And, indeed, the accolades are generally warranted. Although Ma's book reports on work done a decade ago, it is still all too relevant to the problems facing American mathematics education now. Particularly for those not very knowledgeable about elementary school mathematics, there are many valuable insights about the content of elementary school mathematics, the preparation of teachers to teach this mathematics, and the necessary professionalism of teachers if they are to be effective in teaching this mathematics. (In this review I take elementary school to be K-5. I think this is Ma's definition, too, but I don't think she is explicit about this.)

Each of the first four (of seven) chapters deals with a particular subject area of elementary school mathematics (subtraction with regrouping, multi-digit multiplication, division by fractions, the relationship between perimeter and area) and discusses the approaches to each topic by a group of 23 American and 72 Chinese teachers. Through interviews with these teachers, Ma elicited information about the knowledge of the teachers, their approach to teaching the topics, and their abilities to deal with the problems that students have with these topics. Her results may be quickly summarized as showing that on just about every aspect of the teaching of these topics, the Chinese teachers were far superior to the American teachers, with the best Americans in the group barely as good as the weakest Chinese. The single most startling finding is this: When asked to compute 1 3/4 divided by 1/2, only 9 of the 21 American teachers who tried this computation achieved the correct answer, whereas all 72 Chinese teachers did it correctly. Given this result, the most disturbing thing in the book is some data in an appendix that show the generally high degree of confidence that the American teachers had in their own mathematical knowledge. (Of course, Ma's sample is too small to be representative. Nevertheless, I am persuaded that what she reports would generalize to almost all of American education.)

The subsequent two chapters discuss what Ma calls PUFM: Profound Understanding of Fundamental Mathematics.
In the first of these chapters she discusses what PUFM is, and in the second, how it is attained. Although many
Americans believe that elementary school mathematics consists of nothing more than a lot of rules to be learned
(``the algorithms of arithmetic"), surely no reader of this * Monthly* doubts that there is much more to
arithmetic that should be learned by children in elementary school: not just the how but the why, not just individual
procedures but the connections between procedures and the concepts
underlying them, and the value of viewing problems from multiple perspectives. In order for elementary school
teachers to teach these things, they need PUFM that embodies ``an understanding of the terrain of fundamental
mathematics that is broad, deep and thorough". In particular, teachers at each grade level need to understand what
has gone before and what will come after the mathematics they are teaching. Ma requires high standards before a
teacher may be said to have PUFM. Only 8 of the Chinese teachers and none of the Americans qualified; but perhaps
the more important implication is that most of the other 64 Chinese teachers, but very few of the American ones, were
close to attaining PUFM.

In a final chapter (``Conclusions"), Ma suggests what might be done to improve the quality of elementary school teaching in the U.S. She focuses on teacher education as a way to break the ``vicious circle formed by low-quality mathematics education and low-quality teacher knowledge of school mathematics". She says that, "In the United States, it is widely accepted that elementary mathematics is `basic', superficial, and commonly understood ... Elementary mathematics is not superficial at all, and any one who teaches it has to study it hard in order to understand it in a comprehensive way." I'm sure she is correct for Americans broadly but to the extent that she is also correct for the schools and colleges of education that prepare elementary school teachers---and I suspect she is---this is a damning indictment.

This book, then, is well worth reading by anyone interested in elementary school mathematics education. But, in fact, if you are already knowledgeable about the teaching of elementary school mathematics in American schools, there is not much you will learn that you didn't know already, although Ma's lucid presentation itself makes her book a good read. Indeed, the rhapsodic reviews by various research mathematicians [1], [2] can only be explained, I think, by a hidden agenda, one element of which I touch on presently. Often, as Roitman [3] has noted, these reviews "give the unfortunate impression that Ma's is the only work on education that mathematicians need to read." For example, if you don't know that ``the math methods courses that U.S. colleges offer \ldots are light on content" [1] or that the textbook publishing industry, driven as it is by the big bucks available from adoptions in major states, produces some pretty dodgy elementary school mathematics texts, then you haven't been watching what has been going on in U.S. education for decades now.

One of the weaknesses of Ma's book---and something no review I have seen has remarked upon---is that there is nothing in it about the intellectual quality of the American teachers themselves. Were their failures of understanding and approach to the topics discussed in the first four chapters just due to deficiencies in their education and in the texts they used or were these teachers generally not of the intellectual quality we should wish for elementary school teachers? Ma doesn't really try to answer this question. She tells us that the American teachers were considered ``better than average". (Perhaps they were, but if so, we should be all the more worried about the quality of American elementary school teachers.) Anyhow, aside from the fact that some attended a summer mathematics program and the others were involved in a graduate intern program, Ma presents no data to bolster this claim. No doubt such data would have been hard to obtain, but this is a crucial issue to be so neglected.

Indeed, it is an open secret that the quality of elementary school teachers has been declining for three decades or more. (Of course, there are many individual exceptions.) The reasons for this are clear. One is that, as professional jobs have opened up to women generally---and elementary school teachers are still mainly women---the most able women, who used to have few choices beyond teaching and nursing, seldom choose elementary education as a career. This is not just because the financial rewards are far below what can be earned in other spheres, particularly for those who are mathematically talented. It is also because American schools have become increasingly unpleasant workplaces as discipline problems and violence, even in the elementary grades, have become more prevalent. In any case, no matter how good curricula and textbooks are and no matter how much we try to infuse the ethos of PUFM into current and prospective teachers, the overall situation in elementary school mathematics won't improve nearly as much as is needed unless we can attract mathematically talented people to elementary school teaching.

There's nothing much mathematicians or mathematics educators can do about teacher's salaries or about discipline and violence. But there is one important thing that can be done, and Ma's book points directly at it, although reviewers have mentioned it only in passing. You might think from what I've written already that Ma's book is mainly a comparison of American and Chinese methods of teaching mathematics, but it's really much more than that. As Ma notes, most American teachers have a bachelor's degree. (She isn't explicit about the 23 teachers in her study but it is certainly implied that they all have bachelor's degrees.) By contrast, Chinese teachers typically leave school after ninth grade and then have "two or three more years of schooling in normal schools." But the crucial point is this: The Chinese teachers teach only mathematics, that is, they are mathematics specialists, while, like virtually all American elementary school teachers, the American teachers teach all standard subjects to their one class each year.

This, to me, is by far the most important lesson from Ma's book. Unless and until American colleges that educate teachers develop strong programs for elementary school mathematics specialists and schools use these specialist teachers from first grade (perhaps even kindergarten) on, there is essentially no hope of making major improvements in American school mathematics education. Indeed, using mathematics specialists in elementary school and paying them appropriately is probably the only way we can attract the mathematically talented people needed in our elementary school classrooms.

Most readers of this *Monthly* are probably wont to complain about the mathematical preparation of the students
entering their institutions. They often blame this on the high schools. And, indeed, no one would say that there are
no problems in American high school mathematics. But the problems with elementary school mathematics are so
much greater that, by the time students reach high school, many of them are so far behind where they should be that
even the best high school education will not overcome this deficit. If students leave fifth grade with, at best, only
proficiency in pencil-and-paper calculation of the four arithmetic functions, it will be very difficult for all but the best
to overcome their mathematical deficit later.

Middle school, grades 6--8, is another matter. Despite some signs of improvement in recent years, sixth and seventh grades and sometimes even eighth grade are more often than not still mathematical wastelands in which little new material is introduced and almost all the time is spent reviewing what was supposed to have been learned earlier. The excuse for this, when one is given, is often that these are bad years to push kids too hard because they are years when kids spend most of their time thinking from the neck down. The remarkable thing about this argument is that it seems to assume that uniquely in the United States do children go through puberty. The disastrous consequence is that in just those years when kids are most able to absorb lots of new knowledge, they are given very little to absorb.

You will find halfway houses on the road to specialist mathematics teachers in some American elementary schools today. These schools have a teacher designated as a mathematics specialist whose task is to help other teachers with their mathematics teaching. This is a useful idea but it will not make very many good teachers out of inadequate ones. Another suggestion [2] is to partition the current cadre of elementary school teachers into specialists, who would teach only mathematics, and the rest. This is a good idea that could be implemented immediately, but, sadly, there appear to be very few elementary school teachers who would qualify as mathematics specialists.

Of course, the use of specialist mathematics teachers in the elementary grades would represent a major change in a culture in which children have a single teacher for all subjects except for things such as music, art, and gym that are taught once or twice a week. It is often argued that young children need the security of a single teacher who is with them in the classroom except for very occasional subjects elsewhere. There is much to be said for this argument, but it cannot be controlling in view of the crisis that affects American mathematics education. If elementary school children had a different teacher once each day for mathematics, that teacher could come to them rather than having the children change classrooms as with music and gym. This would not be a significant upset. In any case, the bottom line is that, until specialist mathematics teachers are integrated into U.S. education, the U.S. will continue to lag behind other countries in international comparisons. Or perhaps there will be improvement in U.S. results in international comparisons because increasing focus on ``high stakes testing" will result in teaching narrowly aimed at these tests that seldom, if ever, assess significant mathematical learning. In any case, if specialist mathematics teachers are not soon a fixture in U.S. elementary education, all the internecine battles between reformers and traditionalists will not matter much.

The increasing emphasis on standardized, high-stakes testing in American schools is perhaps the most insidious aspect of the current American educational scene. Not only is there no reason whatever to believe that such testing will result in improved mathematics learning, but in addition, it deprofessionalizes teachers by implying that they are not professional enough to assess their students themselves and report appropriately to parents. Who would wish to become a school teacher in such an atmosphere if there were any alternative?

Although Ma's book has been endorsed by reformers, it is the traditionalists, those who wish to reinstall the curriculum that was (supposed to have been) taught in yesteryear, who have most lavishly praised it. One reason for this may be that many traditionalists are just out of touch and think that Ma's truths are new ones, not recognized by the mathematics education community. But there is another reason for the popularity of Ma's book with the traditionalists: Technology, calculators particularly, play no role whatever in the study described in her book. Since many traditionalists believe that calculators rot the brain and are at the heart of the dumbing down of the American mathematics classroom that they (rightly) perceive, an approach that appears to avoid any use of calculators is very attractive to them. Never mind that there is little or no evidence that the use of calculators in schools harms students' mathematical attainment. Never mind also that there is no reason to suppose that the use of calculators need result in a less demanding curriculum [4]. And, finally, never mind that children taught traditional ways of doing arithmetic may wonder why almost no one outside the classroom does arithmetic this way. If you've decided that traditional ways of teaching arithmetic are the only---or anyhow the best---way to teach it, then Ma would seem to be your ally.

But, in fact, there is nothing in Ma's book that could be read as opposing (or supporting) the use of calculators in elementary school classrooms. Her research focused on answering questions about how well teachers understand fundamental mathematics, not about the use of calculators in classrooms. Indeed, it was probably inevitable that Ma's study included no calculator component. Although calculators are not nearly as common in American elementary classrooms as some traditionalists would have you believe, they are certainly much more common than in Chinese classrooms. There I would guess that their use is rare, if only because the cost of calculators, while relatively trivial in the U.S., is not so in China. Issues related to the use of technology in classrooms are simply orthogonal to Ma's research.

Therefore, Ma's book cannot and should not be read as supporting any particular mode of instruction in arithmetic. Instead it should be read as an incisive call for teaching by well-educated teachers who have a profound understanding of the subject matter and who have thought long and hard about the best way to present this subject matter. Indeed, no one like myself who favors calculator use throughout elementary school has any doubt that any mode of teaching arithmetic by such teachers would lead to far better results than are usually attained in American classrooms today.

This book is important, then, not because it espouses a particular curriculum for elementary school mathematics (it doesn't), but because it espouses an approach to teaching and teacher education that is crucial for success in any curriculum. That Ma's book is championed mainly by those with a particular curriculum axe to grind is doubly unfortunate because Ma's message is being distorted and because her most important message, that Chinese elementary school teachers are specialists whereas American teachers are not, may well be lost in the cacophony of the Math Wars.

Acknowledgement: Thanks are due to Judy Roitman and Ed Dubinsky for comments on a draft of this review.

[1] Richard Askey, Knowing and teaching elementary mathematics, Amer. Educato, (Fall 1999) 1-3, 6-8.

[2] Roger Howe, Knowing and teaching elementary mathematics, Notices Amer. Math.Soc., 46 (1999) 881-887.

[3] Judith Roitman, Review of Knowing and Teaching Elementary Mathematics by Liping Ma, AWM Newsletter, (Jan.- Feb. 2000), 20-21.

[4] Anthony Ralston, Let's abolish pencil-and-paper arithmetic, J. Comp. in Math. and Science Teaching, 18 (1999), 173- 194; also available at http://www.doc.ic.ac.uk/~ar9/abolpub.htm.

Department of Computing, Imperial College, London SW7 2BZ ; E-mail: ar9@doc.ic.ac.uk