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.