The fact that we have such a biologically determined representation of number in our brain has many important consequences that I have tried to address in the book. The most crucial one is, of course, the issue of how mathematical education modifies this representation, and why some children develop a talent for arithmetic and mathematics while others (many of us!) remain innumerate. Assuming that we all start out in life with an approximate representation of number, one that is precise only for small numbers and that is not sufficient to distinguish 7 from 8, how do we ever move beyond that "animal" stage? I think that the acquisition of a language for numbers is crucial, and it is at that stage that cultural and educational differences appear. For instance, Chinese children have an edge in learning to count, simply because their number syntax is so much simpler. Whereas we say "seventeen, eighteen, nineteen, twenty, twenty-one, etc..", they say much more simply: "ten-seven, ten-eight, ten-nine, two-tens, two-tens-one, etc."; hence they have to learn fewer words and a simpler syntax. Evidence indicates that the greater simplicity of their number words speeds up learning the counting sequence by about one year! But, I hasten to say, so does better organization in Asian classrooms, as shown by UCLA psychologist Jim Stigler. As children move on to higher mathematics, there is considerable evidence that moving beyond approximation to learn exact calculation is very difficult for children and quite taxing even for the adult brain, and that strategies and education have a crucial impact.

Why, for instance, do we experience so much difficulty in remembering our multiplication tables? Probably because our brain never evolved to learn multiplication facts in the first place, so we have to tinker with brain circuits that are ill-adapted for this purpose (our associative memory causes us to confuse eight times three with eight times four as well as will eight plus three). Sadly enough, innumeracy may be our normal human condition, and it takes us considerable effort to become numerate. Indeed, a lot can be explained about the failure of some children at school, and about the extraordinary success of some idiot savants in calculation, by appealing to differences in the amount of investment and in the affective state which they are in when they learn mathematics. Having reviewed much of the evidence for innate differences in mathematical abilities, including gender differences, I don't believe that much of our individual differences in math are the result of innate differences in "talent". Education is the key, and positive affect is the engine behind success in math.

The existence of mathematical prodigies might seem to go against this view. Their performance seems so otherworldly that they seem to have a different brain from our own. Not so, I claim � or at the very least, not so at the beginning of their lives: they start in life with the same endowment as the rest of us, a basic number sense, an intuition about numerical relations. Whatever is different in their adult brains is the result of successful education, strategies, and memorization. Indeed, all of their feats, from root extraction to multidigit multiplication, can be explained by simple tricks that any human brain can learn, if one were willing to make the effort.

Here is one example: the famous anecdote about Ramanujan and Hardy's taxi number. The prodigious Indian mathematician Ramanujan was slowly dying of tuberculosis when his colleague Hardy came to visit him and, not knowing what to say, made the following reflection: "The taxi that I hired to come here bore the number 1729. It seemed a rather dull number". "Oh no, Hardy", Ramanujan replied, "it is a captivating one. It is the smallest number that can be expressed in two different ways as a sum of two cubes."