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A Practical Guide to Cooperative Learning in Collegiate Mathematics

Nancy L. Hagelgans, Barbara E. Reynolds, Keith Schwingendorf, Draga Vidakovic, Ed Dubinsky, Mazen Shahin, and G. Joseph Wimbish, Jr.
Mathematical Association of America
Publication Date: 
Number of Pages: 
MAA Notes 37
[Reviewed by
Thomas R. Berger
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If you are either interested in or very experienced at using cooperative learning in your college mathematics classroom and wish to see a concise handbook on the subject, definitely, look at this book. If you want references into the literature on cooperative learning, you may wish to jump start your efforts by using the extensive annotated bibliography at the end and the references at the beginning of each chapter.

This is a very concise handbook of practice supported by research. The book tells its story in eight brief chapters spanning only ninety pages. Forty more pages contain documents and references helpful to anyone using the book. The authors have tried to single out those issues that would assist the practice or the understanding of teaching with groups in our mathematics classrooms.

The book provides a wide variety of practices all from a particular point of view. Cooperative learning offers a wider choice of methodologies than the lecture method. From this scope, the authors advocate a method solidly backed by research. The underlying principles are outlined briefly on page 5.

A significant amount of the work of the course should be done in cooperative groups, A positive esprit de corps exists within groups, Team members share a feeling of mutual responsibility for each other, Group membership is permanent and stable, and Group work is included in the evaluation process.

The chapters go on as follows:

Why Use Cooperative Groups?

It is unlikely that anyone will try cooperative methods just because they read this chapter. However, it contains essential information from the research literature for those wondering why these methods receive so much hype. The simple answer is, “They work.” Once beyond that, it is reasonable to wonder why. A short list of answers with references for further reading provides an explanation.

The research on cooperating groups at all ages and in all environments is very compelling. The bibliography here does not reach beyond the classroom to wider settings where groups (such as industrial development or production teams, church committees, volunteer groups, and other assemblages of people) come together to complete a large task. The background is much the same. The groups are small; they work on significant tasks; there is an excitement about the mission of the group; the members share important mutual responsibility; the group must work for an extended period of time; and the whole group is judged (usually by community or organizational standards) for its performance.

The authors discuss the foundations for cooperative learning, but do not address the social process that instructors and students enter when groups are formed. This is the “stuff” of seminars for business leaders who organize their workers into teams. It is equally important to instructors who use cooperative learning. That is, working cooperatively is a socialization process.

Groups have a social dynamic extending through a few phases. Some phases are potentially upsetting. Many of the people I’ve talked with who “tried groups and found they do not work” quit during a troublesome phase without reaching a more beneficial period. Incidentally, similar socialization processes occur with the lecture method. Except for a few students who come forward, the method is sufficiently depersonalized that the suffering individuals do not identify themselves and quietly disappear or fail for reasons we never discover.

How Are Groups Formed?

The authors discuss size, composition, formation, and lifetime of groups usually referred to as basic. They introduce enough methods that anyone ought to find techniques that fit. The appendices contain sample information questionnaires to assist a busy instructor.

What Do Groups Do?

If you don’t lecture, what do you do, especially if there are fifty inquisitive faces formed into groups looking at you? The authors take the point of view that we all know the traditional organization of a course and base their advice upon that. Learning is divided into familiarization activities, classrooms tasks to pull ideas together, and exercises that require group effort outside of class to reinforce concepts. For each of these, the authors give specific examples in classroom-ready form from a wide range of elementary college level mathematics courses.

Because they included computer activities in their courses, the authors also discuss laboratory sessions. When students encounter a new technology, suddenly two learning curves are compounded: the mathematics and the technology. Studies show that there are problems associated with the introduction of technology that fall inequitably on different groups (e.g. males and females). Working cooperatively is not a panacea for solving these problems. The authors do not address these issues.

How Do We Test and Assess Our Students?

This is an area where I have difficulty still. They’ve offered me some very good advice. The authors discuss what they do collectively. They advocate that some testing be in groups: discussing some background and giving sample questions. They discuss group preparation and group and individual test grades. They include the other forms of assessment we can use and the determination of grades: helpful suggestions from a variety of methods and points of view. Appendices include sample participation forms, syllabi, and grading schemes.

How Might a Group Function?

The authors discuss modes of operation of groups, methods of discovering these modes, and the kinds of difficulties that can arise. Most instructors who object to me about cooperative learning feel that some individuals will be slack and others will carry the load, or that a bright student will dominate and overwork for fear that the rest of the group will produce work of low quality. These issues are discussed in the book.

More and more, industry is looking to education to prepare a labor force that will work cooperatively in dynamic teams for extended periods of time. In those teams each person has a pace and a style. Some workers are not terribly productive and will work beside highly efficient people. This is a reality. It happens in the workplace and in the classroom. The authors do intimate that the groups must come to grips with the process of working cooperatively, even if it takes effort, but they do not adequately point out that we must help students to understand that people have varying capacities, that this is all right, and that we can cope. The authors do discuss attitudes and practices that will reduce the probability of difficulties and assist in the resolution of difficulties when they arise.

The information imparted by the authors is strong on practical experience, and therefore quite sound, but lacks the solid background that would come from inclusion of social science research on human beings working in groups and the dynamics of that work. This contrasts with the careful research base they have placed under the learning process.

What Are the Reactions of Students and Instructors?

This is a discussion of reactions both positive and negative, including both students and instructors: brief, organized by topic, and worthwhile. For those who would like a gold mine of such reaction see E. Seymour and N. Hewitt, “Talking About Leaving,” Bureau of Sociological Research, U of CO, 1994: a study that sifts reactions and finds the common threads. Most of us do not like to hear the story told and I have watched as a group of very thoughtful mathematicians deny that any of it was real. The problem here is not our successes, but our failures: there are too many of them, and failure is coming for reasons we do not want. The need to move a far larger portion of the population successfully through mathematics lies at the heart of the present reform movement.

What Are Other People Doing?

Those who get into the “teaching methods” game discover that the variety is unlimited. A list of different organizational schemes for cooperative learning is presented. If you are into “form”, there should be enough here to keep you stimulated for some time to come, plus references to take you beyond that.

For those who can look past cheerleading about small group learning, there is an excellent meta-analysis (a technical term referring to an attempt to statistically pull together a large number of studies) of the research on cooperative small groups: David W. Johnson & Roger T. Johnson, Cooperation and Competition: Theory and Research, Interaction Book Company, Edina, MN, 1989. A surprise: this is not in the bibliography. It was from this meta-analysis that I first learned that human beings mastering repetitive small tasks learn better from lecture type instruction followed by individual drill but that complex tasks are better learned in cooperative fashion. I find this relates very well with the way we traditionally teach and what is traditionally learned as opposed to what we might teach and what might be learned in mathematics. I go further than the authors by assigning problems that usually transcend the capabilities of a single student. A group effort is essential. In this way I can assign interesting mathematical problems and not merely exercises. Further, the students encounter deeper mathematics with only a small number of problem settings to remember after the course is over.

The people engaged in reform that I’ve met are giving 110% to the effort. Most academics I know do give 110% to intellectual effort in some direction. This effort is not always focused on their classroom. I believe this spread of effort is good since there are too many things to be done for all of us to focus just on the classroom. So if reform requires 110% from all instructors then reform is doomed. I don’t think this level of involvement is required. I think books like this one begin the process of making reform easier for the rest of us. In a solid way, based upon wide experience and careful research, with brevity sympathetic to busy instructors, this book provides an easy handbook that points to more deep knowledge elsewhere.

Thomas R. Berger is Carter Professor of Mathematics and Computer Science at Colby College. He has for many years used cooperative learning techniques in his teaching. His e-mail address is

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