
| https://youtu.be/wWfacULP1o0?si=ddBWXFuQMoxxY-Yu
| benreesman wrote:
| That was a belly laugh on a tough day.
|
| Thank you Sir or Madame.
| cchi_co wrote:
| How much of math aversion stems from a chain reaction of
| ineffective instruction
| benreesman wrote:
| According to an excellent mentor: all of it minus epsilon.
| mr_mitm wrote:
| 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.
| benreesman wrote:
| 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 Godel.
| mr_mitm wrote:
| Did you mean to reply to another post? I don't follow at
| all.
| diffeomorphism wrote:
| 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.
| mr_mitm wrote:
| And how would you call the students in the left tail of the
| Bell curve if not bad students?
| diffeomorphism wrote:
| Below average students and as long as the average is high
| enough they are still very competent.
|
| A bad teacher instead gives you a bimodal distribution
| and just doesn't bother teaching those students.
| jyscao wrote:
| > 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 wrote:
| > 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...
| limit499karma wrote:
| > 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...
| vundercind wrote:
| 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".
| limit499karma wrote:
| 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.
| tkgally wrote:
| 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.
| Frummy wrote:
| 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.
| tkgally wrote:
| 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.
| Frummy wrote:
| Thanks for sharing. The reverse direction here, I'm trying
| to go from fuzziness to the exactness of those true/false
| dichotomies, haha. The way I've been attacking mathematics,
| it's like a tree in the forest, one could start with the
| axioms and from the base reach each branch and the leaves
| and fruits. But I've just been walking around the tree,
| looking at the leaves and fruits and branches from
| different directions to see ways of climbing without doing
| a whole lot of climbing. What I mean is I've been thinking
| and reading in an imprecise way a whole lot without
| actually juggling symbols with pen and paper, haha. Or a
| roadtrip analogy, I've done little driving and a lot of map
| ogling. At least I won't miss the turns when I pick up some
| speed.
| plsbenice34 wrote:
| 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 wrote:
| 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 wrote:
| 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.
| namaria wrote:
| > 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.
| katzenversteher wrote:
| I believe charisma, confidence and looks also play a huge
| role.
| DiscourseFan wrote:
| 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
| sonabinu wrote:
| I was actually thinking Jean Paul Satre when I read his answers
| ai4eva wrote:
| 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 wrote:
| 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 wrote:
| >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 wrote:
| > 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?
| Jensson wrote:
| It is easy to learn for some.
| physicsguy wrote:
| 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 wrote:
| 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.
| physicsguy wrote:
| Well, I guess what I mean is that most people have some
| level of general intelligence that when applied correctly
| can generally give good results in most subjects. In
| general the people who do well in school do well in
| everything, even if they have a preference, and as such
| could do well in most of those subjects if they went on to
| further study. The evidence tends to be that in lower
| income countries people push towards subjects more likely
| to bring financial stability than those they prefer which
| bears this out somewhat.
|
| There are some extreme cases of course but I'm not sure the
| general public needs to worry too much about those, most of
| us aren't an Einstein nor do we have learning disabilities.
| j7ake wrote:
| The extreme case does not imply a binary scenario ie that
| there are those that can those that cannot.
|
| Rather, learning ability is a continuum. people have
| varying degrees of ability to learn mathematics. Couple
| this with environmental factors and society generates a
| huge variability in mathematical ability that crosses
| income levels and other demographics.
|
| This view is rejected by many because it is against the
| push for equality.
| fluoridation wrote:
| You get a huge variability if you consider the absolute
| extreme outliers. _Most_ people should be able to reach a
| level of competence where they can understand
| mathematical concepts abstractly and apply that same
| reasoning to other areas, and not feel a visceral
| rejection at the mere idea. I think that 's a modest
| enough standard that a good portion of any given
| population should be able to reach, and yet education is
| failing at achieving that.
| j7ake wrote:
| Your statement is not backed up by data and simply
| wishing it should happen isn't a strong argument.
|
| You probably have a narrow definition of "most people"
| (probably some motivated high school or undergraduate
| student) and too loose with what it means to "understand
| mathematical concepts abstractly".
|
| Take an analogy: imagine professional musicians saying
| that most people should be able to take a piece of music
| and understand its harmonic structure, then apply it to a
| new setting to generate a new piece. Most people will
| reject this idea as absurd.
| fluoridation wrote:
| Where's the data backing up what you said?
|
| >You probably have a narrow definition of "most people"
| (probably some motivated high school or undergraduate
| student)
|
| I was thinking "3-4 out of 5 people you pick on the
| street at random".
|
| >too loose with what it means to "understand mathematical
| concepts abstractly".
|
| Enough that they could recognize whether a mathematical
| concept is applied correctly (e.g. if I have a 2% monthly
| interest, should I multiply it by 12 to get the annual
| interest? Why, or why not?) and conversely to correctly
| apply concepts they already understand to new situations,
| as well as to leverage those concepts to potentially
| learn new ones that depend on them.
|
| >imagine professional musicians saying that most people
| should be able to take a piece of music and understand
| its harmonic structure, then apply it to a new setting to
| generate a new piece. Most people will reject this idea
| as absurd.
|
| Okay, but we're arguing about what is the case, not about
| which idea has more popular support. Since most people
| don't understand thing 1 about composition, why should
| their opinion matter? A skilled composer's opinion on the
| matter should have more bearing than a million laymen's.
| sublimefire wrote:
| > 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.
| cchi_co wrote:
| I do not think that Bessis's argument is entirely "made up"
| barrenko wrote:
| 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 wrote:
| 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 wrote:
| 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.
| agnishom wrote:
| Gentle Reminder that the author of this article used to have a
| wonderful math channel:
| https://www.youtube.com/c/pbsinfiniteseries
| lupire wrote:
| ???
|
| "Mathematician Tai-Danae Bradley and physicist Gabe Perez-Giz
| offer ambitious content ... Previous host Kelsey Houston-
| Edwards "
| gsabo wrote:
| 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 wrote:
| > 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 wrote:
| 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.
| shrubhub wrote:
| But also, some people are stupid, right?
| ajuc wrote:
| The inborn part is how quickly you get results (good or
| bad). Stupidity is the results.
|
| If we spent 50% of time thinking productively - inborn
| thinking speed would matter. But in my estimate even 5%
| is generous.
|
| So it matters far more what kind of feedback you have to
| filter out the wrong results, and how much time you spend
| thinking - than how quickly you can do it.
|
| Also practice helps with speed.
| yawpitch wrote:
| Intellect is like a gas, it will expand to fill its
| container. The container, in humans, is epigenetic and
| social -- genetics only determines how hot or cold your
| gas is, ie how fast and how fluidly it expands, but
| you're _taught_ your limits -- it's best to see _stupid_
| as not how limited you _are_ relative to other but what
| limits you _have now_ and _may_ abandon in the future.
|
| That said, some people received a smaller starting
| container, and might need some help cracking it. That's
| the work of those who think they've found a bigger one.
| shrubhub wrote:
| 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?
| elbear wrote:
| Not even that. It depends on the learned ability to stop
| pushing yourself when your focus is wavering. That's how
| you develop aversion towards the topic. Let your natural
| curiosity draw you to particular topics (that's why you
| might have a winding road through the subject).
| air7 wrote:
| parent comment was a bit tounge-in-cheek but I'll
| continue the sentiment: You're saying that the curiosity
| is "natural" hence one is either born with it or not. I
| think that there is no way around the fact that it will
| be hard and uncomfortable to mimic the progress of
| someone that has an innate inclination towards a subject
| (be it talent or focus or curiosity) artificially.
| card_zero wrote:
| Hey, that doesn't have to be what "natural curiosity"
| means. Besides which it makes no sense to say people are
| born with complex interests. I mean, OK, your genes might
| incline you a certain way, but that's not the same thing.
|
| Being interested in a subject is massively helpful to
| learning it. But interest arises circumstantially, it's
| an emotion. The grim reality that it would be really
| _useful_ to you to learn a certain subject does not
| necessarily make you interested in the subject,
| unfortunately. (Perhaps "financially interested", but
| that's something else.)
| ericd wrote:
| I think there is some natural inclination towards
| abstract thinking versus more grounded in reality, just
| judging based on kids I know. Some of them really enjoy
| playing with ideas in their heads, some enjoy playing
| with things they can touch more. It seems likely that
| those different attractions would express themselves in
| how much they practice different things as time goes on.
| kdfjgbdfkjgb wrote:
| > You're saying that the curiosity is "natural" hence one
| is either born with it or not.
|
| Why does curiosity being natural necessarily mean some
| people are born without it? It could also mean everyone
| (or every average human) is born with it, and overtime it
| gets pushed out of people.
| Retric wrote:
| Some infants explore vastly more than others.
|
| So the minimum might not be zero, but it isn't some fixed
| quantity.
| diffeomorphism wrote:
| 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.
| drbig wrote:
| Thank you for the insight that academic (in a very broad
| sense) bulk-fixed-time approach does in fact produce both
| of the cycles, and the gap indeed only widens with time
| (speaking from personal experience, especially from my life
| as an undergrad student).
|
| Reminds me of my personal peeve that "studying" should not
| be "being taught", studying is pursuit of understanding,
| "being taught" is what happens in primary school (and I'm
| aware I'm simplifying here).
| blackbear_ wrote:
| I would say that you could generalize this even further
| outside of education. A few early successes in life can
| greatly accelerate one's trajectory, while early failures
| could set one many years back. And this happens
| independently of whether those events are due to skill or
| luck.
| jvanderbot wrote:
| Indeed, speed is often read as "smarts" whereas I would
| maintain it's much more often "preparation". We can't on
| one hand believe in the plasticity and retrainability of
| the mind, while simultaneously believing that speed is
| something only a few are born with. On the nature/nurture
| scale, I think it's 20/80 or so - but prodigies and
| geniuses have an _interest_ that keeps them thinking and
| learning 10x or 100x more than other kids, and a little
| bump that lets them get started easier and therefore much
| earlier.
|
| This sets them up for fantastic success very quickly. [1]
| shows a great example of this.
|
| I'm fond of saying "You can do anything you want, but
| wanting is the hard part", because to _truly_ be a
| grandmaster, genius-level mathematician, olympic athlete,
| etc, requires a dedication and amount of preparation that
| almost nobody can manage. Starting late, with emotional
| baggage, kids, and having to spend 5 years relearning how
| to learn? Forget it.
|
| 1. https://danielkarim.com/how-to-become-a-genius-the-
| polgar-ex...
| NoMoreNicksLeft wrote:
| > I'm fond of saying "You can do anything you want, but
| wanting is the hard part", because to truly be a
| grandmaster, genius-level mathematician, olympic athlete,
| etc, requires a dedication and
|
| I was having a problem agreeing with this subthread, and
| I have you to thank for putting it into words that I can
| finally formulate my disagreement against.
|
| Have you never met one of those people for whom they did
| not need to "want"? They could literally phone it in and
| still do better than anyone else, no matter how dedicated
| they were. Even should practice/study be necessary for
| them, they benefited from it to some absurd proportion
| that I couldn't even guess to quantify. I've known more
| than one of these people.
|
| I think most believe they don't exist for two reasons.
| The first is the ridiculous number of television shows
| and movies that depict _motivation_ as being the key to
| success. We 're just inundated with the (unsupported by
| evidence) that this is the means to extraordinary genius.
| Second, I would say that this is the most comforting
| theory. "Why yes, I could have been a gifted whatever or
| a talented something-or-other if I had put the time in,
| but I chose this other thing instead."
|
| Maybe some would say we all need to believe this, that a
| society that doesn't believe in it is harsher or more
| unkind.
| jvanderbot wrote:
| I think I have met those folks. Maybe not. And you're
| welcome!
|
| They're just quick. But the ones I've met, at least, are
| quick to make associations. When I really dig and ask
| them to explain themselves or a concept, they usually
| make analogies to things they know, but I don't. Then I
| have to go learn that thing. Then they try the analogy
| again, but I haven't fully learned it from years of
| making analogies about it.
|
| Years of grad school experience was painful like this,
| until I got to a point 10 years after grad school, after
| a PhD, and well into research, that I "just got" things
| (in my subfield) as well. It's these experiences that
| made me feel that it's 80% preparation and perspiration
| (both of which are dominated by time), and 20% "other"
| mythology. Don't get me wrong, that 20% is what makes a 2
| year old read earlier than others, and getting started
| reading at 2 (and continuing it!) for 4 years before
| starting school _will_ make you light years ahead of your
| peers. The same goes for chess, math, etc etc. There is
| something legendary about Oppenheimer learning enough
| dutch in 6 weeks to deliver a lecture. Or perhaps
| learning to translate his lecture and memorizing it. Who
| knows.
|
| Do we really believe there's a magical "genius" such that
| they can do anything? No, so what are the limits to their
| genius? The limits are defined by what they are a genius
| _at_. This is a tautological definition.
|
| I'm not saying "Anyone at any time can become a genius at
| anything". I'm saying "If you take a kid, start early,
| and cultivate them just right so that you have time to
| realize compounding effects, - you can let them grow into
| basically anything" (probablistically speaking - there
| are learning disabilities and physical issues etc).
| StefanBatory wrote:
| > I think most believe they don't exist for two reasons.
|
| I add third (okay, 2b) - because the pain of coming up
| with the fact other people are better than you at a deep,
| fundamental level is too overwhelming.
| stonemetal12 wrote:
| Bobby Fisher won his first US Championships at 14 against
| people who had been playing chess longer than he had been
| alive. Suggesting they didn't want it more, or practice
| more than some kid is silly.
|
| "We can't on one hand believe in the plasticity and
| retrainability of the mind, while simultaneously
| believing that speed is something only a few are born
| with."
|
| Sure we can, the initial orientation of neurons differs
| between people, so some people need less "plasticity and
| retrainability" to be good at a task. Plasticity is
| physical characteristic like height and varies between
| people.
|
| Initial speed usually isn't that important, but speed of
| learning is important and makes the difference between
| possible and impossible within a human lifetime.
| jvanderbot wrote:
| I think there's a probabalistic argument I'm making
| that's more in line with the article.
|
| Yes - there will be 10x-ers. And that group will have a
| 10x-er iside it, and so on given exponential dropoff of
| frequency of talent. Bobby Fisher is a few std dev above
| even the best, perhaps.
|
| Generally speaking, "You can do anything you want, but
| wanting (enough, and naturally) is the hardest part"
| might need a three standard deviation limit.
|
| Have you heard the phrase: Being average among those who
| practice makes you 9X% among the population? I think
| that's what I'm saying - you can be a top performer if
| you dedicate yourself, especially early enough, but
| almost nobody will.
| matwood wrote:
| I agree with you. I don't think I'm naturally gifted at
| much (I'm just average), but I was taught stubborn hard
| work pretty early on. Unfortunately it took me until my
| 20s to figure out I could be athletic if I applied that
| hard work. I could also be good at programming doing the
| same. I've met people who are truly gifted and it's
| amazing, but I'm pretty decent at the things I worked
| hard at.
| hilbert42 wrote:
| _" Initial speed usually isn't that important, but speed
| of learning is important and makes the difference between
| possible and impossible within a human lifetime."_
|
| Likely so, but is suggest that personality, drive and
| motivation are also very important factors. I know from
| experience that stuff I had little interest in as a
| youngster and that I've still little in I still know
| little about.
|
| Yes, my interests have grown and broadened over the years
| but simply I regard some stuff so irrelevant to my life
| that it's not worth a second thought and I am much better
| off applying my limited number of neurons to matters of
| greater importance and enjoyment.
|
| Of course, no one has the luxury of just learning about
| what one finds interesting and or enjoyable, life's
| knocks and experiences along with utilitarian-like
| imperatives force one to learn stuff they'd rather not
| know about.
| 1980phipsi wrote:
| I find it is good to go back to things you struggled with
| in the past and come at them with a new and broader
| understanding.
| graemep wrote:
| > 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.
| LoganDark wrote:
| > 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.
| yawpitch wrote:
| > a side effect of experiencing the world differently
|
| A side effect for which there is a substantial, lifelong,
| and most importantly _wide_ cost, even if it occasionally
| confers usually small, usually fleeting, and most
| importantly _narrow_ advantage.
| sethammons wrote:
| At such cost with such narrow advantage, why has it
| persisted so pervasively? I would counter that the
| advantage is wider and the cost narrower than your
| current value system is allowing you to accept.
| LoganDark wrote:
| Natural selection doesn't care about cost or advantage,
| only reproduction.
| sethammons wrote:
| It is the sum of costs and advantages that lead to
| reproductive success. The trait is still here and still
| prevalent meaning people are still getting laid and
| starting families and presumably leading fulfilling
| lives.
|
| I'm not sure what you are trying to say.
| LoganDark wrote:
| > I'm not sure what you are trying to say.
|
| I'm saying, if it doesn't ruin lives to the point of
| preventing reproduction, then it stays in the gene pool.
|
| Basically, I'm saying this:
|
| > The trait is still here and still prevalent meaning
| people are still getting laid and starting families and
| presumably leading fulfilling lives.
| vacuity wrote:
| As long as an organism isn't performing too badly, it
| stays in the gene pool. It can persist and even share its
| genes more broadly, if in diluted form, to the other more
| successful organisms. And then some of those mixed-genes
| organisms may occasionally express more strongly, but
| again not enough to affect reproductive success across
| the population.
| LoganDark wrote:
| Yes, there is a significant cost to being built
| differently regardless of perceived advantages (by one's
| self or others). For example, as an autistic, I have to
| cope with finding interaction with non-autistics quite
| difficult for me, even if detail-oriented thinking can
| make certain tasks seem easier to me.
| Malidir wrote:
| >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.
| PittleyDunkin wrote:
| What makes you think that "genius" is nature and not
| nurture? I'd love to see the evidence for this; i'm deeply
| skeptical.
|
| Edit: I don't mean to argue that there aren't genetics
| involved in determining aptitude on certain tasks, of
| course, but the assumption that genius is born and never
| made feels like a very shallow understanding of the
| capacity of man.
| Malidir wrote:
| > I'd love to see the evidence for this; i'm deeply
| skeptical.
|
| Cool, come and have a coffee with me :) I have older and
| younger siblings and was the one randomly blessed.
|
| Whereas most recognised talents are associated with hard
| work and so there is then this visible link, I am a good
| example as I did the bare minimum throughout education
| (and beyond...).
|
| The way my brain processes and selectively
| discards/stores the information it receives is very
| different to majority of the population. I have no
| control over it.
|
| I take zero credit for any of my achievments - I
| regularly meet intelligent people near to retirement who
| have been to a tier 1 university, may have PHDs, worked
| 60 hours a week since they were born, been on course and
| what not and cannot reach the levels I can.
|
| My nurturing was no different to siblings/peers (and was
| terrible!)
|
| Note: I have my weaknesses too, but as a whole, I am
| exceptional. Not through effort!! Completely random -
| neither of my parents are intelligent and nothing up the
| ancestary tree as far as I know.
| dennis_jeeves2 wrote:
| >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 wrote:
| 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.
| sdeframond wrote:
| > 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!
| wtetzner wrote:
| > I respectfully, but strongly, disagree. There's a reason
| most NBA players are over 2 meters tall, and one does not
| become taller with time, dedication nor focus.
|
| Being tall isn't a skill. I suspect you could be skillful
| enough at basketball to overcome the hight disadvantage.
| However, I think most people who might become that skillful
| see the high disadvantage (plus the general difficulty of
| becoming a pro basketball player) and take a different path
| through life. It's also possible that the amount of time
| that would be needed to grow your skill past the height
| disadvantage is too long, so it's not feasible to do it to
| gain a position in the NBA.
| rafaelero wrote:
| Intelligence is also not a skill, but the thing that
| makes you skillful in all cognitive tasks. Just like what
| height does to basketball players.
| nemo wrote:
| >Intelligence is also not skill, but the thing that makes
| you skillful in all cognitive tasks.
|
| Careful with that "all", even the most highly intelligent
| humans still have peaks and deficits in different
| domains.
| samatman wrote:
| It's a matter of the definition. The general factor of
| intelligence, which is measured through various somewhat
| lossy proxies like IQ tests, is exactly the degree to
| which someone exceeds expectation on all cognitive tasks
| (or vice versa).
|
| The interesting finding is that this universal
| correlation is strong, real, and durable. Of course
| people in general have cognitive domains where they
| function better or worse than their g factor indicates,
| and that's before we get into the fact that intellectual
| task performance is strongly predicated on knowledge and
| practice, which is difficult to control for outside of
| tests designed (successfully, I must add) to do so.
| goatlover wrote:
| Height is one physical attribute that helps, and
| professional players are mostly above average height for
| a reason. But also hand-eye coordination and fast-twitch
| muscles help even more. Many basketball players are very
| explosive athletes, because it's a sport with a
| relatively small play area and lots of quick movements
| are needed.
|
| Track and swimming are where innate physical attributes
| have the most obvious benefits. Michael Phelphs had the
| perfect body for swimming. There is no amount of
| trainingg that 99.999% of the population could do to get
| close to what Usain Bolt ran. Most humans could not train
| to run under 4 minutes in a mile or under 2:30 in a
| marathon. They just don't have the right muscular and
| cardiovascular physiology.
|
| Team sports are of course more complicated as other
| qualities come into play that aren't as directly
| physiological.
| nradov wrote:
| Most NBA players are _under_ 2 meters tall. The average
| height is 1.99 meters.
|
| https://www.lines.com/guides/average-height-nba-
| players/1519
| nolamark wrote:
| Since we are being pedantic, your statement may be true
| but it is unsupported by the data you presented. To make
| it simple, let's talk about the imaginary basketball
| league with four players, of unit less heights of 4, 4,
| 4, and 1. The average height is 3.25, yet 3/4 the players
| are taller than average.
|
| A paid promotion of International Median is not Average
| Association.
| benjijay wrote:
| Most people have an above-average number of legs.
| sdeframond wrote:
| What's the average number of legs for humans ?
| cutemonster wrote:
| A bit less than two
| samatman wrote:
| Simpson's Paradox[0] is the reason people are so easily
| seduced by the tempting, but dead wrong, illusion that
| humans are in any sense equal in their innate capacity for
| anything.
|
| Because it turns out that, in the NBA, height _does not_
| correspond with ability! This of course makes sense,
| because all the players are filtered by being NBA
| professional basketballers. A shorter player simply has
| more exceptional ability in another dimension, be that
| dodging reflex, ability to visualize and then hit a ball
| trajectory from the three point line, and so on.
| Conversely, a very tall player is inherently useful for
| blocking, and doesn 't _have to_ be as objectively good at
| basketball in order to be a valuable teammate.
|
| Despite this lack of correlation, when you look at an NBA
| team you see a bunch of very tall fellows indeed. Simpson's
| Paradox.
|
| We see the same thing in intellectual pursuits. "I'm not
| nearly as smart as the smartest programmer I know, but I
| get promoted at work so I must be doing something right.
| Therefore anyone could do this, they just have to work hard
| like I did". Nope. You've already been selected into
| "professional programmer", this logic doesn't work.
|
| [0]: https://en.wikipedia.org/wiki/Simpson's_paradox
| hilbert42 wrote:
| _" 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 wrote:
| 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.
| solarized wrote:
| easy_things -> comfort_zone
| cchi_co wrote:
| This perspective has discouraged so many people from exploring
| their potential
| chipdart wrote:
| > 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 wrote:
| 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.
| chipdart wrote:
| > I think the ML people want to get (a narrow band) of
| stuff done and ivory towered people want to understand a
| prove things. ML is applied mathematic. Both are needed.
|
| I don't agree. First of all, ladder-pulling in math is
| observed at all levels, not only cutting-edge stuff.
| Secondly, it's in applied mathematics where pure math takes
| a queue onto where to focus effort. See how physics drives
| research into pure math.
| theclansman wrote:
| 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 wrote:
| 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.
| doublerabbit wrote:
| 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.
| sethammons wrote:
| My best friend was like that. Couldn't see the practicality
| until he got bit by a geology and water science bug. He
| went from calling me to get help figuring out percentages
| to doing chemistry equations in his head because he "got"
| the applicability.
|
| My brother's mom tutors math. One of her insights with a
| former student was that they were in need of forming some
| number sense. She started by walking them both out to the
| street: "how many tires are there on this street of parked
| cars?" The student, already flummoxed, started panic
| guessing. So she started with counting.
|
| For times tables, have you developed any intuition around
| it? For me, times tables are rectangles composed of unit
| squares and that helps with my intuition. Modern Common
| Core standards in the US focuses a lot on exposing
| different mental models to students. And after seeing the
| same 4x6 enough times your brain will automatically
| associate that with its solution. Instead of calculating,
| it is memorized.
|
| My brain doesn't require car tires, geology, or other
| practical needs: it likes puzzles. I struggle with medical
| stuff and I can feel my brain switching to meh-mode and
| hardly anything sticks. I don't know how many times I have
| been told about the different kinds of sugar and how your
| body uses that energy and I would still have to look it up.
| doublerabbit wrote:
| > For times tables, have you developed any intuition
| around it?
|
| I've tried different approaches. 4x3 being 3x4
|
| But somehow I end up miscounting and giving the answer
| for 4x4 or be off a digit every-time.
|
| A good example was that I was at a brewery last night.
|
| They didn't do pint glasses but glasses of: 1/4, 1/2 and
| 2/3rd's.
|
| I thought 1/2 was more than 2/3rd's so I ordered a 1/2
| rather than thinking I was getting more.
|
| However I was unable to visual how 2/3 is more than 1/2
| when 1/2 is half a pint, or half a glass.
|
| My visual capabilities are great at others but just
| couldn't formulate the equation of 2/3 is more than 1/2.
|
| Very simple stuff, but it just doesn't meld.
| nradov wrote:
| Famously the A&W restaurant chain failed when introducing
| a 1/3 lb hamburger because many customers thought it was
| _smaller_ than a 1 /4 lb hamburger.
|
| https://awrestaurants.com/blog/aw-third-pound-burger-
| fractio...
| tzs wrote:
| > However I was unable to visual how 2/3 is more than 1/2
| when 1/2 is half a pint, or half a glass.
|
| Maybe visualize splitting a pint with a friend. If you
| split the pint into 2 equal parts and each of you gets 1
| of those 2 parts you each get the same amount.
|
| Then visualize splitting it instead into 3 equal parts.
| You get 1 of those parts and your buddy gets 2. There's
| no fractions there so it should be easier to visualize
| that your buddy got twice as much as you did.
|
| Comparing those two visualizations might make it easier
| to see that someone who gets 2/3 of a pint gets more than
| someone who gets 1/2 of a pint.
| khafra wrote:
| > 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_trw wrote:
| >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.
| bamboozled wrote:
| How have you been working on it? Asking for a friend ;)
| bdjsiqoocwk wrote:
| 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 wrote:
| 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
| tgv wrote:
| 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.
| nestes wrote:
| 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.
| chipdart wrote:
| > 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.
| tgv wrote:
| > Not really. There's nothing inherently special about
| people who dedicated enough time to learn a subject.
|
| "You didn't work hard enough." People really blame you for
| that, not for lacking talent.
|
| > So far in human history there were less than 200 people
| who ran 100m in less than 10s.
|
| And many millions have tried. There may be 200 people who
| can run it under 10s, but there are thousands that can run
| it under 11s, and hundreds of thousands that can run it
| under 12s. Unless you've got clear evidence that those
| people can actually run 100m in less than 10s if they
| simply try harder, I think the experience of practically
| every athlete is that they hit a performance wall. And it
| isn't different for maths, languages, music, sculpting (did
| you ever try that?), etc. Where there are geniuses, there
| also absolute blockheads.
|
| That's not to say that people won't perform better when
| they work harder, but the limits are just not the same for
| everyone. So 'capable of mathematical reasoning' either is
| some common denominator barely worth mentioning, or the
| statement simply isn't true. Clickbait, if you will.
| davidbessis wrote:
| 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.
| calf wrote:
| This power law argument immediately reminds me of
| education studies literature that (contrary to the math
| teachers in this thread) emphasize that mathematical
| ability is learned cumulatively, that a student's success
| feeds on itself and advances their ability to grasp more
| difficult material.
|
| As for my own half-baked opinion, I want to say that the
| Church-Turing Thesis and Chomsky's innate theory of
| cognition have something to add to the picture. Homo
| sapiens as a species essentially has the capacity to
| think analytically and mathematically; I want to argue
| this is a universal capacity loosely analogous to the
| theory of universal Turing machines. So what matters is
| people's early experiences where they learn how to both
| practice and, critically, to play, when they learn
| difficult ideas and skills.
|
| Also, as an amateur pianist, most people don't know that
| modern piano teaching emphasizes not fixed limits of the
| student but that many students learn the wrong techniques
| even from well-meaning piano coaches. Just the other day
| I was watching a recent YouTube Julliard-level
| masterclass where the teacher was correcting a student on
| her finger playing technique, presumably this student had
| been taught the wrong technique since childhood. With
| music or sports a coach can visually see many such
| technique problems; with math teaching it of course
| harder.
| cutemonster wrote:
| > document the secret "oral tradition" of top
| mathematician
|
| > 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.
|
| Sounds like people mostly living in different bubbles?
| What do they know about the world?
|
| They aren't hanging out with the kids who fail in school
| because maths and reading and writing is to hard, and
| then start selling drugs instead and get guns and start
| killing each other.
|
| > [they] don't think it's because of their genes
|
| Do you think someone would tell you, if he/she thought it
| was?
|
| I mean, that can come off as arrogant? Wouldn't they
| rather tend to say "it was hard work, anyone can do it"
| and prioritize being liked by others
|
| > Pareto-like distribution like what we see in math
|
| Unclear to me what you have in mind. If there's a graph
| it'd be interesting to have a look? I wonder whats on the
| different axis, and how you arrived at the numbers and
| data points
| llm_trw wrote:
| 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.
| tgv wrote:
| Strangely enough, you'd be hard pressed to find a
| mathematician who doesn't know what a square root is.
|
| And I didn't mention genetics. Nature is complicated.
| llm_trw wrote:
| You'd also be hard pressed to find one who doesn't know
| how to flush a toilet.
|
| Neither trivia has anything to do with being good at
| mathematics as done by mathematicians.
| tgv wrote:
| Are you an LLM? You brought up the point of
| mathematicians not knowing what a square root is
| yourself. Anyway, the square root is is so many levels
| below maths as done by mathematicians, it's laughable.
| Tainnor wrote:
| Square roots are not some "mathematical trivia", they're
| amongst the most fundamental operations in mathematics.
| llm_trw wrote:
| In arithmetic. There is a lo more to math than
| arithmetic.
| theclansman wrote:
| Anybody can do everything if we restrict everything to things
| everyone can do.
| bloqs wrote:
| 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.
| vacuity wrote:
| I think most people can become fairly skilled in useful
| fields if educated properly, and the people who can't are a
| small group that can be cared for. I agree that even in a
| better education system, people aren't all going to be
| equally skilled in the same fields, just that most people
| can contribute something of value.
| sourcepluck wrote:
| Imagine our world was extremely similar to how it is now in
| any way you'd care to imagine, except two things were
| different.
|
| 1. Everyone (young, old, poor, rich) thinks that maths is
| interesting and fun and beautiful and important. Not
| important "to get a good job" or "to go to a good college"
| or "to be an impressive person", but rather important
| because it's fun and interesting. And maybe it also helps
| you think clearly and get a good job and all these
| practical things, but they're secondary to the tremendous
| beauty and wondrousness of the domain.
|
| 2. Everyone believes that barring actual brain injuries
| people can learn mathematics to a _pretty high_ level. Not
| Ramanujan level, not Terrence Tao, not even a research
| mathematician at one of the smaller universities, but a
| level of extreme comfort, let 's say a minimum level of
| being able to confidently ace the typical types of exams 17
| and 18 year olds face to finish secondary school in various
| countries.
|
| Would you claim that in that world - people think maths is
| great, and that anyone can learn it - we'd see similar
| levels of ability and enjoyment of mathematics?
|
| My claim is that we don't live in "Math-World", as
| described above, but "Anti-Math-World". And further, that
| anyone suggesting things have to be the way they are in
| Anti-Math-World is not only wrong, but also fundamentally
| lacking imagination and courage.
|
| Kids are told week in week out that maths is stupid, that
| they are stupid, that their parents themselves are stupid,
| that the parents hated maths, that the teachers are stupid,
| and then when they end up doing poorly, people say: "ahhh,
| some kids just aren't bright!"
|
| Parents who like things like learning and maths and reading
| and so on, have kids that tend to like those things. And
| parents that don't, usually don't. Saying that this somehow
| tells us something concrete and inalterable about the
| nature of the human brain is preposterous.
|
| It's a card that's used by grown-ups who are terrified by
| the idea that our education systems are fundamentally
| broken.
| hilbert42 wrote:
| _" Kids are told week in week out that maths is stupid,
| that they are stupid. ...."_
|
| Come on, how often are kids exposed to such stupid talk?
| I suspect very infrequently.
|
| My grandmother, who wasn't stupid by any means but who
| knew only basic arithmetic, would never have uttered such
| nonsense.
|
| And I'd stress, like many of her generation and
| background, her knowledge of mathematics was minimal, if
| she'd been ask what calculus was she'd likely have been
| perplexed and probably have guessed it to be some kind of
| growth on one's foot.
| Barrin92 wrote:
| 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.
| antegamisou wrote:
| > _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 wrote:
| 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 wrote:
| > 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 wrote:
| > 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 wrote:
| 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 wrote:
| > 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.
| ericmcer wrote:
| 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.
| te_chris wrote:
| 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.
| namaria wrote:
| 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.
| magicalhippo wrote:
| 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_co wrote:
| Effort, combined with the right motivation, can overcome most
| perceived barriers
| magicalhippo wrote:
| 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.
| gammalost wrote:
| Well, isn't that a summary of most things? Most things
| worth learning are hard, but many things _not_ worth
| learning are also hard. So we have to prioritize what hard
| things are worth learning. Math is low on the list for many
| people for (I think) understandable reasons.
| magicalhippo wrote:
| What I meant was I think it's detrimental to be priming
| the kids with a negative view, or nurturing any negative
| views.
| sethammons wrote:
| 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
| vundercind wrote:
| > 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_.
| cchi_co wrote:
| I totally agree! The barriers many of us face with math are less
| about ability and more about how we've been taught to approach
| it. All it took was for me to change my math teacher at school,
| and boom. Love, but at second sight. And curiosity and
| persistence can unlock more than just numbers
| block_dagger wrote:
| Statistical (Bayesian) thinking is an extremely underrated way of
| thinking of almost everything.
| DiscourseFan wrote:
| Frankly, its overrated. Now you can adjust your priors.
| ai4eva wrote:
| 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.
| vundercind wrote:
| 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 wrote:
| 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.
| quus wrote:
| 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.
| guerrilla wrote:
| 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.
| quus wrote:
| 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.
| guerrilla wrote:
| It doesn't need to be for me to be right. These are false
| constraints you're trying to put on it. Mathematics in
| moderation can benefit everyone. This claim stands.
| defrost wrote:
| 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).
| quus wrote:
| 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.
| defrost wrote:
| What are the _costs_ (in your estimation at least) to
| "regular people" (regular by your metric) of _not_ engaging
| in easy bake low level "mathematical thinking".
|
| * How many have a lower return on { X } through not
| understanding compound interest, tax brackets, leveraging
| assets, etc.
|
| * How many have steady net losses through "magical
| thinking" wrt gambling, betting, hot stock tips.
| quus wrote:
| You're sort of making my point -- there are people out
| there who think math education sets the mind free and
| opens the gates of higher cognition, and then others
| talking about hum drum stuff like tax brackets and
| compound interest. If the benefits really just amount to
| a few units of pre-algebra content, that would be
| disappointing.
| purplethinking wrote:
| 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.
| dboreham wrote:
| Careful there. They'll start voting logically..
| kristopolous wrote:
| 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.
| katzenversteher wrote:
| 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?
| vundercind wrote:
| 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.
| atribecalledqst wrote:
| 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)
| kristopolous wrote:
| 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.
| openrisk wrote:
| 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.
| agentultra wrote:
| 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...
| openrisk wrote:
| 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.
| vundercind wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| > 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.
| bmitc wrote:
| > 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 wrote:
| > 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.
| KevinMS wrote:
| But what is the difference between math talent and plug-n-chug
| math talent? That seems to be the most significant filter.
| LoganDark wrote:
| 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 wrote:
| 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..
| jonplackett wrote:
| 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 wrote:
| 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 wrote:
| Check out Susan Rigetti's guide:
| https://www.susanrigetti.com/math
| webdev1234568 wrote:
| 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!
| rnewme wrote:
| 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 inaEUR"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 wrote:
| Possibly making a small, focused game!
|
| This is how I got back into learning maths... through necessity
| and immediate application.
| ChaitanyaSai wrote:
| "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
| rnewme wrote:
| 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.
| ChaitanyaSai wrote:
| Tiring because the answer wasn't revealed? That was the whole
| point :) It's the path, not the summit.
| GrantMoyer wrote:
| Does the quip have to do with someone asking what the quip
| is?
| jojobas wrote:
| - Hey teach, will I really need all these logarithms, derivatives
| and vectors in my adult life?
|
| - No, but the smarter kids might.
| practal wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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/
| nestes wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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.
| atribecalledqst wrote:
| 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.
| trialAccount wrote:
| trial
| revskill wrote:
| 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 wrote:
| 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
| bmitc wrote:
| 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_trw wrote:
| 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.
| vundercind wrote:
| 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 wrote:
| 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.
| crispyambulance wrote:
| 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.
| bunderbunder wrote:
| 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 wrote:
| 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.
| fluoridation wrote:
| 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 wrote:
| 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.
| fHr wrote:
| 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.
| Malidir wrote:
| 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 wrote:
| 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 wrote:
| That's a very important thougt and I belive the world would be
| better, if more people would connect with their mathematical
| side.
| musgravepeter wrote:
| https://www.smbc-comics.com/comic/grind
| agentultra wrote:
| 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.
| trialAccount wrote:
| The author: Would love to participate but account creation seems
| to be broken on Hacker News
| https://x.com/davidbessis/status/1859561768915173466
| macintux wrote:
| If someone still on X could send them the hn@ycombinator.com
| support address, that'd be useful.
| mbbbackus wrote:
| 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.
| sourcepluck wrote:
| > 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!
| Shosty123 wrote:
| 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."
| Nevermark wrote:
| I have so wanted such resources for years. I have found
| some and should make a list.
|
| The first time the difference between understanding some
| math, and _understanding_ what the math meant, was after
| high school Trig. The moment I started manually programming
| graphics from scratch, the circle as a series of dots,
| trigonometry transformed in my mind. I can 't even say what
| the difference was - the math was exactly the same - but
| some larger area of my brain suddenly connected with all
| the concepts I had already learned.
|
| While ordering the "Mathematica: A Secret World of
| Intuition and Curiosity" I came across these books, which
| looked very promising in the "learning formal math by
| expanding intuition" theme, so I bought them too:
|
| Field Theory For The Non-Physicist, by Ville Hirvonen [0]
|
| Lagrangian Mechanics For The Non-Physicist, by Ville
| Hirvonen [1]
|
| The Gravity of Math: How Geometry Rules the Universe, by
| Steve Nadis, Shing-Tung Yau [2]
|
| Vector: A Surprising Story of Space, Time, and Mathematical
| Transformation, by Robyn Arianrhod [3]
|
| [0] https://www.amazon.com/dp/B0CN7HMTJN
|
| [1] https://www.amazon.com/dp/B0CN7HMK38
|
| [2] https://www.amazon.com/dp/1541604296
|
| [3] https://www.amazon.com/dp/0226821102
|
| Excited to read each (based on their synopses & ratings),
| and if I will get compounding fluency across both math and
| physics between all five books.
| Shosty123 wrote:
| Burn Math Class follows that tradition, although it
| starts pretty basic, so it requires some patience.
|
| https://a.co/d/fZnWUU8
| auxbuss wrote:
| If you're interested in how vector calculus developed, and
| who was instrumental, all the way from Newton/Leibnitz to
| Dirac or so, by way of Hamilton, Maxwell, Einstein and
| others, then Robyn Arianrhod's 'Vector' is brilliant.
|
| But be warned, it gets progressively harder, along with the
| concepts, so unless you're conversant with tensors, at some
| point you will have to put on your thinking cap.
|
| The reviews on Goodreads - including my own - are worth
| reading to get a flavour:
| https://www.goodreads.com/book/show/202104095-vector
| gravypod wrote:
| 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 wrote:
| 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).
| ericmcer wrote:
| 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.
| hosh wrote:
| 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.
| FractalHQ wrote:
| I would love to learn about some of these anti-pattern
| proofs if you have any examples or references you can
| share!
| chankstein38 wrote:
| 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 wrote:
| Agree. The article turned me off as well. No specific
| example, felt like an ad.
| burnte wrote:
| Yeah, I quit reading it because it didn't talk about the
| book, it felt like a meta article.
| dfxm12 wrote:
| 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 wrote:
| 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 :)
| delta_p_delta_x wrote:
| 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.
| itissid wrote:
| "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.
| itissid wrote:
| 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.
| itissid wrote:
| 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
| notepad0x90 wrote:
| 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.
| itissid wrote:
| 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
| notepad0x90 wrote:
| I don't think UBI is feasible, it is anathema to the human
| condition to be content with the bare minimum. if not for
| our own selves, we would want the best life possible for
| our loved ones (present or future). my "UBI" would be a
| couple of million dollars.
|
| I want to learn those subjects because I enjoy learning and
| understanding. Life should be lived with knowledge applied
| through wisdom.
|
| > Could you volunteer me how much time you spend on it? And
| how is your day job?
|
| On entertainment? I can't tell you, I like to watch a movie
| or a tv show whenever I have time for it. There are more
| enjoyable pursuits in life, and most worthy pursuits
| involve adversity and require perseverance.
| diffeomorphism wrote:
| > I don't think UBI is feasible, it is anathema to the
| human condition to be content with the bare minimum.
|
| Hm? That is exactly why UBI works, no? As the name
| indicates, the bare minimum is taken care so that you can
| work towards "the best life possible for our loved ones"
| without worrying about starving, sickness or
| homelessness.
|
| In contrast, if you were wrong and people would be
| content with the bare minimum, then UBI would be a bad
| idea. Though then they could just commit some felonies
| and be content with having a cell, bread and water for
| the rest of their lives, no?
| notepad0x90 wrote:
| I posted a more detailed sibling comment on this thread,
| but that's not why UBI works. it just shifts what the
| "bare minimum" is. Most Americans aren't fighting to get
| the best tent spot while out on the streets because they
| can't afford housing or begging for food on the streets.
| UBI isn't solving that, except as a welfare replacement
| for a small percentage of the population (and not a great
| replacement either). maybe with UBI, everyone who lives
| in a crappy apartment can now afford a nicer apartment,
| but costs for the nicer apartment would naturally go up
| as well. In other words, most people won't quit their
| jobs because of UBI, they would just temporarily afford
| nicer things. Those that do quit their jobs can not work
| and not worry about starving, but that's not a new
| condition. if you don't want to work in America, you
| won't starve. maybe housing would be a problem but the
| people for whom housing would be a problem if they
| stopped working are not typically the same people who
| would be content with the cheapest/worst livable
| condition (UBI).
|
| In short, it is silly to expect UBI to be a means by
| which people would work only if they want to work, and
| they would pursue their passions and dreams instead. That
| kind of a society I think is possible, but it would also
| have to reach a level of wealth where money itself is not
| required (think star trek).
| euvin wrote:
| > my "UBI" would be a couple of million dollars.
|
| There's a difference between what your human brain would
| become accustomed to (which you'd be right, it'd scale up
| and up forever) versus what would allow the base level of
| health and opportunity. As in, not having to worry about
| eating the next day.
|
| And because you're right that human brains strive for
| more wealth, UBI should grant you the opportunity to
| pursue it without fear of failure.
| notepad0x90 wrote:
| UBI can be used as a replacement for existing welfare
| programs, but you're not pursuing arts or starting a
| business on it. My point was, people will still
| prioritize earning more money when on UBI instead of
| pursue their passions because it won't be enough. UBI is
| not a safety net, if a middle class salary person fails,
| they would have to work hourly lower wage work, that's
| why they keep working their middle class job, it isn't
| because they fear starvation or losing their shelter.
|
| UBI would relieve stress for the lowest earning people,
| but it won't result in pursuit of passion for most
| people. economically speaking, because most people can
| afford certain things (like rent) the price of those
| things will go up, things are priced based on what
| potential consumers are willing to pay. If rent costs
| $1000 for a specific type of unit, but suddenly everyone
| on UBI can afford that easily, the landlord would raise
| the rent, the cost of things won't remain static when
| wages rise for a large portion of the population.
| Increased demand without increased cost is loss of
| potential revenue. The quality of life for people on UBI
| would be barely surviving, and UBI would need to increase
| constantly to keep people from becoming homeless or
| starving.
|
| This is the "Cobra effect" embodied. It provides a
| perverse incentive. healthcare in the US is out of
| control for this reason. health care providers keep
| increasing cost, because the patient is not the client,
| the insurance company is, so long as everyone is getting
| insurance, the cost of care is the maximum reliably
| predictable pay out by the insurance company. Not
| increasing cost is just bad business. You will have to
| also force all kinds of businesses from raising prices if
| it can work, and even when it works UBI will result in
| subsidizing low-wager workers for businesses, because
| they'll still have to work some job to afford anything
| outside of food,shelter and the basics.
|
| A practical alternative to UBI is a local tax on
| businesses, kind of like a property tax but this tax is
| based on an inverse of an assessment of wages, rent,
| welfare pay out and other social conditions in the area.
| the higher wages are, lower rent is,etc.. the lower the
| tax is, it might even result in a credit. An inverse of a
| perverse incentive like UBI. Unemployment would also be
| partially funded through this, the unemployed would
| forever get a UBI like pay out so long as they are
| pursuing education or work of some kind based on what is
| in demand in their area. Businesses get a healthy talent
| pool to choose from and cost of living is balanced.
| nradov wrote:
| 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.
| itissid wrote:
| And how far would that get one doing that? 1, 5, 10, 40
| years?
| ahoka wrote:
| Perhaps some, but why is that a problem? There are already
| people who do this.
| intelVISA wrote:
| Is that not better than the current alternative where they're
| forced to become grifters?
| nradov wrote:
| No one is forced to become a grifter.
| manvillej wrote:
| I would perfect my video gaming art. Teenagers in their
| basement would fear me after school.
| intelVISA wrote:
| 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 wrote:
| Yeah, but we are all in software so for us economy is not a
| concern. All of us should learn maths.
| lifter3101 wrote:
| 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 wrote:
| 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.
| hilbert42 wrote:
| _" 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 Evariste 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 wrote:
| 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.
| lupire wrote:
| 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.
| davidbessis wrote:
| 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.
| samatman wrote:
| > _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.
| davidbessis wrote:
| 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.
| samatman wrote:
| > _I think you really are confused_
|
| I have no respect at all for people who conduct themselves
| this way.
| alganet wrote:
| > 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 wrote:
| 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.
| davidbessis wrote:
| 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 wrote:
| What an incredibly neuro-typical thing to claim
| johnp314 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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.)
| aleden wrote:
| 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 wrote:
| 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.
| aleden wrote:
| 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.
| javier_e06 wrote:
| 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 wrote:
| 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 wrote:
| 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, optimized 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.
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