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Bertrand Russell, a famous philosopher, mathematician, historian, logician, and all around super smart and sexy guy, once wrote:

“It must have required many ages to discover that a brace of pheasants and a couple of days were both instances of the number 2: the degree of abstraction involved is far from easy.” — Introduction to Mathematical Philosophy by Bertrand Russell, page 3

Really? Cause it sounds pretty easy to me. So how does all this squishy stuff inside our skulls enable us to do math, anyway? Well, according to phrenology, a pseudoscience of the 1800s, the math organ, the part of the brain that does calculations, is right about… here. (Pointing to just behind my.)

The phrenologists thought you could predict a person’s abilities by feeling the bumps on their skull. And they found a bunch of people who were good at math, who also had really big bump in the same place, so they figured: ah, that must be the math organ! Seems legit to me.

There is some truth to it, though. Modern neuroscience has found that there is a specific part of the brain that seems to control our intuitive ability to do things like estimating how many marbles are in a jar, converting Farenheit to Celsius, and other fun stuff like that. That intuition is usually thanks to the approximate number system, usually just called our number sense. It seems to be located in the parietal lobe — more specifically, in a structure called the intraparietal sulcus. I hope I’m saying that right. I’m not exactly a neuroscientist… Ooh! But I can quote one!

“We now know that the human brain has evolved specialized circuits to exploit the fact that much of the perceptible world is countable. This is why neurological damage can affect numbers and nothing else, and why people are born dyscalculic.” — The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene, page 351

Just like some people are dyslexic, meaning their brain has trouble with letters, a similar percentage of people are dyscalculic — their brains have trouble with numbers. One study found that children who are dyscalculic actually have less grey matter in the intraparietal sulcus. And a handful of people even have what’s called arithmetic epilepsy. Their epileptic seizures are triggered whenever they do math. (I know I shouldn’t say this, but a really small part of me wishes I had that, just because it would’ve been the perfect excuse not to do my math homework!)

Anyway, it seems we have a calculator built into our brains. But unlike a real calculator, our brain calculators only handle estimates. The more objects there are, the less accurate your estimation becomes. This has been called the magnitude effect. And there’s also what’s called the distance effect: the closer together two numbers are, the harder it is to tell them apart.

All that stuff might help to explain why we can only perceive up to about four objects at a time without counting them. There’s even a name for that ability:

Subitizing: the “ability to take in the numerosity of a visual array of objects at a glance, and without counting” — What Counts: How Every Brain is Hardwired for Math by Brian Butterworth, page 113

Now once there are more than about four objects, we have to either count or estimate. And it turns out that everyone’s estimating ability, or number sense, is different. Professor Justin Halberda created a test that measures your number sense. You see a bunch of blue and yellow dots for only an instant, and you have to guess which color has more dots. Since you don’t have time to count them, you’re forced to rely on your number sense.

(Taking the number sense test at Panamath.org.) OK, let’s see how I do… I feel like I’m just guessing randomly, but I seem to be getting most of them correct! Correct, correct, correct… aww, darn.

(Looking at the results.) Dude… I’m awesome at this! I wasn’t expecting this. I’m terrible at arithmetic! Oh, OK, hang on. This explains it. I’m really good, but I’m also really, really slow. That actually explains a lot.

The very latest research, just published in January 2012, created an artificial brain that taught itself how to estimate the number of objects in an image without being preprogrammed to do so. Wow. As the lead researcher says,

“It answers the question of how numerosity emerges without teaching anything about numbers in the first place.” — Marco Zorzi, quote from Neural network gets an idea of number without counting by Celeste Biever, New Scientist

So if an artificial neural network can teach itself to estimate the number of objects, regardless of size or color or stuff like that, then clearly abstraction thinking isn’t that difficult after all, and it doesn’t take ages to discover that a brace of pheasants and a couple of days are both instances of the number two. Sorry, Bertrand Russell. Despite your sexy mustache, I guess you were wrong. That’s OK, though. I forgive you.


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16 thoughts on “Math in the Brain: Number Sense

  1. Hi
    Nice video dear Liz :)
    I’m in believe that our brain can automatic calculate some simple mathematics problems without thinking, but just there is a problem! and it’s our ability to access this data in our brain!
    If one day human be able to access this parts of brain, he/she no need to solve problems on papers (or speaking and thinking silent….)
    This is just a theory ;)

    (excuse me if i use bad grammar, my language is not English… ;) )

    • Anything’s possible, I guess. Did you take the Panamath.org number sense test? It does feel like your brain is estimating the number of dots automatically, like you have no control over it. Crazy stuff.

      • Yeap, and I’ve got 0.14 Weber fraction. It was realy crazy and funny, I like it ;)

        I belief that “Just Impossible is Impossible“.

  2. Hey! Nice video! I actually enjoyed watching it, although, its not info that is relevant to many people, maybe neurosurgeons, or math nerds. Well not practial math nerds like game designers or programmers but you know what I mean. Its rare on t’interveb we find anything interesting and educational, keep up the good work, and eh make with the relevant!!

    • Thanks. :) Glad you thought it was interesting. I mean, I certainly think it’s interesting! Relevance be damned! Who needs relevant stuff? I do relevant stuff all day long. So now I’m making myself learn irrelevant stuff and I’m having so much fun with it! But OK, OK, I promise I will get around to more practical stuff eventually. What sort of things do you spend your time learning?

  3. Hello Liz, nice videos! Thanks for taking the time to put them together.

    I have nothing for or against you or Bertrand Russell, but I do not think your scenario and Mr. Russell’s are fully comparable.

    You see, the ability to extract a rough numerical value from examining reality at any one instant is relatively simple. It implies parallel processing of many signals coming simultaneously from many sensors and brains in general are good at parallel processing.

    However, arriving to the conclusion that “today plus yesterday make two days” requires a series of successive examinations of reality (noticing one sunrise, one sunset -signaling the end of day 1- and a second sunrise), exctracting an abstract concept out of them -the notion of a day- and adding both days in order to get the number two). That implies storing abstract concepts in memory for at least 24 hours before adding them together.

    In any computer system, and the brain is a very complex computer, series processing requires a memory buffer and mechanisms to synchronize it that parallel processing does not. And even though human brains are, in general, fairly good at series processing (most of us can retell events lived in the past in a reasonably accurate chronological order) that does not necessarily mean it is as simple as estimating the number of objects in view at one glance.

    Do you think this might the difference Mr. Russell was trying to point out?

    In any case, thanks again for your educational efforts. All the best.

    • Hi Diego, thanks for sharing your thoughts! I love reading longer, thoughtful comments. :) I actually love that quote from Bertrand Russell, just because it’s a great reminder not to take number and language for granted. I definitely don’t think Russell intended for it to be interpreted literally, though. Replacing “a couple days” with “a couple shoes” would make the same point.

      I just wanted to point out that between 1919 (when he wrote Introduction to Mathematical Philosophy) and today, we have an entirely different understanding of how the mind works, and research indicates that abstract thinking is much simpler and developed much earlier than previously thought. The following video I did on number sense in children has examples of that, like addition in nine-month-old infants. And I’m still working on a video about the research done on animals, which also indicates that the ability to recognize number across different objects actually developed very early on.

      I’m sure you’re right, though, that the ability to recognize the number of a group of objects seen all at once predates the ability to remember that number, and then recall it and add onto it later. I appreciate your attention to detail, so thanks for pointing that out!

  4. Liz,
    Actually been giving this some thought because of a case I am working on at work, and came across something I had almost completely forgotten about. The Lebombo Bone and other artifacts clearly demonstrate number sense and its significance in early modern human society and cultural. These artifacts are more than 35,000 years old.

  5. I took the test and scored 0.16. I didn’t know what to expect so I’m not surprised.

    I admire your ambition to learn everything. I want to do that myself.

    I have a theory; I call it “recursive learning”. Recursive learning involves the process of learning how to learn faster. You look for similarities between what you know (anything!) and what you want to learn then you form associations between them. The subject matter do not have to be within the same field of study.

    I’ve tried the theory on myself with some success. For example, I learnt Java by associating it with the family structure. Then I learnt another programming language by associating it with Java and the family structure.

    Is this theory new?

  6. Pingback: Class #1: Monday September 17 « math04fall2012

  7. I just want to mention that I got a Weber fraction of 0.22. It is also true that I have been really good at non-arithmetical math (I mean, proving theorems and stuff like combinatorics and real analysis), and also really poor at arithmetic (this meant that I would often score low simply because of poor calculations as we don’t have calculators at school in my country; and hence had to be real good at the non-arithmetical stuff so that I could spend much more time during the exam on the arithmetic).

    • I just noticed the accuracy vs response time issue – it would happen with me that even after spending a lot of time on calculations in the exam, I would still make plenty of mistakes! So, my strategy was to repeat calculations and cross-check them by solving the same question using different methods.

      • Oh, and just to mention, my response time in the test was 1685 ms (and the poor performance red line was at about 1264 ms!). I guess I am inaccurate AND really slow!

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