This book, volume 55 of the MAA Notes series, is a magnificent work. Seventeen authors and four editors joined forces to summarize the collective wisdom of nearly 150 mathematics faculty who have participated in the MAA Project CLUME (Cooperative Learning in Undergraduate Mathematics Education) workshops, which began in 1995. The authors also draw upon results of the official CLUME survey, which was administered in 1997 and to which 114 mathematics instructors responded.

The main body of the book consists of 100 pages divided into seven chapters. Each chapter deals with a particular topic in the cooperative learning of mathematics and is written by multiple authors. One might expect that this team style would lead to clumsy and disjointed writing. The authors have, however, done a masterful job in coordinating their efforts, being very thorough yet avoiding redundancies. Clearly, these authors are themselves good collaborators. Every chapter was "written and re-written by small groups of authors and then critiqued by the larger group", and it took nearly four years to complete the volume.

The first chapter gives an overview of cooperative learning in mathematics. Why should an instructor use cooperative learning at all? A host of reasons are offered. First, small groups offer a "social support mechanism", and students often feel comfortable asking questions of their peers which they wouldn't ask the instructor. Another reason is that students are more likely to see multiple correct ways of approaching a problem. In a group, students may be able to solve problems which are more complex and thought-provoking than the problems they can solve individually. These reasons and others are analyzed in enough depth to be enlightening without overwhelming the reader in a sea of details. Another particularly interesting feature of this chapter is a series of case studies in large scale implementations of cooperative learning in math by colleges and universities. Although one university found that "the fact that students work together on problems seems to detract from their ability to solve problems individually on tests", most institutions have apparently been very successful in implementing these programs.

Chapter Two is entirely devoted to "practical ways to develop a social climate conducive to cooperative learning in the classroom". For example, how should groups of students be formed: by the students themselves, by random selection, or should the instructor choose the groups? Instructors can choose groups based on math ability (measured perhaps by standardized test scores or previous math grades), or compatible class schedules, or even by personality inventories such as the Meyers-Briggs test. Pros and cons of each method are discussed. In this case, the authors conclude that there is widespread support among math instructors for each of the three major methods of group selection. Occasionally the authors do give specific advice (for example, there is wide consensus that five students in a problem solving group is too many) but in general they are refreshingly nonjudgmental and very slow to impose their own preferences on the reader.

Chapter Three describes many classroom strategies for cooperative learning. For example, the "Groups/Pairs Exchange" method works like so: "Each group or pair of students is asked to investigate a mathematical object. The example is then passed along to a second group or pair who responds in some way to the item received. The response is then returned to the original pair and the results are reviewed. The second group can then pass their work to a third group, which does some further work. If appropriate, they can continue to a fourth group, etc." Sample exercises are provided for each of the most commonly taught college math classes: math for elementary education, statistics, discrete math, precalculus, calculus, linear algebra, and other courses. The many specific examples make this book really practical, and I would think that any instructor wanting to try these strategies could find ideas for several activities which are appropriate for his or her classes.

There is a chapter which provides useful ideas on how an instructor might assess students individually (for grading) when collaborative learning is a component of the course. Two chapters deal with educational theory and how it applies to group learning. There is even a chapter with suggestions for conducting faculty development workshops on this topic. There are plenty of helpful examples, anecdotes, and case studies throughout the book.

An appendix lists the responses to the 1997 CLUME survey numerically tabulated, along with summaries of respondents' additional comments. Finally, a substantial bibliography is provided. It is conveniently arranged into a section on further reading for instructors, and a section on textbooks and course materials that work well in a cooperative classroom.

Overall, this book is a fantastic resource for any college mathematics instructor who uses cooperative learning or is interested in incorporating it into his or her classroom. It's packed with practical, usable information, and very comprehensive. I highly recommend it.

Andrew Perry (andy@perry.net) is Assistant Professor of Mathematics at Springfield College in Springfield, MA.