The Number Sense: How the Mind Creates Mathematics by Stanislaus Dehaene, Oxford University Press 1997
This book, written by a noted neuropsychologist, explores the new field of mathematical cognition. That is, it attempts to root our understanding of the development of mathematics in the biology of the brain. It is one of those rare books written by a pioneering researcher in a scientific field who is also an excellent writer – in English as well as presumably in his native French. I think it is particularly valuable for those of us in education, because in order to teach mathematics we must understand how children actually acquire mathematics. While there is much to learn here, I also found much to disagree with, and I will deal with these points below. Perhaps the major drawback to the book may be its date of publication, since Dehaene indicates that the ten years following the writing of the book promise to be a time of unparalleled scientific advance in the field.
The book is organized into nine chapters:
Chapter 1, “Talented and Gifted Animals”, discusses scientific research that shows that many animals have innate primitive arithmetic skills, which enable them to add, subtract, and compare small integers. Calculations and comparisons of numbers become less accurate as the numbers involved increase beyond three.
Chapter 2, “Babies Who Count”, sets forth the contention, supported by ingenious research, that shows that, similar to animals, human babies as young as a few days old also have innate arithmetic skills, enabling them to understand and manipulate small integers.
Chapter 3, “The Adult Number Line”, discusses the conception that human adults have of number. Much of this chapter has to do with discovering the extent to which we can manipulate numbers very quickly, that is, without visible thought.
Chapter 4, “The Language of Number”, discusses the ways different cultures name numbers, and the effect this has on calculating abilities.
Chapter 5, “Small Heads for Big Calculations”, applies the results covered in the previous chapters to the difficulties of teaching arithmetic to children.
Chapter 6, “Geniuses and Prodigies”, presents case studies of a number calculating prodigies and mathematical geniuses, and attempts to show that their abilities are not different in kind from that available to any intelligent adult.
Chapter 7, “Losing Number Sense”, discusses the relationship between brain function and number sense as revealed by studying people who have lost various parts of their number sense due to lesions in particular parts of their brains, or to other brain injury.
Chapter 8, “The Computing Brain”, shows how modern advances in brain research cast light on relationship between calculation and the brain. The tools of positron emission tomography (PET) and electro- and magnetoencephalograpy are described, and some results obtained by applying these tools to mathematical cognition are discussed.
Chapter 9, “What Is a Number?” moves into the philosophy of mathematics. Dehaene tackles questions such as the merits of the formalist, Platonist, and intuitionism theories of mathematics, and the relationship between mathematical truth and reality.
A book of this wide coverage is bound to be controversial. I recommend reading it yourself and making up your mind about some of the controversial issues, but I’d like to bring up a few places where I disagree with the author.
It seems to me that one of the dangers of neuropsychology is that of reductionism, and although Dehaene is a sophisticated thinker I don’t think he escapes this.
I take issue with his apparent assumption, which seems unsupported by data, that ability to perform arithmetic calculations is strongly correlated with the ability to do higher mathematics. Among mathematicians I have known, some excel at arithmetic, some are poor, and many are in between. The type of thinking that is involved in geometry, for example, seems to have little to do with arithmetic ability.
I find particularly problematic his discussion of mathematical geniuses, for several reasons. First, he lumps together the self-taught Indian mathematical genius Ramanujan with autistic super-calculators and idiot savants. To me, this is as if one compared Shakespeare with a pre-typewriter clerk who filled thousands of pages of commercial transactions. Both men may have had unusual ability to produce fast legible handwriting, but we would only call one a genius. Second, Dehaene makes clear that he believes that anyone could be a super-achiever in mathematics or arithmetic if they devoted enough time and effort to the enterprise; that there is nothing special about the brain (or mode of thinking) of the genius. This is speculation, and I prefer the opposite speculation of Oliver Sachs, whose prime-number generating autistic twins seem not to calculate but rather to see the integers “directly, as a vast natural scene” or Ramanujan, who described his own mathematical discoveries as being handed to him by a Hindu god while he slept. Non-believers can imagine that Ramanujan’s unconscious mind allowed him to make his discoveries operating in a way that might be totally different from his conscious mind.
Another oversimplification is Dehaene’s belief that young Oriental students do better than Western students at learning mathematics because the Eastern languages have shorter more user-friendly names for the digits. He seems to not consider the cultural differences that lead Oriental families to value hard academic work more than Occidental families do, which by itself is enough to explain differences in achievement.
In terms of pedagogic implications, Dehaene’s research has led him to the belief that the human brain is not well designed for calculation: “Ultimately, [innumeracy] reflects the human brain’s struggle for storing arithmetical knowledge”. He therefore feels that “by releasing children from the tedious and mechanical constraints of calculation, the calculator can help them to concentrate on meaning.” This is a position I have long shared; however I am now teaching middle-school mathematics teachers, and they mostly report that their students, who have grown up using calculators, are grossly innumerate. Since many algorithms of elementary algebra have counterparts in arithmetic algorithms, these students are not able to progress in algebra. I now advocate getting children to a state of competence in calculation before letting them use the calculator freely. However, I agree with Dehaene on the usefulness of concrete computational representations (manipulatives) in the classroom.
Dehaene gives a good description of the basic theories of mathematical epistemology: Platonism (mathematical objects have a reality, and the mathematician discovers this reality rather than inventing it), formalism (mathematics is about the formal manipulation of strings of symbols following basic laws of logic), and intuitionism (mathematics is a construction of the human mind, so that alien intelligences would create different a mathematics different from the human one.) He comes down for intuitionism, but it seems to me that his dismissal of Platonism is entirely too glib. He asks, rhetorically, “If these [mathematical] objects are real but immaterial, in what extrasensory way does a mathematician perceive them?” I would argue that they are perceived in the same way that we perceive a coherent world from the streams of sense data that enter our brains. We create our mental worlds, and this seems to be true whether or not the basis of the world is “material” or whether is grounded in ideas. Both the material and mathematical mental worlds are subject to laws of internal consistency, and both are subject to judgment by members of a community.