Knowing and Teaching Elementary Mathematics

The United States has always been a nation of pragmatism—the only philosophical system native to our shores even went by that name. What works is what's best, it's believed, and this has been applied to everything from politics and religion to vocation and education. Particularly education, a field in America dominated more by system and method than deep thought and reflection.

Liping Ma's Knowing and Teaching Elementary Mathematics explores this tendency by comparing American and Chinese teachers's understanding of elementary arithmetic principles. Originally Ma's doctoral dissertation, the book is in its second (anniversary) edition, and has sold 70,000 copies, no mean feat for a technical book about math education.

Unlike too many academic books about education, Ma's is based on original research. Her ideas aren't simply formed from browsing preexisting works, but rather are the result of actual interviews and comparison of data. 23 "above-average" teachers in the United States and 72 Chinese teachers from mediocre to excellent schools provided the interviews.

Any good subtitle will convey the author's bias, and that is certainly true for this book: "Teachers' Understanding of Fundamental Mathematics in China and the United States." It might not be obvious, but it at least hints at the fact that Ma believes teachers in her homeland are more equipped to teach elementary math than their counterparts in the U. S.

After reading her findings, it would be hard to disagree. The first four chapters are in-depth accounts of conversations held with both groups concerning specific aspects of teaching subtraction, multplication, division of fractions, and the relationship between perimeter and area.

In each case, U. S. teachers approach problems from a purely methodological standpoint—students are told how to "do" the problem without being told why. The Chinese teachers (who've received less formal teaching than the Americans), on the other hand, explain how to solve problems based on thorough understanding of the principles behind them.

Chapters five and six look at what Ma calls Profound Understanding of Fundamental Mathematics (PUFM), and when and how such knowledge is obtained. The vast majority of the Chinese teachers had such knowledge, while the Americans did not, and she asks poignant questions while providing thought-provoking answers.

The final chapter offers insights and suggestions for strengthening PUFM among teachers. An important essay by Yanping Fang and Lynn Paine titled "Bridging Polarities: How Liping Ma'sKnowing and Teaching Elementary Mathematics Entered the U. S. Mathematics and Mathematics Education Discourses" is included in this anniversary edition.

One of Ma's fundamental assertions is that there is a direct relationship between how elementary mathematics are taught and how well the teacher understands the concepts he or she is trying to convey. U. S. teachers, she found, are much more likely to see fundamental math as "basic," a random collection of facts to be memorized, rather than a coherent discipline relying on relationships between the different functions to be understood.

Because Chinese teachers were trained to view elementary math in the second way, as a related web, they more fully understood the rationale behind algorithms and were better able to explain and teach those functions and algorithms as a result. If teachers are able to provide the why of math to students, those students are more likely to succeed at the elementary level and less likely to falter when studying more advanced mathematics.

This isn't what most would call a fun book, but it has its own appeal. There's a lot of technical academic jargon, and any non-professional will doubtless need to look up words and terms, but Ma's findings unfold like a kind of educational thriller, as we learn more and more about how math is taught, and why those methods are good or bad.

Don't rush out to buyKnowing and Teaching Elementary Mathematics thinking it's some kind of lighthearted mystery novel. Ma doesn't embellish her text, and as many online reviewers have noted, it can be dry at times. But for those wanting to know how better to master arithmetical knowledge in order to pass that knowledge on will find this book indispensible and fascinating.

Review by C. Hollis Crossman C. Hollis Crossman used to be a child. Now he is a husband and father, teaches adult Sunday school in his Presbyterian congregation, and likes weird stuff. He might be a mythical creature, but he's definitely not a centaur.Read more of his reviewshere.