Brian Butterworth's central thesis, which I support, is that virtually all of us possess within our brains a "number module" that is in some sense wired in. I'm less happy with the idea that it is specified by some particular bit of the genome. I would expect it to be built, along with the rest of the brain, by a whole suite of genes, most of which do a lot of other jobs.
Our number module is a "mathematical start-up kit that gives us a rapid intuitive grasp of numbers up to about five". The evidence in its favour is pretty convincing. Curiously regular marks cut into the 20,000-year- old Ishango Bone, found on the shore of an African lake, suggest an ancient ability to handle small numbers. And a series of clever experiments suggest that babies as young as one week possess the start-up kit.
Where The Mathematical Brain and I start to part company is quite near the end of the story - the question of a "talent" for mathematics. We drift further apart when it comes to educational prescriptions. These differences arise because the author stretches his thesis too far, claiming that "Nature, courtesy of our genes, provides the piece of specialist equipment, the number module. All else is training."
Twaddle. This absurd contention rests largely on a study carried out by Anders Ericsson and colleagues, showing that in mathematics, music, chess and swimming, "the highest levels of performance and achievement require around 10 years of intense prior preparation". The results of this study are not in dispute, but the conclusion drawn from them - that "talent" does not exist but is merely the result of solid application - is fallacious.
The reason is a flaw in the experimental design. Take a sample of people who have achieved very high standards in some field, and a sample of people who have not. Find out how much time each spent practising that activity; then compare. The group of high-achievers did far more work than the others.
The mistake is the choice of control group. It should have been people who tried very hard but did not reach high standards. If the theory is right, there shouldn't be any. There are very few, but for a different reason. If you're not getting anywhere, you normally give up.
Nonetheless, such people do exist, suggesting that willingness to devote a lot of effort to a topic may be a consequence of talent, not a cause. More realistically, each can feed off the other. The alleged causal link from hard work to "talent" fits the Protestant work ethic beautifully, and I'm sure that the morons who saddled British higher education with the Teaching Quality Assessment would find it congenial.
When I was at college the students who worked hardest at mathematics were those who did worst. I goofed off most of the time and got a comfortable first; my friend Chris worked 12 hours a day and scraped a third. I'm not decrying hard work - but you need more than that to reach the heights.
As for educational prescriptions, I give one example only: the hoary topic of "times tables". The book's discussion assumes that the only choice is between "rote learning" - endless repetition of something you never understand - and "understanding", in which it is important to know that 6x7 = 7x6 but not necessary to know that the answer is 42.
We've been there, and it doesn't work. Anyone who teaches mathematics is aware that students need a repertoire of basic operations that have been internalised, able to run "in the background" while the thinking part of the brain attends to the important issues.
The multiplication table happens to be one of them. It's a pity that it became an icon of back-to-basics right-wing twerps; despite that, it's important. Small children have good memories and love repetitive chanting; "six sevens are forty-two" is great fun and soon becomes indelible. Later, they can understand what it means. Rote learning and understanding are not mutually exclusive. Used judiciously, they reinforce each other.
The majority of this book, though, is gripping stuff. There's some overlap with Stanislas Dehaene's The Number Sense, and I would have preferred more about the brain and less about the history of number systems. However, The Mathematical Brain is a book to be proud of, and if I had agreed with everything in it, I would have been really worried.
The reviewer is professor of mathematics at Warwick University