Are all brains good at math?

https://nautil.us/are-all-brains-good-at-math-26556/

by dnetesn • 4 years ago

35 67 4 years ago

carapace2• 4 years ago

I'm firmly of the opinion that kids are naturally born geniuses, and that our [lack of] child-rearing and educational systems traumatizes them into dullards before they get a chance to grow into their full potential.

Clearly, it's possible to do much better than the status quo, E.g.:

"Squeak Etoys, Children & Learning", Alan Kay

The projects developed in Squeak use a different and simpler kind of incremental mathematics that allows very young children to understand some of the key ideas in calculus. We have found that more than 90% of 5th graders not only understand “Galilean gravity” but are able to derive the mathematical “formulas” (actually 2nd order differential equations) using this alternative mathematics, and author a simulation that matches up very well with the experimental data.

http://www.vpri.org/pdf/rn2005001_learning.pdf

Or László Polgár

László Polgár (born 11 May 1946) is a Hungarian chess teacher and educational psychologist. He is the father of the famous Polgár sisters: Zsuzsa, Zsófia, and Judit, whom he raised to be chess prodigies, with Judit and Zsuzsa becoming the best and second-best female chess players in the world, respectively.

https://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3_Polg%C3%A1r

jayant_kaushik• 4 years ago

Just read the wiki. What a great inspiration Mr. Polgar. I have quiet a few nephew and nieces (all below 10) and that allows me to see the pattern. I see that we need to create a list of skills that a kid should know by a certain age and teach them- it sky rockets their level of confidence and they are able to achieve more faster in any sphere of life. Specialising in a few areas sounds good and till now I have thought about making my kids specialise in astronomy as I think by the time they turn 20, we would have landed on Mars and lots of ISS and talented people would be required to run them / go there. The second option is - AI ( but don’t know if we attain AGI, would the world need any talent in this space) So yes, I think about fatherhood even though I’m not there yet. Talking about education system, it’s we the people who have created it to outsource the job of rearing the kid to the schools. So they can learn the skills demanded by the world to survive. We do that to focus on our career, dreams, or just to chill. And schools are doing a terrible job world over. In

ellyagg5• 4 years ago

Probably true. My daughter finished 2nd grade behind in math and with a deep dread of it. I had been too busy to realize what was happening.

So I bought the California Common Core books for 3rd grade to understand the requirements. In our 10 minute car rides to and from camp every day, I was able to teach her all of it, without books or writing anything down.

The way they teach this in schools is just so far from optimial it's heartbreaking. To give one concrete example, a huge part of 3rd grade here is memorizing the times and divide tables. After some quick experiments, it was obvious that all she'd need to do was learn to count each number series, which is MUCH easier than memorizing flashcards, and can be done anywhere. Once you can count the number series, times and divide problems fall out naturally with practically no memorization needed.

The second big thing this year is word problems for those math facts. These word problems are so easy if you give the child the vocabulary and concepts to break them down in into their key parts. Instead, schools just leave it up to the courage and interest of the student to keep hitting their heads against the wall.

We're on to 4th grade math now and she's a week into 4th grade. I'm keen to check the Alan Kay link. :)

boppo1• 4 years ago

learn to count each number series, which is MUCH easier than memorizing flashcards, and can be done anywhere. Once you can count the number series, times and divide problems fall out naturally

Can you describe this process a little more clearly?

hinkley2• 4 years ago

I knew a kid who would practically melt down doing math homework a 8 years old. Originally contemplated a career in music or art. Somewhere along the way something clicked, and now they’re applying to engineering programs for college.

I was also about eight when I figured out how to enjoy math, and I think it had something to do with realizing the teachers aren’t authorities. There is no “way” to teach things and you don’t have to like their strategies for teaching it.

That love for math came at the same time I started running a dialog in my head, stepping out of myself, observing my own mind, and being my own coach.

Years later I encountered this idea in the context of meditation and mindfulness, and while I had always felt like maybe I was doing something others were not, the ways that mindfulness sometimes belabors the point made me realize that maybe there are quite a lot of people who don’t figure this out.

ellyagg5• 4 years ago

*week into 3rd grade, I should have said

tpmx2• 4 years ago

Are dogs also naturally born geniuses and its our lack of pup-rearing and educational systems that traumatizes them into dullards before they get a chance to grow into their full potential?

oblak2• 4 years ago

Unlikely, though dogs can appear stupid due to being dogs, hence having different drives in life. This may apply to anything besides dogs.

stephendause• 4 years ago

I'm firmly of the opinion that kids are naturally born geniuses

Do you mean that, relative to kids' intelligence today, everyone could do a lot better but not necessarily equally better? Or do you mean that everyone is born with roughly the same ability and that only some people succeed due to our child-rearing and educational systems?

carapace2• 4 years ago

Both. I don't actually know, of course, and there are obviously people who are born idiots, or savants, or both, eh? But generally speaking, yes, every more-or-less normal child could do a lot better. Could they all do equally better? Is everyone born with roughly the same ability? I believe so for two reasons:

First, the ability to learn new abilities is the defining characteristic of being a human being. Humans are universal learners in the same way that Turing Machines are universal machines. So yeah, it seems to me that any human should be able to learn to do anything any other human can do (modulus raw physical parameters, of course. But even there there's a lot of flexibility.)

Second, I believe that the spark of genius is a metaphysical or spiritual thing that is universally available to all people, you just have to learn how to "tap into it". Since genius is not intrinsic to the individual (no more than gravity or the Sun) we all have equal access to it, although not all of us realize it or make use of it in our daily lives.

annyeonghada2• 4 years ago

I'm firmly of the opinion that kids are naturally born geniuses

Well, your opinion goes against the basics of genetics and reality.

and that our [lack of] child-rearing and educational systems traumatizes them into dullards before they get a chance to grow into their full potential.

Incredible how that only some people get to be traumatized and others do perfectly fine.

László Polgár

What is never mentioned in László Polgár's story is that he made the sister choose the activity they already enjoyed and were good at (natural talent, in other words), and by doing that he completely invalidates the results he would have liked to obtain: I bet that if he were to force the sister to study Physics or Chemistry or Literature or whatever he would have not achieved the same "extraordinary" results.

blackbear_• 4 years ago

Is any of this specific to math, though? Or is it just that math "proficiency" is easier to measure?

mrkramer• 4 years ago

No, at least mine is not.

erdos4d• 4 years ago

Oh hell no. There is no other subject I am aware of that has the skill variance math does. Some people you cannot teach, others you cannot catch.

tpmx2• 4 years ago

No.

samatman• 4 years ago

Succinct. I'll add that it's cruel to pretend that all are.

Raidion• 4 years ago

I would assume (like everything else) most people fall on a Bell curve, where there is higher level math that's inaccessible to most people, but the math for daily use is (almost by definition) pretty useful.

I definietly think there are better ways to approach math, some like the theoretical perspective, some prefer the engineering approach "I need to solve this problem, and these are some tools to help us do so.

I'm not "good" at math (and I could dig out old college transcripts to prove it), but I do have a math minor because I really enjoyed the experimental engineering/physics/comp sci stuff, but found math classes on their own really boring. I feel a lot of people would benefit a more holistic approach to math, starting with algebra/geometry. It's just a mess of graphs and equations, and I struggled to find a reason to care, though I was fortunate to have a brain (and the cultural conditioning) that made those abstract "puzzles" interesting.

I got a B in Linear Algebra, and I still can't describe why you'd need that in the real world, while calculus/diff eq/discrete math were clearly tied to physics/thermodynamcis/computer science problems I knew.

To bastardize a famous quote: If you wish to build mathematicians, do not divide the men into teams and send them to the forest to graph equations. Instead, teach them to long for the vast and endless understanding of a problem that they find interesting.

sigstoat• 4 years ago

I got a B in Linear Algebra, and I still can't describe why you'd need that in the real world

seems odd in a world full of 3D graphics and machine learning, which are built on heaping piles of linear algebra.

eindiran• 4 years ago

I think this is one of the big advantages of a broad undergraduate education in math; the math that I use frequently looks very different from the math used by people in <insert other field>. I have essentially never used any calc/analysis/topology/geometry/number theory/etc stuff in the real world, but graph theory, stats, linear algebra, etc have come up a LOT. Which stands at odds with a lot of my college friends who need a completely different set of tools in their current work.

ivan_ah• 4 years ago

I got a B in Linear Algebra, and I still can't describe why you'd need that in the real world, [...]

Here is a short list of applications of linear algebra:

   - Balancing chemical equations
   - Input–output models in economics
   - Electric circuits
   - Graphs
   - Least squares approximate solutions
   - Computer graphics
   - Cryptography
   - Error-correcting codes
   - Fourier analysis

You can read about these in Chapter 7 of my book in LA. Here is an extended preview of it here: https://minireference.com/static/excerpts/noBSLA_v2_preview....

LunaSea• 4 years ago

I think that what OP meant by "real world" is something closer to "daily life" use.

Something you'd encounter as Mr. random.

cat_man• 4 years ago

I got a B in Linear Algebra, and I still can't describe why you'd need that in the real world, while calculus/diff eq/discrete math were clearly tied to physics/thermodynamcis/computer science problems I knew.

I thought this was an interesting comment, because I personally believe linear algebra is one of the most applicable topics in math and relates a lot to the topics you contrasted it with. For example, in multivariable calculus, derivatives of functions with multiple variables end up being linear maps, and understanding properties of those maps and how they're transformed helps a lot with understanding the properties of derivatives and how to apply them. Differential equations are solved in practice by approximating them as linear systems and solving those equations, so again, understanding linear algebra helps a lot there (e.g., eigenvalues are intimately connected to the ways differential equations behave). I'm not so familiar with discrete math, but I do know there are connections between linear algebra and some areas of graph theory (not sure how critical they are to those areas, though).

That's not meant as a criticism of your comment, because I think the way linear algebra courses are taught doesn't do much to make those connections clear. Intro courses focus a lot on mechanical problem solving and do a poor job of motivating concepts (e.g., I remember eigenvalue problems showing up mostly out of nowhere). In courses beyond the introductory level the presentation and focus is more abstract and does little to demonstrate why you would care beyond intrinsic interest. I think if more motivation or context were provided, it would help encourage those more interested in applications than math for math's sake to go deeper into a topic that can be very useful in a lot of applied areas.

pigtailgirl• 4 years ago

https://wikipedia.org/wiki/Dyscalculia

-- from the article: Of course, some kids do have clinical learning disabilities that are not caused by anxiety, cultural myths, or poor teaching. Developmental dyscalculia, a clinical term for someone who has a math-related learning disability, affects about 3 to 7 percent of the population. This is the lesser-known analog of dyslexia, a learning disability related to reading, and the two conditions often occur together. --

knaik944• 4 years ago

I recognize the purpose of learning fundamentals but I think we do a great disservice to students learning mathematics by not emphasizing the power of computers and calculators in modern arithmetic. Many people with math anxiety see mathematical skills to be the same as mental arithmetic skills. Textbooks still focus things like von Neumann's ability ability to multiply large digit numbers in his head as a proxy for his math proficiency. While impressive, the technologies built on top of von Neumann's work make that kind of skill obsolete.

I firmly believe that all brains are good at math but not everyone has had a teacher who will spend the time to teach the cognitive tools required for it. If we stop teaching calculators as "cheating" then a lot more students will be able to focus on the understanding the underlying concepts and will feel less scared about approaching new mathematical problems.

buscoquadnary• 4 years ago

I'd argue that not all brains are good at math but that they can be trained to be good at math.

That's the problem we treat math like it is some innate talent, it's not it just requires certain ways and forms of thinking. Example: I was decent at math in HS, I didn't do very well because I didn't like doing the homework and quite frankly most people good at math aren't going to end up being public school teachers, so the explanations were always lacking.

Well I went on learning programming on my own got a job as an SWE, then a couple of years later ran into a situation where I had to learn some new math. It was world's easier this time, because of my work with programming it all jelled immediately and suddenly clicked it all became very logical. I don't believe I suddenly became better at math, but rather I had developed certain thought patterns and ways of thinking that could be applied to mathematics.

syzar5• 4 years ago

Not everyone can learn calculus or even basic algebra. A person with a severe intellectual disability isn’t going to be able to learn these subjects. Between severely intellectually disabled and Einstein is a boundary between those who can and those who can’t. I don’t know where the boundary is but it exists.

CivBase2• 4 years ago

Computers and calculators should absolutely be embraced by modern math education. Once someone thoroughly understands long division, there's no sense in forcing them to tediously repeat every division operation by hand. But it's still imperative that students learn conceptually how and why mathematical processes work.

Math is different from most other primary school subjects in that it builds on itself year after year. There is a danger to letting computing devices take over before the students fully grasp a math concept. If they fail to learn a concept, moving on with a calculator as a crutch may inhibit them from understanding the next math concept.

syzar5• 4 years ago

Students don’t fully grasp concepts at the early stages and they never will. Very few students who master the quadratic formula understand that this formula allows you to factor second degree polynomials. Many students will correctly solve x^2+x+1 = 0 while simultaneously believing that this polynomial doesn’t factor. Here’s a simple problem that tests understanding,

A second degree polynomial with leading coefficient 3 has zeros of -1 and 2. Find all the terms of the polynomial.

Most students can’t do this. Even most calculus students can’t do it.

We teach algorithms like long division and the quadratic formula because they are relatively easy computations to learn but they don’t in any way lead students to fully grasping a concept. It’s only with a certain level of mathematical maturity that one is able to understand the full import of even basic concepts.

I can walk into pretty much any first semester calculus class and ask students to write down an example of an equation with no solution. A large majority will fail to do so. It doesn’t occur to them that 0=1 is such an example. They’ll play around with x’s in various complicated looking expressions. Even something as basic and fundamental as the meaning of an equation eludes people at this level even though they have been dealing with equations for years.

Well, such is my experience teaching math at a community college for over 20 years.

jrgoff• 4 years ago

I'm curious what value you consider understanding long division to hold? In my mind it has some value in training how to follow an algorithm and obviously has some value in being able to divide arbitrary integers (and beyond). But of course as you point out there's not much point in forcing long division when there are ubiquitous computers available to do the job better. And while I think following an algorithm carefully can be a valuable skill, if that is the primary goal then I'm not sure teaching long division is a particularly good way to go about it.

I agree with your general point that math curriculum builds on itself and if students fall behind it can make things worse and worse for future classes. But in general I think, at least when I was in school, there was probably too much focus on algorithms and not enough on conceptual understanding. I think part of the reason is that algorithms historically were necessary to know before calculators were ubiquitous, and also I think algorithms are much easier to teach and evaluate so there's probably a tendency to focus on them for that reason.

scythe• 4 years ago

I think we do a great disservice to students learning mathematics by not emphasizing the power of computers and calculators in modern arithmetic.

But all of the students who came to the help desk in college were practically addicted to their overpriced graphing calculators. A decent fraction of the time I would look over their work, they'd be on their last try in WebAssign (may it be cast into the fire) with a scared look on their face, and I would say "you entered this part into the calculator wrong". I knew immediately because I learned to estimate these things. You don't have to even know the multiplication algorithm; you can stick to multiplying the first digit, adding exponents in scientific notation, etc. Learning to work with the calculator instead of behind the calculator will make you use it more effectively.

The multiplication table has 45 entries that aren't duplicates or multiples of 1 or 0. Kids memorize longer things all the time. The brain is definitely not the limiting factor.

Textbooks still focus things like von Neumann's ability ability to multiply large digit numbers in his head as a proxy for his math proficiency.

I don't recall ever seeing that in a textbook. There is a famous story about JvN summing an infinite series in his head, but very few people can do that or would be expected to. He also estimated the yield of Trinity within a factor of four by just looking at it. One should never compare themself to von Neumann.

I honestly believe the biggest factor in math anxiety is osmosis. Their parents are afraid of math, their teachers (often!) are afraid of math, and there's a robust body of evidence in developmental psychology that children inherit the worries of their guardians.

boboralice• 4 years ago

My husband is terrified of mathematics but he's a great programmer. I'm no genius when it comes to the subject because it's not my passion. I've only used degree-level math a couple of times in the (mumble) years since I graduated and I had to dig the textbooks out and revise it every time.

My husband and I went to uni together. I dragged him through the mandatory maths units in the computer science degree by reframing the equations we were working on as "a bunch of steps, just like code" and he got through the unit, but our lecturer was a horrible, impatient person and managed to reinforce his fear of the subject.

A lot of people I know have similar stories. Maths and physics seem to attract teachers who think (out loud) that students who don't intuitively "get it" are beneath them.

theteapot• 4 years ago

Textbooks still focus things like von Neumann's ability ability to multiply large digit numbers in his head as a proxy for his math proficiency.

Really? Which textbooks are these? Done high school, uni. Never heard that, or really much about von Neumann outside a few passing mentions in advanced topics. Sounds like some really strange math textbooks.

abdulhaq2• 4 years ago

Some people will learn maths much faster than others. A person of IQ 140 will learn calculus in much less time than someone with an IQ of 70. Even if everyone had infinite time I think we would all hit a limit, and that limit would span a very large range of ability. Hence the answer is no, not all brains are 'good' at math.

yboris• 4 years ago

It seems you're answering a narrow question focusing on "all". At this point, you may as well argue "there are people in persistent vegetative state, so 'no, not all brains are good at math'". This doesn't seem like a charitable interpretation of what the focus is on. If we are looking at the 95% of the population, excluding the obvious examples, your argument might not apply very much.

I think the claim is also contingent on what is meant by math (how far do you need to go to be proclaimed "good" at it) - and I doubt calculus is a requirement.

For your argument to work, you'd need to confront whether in fact at least 95% of the general population would be capable of understanding enough math for (and here we can differ on what "good" is, but I'd say something like) general understanding of algebra and its application towards regular life activities (e.g. understanding percents, exponential growth, loan interest payments, solving for an unknown, etc).

abdulhaq2• 4 years ago

Yes it comes down to 'good' meaning having achieved a certain level, or 'good' implying the existence of a significant spread in ability across e.g a central 95% of people. I've chosen the second meaning, and do believe the spread remains significant.

zmgsabst• 4 years ago

At my university, almost all of the math majors and a fair number of engineering students graduated high school with calculus credits.

I think the person is making a fair distinction that most brains are okay at math — and some at each end of the bell curve are good or bad at math.

- - - - -

A few asides:

- really understanding exponential growth requires derivatives/rates of change

- we do a bad job at conveying math subjects in education; you can teach kids the basics of abstract algebra and calculus if you tie the ideas to familiar concepts

- we should definitely rework the curriculum in light of computers

desindol• 4 years ago

IQ doesn’t help you understand higher math concepts if you got burned by bad mathematical education early in your life.

bitwize• 4 years ago

My dog divides the food in her dish cleanly in half, eating one half now and saving the other half for later. I read somewhere that dogs can count up to about four, so taking half of something is within their bailiwick.

I suspect that brains, human and otherwise, are better at math than we usually give them credit for.

RcouF1uZ4gsC• 4 years ago

I think an analogy to writing is helpful.

Universal literacy is a very recent phenomenon. In the past, only a few elites in society could read and write.

However, now the expectation is that everyone should be able to read and write and in developed countries there is above 90% literacy rates.

Just like with math, there is a continuum of skills. Not everyone will be able to write like Hemingway or Faulkner or Proust. But, we can and should expect a certain level of competency from everyone.

Same with math. If being good at math is being like Ramanujan or Noether, then I suck at math. However, there are a lot of stuff I know about math and can use it to solve a lot of problems.

Once we make a certain level of mathematical competence expected and gear our educational system to achieve that, this question becomes moot.

Yes, some brains are better at math than others. Some brains are better at writing than others. Some bodies are better at running than others.

But we shouldn't let that stop us from having everyone achieve competence. My kid will never be Usain Bolt, but I will do my best to make sure he can walk and run and jump.

butUhmErm• 4 years ago

I’ve been thinking about this a lot lately; human languages are only 5,000 years old. Humanoid spatial reasoning, intuiting “enough heat, food, water” is a set of fundamental problems we evolved to survive.

I’d say it’s more than just “all brains” and say “all bodies”. Maybe our brains gave us the ability to generate syntax to capture meaning but our bodies play a role in letting us know when we are “cold” as a response to external measure. A body is a gradient of spatial points, matter interacting with fields; not just the brain.

Rigid physical models of the past must give way to updated awareness, an idea many have understood throughout time. People missing 80% of their brain have gone on to live full lives, graduate from higher education, succeed in a career. The brain is not the center of our awareness, merely a component.

jongjong• 4 years ago

I think it's mostly about how it's taught.

As much as I like math, I really don't like math education. It's one of those subjects which is simple to understand conceptually but the terminology used to explain math is both overcomplicated and inadequate.

The way people typically use English is too ambiguous to describe something which requires as much precision as math... And people who are good at math aren't particularly good at communication. Also, mathematicians like to keep inventing new words instead of reusing existing words; reading math articles on Wikipedia often requires a LOT of clicking through definitions to make sense of really simple concepts. Ironically though, when it comes to symbols, mathematicians love to reuse the same symbol for different things (depending on the specific field).

sinenomine• 4 years ago

There is an interesting, relatively novel body of research studying electrophysiological and genetic correlates of mathematical talent. And among it one can find preliminary evidence showing that even after controlling for general intelligence, there are are some nontrivial differences between individuals with mathematical talent and controls:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4052339/pdf/fnh...

https://journals.plos.org/plosbiology/article?id=10.1371/jou...

Looks like mathematical aptitude really takes a different architecture of the brain.

exolymph• 4 years ago

Betteridge's law strikes again! Anyone who's done math tutoring knows that the answer is no.

oblak2• 4 years ago

It doesn't take tutoring to realize that not all people can be good at thing.

South Park had a nice episode about Kyle getting good basketball

avnigo• 4 years ago

Here's a fascinating related Numberphile video with Professor Brian Butterworth on the way our brains see numbers:

https://www.youtube.com/watch?v=D1sPBCxlDQQ

mjreacher• 4 years ago

One thing I think would be interesting to research is how different people are good in different areas of math. There's lots of anecdotal evidence[0] of people saying they tend to be good at either more algebraic subjects (say group theory or algebraic number theory) or more analytical ones (PDEs or measure theory) likewise I would extend this to also include geometric/topological subjects.

In general especially as once gets into more advanced mathematics the abstraction level gets much higher quite quickly so it would interesting to know if there is any connection between brain structures and ability to grasp abstract mathematics. As an example I know a physicist who can tell you everything about general relativity and has a solid grasp of differential geometry but one time I was trying to teach him some mathematical logic and he just couldn't grok it no matter how many hours we spent on it. Likewise subjects that use category theory heavily (homological algebra/algebraic geometry for example) are known to be difficult to understand even for professional mathematicians in other areas.

[0] https://bentilly.blogspot.com/2010/08/analysis-vs-algebra-pr...

neosat• 4 years ago

Rather than generalizing ability to "Math" as a whole, I think it might be more meaningful to state that individuals may differ in the type of problem solving their brain is better at (or at least they enjoy more and thus with effort get better at it). Enjoying and being good at abstract mathematical concepts of sets and number theory might be different than being good at geometrical manipulations or calculus. Some capabilities required for certain types of math problems might cluster closer with problem solving capabilities in other domains (e.g. chemistry or computer science or even writing) than they do other capabilities in math.

Are all brains good at math?

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