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It think this is silly on multiple accounts. I'll claim that there's not real thing such as a "language brain" or "math brain." I'll also claim that most people

It think this is silly on multiple accounts. I'll claim that there's not real thing such as a "language brain" or "math brain." I'll also claim that most people don't know what math is, and that their evidence supports a "math brain". Math isn't about calculations/computations, it is about patterns. You get to algebra and think "what are these letters doing in my math" but once you get further you think "what are these numbers doing in my math?" A great tragedy we have in math education is that we focus so much on calculation. There's tons of useful subjects that are only taught once people get to an undergraduate math degree or grad school despite being understandable by children. The basics of things like group theory, combinatorics, graphs, set theory, category theory, etc. All of these also have herculean levels of depth, but there's plenty of things that formalize our way of thinking yet are easily understandable by children. If you want to see an example, I recommend Visual Group Theory[0]. Math is all about abstraction and for some reason we reserve that till "late in the game". But I can certainly say that getting this stuff accelerates learning and has a profound effect on the way I think. Though an important component of that is ensuring that you really take to heart the abstraction, not getting in your own way by thinking these tools only apply in very specific applications. A lot of people struggle with word problems, but even though they might involve silly situations like having a cousin named Throckmorton or him wanting to buy 500 watermelons, they really are part of that connection from math to reality. This is why "advanced" math accelerating my learning, because that "level" of math is about teaching you abstractions. Ways to think. These are tremendously helpful even if you do not end up writing down equations. Because, math isn't really about writing down equations. But we do it because it sure helps, especially when shit gets complicated. [0] https://www.youtube.com/watch?v=UwTQdOop-nU&list=PLwV-9DG53NDxU337smpTwm6sef4x-SCLv https://www.youtube.com/watch?v=UwTQdOop-nU&list=PLwV-9DG53N...

danielmarkbruce 2y ago
100%. I have 2 kids and what they learn at school is absurd... I have to put them in an extra class outside school where they learn a lot of the more abstract stuff, albeit as you say at a level which makes sense (not too deep).
- godelski 2y ago
  Yeah, certainly we'd need to adapt classes to match their level, but we've tried before[0]. While it "failed" it also seemed pretty successful, especially in France and the USSR. I mean is it any surprise kids get bored in math? They spend years learning the same thing. You spend years learning addition and multiplication (subtraction and division are the same operators). I sure as hell hated math as a kid. Especially doesn't help that it is always taught by someone who's also dispassionate about the subject. You really can get a process going where most middle schoolers are doing calculus and linear algebra (and advance ones are doing this in elementary). It isn't as far fetched as many would believe. [0] https://en.wikipedia.org/wiki/New_Math https://en.wikipedia.org/wiki/New_Math
  - Nevermark 2y ago
    > I mean is it any surprise kids get bored in math? Indeed. Kids did not evolve to learn things because they would be useful years down the road. But they did evolve to inhale knowledge at very high rates wherever it extends their capabilities in the moment. My take on this is that math should be tied to kids crafting, experimenting, and learned directly in the context of its fun and creative uses. Geometry screams out to be developed within a context of design, crafting, art and physical puzzles. Algebra, trigonometry, calculus ... they all have direct (and fun) uses, especially at the introduction stage. The availability for graphics software, sim worlds, 3D printing, etc. should make math, from the simplest to most advanced, more fun and immediately applicable than ever. Boredom with math is a failure to design education for human beings. (Then there is the worst crime of all - moving kids through math in cohorts, pushing individuals both faster and slower than they are able to absorb it. Profound failure by design.)
    - godelski 2y ago
      In general I agree but > Indeed. Kids did not evolve to learn things because they would be useful years down the road. But they did evolve to inhale knowledge at very high rates wherever it extends their capabilities in the moment. I strongly disagree with this. Children play. Most animals play. It is not hard to see how the skills learned through play lead to future utility despite potentially none at the time. We invent new games, new rules, and constantly imagine new worlds. The skills gained from these often have no utility beyond that that is self-constructed. Though many do have future rewards. I think we often miss those though because that path is through generalization. Learning how to throw a ball can help with learning physics, writing, driving, and much more. They don't perfectly transfer but it'd be naive to conclude that there aren't overlaps. I think the truth is that as long as we are unable to predict the future, we really can't predict what skills will be useful and what specific knowledge should be pursued. We can do this in a more abstract sense as we can better predict the near future than far but that also means we should learn creativity and abstraction, as these allow us to adapt to changes. And that is why I believe we evolved these methods, and why you see things like play in most creatures.
belinder 2y ago
Do you recommend any other resources for children to learn the basics of those theories?
- PartiallyTyped 2y ago
  A book of abstract algebra is very approachable!
- godelski 2y ago
  Unfortunately I don't know any direct resources. I really do hope they are out there and some will share. But if you're willing to hunt, I know that this idea was attempted before[0]. France and USSR had better success than the US. I'm sure there are still people working in this direction. I don't have children, but fwiw I've taught my nieces and nephews algebra and even some of calculus before they were 10 just in visiting time during vacations. They were bored and it seemed more fun than talking about the drama, politics, and religion that the rest of my family likes to spend most of their time on. Kids were similarly disinterested in that stuff lol. I've also seen my god{son,daughter} be able to learn these types of skills, so I'm highly confident it is doable. [0] https://en.wikipedia.org/wiki/New_Math https://en.wikipedia.org/wiki/New_Math
mrexroad 2y ago
Thank you. I’ve long held that calculus, formal logic, combinatorics, etc. should be started in elementary school, separate from arithmetic. Even after stripping away the numbers and procedural knowledge, there is tremendous real world value in simply understanding many of the concepts. Plus, (and I say this half-jokingly) won’t we all be using llms to vibe-math everything that needs math-ing anyway?
vendiddy 2y ago
It took me a long time to realize math consisted of abstractions. I was never told, for example, that matrices are a useful abstraction (shorthand?) for representing linear equations. Or that imaginary numbers were an invented abstraction that made certain calculations easier. It would have been nice to learn it from the perspective of how those abstractions came into being and why they're useful.
- Nevermark 2y ago
  Even if we narrow math to numbers, what many people thing of as "math", we quickly move from arithmetic to patterns. The patterns of repetitions of single things, repetitions over different things, repetitions between different things, repetitions over repetitions, ... Natural/counting numbers are just the simplest patterns of repetition.
- Jensson 2y ago
  Numbers are an abstraction, not just "imaginary" numbers, all numbers are imaginary they don't exist in reality.
  - sieabahlpark 2y ago
    [dead]
- bee_rider 2y ago
  > I was never told, for example, that matrices are a useful abstraction (shorthand?) for representing linear equations. Congratulations on overcoming the (very weird) choice of whoever taught your intro to linear algebra class.
  - godelski 2y ago
    I'm not sure how to interpret this comment, it could easily be read in multiple ways. (Depends on priors) Regardless, it's a shockingly common occurrence. I'd agree it's not the right way, but it is also a common way. That perspective might define those priors
PartiallyTyped 2y ago
I am willing to bet good money that teaching pure mathematics like group theory, abstract algebra, or stuff like big Oh and combinatronics will do our kids a lot more good and advance them far more than calculus or geometry as taught in schools will. I have the same experience as you did, in that studying by myself abstract algebra accelerated my learning and reasoning skills.
- godelski 2y ago
  I think differential equations was where the switch from "I'm good at math" to "I think I want to actually know more of this" happened. I changed my major from engineering to physics. But agreed, abstract algebra was another big change and after that I took every class my university would let me and the chair of the department even made some new ones for me and some others. I don't think this experience is uncommon (among people who get to these levels). Which is why it is really sad. Especially given how linear algebra and abstract algebra are in a lot of ways easier than calculus. I also think they should be taught earlier purely due to the fact that they teach abstraction and reasoning.
lo_zamoyski 2y ago
Math is about form and traditionally the science of quantity. It is called a formal science for this reason. Language also concerns form. Grammar has form. Concepts are forms. Math is language. 'Everything' is language. Language is the image of reality. In the beginning was the Logos...
- godelski 2y ago
  > Math is language. Facts. I think it is hard to disagree with this, and it seems like a rare opinion to be held by people at high levels. > Language is the image of reality. An image?[0] ;) [0] https://ncatlab.org/nlab/show/image https://ncatlab.org/nlab/show/image
  - seadan83 2y ago
    Math major here. 'is' is a reflexive relationship. If math is language, therefore language is math. I believe clearly language is not math, therefore math is not language. Math is described with language, Math itself is not language. It is a 'has-a' relationship vs a 'is-a' relationship.
    - godelski 2y ago
      > Math major here Forgive me if I doubt, but your comment would strongly suggest otherwise, along with this one[0]. The reason for doubt is a failure in fairly basic logic. Your claim is: ¬(A ↦ B) ⟹ ¬(B ↦ A) I'd expect anyone willing to claim the title of "math major" is aware of structures other than bijections and isomorphisms. The mapping operator doesn't form an abelian group. [0] https://news.ycombinator.com/item?id=43873351 https://news.ycombinator.com/item?id=43873351
      - seadan83 2y ago
        The word "is", typically refers to the logical equals operator (written as "="). Let A = Math, and let B = Language, the phrase "Math is Language" is therefore "A = B". My claim is that "B != A", which implies "A != B" because "!=" is reflexive. "A != B" then contradicts the claim "A = B". In words: because "language is not math", therefore it cannot be true that "math is language". This is not number theory, not set theory, just logic. I don't see how the properties of Abelian groups applies. I do suspect that non-abelian groups is a case where "¬(A ↦ B) ⟹ ¬(B ↦ A)" is false, and would be a contadiction to my counter claim if "¬(A ↦ B) ⟹ ¬(B ↦ A)" was my counter-claim. No, my counter claim is simply that because "B != A" therefore "A != B" The statement "¬(A ↦ B) ⟹ ¬(B ↦ A)" is quite different, and I'm not 100% sure whether you are mixing set and logic notations? [0][1] It does seem you are bringing in number theory concepts, or we are misunderstanding one another? I assume "↦" is the set mapping operator and "⟹" is logical implies, and "¬" is logical not. "¬(A ↦ B) ⟹ ¬(B ↦ A)" could potentially be phrased as: "If the element A cannot be mapped to B, then the element B cannot be mapped to A". I would agree that statement is not 'generally' true. Could you please clarify so we are not talking past each other. [0] Per google: The symbol "↦" is a mapping symbol, typically used to indicate a function or relationship where one element maps to another. It's not a standard logical operator in the same way that symbols like &&, ||, or ¬ are. Instead, it represents a directional relationship between sets or elements, often seen in set theory and mathematical notations [1] https://simple.wikipedia.org/wiki/Implication_(logic) https://simple.wikipedia.org/wiki/Implication_(logic)
NickM 2y ago
I think you might be missing the point of the article: the study being cited isn't trying to establish the existence of a "language brain" or a "math brain", that's just the way the headline editorialized it to help people understand the conclusions. The conclusion of the study was that linguistic aptitude seemed to be more correlated with programming aptitude than mathematical aptitude, which seems fairly interesting, and also fairly unconcerned with which specific physical regions in the brain might happen to be involved.
- godelski 2y ago
  I understood it. > The conclusion of the study was that linguistic aptitude seemed to be more correlated with programming aptitude than mathematical aptitude And this is what I'm pushing back against and where I think you've misinterpreted. > They found that how well students learned Python was mostly explained by general cognitive abilities (problem solving and working memory), while how quickly they learned was explained by both general cognitive skills and language aptitude. I made the claim that these are in fact math skills, but most people confuse with arithmetic. Math is a language. It is a language we created to help with abstraction. Code is math. There's no question about this. Go look into lambda calculus and the Church-Turing Thesis. There is much more in this direction too. And of course, we should have a clear connection to connect it all if you're able to see some abstraction.
  - seadan83 2y ago
    > Math is a language. Language is not math, therefore math is not language.
    - godelski 2y ago
      Logic doesn't follow. There is no problem with A -> B ∧ B -/-> A Here's an example. "I live in San Francisco" would imply "I live in the US". But "I live in the US" does not mean "I live in San Francisco". Here's a more formal representation of this: https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection https://en.wikipedia.org/wiki/Bijection,_injection_and_surje...
necovek 2y ago
To be fair, the original study uses "numeracy", and the correlation numbers are almost exactly like the ones for "language aptitude". At the same time, the study excluded "five participants ... due to attrition (not completing the training sessions), and one participant ... because he was an extreme outlier in learning rate (>3 sd away from the mean)." I mean, if you are to exclude 15% of your subjects without looking at their aptitude (maybe they didn't do it because it was too hard to pass the training tests to move to the next lesson, yet their language aptitude is high?), with only 36 subjects of which 21 are female (it's obvious programming is male dominated, so they only had 15 males: maybe it doesn't matter, but maybe it does), how can you claim any statistical significance with such small numbers?
- godelski 2y ago
  > the original study uses "numeracy" That's fair, although I don't think it changes my response. And the article still really leads to the wrong conclusions. You want to teach children abstraction and reasoning? You teach them math. Not numeracy, math.
apeescape 2y ago
Finland tried teaching maths to children using Group Theory in the 70s [1], but the results weren't that good; it proved to be too abstract for young kids. Ultimately, I believe basic algebra and geometry are the most important takeaways from math classes for most people. [1]: https://www.hs.fi/tiede/art-2000004823594.html https://www.hs.fi/tiede/art-2000004823594.html (sorry, it's in Finnish and behind a paywall)
- hirvi74 2y ago
  I am no expert in pedagogy of mathematics, and I am sure someone will correct me if I am wrong, but I think there was/is a Russian academic program in which students were/are basically only taught algebra in an iteratively increasing manner. As it was explained to me, one wouldn't take a "Calculus I" class as a prerequisite for say an entry-level engineering course. One typically had such a strong foundation of algebra, that when encountering a problem that required calculus, the student would just learn the necessary calculus at that point in time. In other words, with such a strong algebraic background, other aspects of math, within reason, were much easier to grok.
- godelski 2y ago
  > the results weren't that good; it proved to be too abstract for young kids You cannot make that conclusion as a result of the evidence. Yes, the evidence might support that conclusion, but there are many others that also could. For example, they could have just been really bad at teaching. This even seems like a likely one as it is difficult to perform such a reformulation and to do so broadly and quickly. The other reason I'm willing to accept alternative conclusions is that France and the USSR had far more success than Finland (or even America). Their success contradicts a claim that "[it is] too abstract for young kids". You'd need to constrain it to something like "[it is] too abstract for Finish kids" which I think both of us would doubt such a claim.
  - apeescape 2y ago
    Yeah fair point. However, curious of what you base your claim "France and the USSR had far more success" on?
florbnit 2y ago
> It think this is silly on multiple accounts. I'll claim that there's not real thing such as a "language brain" or "math brain." It seems plainly obvious that this language just means “areas of brain that activate when dealing with math problems” vs “areas of brain that activate when dealing with language problems” and yes there is hard evidence that there is a difference between them.
- godelski 2y ago
  Please reread my comment in full. I'm willing to bet we disagree on the definition of math. I'll strongly insist that the one I'm using is common among mathematicians. If you'd like to retort by saying it's about semantics then congrats, we're on the same page (and can be verified by reading you sibling comments and/or my replies to some of them)
  - florbnit 2y ago
    It has nothing to do with the definition of math. If you put people in an MRI and give them random problems then you’ll notice a distribution in the patterns of brain activity that are distinct. Some of these groupings we’ll call “math like” because the problems are more like mathematics than linguistics. And in the other distribution we’ll call them “word like” because the problems fall into the category we associate with word problems. The fact that we see separate patterns of brain activity is the interesting part not any semantic argument about what you or mathematicians feel should be defined as being math or not.
jeffhuys 2y ago
Interesting to me how Wernicke’s Aphasia works then (which I have occasionally; my epilepsy stems from there + broca’s area). Could you explain that?
- godelski 2y ago
  What is the question asking? Why do you have Aphasia? I'm not sure what the relationship here is. Or are you suggesting you're good at math but not good at language? I feel weird answering even if I infer the right question because it feels tautological. If you have Wernicke's Aphasia does it not create the possibility that I already have but your condition resulted in misunderstanding. Given the condition does it not create a high probability that a response will similarly be misunderstood? Is not Anosognosia quite common? Maybe I'm really misunderstanding but honestly I'm not sure what you're asking
atomicnumber3 2y ago
Just anecdotally based on my 2000something kid high school, the copious amount of tiering of math classes seemed to indicate to me that either we're really bad at teaching math to 80% of people, or only 20% of people will be able to handle precalculus. We had basically 4 tracks: one ended with you doing algebra 1 in senior year, another ended with you doing trig in sr year, yet another that ended with trig (no precalculus), and then one that ended with you doing trig and precalc. That final class then had further subdivisions that were too small to have their own full classes: some kids just did precalc, some did calc 1 and took the AP Calculus AB exam and/or IB Math SL, while even even smaller group took AP Calculus BC and/or IB Math HL. The total number of kids who took the AP Calc AB exam in my year was 20ish, out of a graduating class of 500-600.
- godelski 2y ago
  In either case it would seem to suggest we should radically change the system
- serial_dev 2y ago
  This system sounds extremely complicated. > the copious amount of tiering of math classes seemed to indicate to me that either we're really bad at teaching math to 80% of people, or only 20% of people will be able to handle precalculus Or maybe it indicates that the people designing this system should be fired? Job security through complexity? (Or maybe I’m just biased by the system I know… I’m just asking questions)
  - godelski 2y ago
    > … I’m just asking questions I'm not disagreeing, but just wanted to point out that this phrasing is commonly used by bad faith actors. I'm not saying you're using it this way and I legitimately do not think you are. But I wanted to point it out because I think your comment could be interpreted another way and this phrase might be evidence for someone to make that conclusion. The classic example is conspiracy theorists. But lots of bad faith actors also use it to create leading questions that generally ignore or lead away from important context. Again, I do not think you're doing this. The rest of your comment makes me think you're acting in good faith.
HarHarVeryFunny 2y ago
> I'll claim that there's not real thing such as a "language brain" or "math brain. Did you even read beyond the silly headline? The article itself is about pre-testing subjects on a range of capabilities from problem solving ability to second (foreign) language learning ability, and then seeing how these correlated to the ability of the test subjects to learn to code. The results were basically exactly what might be expected - people who learned Python the quickest were those who scored the best at learning a second language, and those who learned to wield it the best were those who had scored the best at problem solving. Not surprisingly math ability wasn't much of a predictor since programming has little to nothing to do with math.
- ubercow13 2y ago
  I think the above comment stands. The point is, what do they consider maths ability? High school maths has very little to do with programming but university-level maths certainly feels very similar to it in structure. Many of my (good) classmates in my maths degree were very bad at things like mental arithmetic. So maybe "maths ability" (defined some way) isn't very important for being good at "maths" (proper).
- godelski 2y ago
  > Did you even read beyond the silly headline? Yes. I'll also refer you to the HN guideline on this manner. You're welcome to disagree with me but you must communicate in good faith and unless you have a very specific reason for thinking I didn't "RTFM" then don't make the accusation. I'm happy to continue discussing, but only on those terms. In fact, I think we're in far more agreement than your tone suggests. But I think you missed the crux of my point: math isn't number crunching
  - HarHarVeryFunny 2y ago
    Sheesh - why so touchy?! Your response opened with addressing the headline, which anyone who had "RTFM" (RTFA) would have realized was unrelated to the body of the article. You then veered off into a tangent about the nature of math which again was not addressing the content of the article. The underlying nature article, linked from the posted story, makes it even more clear what is being discussed, with the abstract stating: > This experiment employed an individual differences approach to test the hypothesis that learning modern programming languages resembles second “natural” language learning in adulthood. https://www.nature.com/articles/s41598-020-60661-8 https://www.nature.com/articles/s41598-020-60661-8
hydrogen7800 2y ago
If we taught music the way we teach math. No wonder so many people are "bad at math". “Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely. One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way.” https://en.m.wikipedia.org/wiki/A_Mathematician%27s_Lament https://en.m.wikipedia.org/wiki/A_Mathematician%27s_Lament
- thomasikzelf 2y ago
  Of course Brett Victor has the essay on his website: https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician's_Lament.pdf https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician'... It is a great read!
- seba_dos1 2y ago
  Frankly, with all the mess that music notation is and the way it's often being taught, this may actually be a slight upgrade over status quo. I have never experienced more "that's just how it is" than in music.
thaumasiotes 2y ago
> most people don't know what math is It's surprisingly common. Case in point: "The unreasonable effectiveness of mathematics in the natural sciences". A normal person wouldn't be surprised that describing how something works is a good way to understand it.
LegionMammal978 2y ago
By the very expansive definition that you advocate for, what exactly do we do that isn't math? One could argue that all we do is turn thoughts, senses, and memories into further thoughts and appropriate actions, which is applying a pattern, which is math. But at that point the definition is too broad to be helpful for anything but playing word games.
- godelski 2y ago
  There are no rules. On the other hand, code is very formal. It is a set of rules. It has structures. There is wrongness and you can do verification that is not the result of opinion nor is it subjective. But yes, math is very broad. But it doesn't cover everything. At least not at this time.

danielmarkbruce 2y ago

100%. I have 2 kids and what they learn at school is absurd... I have to put them in an extra class outside school where they learn a lot of the more abstract stuff, albeit as you say at a level which makes sense (not too deep).

godelski 2y ago
Yeah, certainly we'd need to adapt classes to match their level, but we've tried before[0]. While it "failed" it also seemed pretty successful, especially in France and the USSR. I mean is it any surprise kids get bored in math? They spend years learning the same thing. You spend years learning addition and multiplication (subtraction and division are the same operators). I sure as hell hated math as a kid. Especially doesn't help that it is always taught by someone who's also dispassionate about the subject. You really can get a process going where most middle schoolers are doing calculus and linear algebra (and advance ones are doing this in elementary). It isn't as far fetched as many would believe. [0] https://en.wikipedia.org/wiki/New_Math https://en.wikipedia.org/wiki/New_Math
- Nevermark 2y ago
  > I mean is it any surprise kids get bored in math? Indeed. Kids did not evolve to learn things because they would be useful years down the road. But they did evolve to inhale knowledge at very high rates wherever it extends their capabilities in the moment. My take on this is that math should be tied to kids crafting, experimenting, and learned directly in the context of its fun and creative uses. Geometry screams out to be developed within a context of design, crafting, art and physical puzzles. Algebra, trigonometry, calculus ... they all have direct (and fun) uses, especially at the introduction stage. The availability for graphics software, sim worlds, 3D printing, etc. should make math, from the simplest to most advanced, more fun and immediately applicable than ever. Boredom with math is a failure to design education for human beings. (Then there is the worst crime of all - moving kids through math in cohorts, pushing individuals both faster and slower than they are able to absorb it. Profound failure by design.)
  - godelski 2y ago
    In general I agree but > Indeed. Kids did not evolve to learn things because they would be useful years down the road. But they did evolve to inhale knowledge at very high rates wherever it extends their capabilities in the moment. I strongly disagree with this. Children play. Most animals play. It is not hard to see how the skills learned through play lead to future utility despite potentially none at the time. We invent new games, new rules, and constantly imagine new worlds. The skills gained from these often have no utility beyond that that is self-constructed. Though many do have future rewards. I think we often miss those though because that path is through generalization. Learning how to throw a ball can help with learning physics, writing, driving, and much more. They don't perfectly transfer but it'd be naive to conclude that there aren't overlaps. I think the truth is that as long as we are unable to predict the future, we really can't predict what skills will be useful and what specific knowledge should be pursued. We can do this in a more abstract sense as we can better predict the near future than far but that also means we should learn creativity and abstraction, as these allow us to adapt to changes. And that is why I believe we evolved these methods, and why you see things like play in most creatures.

belinder 2y ago

Do you recommend any other resources for children to learn the basics of those theories?

PartiallyTyped 2y ago
A book of abstract algebra is very approachable!
godelski 2y ago
Unfortunately I don't know any direct resources. I really do hope they are out there and some will share. But if you're willing to hunt, I know that this idea was attempted before[0]. France and USSR had better success than the US. I'm sure there are still people working in this direction. I don't have children, but fwiw I've taught my nieces and nephews algebra and even some of calculus before they were 10 just in visiting time during vacations. They were bored and it seemed more fun than talking about the drama, politics, and religion that the rest of my family likes to spend most of their time on. Kids were similarly disinterested in that stuff lol. I've also seen my god{son,daughter} be able to learn these types of skills, so I'm highly confident it is doable. [0] https://en.wikipedia.org/wiki/New_Math https://en.wikipedia.org/wiki/New_Math

mrexroad 2y ago

Thank you. I’ve long held that calculus, formal logic, combinatorics, etc. should be started in elementary school, separate from arithmetic. Even after stripping away the numbers and procedural knowledge, there is tremendous real world value in simply understanding many of the concepts. Plus, (and I say this half-jokingly) won’t we all be using llms to vibe-math everything that needs math-ing anyway?

vendiddy 2y ago

It took me a long time to realize math consisted of abstractions. I was never told, for example, that matrices are a useful abstraction (shorthand?) for representing linear equations. Or that imaginary numbers were an invented abstraction that made certain calculations easier. It would have been nice to learn it from the perspective of how those abstractions came into being and why they're useful.

Nevermark 2y ago
Even if we narrow math to numbers, what many people thing of as "math", we quickly move from arithmetic to patterns. The patterns of repetitions of single things, repetitions over different things, repetitions between different things, repetitions over repetitions, ... Natural/counting numbers are just the simplest patterns of repetition.
Jensson 2y ago
Numbers are an abstraction, not just "imaginary" numbers, all numbers are imaginary they don't exist in reality.
- sieabahlpark 2y ago
  [dead]
bee_rider 2y ago
> I was never told, for example, that matrices are a useful abstraction (shorthand?) for representing linear equations. Congratulations on overcoming the (very weird) choice of whoever taught your intro to linear algebra class.
- godelski 2y ago
  I'm not sure how to interpret this comment, it could easily be read in multiple ways. (Depends on priors) Regardless, it's a shockingly common occurrence. I'd agree it's not the right way, but it is also a common way. That perspective might define those priors

PartiallyTyped 2y ago

I am willing to bet good money that teaching pure mathematics like group theory, abstract algebra, or stuff like big Oh and combinatronics will do our kids a lot more good and advance them far more than calculus or geometry as taught in schools will. I have the same experience as you did, in that studying by myself abstract algebra accelerated my learning and reasoning skills.

godelski 2y ago
I think differential equations was where the switch from "I'm good at math" to "I think I want to actually know more of this" happened. I changed my major from engineering to physics. But agreed, abstract algebra was another big change and after that I took every class my university would let me and the chair of the department even made some new ones for me and some others. I don't think this experience is uncommon (among people who get to these levels). Which is why it is really sad. Especially given how linear algebra and abstract algebra are in a lot of ways easier than calculus. I also think they should be taught earlier purely due to the fact that they teach abstraction and reasoning.

lo_zamoyski 2y ago

Math is about form and traditionally the science of quantity. It is called a formal science for this reason. Language also concerns form. Grammar has form. Concepts are forms. Math is language. 'Everything' is language. Language is the image of reality. In the beginning was the Logos...

godelski 2y ago
> Math is language. Facts. I think it is hard to disagree with this, and it seems like a rare opinion to be held by people at high levels. > Language is the image of reality. An image?[0] ;) [0] https://ncatlab.org/nlab/show/image https://ncatlab.org/nlab/show/image
- seadan83 2y ago
  Math major here. 'is' is a reflexive relationship. If math is language, therefore language is math. I believe clearly language is not math, therefore math is not language. Math is described with language, Math itself is not language. It is a 'has-a' relationship vs a 'is-a' relationship.
  - godelski 2y ago
    > Math major here Forgive me if I doubt, but your comment would strongly suggest otherwise, along with this one[0]. The reason for doubt is a failure in fairly basic logic. Your claim is: ¬(A ↦ B) ⟹ ¬(B ↦ A) I'd expect anyone willing to claim the title of "math major" is aware of structures other than bijections and isomorphisms. The mapping operator doesn't form an abelian group. [0] https://news.ycombinator.com/item?id=43873351 https://news.ycombinator.com/item?id=43873351
    - seadan83 2y ago
      The word "is", typically refers to the logical equals operator (written as "="). Let A = Math, and let B = Language, the phrase "Math is Language" is therefore "A = B". My claim is that "B != A", which implies "A != B" because "!=" is reflexive. "A != B" then contradicts the claim "A = B". In words: because "language is not math", therefore it cannot be true that "math is language". This is not number theory, not set theory, just logic. I don't see how the properties of Abelian groups applies. I do suspect that non-abelian groups is a case where "¬(A ↦ B) ⟹ ¬(B ↦ A)" is false, and would be a contadiction to my counter claim if "¬(A ↦ B) ⟹ ¬(B ↦ A)" was my counter-claim. No, my counter claim is simply that because "B != A" therefore "A != B" The statement "¬(A ↦ B) ⟹ ¬(B ↦ A)" is quite different, and I'm not 100% sure whether you are mixing set and logic notations? [0][1] It does seem you are bringing in number theory concepts, or we are misunderstanding one another? I assume "↦" is the set mapping operator and "⟹" is logical implies, and "¬" is logical not. "¬(A ↦ B) ⟹ ¬(B ↦ A)" could potentially be phrased as: "If the element A cannot be mapped to B, then the element B cannot be mapped to A". I would agree that statement is not 'generally' true. Could you please clarify so we are not talking past each other. [0] Per google: The symbol "↦" is a mapping symbol, typically used to indicate a function or relationship where one element maps to another. It's not a standard logical operator in the same way that symbols like &&, ||, or ¬ are. Instead, it represents a directional relationship between sets or elements, often seen in set theory and mathematical notations [1] https://simple.wikipedia.org/wiki/Implication_(logic) https://simple.wikipedia.org/wiki/Implication_(logic)

NickM 2y ago

I think you might be missing the point of the article: the study being cited isn't trying to establish the existence of a "language brain" or a "math brain", that's just the way the headline editorialized it to help people understand the conclusions. The conclusion of the study was that linguistic aptitude seemed to be more correlated with programming aptitude than mathematical aptitude, which seems fairly interesting, and also fairly unconcerned with which specific physical regions in the brain might happen to be involved.

godelski 2y ago
I understood it. > The conclusion of the study was that linguistic aptitude seemed to be more correlated with programming aptitude than mathematical aptitude And this is what I'm pushing back against and where I think you've misinterpreted. > They found that how well students learned Python was mostly explained by general cognitive abilities (problem solving and working memory), while how quickly they learned was explained by both general cognitive skills and language aptitude. I made the claim that these are in fact math skills, but most people confuse with arithmetic. Math is a language. It is a language we created to help with abstraction. Code is math. There's no question about this. Go look into lambda calculus and the Church-Turing Thesis. There is much more in this direction too. And of course, we should have a clear connection to connect it all if you're able to see some abstraction.
- seadan83 2y ago
  > Math is a language. Language is not math, therefore math is not language.
  - godelski 2y ago
    Logic doesn't follow. There is no problem with A -> B ∧ B -/-> A Here's an example. "I live in San Francisco" would imply "I live in the US". But "I live in the US" does not mean "I live in San Francisco". Here's a more formal representation of this: https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection https://en.wikipedia.org/wiki/Bijection,_injection_and_surje...

necovek 2y ago

To be fair, the original study uses "numeracy", and the correlation numbers are almost exactly like the ones for "language aptitude". At the same time, the study excluded "five participants ... due to attrition (not completing the training sessions), and one participant ... because he was an extreme outlier in learning rate (>3 sd away from the mean)." I mean, if you are to exclude 15% of your subjects without looking at their aptitude (maybe they didn't do it because it was too hard to pass the training tests to move to the next lesson, yet their language aptitude is high?), with only 36 subjects of which 21 are female (it's obvious programming is male dominated, so they only had 15 males: maybe it doesn't matter, but maybe it does), how can you claim any statistical significance with such small numbers?

godelski 2y ago
> the original study uses "numeracy" That's fair, although I don't think it changes my response. And the article still really leads to the wrong conclusions. You want to teach children abstraction and reasoning? You teach them math. Not numeracy, math.

apeescape 2y ago

Finland tried teaching maths to children using Group Theory in the 70s [1], but the results weren't that good; it proved to be too abstract for young kids. Ultimately, I believe basic algebra and geometry are the most important takeaways from math classes for most people. [1]: https://www.hs.fi/tiede/art-2000004823594.html https://www.hs.fi/tiede/art-2000004823594.html (sorry, it's in Finnish and behind a paywall)

hirvi74 2y ago
I am no expert in pedagogy of mathematics, and I am sure someone will correct me if I am wrong, but I think there was/is a Russian academic program in which students were/are basically only taught algebra in an iteratively increasing manner. As it was explained to me, one wouldn't take a "Calculus I" class as a prerequisite for say an entry-level engineering course. One typically had such a strong foundation of algebra, that when encountering a problem that required calculus, the student would just learn the necessary calculus at that point in time. In other words, with such a strong algebraic background, other aspects of math, within reason, were much easier to grok.
godelski 2y ago
> the results weren't that good; it proved to be too abstract for young kids You cannot make that conclusion as a result of the evidence. Yes, the evidence might support that conclusion, but there are many others that also could. For example, they could have just been really bad at teaching. This even seems like a likely one as it is difficult to perform such a reformulation and to do so broadly and quickly. The other reason I'm willing to accept alternative conclusions is that France and the USSR had far more success than Finland (or even America). Their success contradicts a claim that "[it is] too abstract for young kids". You'd need to constrain it to something like "[it is] too abstract for Finish kids" which I think both of us would doubt such a claim.
- apeescape 2y ago
  Yeah fair point. However, curious of what you base your claim "France and the USSR had far more success" on?

florbnit 2y ago

> It think this is silly on multiple accounts. I'll claim that there's not real thing such as a "language brain" or "math brain." It seems plainly obvious that this language just means “areas of brain that activate when dealing with math problems” vs “areas of brain that activate when dealing with language problems” and yes there is hard evidence that there is a difference between them.

godelski 2y ago
Please reread my comment in full. I'm willing to bet we disagree on the definition of math. I'll strongly insist that the one I'm using is common among mathematicians. If you'd like to retort by saying it's about semantics then congrats, we're on the same page (and can be verified by reading you sibling comments and/or my replies to some of them)
- florbnit 2y ago
  It has nothing to do with the definition of math. If you put people in an MRI and give them random problems then you’ll notice a distribution in the patterns of brain activity that are distinct. Some of these groupings we’ll call “math like” because the problems are more like mathematics than linguistics. And in the other distribution we’ll call them “word like” because the problems fall into the category we associate with word problems. The fact that we see separate patterns of brain activity is the interesting part not any semantic argument about what you or mathematicians feel should be defined as being math or not.

jeffhuys 2y ago

Interesting to me how Wernicke’s Aphasia works then (which I have occasionally; my epilepsy stems from there + broca’s area). Could you explain that?

godelski 2y ago
What is the question asking? Why do you have Aphasia? I'm not sure what the relationship here is. Or are you suggesting you're good at math but not good at language? I feel weird answering even if I infer the right question because it feels tautological. If you have Wernicke's Aphasia does it not create the possibility that I already have but your condition resulted in misunderstanding. Given the condition does it not create a high probability that a response will similarly be misunderstood? Is not Anosognosia quite common? Maybe I'm really misunderstanding but honestly I'm not sure what you're asking

atomicnumber3 2y ago

Just anecdotally based on my 2000something kid high school, the copious amount of tiering of math classes seemed to indicate to me that either we're really bad at teaching math to 80% of people, or only 20% of people will be able to handle precalculus. We had basically 4 tracks: one ended with you doing algebra 1 in senior year, another ended with you doing trig in sr year, yet another that ended with trig (no precalculus), and then one that ended with you doing trig and precalc. That final class then had further subdivisions that were too small to have their own full classes: some kids just did precalc, some did calc 1 and took the AP Calculus AB exam and/or IB Math SL, while even even smaller group took AP Calculus BC and/or IB Math HL. The total number of kids who took the AP Calc AB exam in my year was 20ish, out of a graduating class of 500-600.

godelski 2y ago
In either case it would seem to suggest we should radically change the system
serial_dev 2y ago
This system sounds extremely complicated. > the copious amount of tiering of math classes seemed to indicate to me that either we're really bad at teaching math to 80% of people, or only 20% of people will be able to handle precalculus Or maybe it indicates that the people designing this system should be fired? Job security through complexity? (Or maybe I’m just biased by the system I know… I’m just asking questions)
- godelski 2y ago
  > … I’m just asking questions I'm not disagreeing, but just wanted to point out that this phrasing is commonly used by bad faith actors. I'm not saying you're using it this way and I legitimately do not think you are. But I wanted to point it out because I think your comment could be interpreted another way and this phrase might be evidence for someone to make that conclusion. The classic example is conspiracy theorists. But lots of bad faith actors also use it to create leading questions that generally ignore or lead away from important context. Again, I do not think you're doing this. The rest of your comment makes me think you're acting in good faith.

HarHarVeryFunny 2y ago

> I'll claim that there's not real thing such as a "language brain" or "math brain. Did you even read beyond the silly headline? The article itself is about pre-testing subjects on a range of capabilities from problem solving ability to second (foreign) language learning ability, and then seeing how these correlated to the ability of the test subjects to learn to code. The results were basically exactly what might be expected - people who learned Python the quickest were those who scored the best at learning a second language, and those who learned to wield it the best were those who had scored the best at problem solving. Not surprisingly math ability wasn't much of a predictor since programming has little to nothing to do with math.

ubercow13 2y ago
I think the above comment stands. The point is, what do they consider maths ability? High school maths has very little to do with programming but university-level maths certainly feels very similar to it in structure. Many of my (good) classmates in my maths degree were very bad at things like mental arithmetic. So maybe "maths ability" (defined some way) isn't very important for being good at "maths" (proper).
godelski 2y ago
> Did you even read beyond the silly headline? Yes. I'll also refer you to the HN guideline on this manner. You're welcome to disagree with me but you must communicate in good faith and unless you have a very specific reason for thinking I didn't "RTFM" then don't make the accusation. I'm happy to continue discussing, but only on those terms. In fact, I think we're in far more agreement than your tone suggests. But I think you missed the crux of my point: math isn't number crunching
- HarHarVeryFunny 2y ago
  Sheesh - why so touchy?! Your response opened with addressing the headline, which anyone who had "RTFM" (RTFA) would have realized was unrelated to the body of the article. You then veered off into a tangent about the nature of math which again was not addressing the content of the article. The underlying nature article, linked from the posted story, makes it even more clear what is being discussed, with the abstract stating: > This experiment employed an individual differences approach to test the hypothesis that learning modern programming languages resembles second “natural” language learning in adulthood. https://www.nature.com/articles/s41598-020-60661-8 https://www.nature.com/articles/s41598-020-60661-8

hydrogen7800 2y ago

If we taught music the way we teach math. No wonder so many people are "bad at math". “Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely. One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way.” https://en.m.wikipedia.org/wiki/A_Mathematician%27s_Lament https://en.m.wikipedia.org/wiki/A_Mathematician%27s_Lament

thomasikzelf 2y ago
Of course Brett Victor has the essay on his website: https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician's_Lament.pdf https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician'... It is a great read!
seba_dos1 2y ago
Frankly, with all the mess that music notation is and the way it's often being taught, this may actually be a slight upgrade over status quo. I have never experienced more "that's just how it is" than in music.

thaumasiotes 2y ago

> most people don't know what math is It's surprisingly common. Case in point: "The unreasonable effectiveness of mathematics in the natural sciences". A normal person wouldn't be surprised that describing how something works is a good way to understand it.

LegionMammal978 2y ago

By the very expansive definition that you advocate for, what exactly do we do that isn't math? One could argue that all we do is turn thoughts, senses, and memories into further thoughts and appropriate actions, which is applying a pattern, which is math. But at that point the definition is too broad to be helpful for anything but playing word games.

godelski 2y ago
There are no rules. On the other hand, code is very formal. It is a set of rules. It has structures. There is wrongness and you can do verification that is not the result of opinion nor is it subjective. But yes, math is very broad. But it doesn't cover everything. At least not at this time.