Are all brains good at math?
nautil.us
nautil.us
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.
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.
Anyway, this all led to me doing very poorly in math at school, and it’s entirely because my brain just isn’t good at it. Luckily these days the educators are smarter and my son has always been allowed a calculator for math class.
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.
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.
It's interesting that you picked 0=1 as your example, because I'd argue it stretches the definitions of "equation" and "solution" into semantic triviality. It's more of a falsehood than an "equation", since the two sides are trivially defined as not equal, and there's no variables to "solve". Using that as example exists somewhere between sophistry and pointing out the absurdity that mathematical definitions for terms technically hold even in trivially untrue situations. That's not how normal human communication works, and not recognizing that divide probably goes a long way in explaining the "inability" you see in students.
In other words, maybe you should have just used "0x=1" as your example :P
Suppose I said solve
x = x+1
You then subtract x from both sides and end up with
0 = 1
Then you conclude that the original equation has no solution. I’m guessing that you wouldn’t realize that the reason we conclude that the original equation has no solution is because the two equations
x = x + 1
and
0 = 1
have the same solution set since adding the opposite of x to both sides is a solution set preserving operation. It transforms a given equation into a new equation with the same solutions and clearly 0=1 has no solution. That is, 0=1 is a perfectly valid equation.
The larger point, that is missed by people, is that an equation in essence is asking for one to find the instances when two expressions are equal. To find an example of an equation with no solution just find two expressions that are never equal to each other.
Respectfully, you've got this backwards. An equation, by definition, is an assertion that two expressions are equal. 0=1 is a logically consistent assertion, but it happens to be false. Most students will intuitively have trouble with the idea that you want them to make a false statement, even if they don't realize that, because their whole schooling has taught them the opposite.
The issue is precisely that we are teaching those students that that "an equation in essence is asking for one to find the instances when two expressions are equal". Mathematical statements don't "ask" anything, they simply are. That's a pedagogical definition, not a mathematical one, and by teaching students that, you're teaching them how to pass a math test rather than teaching them math. And there's no blame on you for that, since you're paid to teach students to pass math tests. But framing it that way doesn't teach them math, it teaches them how to guess the teacher's password[1]. It's a focus on getting an answer rather than understanding the actual axioms.
So of course students don't come up with an equation with nothing to solve, because you've taught them equations are things that only exist as things with unknowns to solve.
It might be obvious to someone who already is extremely well versed in mathematics that 0=1 is "an equation without a solution". But it's unfair to expect students who don't already have that answer to derive it, because they're working off of the wrong axioms. It's a communication failure, not a mathematical one.
[1]https://www.lesswrong.com/posts/NMoLJuDJEms7Ku9XS/guessing-t...
x^2 + x + 1 = 0
And say solve it we are definitely not asserting that the two expressions are the same. Indeed they are not the same polynomials and if your view were correct we wouldn’t spend time teaching how to solve the equation. There are values for which the two polynomials evaluate to the same number. Those are the solutions.
EDIT: In mathematical logic class one talks about predicates and you learn to think of equations as assertions that two expressions are the same. However, as people typically use and think about math they don’t think in these terms. Indeed, the graphical interpretation of an equation in one variable lends itself to the idea that solving an equation, in essence, is finding values of x that make two functions have the same value.
It is also equally clear that you haven’t taught basic mathematics to innumerate students. When students are taught to solve basic linear equations we include in our instruction that they can encounter situations like:
x+1 = x
And that they can see there is no solution because they reduce the equation to solving 0=1 and that equation has no solution.
You are in an absurd position when you think
0x = 1
is an equation but that
0=1
is not. I doubt that when you simplify:
x^2-2x - (x^2 -2x)
You write 0x^2 + 0x. What I wrote about solving equations has an important word in it. Namely “essence”. In essence…. I was not providing a mathematically rigorous definition. Indeed, the rigorous definition is far beyond the scope of students of basic mathematics. So we have to teach them the essence of things.
> we are definitely not asserting that the two expressions are the same.
Correct. Not the same, equal. Because that's definitionally what the equals sign means. "A=B" is a symbolic representation of "'The expression A' equals 'The expression B'". I hope we can agree on that?
>if your view were correct we wouldn’t spend time teaching how to solve the equation.
What I actually said implies the exact opposite. You teach how to solve equations because that's the use case for equations as tools. That's not a bad thing, it's an extremely useful thing to teach.
But teaching how a tool is used is not the same as teaching the fundamentals of what a tool is. It can help in that goal, certainly, (and might even be required as a prerequisite) but it's not the same. It's exactly like you said:
>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 not fair to blame students' "innumeracy" for not being able to derive "0=1" as "an equation without a solution", because they've successfully learned the thing that they were actually taught, that equations are "things with unknowns that we have to solve for". Of course generating a solution that has neither unknowns nor a solution is foreign, because everything they've learned about it as a tool goes against that.
(It's worth noting that there's another reason that the teachers teach this, one that's perhaps even more important for the school system; it's an easy thing to evaluate student understanding of. You can easily test whether a student can "solve" an equation, and return the correct answer. It's something you can get immediate, iterative feedback on. You can't really test if they actually grok a definition, because they can just parrot a definition with no understanding.)
Fundamentally, my argument is about language, not mathematics. You're saying that students aren't able to derive answers based on the definitions of terms, but not only are those definitions wildly divergent from their English meaning, they're divergent from the actual definitions the student are learning via practice.
Take, for example:
>There are values for which the two polynomials evaluate to the same number. Those are the solutions.
Values of x, you mean but didn't say. Because it's so heavily implied in the existence of an "equation to solve" that unknown quantities you are solving for are the ones written in the equation itself, that it's not even worth mentioning. But it's precisely this linguistic assumption that obscures what an equation actually is to students.
I was taught that using the quadratic formula to find roots is "cheating" when I tried using it before we had covered it in class. What exactly does "completing the square" test for other than your skills at mental arithmetic?
The reason why many student believe that polynomial doesn't factor is because teachers do a lot of hand waving when it comes to explaining what it means to have no rational roots of a polynomial. Few teachers will take the time to teach the foundations of the cartesian coordinate system and how complex solutions don't map easily on the typical plane of rational numbers. All students learn is if there's an "i" the answer is "no solution".
Zeros being the solutions of functions is a question on the finding roots of a polynomial chapter in basically every single high school algebra 2 class. It's a prerequisite to learning how to graph second order polynomial functions. Many students learn and forget how to do it before reaching calculus, let alone college.
I genuinely haven't done long division in the last decade. I struggled to help my younger cousin with it recently and had to relearn myself because it's such a useless algorithm in the age of computers. Certain multiples and powers I remember, because of how often I come across the numbers, but in general I will choose a calculator every time. I would even choose a calculator to double check my own work with a paper and pencil. At this point what is the value in doing the work by hand? In many cases a decimal to the hundredths is required as well.
When I hear the question, write down an example of an equation with no solution, my intuition and experience doesn't lead me to writing an incorrect equation. It leads me to think about writing a polynomial function that I know will have complex roots because I was taught the answer to that is "no solution", or writing a system of equation where x is a specific number while at the same time having an equation where that specific number can't be a part of the domain. More fundamentally, a system of equation with no solution is one where the two lines that are graphed are parallel.
I have to admit, my experience is a little biased as I was placed in accelerated math since elementary school. It wasn't difficult for me or my peers. My math class senior year was fundamentals of multivariable calc and linear algebra as a senior in high school, having finished AP calc bc the year before. I was far from the only one in that situation, there were at least 60 of us that year, some seniors and some juniors. I can't say I have experience teaching a full class but I have been tutoring high schoolers in math for over 8 years. Many of my students have also been in accelerated math, but not all of them. I don't think anyone tested out of multivariable calculus, but I did have a friend who tested out of linear algebra at reputable universities.
I know that I have some time and experience left before I feel confident in my own mathematical maturity, but I'd like to imagine I'm somewhat good at math. At the very least I wouldn't consider myself bad at math, even though I still feel like I am at the early stages of learning in specific branches of math.
By the time one reaches calculus they have been taught that complex solutions are valid solutions. They just aren’t real solutions. Therein lies one of the problems teachers of mathematics have. Conveying the concept of the answer depending on what the current algebraic object one is working on. We have to hand wave do some brain washing because the nuances involved are far too complicated for the students to understand at this level.
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.
The point is not for students to learn how to follow an algorithm (or master doing it quickly), but that they learn why the algorithm works and how to interpret the result.
The algorithm matters and I argue that once students can naturally derive the algorithm on their own, there's no place for human calculations. Computers and calculators should be used. We're no longer living in a society where students will have to worry about "what if you don't have a calculator on you". It seems as weird as memorizing unit conversions to me.
What's interesting is that the purpose of implementing the common core standard was exactly what I am arguing for. A shift away from algorithmic memorization to a number sense way of doing math. The failure lies with the teachers not being able to teach well much more than students not learning.
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.
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.
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.
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).
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
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.
seems odd in a world full of 3D graphics and machine learning, which are built on heaping piles of linear algebra.
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
Something you'd encounter as Mr. random.
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.
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.
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.
-- 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. --
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...
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.
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. :)
Can you describe this process a little more clearly?
I am generally very good at taking tests but that was learned behavior. My origin story was a conversation about timed math tests with multiplication tables, and I was hit with the twin observations that you don’t have to do the answers in order, and if you can’t recall 7x9 but you can recall 9x7 it’s the same answer.
All of a sudden I wanted to know what other tricks there were and boy did I look for all of them.
First, you just learn to count them fluently. After that, you already know all the anwers, which is a huge head start.
I would have her count up and down. So now, just to learn the counting series she also had to do a bunch of addition and subtraction too.
After she could count the series, she could actually guess many of the answers the first time through a set of flashcards. And it only took a couple repetitions to get them all right.
And then, if you forget one, you can landmark off of easier ones, likes 5s and 10s. Since you deeply understand that 6x6 is just one more 6 past 5x6, you go 30+6.
It was fun and cute to see her working out the answers when she didn't know it off the top. And even if you have to count from 0, you're quick at it so it's not a big deal.
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.
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?
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.
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.
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).
I suspect that brains, human and otherwise, are better at math than we usually give them credit for.
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.
South Park had a nice episode about Kyle getting good basketball