I'm going to assume anyone reading this has already read the other reviews in amazon.com. This leaves me free to comment on problems with the book rather than provide a synopsis.

The first two sentences in the preface to "What Counts" explain the basic fact, I am not particularly good at maths or calculation."

Butterworth proves this often enough for it to be a very good reason why he shouldn't have written of flaws, only someone who has no feel for mathematics could write a book containing many typos of the form a^2 + b^2 = (a - b)(a + b).

o He's discovered a new and amazing correspondence with any subset that is neither the whole set nor the empty set." Imagine, there's a one-to-one correspondence between the integers and the set {0,1}. Well, no there isn't.

o He's made the equally exciting discovery that the rationals between 0 and 1 are uncountable. It is revealed on page 339 that the points on the real line are uncountable "because there is points." Since the argument applies to the rationals, they too must be uncountable. Sigh.

Here are some specifics to illustrate other problems in "What Counts".

o The discussion of cognitive archaeology is highly speculative and frequently unconvincing. For example, he speculates that counting lunar phases is important to women so they'll know when their baby is due. This isn't of value without a citation of "primitive" peoples who do this.

o Butterworth seems to believe that math is the same as arithmetic, though of course he does know better. The book is almost exclusively about our "natural ability" to add, subtract, multiply, and divide. Geometry, the other "basic" mathematics, is almost completely ignored. The omission is a major deficiency.

o He also has a very strong opinion that there is no such thing as a mathematical gift. Rather, it's a manifestation of interest, good teaching, and hard work. The argument is made quite intensely, but not convincingly, and probably would almost universally be disputed by mathematicians (which doesn't prove it wrong, of course). What is convincing and should have been the point of the discussion is that we could be doing a much, much better job of teaching mathematics. (The previous reviewer has correctly pointed out the value of Butterworth's critique.)

o The appendix contains a less-than-satisfying discussion of Godel's Incompleteness Theorem, which has no apparent purpose other than to dazzle and confuse the naive reader.

There's quite a bit more that's objectionable, but the point should have been made adequately with this list.

On the other hand, the quote from Oliver Sacks on the dust jacket about how the book "solicits the reader's own thoughts" is correct. I came away from the book with ideas for dozens of experiments and possible research areas. Of course, since my background is mathematics and not a cognitive neuropsychology, I can't comment the non-mathematical assertions but can only hope them to be accurate.

The book is valuable as it has nuggets of great interest and the subject matter is fascinating. There aren't many popular books covering this material, so I'm giving it 3 stars. Good editing and minor collaboration with someone who is "good at maths" could turn it into a 5 star book

"What Counts" is a necessary rebuttal to the idea of mathematical giftedness or genius in general which pervades our culture and manifests itself in Hollywood movies like "Good Will Hunting" and, more tragically, in our system of education. The author confronts both of these issues in detail.

For example, on Hollywood's prodigy Will Hunting he challenges anyone to come up with a real life example of this character which would be a counter example to his premise which states that higher mathematical learning/ability is a result of zeal, hard work (10 years for truly great achievements), and exposure to the necessary culture, i.e. teachers and books.

As Butterworth explains, Will Hunting seemingly has no zeal for anything but girls and spends most of his time in bars yet he knows all about and comprehends arcane mathematical concepts and myriad other subjects.

Mathematicians may like to hang on to the idea of their own giftedness for the sake of their egos and most people who see "Good Will Hunting" think the character is believable so this book is a definite challenge to a popular myth.

Except for the chapters dealing strictly with mathematics which are not necessary (and hence the lack of 5 stars) this book may inspire people to work hard instead of making excuses.

Look for more on this subject from author/mathematician Keith Devlin with his book (coming out in August) "The Math Gene: Why Everybody Has It, but Most People Don't Use It."

In this highly readable book, Prof. Brian Butterworth (a cognitive neuroscientist at the University of London) argues persuasively for a new comprehension of the development and exercise of mathematical ability. Proponent of a separate center for mathematical intelligence, Butterworth nevertheless argues that the existence of a biological 'numerical center' means that nearly everyone has the capacity to become highly proficient at mathematics and mathematical thinking. Especially interesting to me was his demonstration of the futility of rote learning--and his trenchant dissection of the educational causes of most people's mathematical anxieties and related math difficulties... I've read widely on this topic, and have heretofore remained unenlightened. In addition to advancing a new basis for the way we must view math skills and teach them, Butterworth writes cogently and compellingly, adducing powerful evidence for his findings from provocative new research. This is an optimistic book. It makes clear that, Hollywood be damned, Will Hunting lives in all of us.