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Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States

Liping Ma
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
P. N. Ruane
, on

This is the 10th anniversary edition of a book that was first published in 1999. During this period, 70 000 copies of the book have been sold, and it seems to have caused a resumption of what is referred to in the American media as ‘the math wars’. This sort of pedagogical controversy is not new, and it is certainly not confined to the USA. Moreover, such disputes long precede the research upon which this book is based.

Anyway, the changes incorporated in this edition include an updated preface and a revised introduction, but there have also been additions to the appendices. Otherwise, the contents of the book are essentially unaltered, and they were thoroughly described in two excellent reviews of the first edition. Those reviews also place Ma’s research findings in the general context of the problems facing mathematics education in the USA and they can be retrieved by means of the following links:

The central message of Liping Ma’s book is that, compared to their American counterparts, Chinese teachers have a deeper and more coherent knowledge of elementary mathematics, which will be reflected in the relative mathematical achievement of Chinese and American children. But the book focuses mainly upon teachers’ knowledge of, and attitudes towards, elementary mathematics. In this context, we are told that Chinese teachers operate under much more favourable conditions than teachers in US elementary schools. Two principal differences are as follows:

  1. Chinese teachers are specialists, and teach only mathematics, but American teachers, being generalists, are required to teach a range of different subjects.
  2. In China, mathematics teachers are allocated a good amount of time for preparation of lessons, and they operate within a national curriculum that offers clear guidelines and relevant teaching materials. Conversely, U.S. teachers spend the whole day in front of a class. Working in relative isolation, they tend to rely too heavily upon commercially produced textbooks.

It might seem, however, that Chinese teachers are disadvantaged by the fact that their training involves far less time in higher education. Typically, they do not even attend high school and receive only two or three years of teacher training. On the other hand, American teachers will have finished high school, completed a Bachelor’s degree and then undertake additional teacher training. However, measured solely in terms of understanding of elementary school mathematics, Chinese teachers came out best in Ma’s research.

In relation to point 2 above, I would imagine that the provision of textbooks for elementary schools in the USA is a hugely profitable commercial venture, which proves to be a constraining factor with respect curriculum development. Another stultifying phenomenon is the long-standing internecine tension between progressives and traditionalists.

The extreme traditionalist view is that knowledge should be didactically transmitted to obediently attentive pupils, who engage in closed-ended practice exercises and rote learning. The ultra-modernist believes that pupils should learn at their own individual pace, and partly by investigative activities, with the teacher being the facilitator of such a process.

The balanced view is that, within a well planned and adequately resourced curriculum, mathematics lessons should incorporate a range of teaching styles according to what is being taught. Lessons should include a mixture of exposition, and individual and cooperative activities. There should be discussion of pupils’ ideas, including their errors and misconceptions. Some lessons will be relatively formal and others will be less so, but activities should be challenging and practice exercises should, where appropriate, include an investigational element. Each lesson, or lesson sequence, should have definite aims and objectives based upon conceptual analysis of the relevant mathematics.

From this book it appears that this ‘balanced view’ is reflected in the approach to mathematics teaching in Chinese elementary schools — although the lessons are referred to as ‘carefully prepared lectures.’ Furthermore, in Chinese schools, mathematics lessons occupy about 60% of the teaching day, which I believe to be far in excess of the amount of time spent on this subject in British and American schools.

It seems to me that the research undertaken by the author was, in fact, more of a pilot study, because it is based upon the outcome of interviews with only 23 teachers from elementary schools in the USA and 72 teachers from Chinese schools. The American teachers were classed as above average, but the Chinese teachers were of more varied ability. They came from urban and rural schools that ranged from stronger to weaker.

So, given that the populations of the USA and China are respectively 330 million and 1.34 billion, Ma’s sample groups are exceedingly small. But her conclusions seem all the more plausible due to the differences in the working conditions between American and Chinese teachers (described above). Their credence is also enhanced by the fact that children from many Asian countries attain higher mathematical standards than those in the USA and various European countries. (But his can be partly explained in terms of cultural differences as outlined, for example, in the link

The core of this book is contained in its first four chapters, which are respectively devoted to the analysis the teachers’ mathematical understanding of the following ‘problems’:

  1. 52 – 25.
  2. 123 x 645.
  3. 1¾ ÷ ½.
  4. A hypothetical situation, which I paraphrase as follows: ‘One of your students comes to class very excited. She tells you that she has discovered that, as the perimeter of a closed figure increases, so does its area. How would you respond to this student?’

Each of the chapters includes transcripts of interviews with Chinese and American teachers, and their responses are carefully analysed. Overall, the conclusion is that the group of Chinese teachers easily outperformed those from the USA. This was in respect of both personal mathematical understanding and understanding as to how these ideas might be taught.

To this extent, the book successfully represents the research undertaken by its author, and the last three chapters place these issues within a more general discussion of teachers’ knowledge of what Ma calls ‘Profound Understanding of Fundamental Mathematics’ (PUFM).

For readers who are not specialists within the realm of mathematics education, the notion of PUFM may seem to be innovative, but this is not so. For example, the book by Richard R. Skemp [1] provides a much earlier and far more coherent account of the psychology of learning mathematics. He was, first and foremost, a mathematician who constructed the best psychological analysis of what it means to understand mathematics, and his book encapsulates and expands upon all of the ideas propounded by Ma.

These comments should not be seen as disparaging Ma’s ideas on PUFM, but merely to put her ideas in the wider context of mathematics education. In that respect, I also wonder about the nature of the Chinese elementary school mathematics curriculum. For one thing, her four questions seem to concentrate on the learning of standard algorithms as opposed to more general underlying mathematical processes.

On the very day on which I saw the four ‘test’ problems in her book, I happened to be visiting friends for an evening meal. One was a 90 year old retired bank manager, two were senior managers in the electronic industry and two were retired head teachers from elementary schools here in the UK.

Without use of pencil and paper, all except the retired bank manager were able to compute 52 – 25 in their heads, and their methods were ingeniously different. None of them could work out 1¾ ½, and both engineers said that they would never express any calculation using the notation of fractions. All of the guests said that they would use a calculator for the multiplication 123 x 654, and two of them began to do it on their ever-present mobile phones.

The fourth of Liping Ma’s problems was connected with the relationship between area and perimeter, and a very special case of this was represented symbolically. This topic, of course, can be explored quite deeply by students as young as 6 years old, by employing what is referred to in this book as ‘manipulatives’. For example, take a closed loop formed by pipe-cleaners (say). How many bottle tops fit in the loop? Change the shape of the loop. How many bottle tops now? Etc.

Well, I make these points to illustrate something that appears to be outside the scope of Ma’s book. The main point is that, when mathematical ideas are formalised too early in students’ education, the problem of learning and teaching them is made unnecessarily difficult. Research undertaken here in the UK in the early 1980s shows that this has unforeseen consequences. These are partly manifested in the phenomenon of subterfuge ‘child methods’. Sometimes these are ingenious, but they are usually inefficient. Anyway, a formal approach to the teaching of division of fractions should not be part of the elementary school curriculum — whether in the UK, USA or China.

Despite the fact that many of the theoretical aspects of Liping Ma’s research are not new, she is to be congratulated on writing a book that has brought discussion of them to the forefront of national debates on mathematics teaching. What she says also applies to mathematics teaching in British schools, where politicians have the power to inflict their opinions on any aspect of the state education system. Indeed, at the moment of writing, a senior government minister has dictated that the glorious history of the British Empire should once more be a focal point of the school history syllabus.

Long live the Boston Tea Party!


[1] The Psychology of Learning Mathematics, by Richard R. Skemp (Penguin Books). First published in 1972, but with many later editions.

Over a forty-year period, Peter Ruane taught in elementary schools, secondary schools, and various universities in which he was involved with the training of mathematics teachers in the UK and Canada.

Author’s Preface to the Anniversary Edition

Series Editor’s Introduction to the Anniversary Edition

A Note about the Anniversary Edition




1. Subtraction With Regrouping: Approaches To Teaching A Topic

2. Multidigit Number Multiplication: Dealing With Students’ Mistakes

3. Generating Representations: Division By Fractions

4. Exploring New Knowledge: The Relationship Between Perimeter And Area

5. Teachers’ Subject Matter Knowledge: Profound Understanding Of Fundamental Mathematics

6. Profound Understanding Of Fundamental Mathematics: When And How Is It Attained

7. Conclusion



New to the Anniversary Edition: Journal Article #1

New to the Anniversary Edition: Journal Article #2

Author Index

Subject Index