Besides an overly enthusiastic usage of the phrase "in the final analysis," this book is near-flawless. Thorough, accurate, insightful, useful, even damned funny in parts . . . call me a Dehaenophile through and through!

It will appeal most to those interested in the intersection between neuroscience and math education. ( )

great fascinating material, cool experiments, not great writing though. ( )

On average, Chinese language speakers can remember more numbers because their WORDS for numbers are far shorter (10 vs. 7 for English speakers). Animals, including rats, pigeons, raccoons, and chimpanzees, can perform simple mathematical calculations, and human infants have a rudimentary number sense. How much is learned, if not all, and how is it learned optimally? ( )

This book's subtitle (How the mind creates mathematics) is a clear description of the leitmotif of the book: to understand the neurological basis of elementary mathematical calculations. The author is a cognitive neuropsychologist (with a first degree in mathematics) and his main thesis is that Evolution has endowed humans (and other higher animal species) with an innate ability for intuitive counting which, coupled with the human capacity for language, is the basis of the unique mathematical capacity of the human species. In support of this thesis Dehaene amasses an extraordinary variety of evidence, namely a number of very inteligently designed animal experiments, as well as psycologist's tests with humans, even amazing experiments with babies as young as five months that clearly established the erroneous nature of some aspects of Piaget's construtivist theory of child development. These, together with evidence from modern brain imaging techniques and clinical data about several types of brain lesions, helps to build a very compelling case about the physiological mechanism behind the human ability to do mathematics. In the last chapter, Dehaene allows himself a more philosophically minded speculation about the implications his analysis and conclusions have for pedagogical matters, as well as for the philosophical debate among Platonists, Formalists, and Intuitionists on the foundations of mathematics, with some surprisingly reasonable arguments in favor of (a mild version of) intuitionism. Summing up: reading this book was a wondrous experience and I am sure I will often return to parts of it in the future. Nobody interested in Mathematics, its teaching, or the mechanisms of brain functioning, should miss this book! ( )

Dehaene is a psychologist specializing in the neurobiology of mathematical acquisition, his book is a record of many of the facts that have been discovered concerning the way in which people learn mathematics, they way they organize its ideas in our minds, the way math is retrieved from memory. (All of these are approached from the point of view of neurobiology.) At its most basic level, our sense of mathematics is very little advanced beyond that of many animals, who share with us a precise sense only of the numbers 1, 2, and 3; beyond this is a roughly-reckoned haze of numeric quantities. Dehaene compares our mental conception of number with an "accumulator" with approximate graduations allowing us to give rough estimates of large quantities, but which fails to give precise values for these same quantities.

A few snippets:

1. Even as soon as a few days after birth, babies are able to discern between the numbers 2 and 3. (See p. 50.)

2. We (adults included!) are susceptible to "the magnitude effect": it's harder for us to discern the difference between 90 objects and 100 than it is the difference between 10 objects and 20. Various factors (symmetry, density, etc.) militate and mitigate this effect. (See pp. 71 ff.) Similar to this is the "distance effect," positing that it's more difficult to discern quickly between 8 and 9 than it is between 8 and 19.

3. Studies show that when asked to compare numbers, such as 5 and 7, and state which is the larger, instead of behaving reflexively and answering based upon our knowledge that the symbol "7" represents a larger quantity than the symbol "5," we instead convert each of these abstract digits into collections of the requisite number of objects before performing the comparison on these collections. (See pp. 75 ff.)

4. We have a tendency to "compress" numbers as they grow, storing them in our minds as though on a logarithmic scale. One corollary of this behavior is that when asked to provide a random sample of numbers in a certain range, people will tend to elect an overrepresentation of smaller values, as though these were more widely spaced than their larger compatriots. (See pp. 77 ff.)

5. Since adults compute sums and products (for example) by retrieving the resultant quantity from a memorized table, those whose native languages have exceedingly short names for the ten numerals (like Chinese and Japanese) are able to more efficiently memorize the desired sums and products, and so perform much more quickly and with fewer errors than their counterparts speaking other tongues. (See pp. 130 ff.)

These are just a few of the fascinating facts this book has taught me about the development and refinement of mathematical thought processes in and by the human mind. Ultimately, one of Dehaene's primary points is summed up nicely on pp. 118-119: "Although our knowledge of this issue is still far from complete, one thing is certain: Mental arithmetic poses serious problems for the human brain. Nothing ever prepared it for the task of memorizing dozens of intermingled multiplication facts, or of flawlessly executing the ten or fifteen steps of a two-digit subtraction. An innate sense of approximate numerical quantities may well be embedded in our genes; but when faced with exact symbolic calculation, we lack proper resources."

The final chapter considers more philosophical matters, comparing the Platonist, formalist, and intuitionist schools of mathematical thought, and indicating the repercussions these movements have had on (primarily 20th century) mathematics.

This was a positively fascinating read, I recommend it to anyone with an interest in the psychology of learning, or in the philosophy of mathematics. ( )