It's hard to improve on THE TEACHING GAP, by James Hiebert and James Stigler, if what you want is a brief, clear, and compelling explanation of how and why Japan does so much better than America in educating its children in mathematics.
Chucklesbythebay is quite right: the Japanese have a qualitatively different (and mathematically more satisfactory) understanding of the whole enterprise. But there's more to it than that, unfortunately; for one thing, they also have a vastly more qualified teacher corps.
Why is this? It's only partly because teachers of mathematics in Japan are (relatively speaking) better paid than their American counterparts. It's also because Japanese mathematics teachers work in a system that requires them continually to refine their pedagogical skills -- and it provides them both the time and the means to do so.
This system is called "lesson study," and Hiebert and Stigler outline its main features in THE TEACHING GAP. (A more detailed exposition is available in the fascinating omnibus volume, TEACHING AND LEARNING IN JAPAN -- and, of course, in the landmark work KNOWING AND TEACHING ELEMENTARY MATHEMATICS, by Liping Ma.) Hiebert and Stigler also offer suggestions about how Japanese-style lesson study might be imitated successfully here in the US.
But the trouble, as far as I can see, is that the sort of pedagogy the Japanese practice in their math classrooms requires a teacher corps that really knows its stuff -- which, sadly, is just plain not what we now have here in the US.
Deborah Ball of the University of Michigan has documented the awful truth: almost fifty percent of the (hundreds of) Midwestern math teachers she studied were unable to name any number between 3.1 and 3.11. And Liping Ma's book gives discouragingly persuasive evidence that most American elementary and middle school math teachers cannot divide 1 3/4 by 1/2. We have lots and lots of math teachers out there who simply do not know their subject.
Of course it makes no sense to blame the teachers for this, for where were they to have learned it? The very same system that is now providing us such unsatisfactory results also educated the teachers to whom we're entrusting our kids.
It might seem that the answer to this problem is a vigorous program of professional development. Perhaps if we train our present teacher corps agressively enough, we can break the vicious circle of incompetence and frustration in which they, and our children, now find themselves trapped.
But the task is vast, and even if we could find some way to pay for it, we'd still be stymied. Why? Because we have far fewer qualified professional development personnel than we need for the job. Good professional development work is very subtle and difficult, and those practitioners who can effectively explain and demonstrate how to teach math "a la Japonaise" are extremely few indeed.
Moreover, as Chucklesbythebay notes, there is also the pressing need to change the whole American attitude toward mathematics -- the very definition of the word in our national discourse -- which, alas, promises to be extremely hard work.
To a mathematician like myself, it's obvious that mathematics is an art form, like music, poetry, and painting. And as such, it calls for much more than mere technical mastery. A pianist who succeeds only in hitting all the right notes accomplishes nothing, for art is not virtuosity. We want to be moved and enlightened by a musical performance, and this can only happen if its author has invested it with that peculiar mix of feeling and understanding that alone can reveal meaningful artistic truths.
Although it requires great discipline, art is, at bottom, a kind of sophisticated play. And that is exactly how mathematics is viewed by teacher and student alike in the Japanese elementary school classroom. It's how we must all learn to view mathematics here in America, too, if we wish to become competitive with the best the world has to offer.
