## Number Sense: What we can’t do? or What we can see

A number of websites have reported on a recent study, that correlated innate number sense with mathematical ability. A concise report of the study can be found in the Johns Hopkins University Gazette, published by the institution where the study was done. The study’s results confirm a correlation between the strength of ones number sense and the development of other mathematical abilities. New to this finding, however, was that participants in the study were young children. None of them had yet had the formal schooling that could obscure the correlation or confuse its interpretation. The article entitled, *You Can Count On This,* says clearly:

According to the researchers, this means that inborn numerical estimation abilities are linked to achievement (or lack thereof) in school mathematics.

A number of things went through my head as I read the report (and responses to it) and I would like to say something about a few of them.

First, I think it is dangerous to express these observations in a way that suggests that one can be born with or without mathematical ability. One of the reports I saw on the study took the results in exactly this direction. The article was called *People Are Born Good or Bad at Maths* and says this:

Now the researchers, who carried out tests on children too young to have been taught mathematics, found that people are either born with a mathematical brain or not, news paper reported.

This is not a view that should be encouraged. And there is good reason to be reserved about the correlation observed. Mathematics is expressed with number, but it is not defined by number. Number sense is not the beginning of mathematics (and hence mathematical ability). It is true that one of our earliest expressions of mathematics happened in the symbolic representation of quantity. But, as we all know, this was not its only early expression. The other was spatial, and took the form of geometry. Only with the persistent mingling of these, could we profit from the vitality we find in every instance of their overlap. This is the energy that brings life to modern mathematics.

It is more likely that mathematics rests on manifold, interwoven, perceiving mechanisms. My hunch is that these are actually reflected in the reasoning that builds mathematical structure. It does seem that the idea of equivalence begins with counting, but equivalence is found in spatial relationships as well (like the simple idea of congruence) and mathematics’ many powerful extensions of equivalence would not be possible without, at least, these two notions.

Mathematics is challenging, sometimes very difficult. Some problems will even seem impossible. It should not be made easier for students, at various levels of instruction, to decide that they’re just not built for it. A teacher of mine once said that doing mathematics relies on ones willingness to look stupid. We don’t understand enough about ourselves, or even about what mathematics is, to think too broadly about a correlation between number sense and some facility with *numerical* ideas. I like the words Richard Courant chooses to describe the basic elements of mathematics (in the well-known *What is Mathematics?)*

Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.

I also want to draw attention to a NY Times report on a related study from a few years ago. This article was given a different kind of title. It’s called *Gut Instinct’s Surprising Role in Math* and describes a reservation on the part of investigators:

The researchers caution that they have no idea yet how the two number systems interact. Brain imaging studies have traced the approximate number sense to a specific neural structure called the intraparietal sulcus, which also helps assess features like an object’s magnitude and distance. Symbolic math, by contrast, operates along a more widely distributed circuitry, activating many of the prefrontal regions of the brain that we associate with being human. Somewhere, local and global must be hooked up to a party line.

In his book *The Lightness of Being*, Frank Wilczek compares the expressions of Einstein’s first and second law. His point is that “different ways of writing the same equation can suggest very different things, even if they are logically equivalent.”

Einstein’s first law is: *E = mc² * His second law is: * m=E/c²*

The first one, Wilczek argues, suggests the possibility of getting large amounts of energy from small amounts of mass (images of bombs and nuclear power plants). The second law, however, suggests that we might be able to understand how mass arises from energy, or that we may have some handle on the creation of mass (which has the attention of high energy experiments going on right now).

In this light, I’d like to end this with another quote from that NY Times article from a few years ago, where researchers are looking at the same pieces of information and seeing something slightly different. They say:

What’s interesting and surprising in our results is that the same system we spend years trying to acquire in school, and that we use to send a man to the moon, and that has inspired the likes of Plato, Einstein and Stephen Hawking, has something in common with what a rat is doing when it’s out hunting for food. I find that deeply moving.

BTW, I loved your reference to the article about plants doing division. Is that how our brains do division?

They’re not that much different: plants are networks of cells, and brains are networks of neurons. If only we had the math to compare them, like Newton inventing calculus by seeing the mathematical similarities between the orbit of Mars and a falling apple.

Are we on the verge of a new type of math? Just a thought…..

Well said. I totally agree with everything you recently wrote. As a public school teacher, I fight the “I’m just not good at math” bullshit every school-year. It is especially discouraging when I hear it from a cheer-leader type who is probably loaded with talent, and who I observe secretly showing enjoyment at problem solving, but doesn’t pursue the subject because of the cultural problem you addressed above.

I got defensive because the articles you quote concern research into what appears to be a genuine math disability akin to dyslexia, accordingly called ‘Dyscalculia.’ A great pop-sci article in Discover Magazine (July/Aug 2013) quotes a leading neuroscientist as saying:

“A lot of people say ‘I’m not good at math’ because they couldn’t handle pre-calculus or something….People with dyscalculia struggle to tell you whether seven is more than five.” (Neuroscientist Edward Hubbard, University of Wisconsin-Madison).

I am impressed with the vast research you’ve done and pondered upon on your site, but I’m not sure you read the articles you quote above? Natalie Angier (the NY Times article you quote above about the rat) is very clear about how much we understand about numerical cognition, but none of that is in evidence in your writing.

For example, you write that math “has emerged from some very fundamental biological processes…” But neuro-scientists investigating Numerical Cognition can be more precise than that:

1. There are numerical abilities we share with animals: e.g., approximating the sizes of two or more quantities (which tree should I climb? Which has more fruit? Which Wildebeast is smaller? Am I an outnumbered baboon?). Brain scans indicate that this basic “number sense” is computed in the same region as the somatosensory system (e.g., temperature, pain, touch and body perceptions. Essential processes for most vertebrate brains).

2. There are math abilities that no animal can do: e.g., arithmetical operations and beyond. Brain scans hint that math more ‘advanced’ than number sense is computed in the same centers as our language.

This explains why students suffering from Dyscalculia can often do algebra, but struggle — and often fail — to correctly make change.

But I don’t mean to be contentious. We are on the same side. As evidence: Dyscalculia affects about as many people as dyslexia, but the latter is studied much more. Corroborating our mutual frustration with living in an anti-math culture:

“One reason for that disparity [between dyscalculia and dyslexia research] may be the belief that literacy is more important than numeracy. “People freely admit at dinner parties that they are poor at math, while few would admit that they are a poor reader….] (Daniel Ansari, quoted in article above).”

Finally, I totally agree that math is about more than numbers. But if a kid can’t add or subtract because of a disability, s/he is much less likely to enjoy calculus later on. The sooner we can recognize a disability, the sooner we can develop interventions to overcome it. But to recognize a disability, you have to admit that it exists: i.e., a particular child is born with strong number sense, or else they are not. That’s how science progresses. No?

Thanks for your thoughts. It’s important to me that I not be misunderstood. I have spent a good deal of time making the case that mathematics has emerged from some very fundamental biological processes – perceiving, navigating and learning processes – by looking at studies on number sense, the perception of magnitude, the patterns in food foraging behaviors, the patterns in how the eye searches, even arithmetic operations that appear to be happening in plants. Studies provide data. The interpretation of the results of a study relies, to some extent, on understanding how the data was collected, as well as how the data can be understood. I don’t want to suppress findings, nor do I want to irresponsibly discredit them. I only wanted to make the point that given the general distaste for mathematics that exists today, reports on these studies should be fairly precise. I have little doubt about the relationship between number sense and mathematical ability. But mathematics is not only about numbers. And the “I’m just not good at math,” response to the difficulty that one might have learning it is doing a lot of harm. We know very little about how mathematics has actually happened, what it is, where it comes from, or why it is so effective.

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As a public-school math teacher, I totally understand your hesitancy to ascribe mathematical abilities as ‘innate.’

However, as a former chemist I am disturbed at the language you use to describe your reaction(s) to the research. Your writing strikes me as the views of a theologian or politician rather than a mathematician.

For example, in reference to the article ‘People Are Born Good or Bad at Maths,’ you wrote “This is not a view that should be encouraged.” Should it be suppressed? Are the results of this paper a moral issue? Scientifically speaking, the results are not ‘views,’ they are empirical findings. Like proving theorems, empirical findings are either true, or they are false. Either math ability is innate, or it is not. I realize that comparing psychological findings to the rock-solid axioms of mathematics is tenuous, but your references to Einstein lead me to think you understand the relationship of a theorem to a theory.

Further, you write “it is dangerous to express these observations in a way that suggests that one can be born with or without mathematical ability.” Again, are you saying that some scientific observations should be suppressed, while others should be allowed? Violent criminals are dangerous; guns are dangerous; hydroflouric acid is dangerous; we are speaking here of scientific research.

Anyway, I’d like to continue this discussion. You write very well and have very important things to say. But I want to see if you’re interested in continuing the discussion before I spend anymore time on it!

I agree that it would be very difficult to sort these things out, but what’s called an ‘approximate number sense’ is a purely visual or auditory discernment of the difference between say 1, 2 and 3 things. It has been observed in very young children and in animals other than human ones. I think there is value in looking at how an intellectual activity is tied to biological mechanisms,and even to other creatures. It can put many things in a new perspective. I also think it’s important, however, not to use these ideas to separate people according to some ill-conceived judgment of their abilities.

The article about the study mentions that the children involved were too young to have significant formal math training. I don’t know that that has any bearing on the outcome of the study. Parents guide their children’s learning from a very early age.

I’m willing to bet I would have performed better on the test as a child than some of my peers because of the “games” my father would play with me. He would give me a pile of buttons and have me sort them and count them and then combine groups and take away groups. I was probably 3 or 4 years old at the time.

We, as humans, learn so much during in our first few years of life, before we ever enter a classroom. Just because children don’t receive formal training in math, you can’t make the assumption that their knowledge and abilities are innate. There’s a lot children are exposed to that will help them “learn” number sense indirectly.