Knowing and Teaching Elementary Mathematics by Liping Ma

A Book Review by Chita DuvalAMTRA Newsletter - 2000


In the conclusion of her book, Knowing and Teaching Elementary Mathematics, Liping Ma writes, "while we want to work on improving students’ mathematics education, we also need to improve their teachers’ knowledge of school mathematics." Struck by the consistent underperformance of U.S. students in national and international tests, such as the Third International Mathematics and Science Study (TIMSS), Ma set about investigating teachers’ understandings of elementary school mathematics. In a comparative study of 23 U.S. teachers and 72 Chinese teachers of comparable levels of experience, she found that unlike their Chinese counterparts, these U.S. teachers—both experienced and new teachers—lacked a "profound understanding of fundamental mathematics (PUFM)."
In order to arrive at this conclusion Ma interviewed individual teachers, presenting them with four problems: subtraction with regrouping, multidigit multiplication, division by fractions, and perimeter and area of a closed figure. The first four chapters contain lengthy discussions of how each teacher approached solving and interpreting each problem. The results of these interviews provide insights into how both U.S. and Chinese teachers approach teaching and explaining these types of problems. The results were not encouraging. For example, given the problem 1 ¾ ¸ ½, only 43% of the U.S. teachers were able to successfully calculate the answer, and almost all failed to come up with a correct interpretation of dividing by a fraction.
This book is extremely provocative in its suggestion that U.S. teachers’ lack PUFM. According to Ma (and others whose research she quotes) U.S. teachers teach procedurally, not conceptually. However, students should not view elementary mathematics as "¼ an arbitrary collection of facts and rules in which doing mathematics means following set procedures step-by-step to arrive at answers." Not only must students and teachers know the "how" of mathematics, but also the "why."
As math teachers, it’s important to read this book and evaluate our own understandings. It also seems important to include discussions of these issues in each of our districts as we review new standards in mathematics. Ma’s research is compelling, and we should pay careful attention to her findings and their implications for our profession.

Ma, Liping, Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States, Lawrence Erlbaum Associates, Publishers, Mahwah, New Jersey, 1999.