This article discusses several cooperative learning techniques, principally for formative assessment. These include ways to help students learn the importance of clear definitions and several review techniques (comment-along, teams-games-tournaments, and jigsaw).
Background and Purpose
From 1993-1995, I taught at Shippensburg University, PA, a conservative, rural, state university with about 6000 students. I used cooperative learning in a geometry course for students most of whom were intending to become teachers, but this method could be used with most students. The text was Greenberg .
The NCTM Standards  urge teachers to involve their students in doing and communicating mathematics. The responsibility for learning is shifted to the students as active participants and the instructor's role is changed from dispenser of facts to facilitator of learning.
I chose to use the cooperative learning format for two reasons: I felt my students would learn the content better in a more interactive atmosphere and they may need to know how to teach using these methods. I hoped experiencing these new methods would help them in their future teaching careers. Since most teachers teach the way they were taught, this would be one step in the right direction.
I use heterogeneous groups (based on sex, ability, age, and personality) of four students, or five when necessary. I use a variety of cooperative activities, including team writing of definitions, Comment-Along, Teams-Games-Tournaments, and Jigsaw. I'll describe each briefly.
One of the first activities the students work on is writing definitions. Each group receives a geometric figure to define, and then exchanges definitions (but not the original figure). The new group tries to determine the shape from the definition given to them. In whole class follow-up discussion, we try to verify if more than one shape can be created using the definition, and in some cases they can. Students are thereby able to understand the difficulty and necessity of writing clear definitions.
To be successful in mathematics, students must be able to decide if their judgments are valid. Comment-along gives them practice with this. This activity begins as homework, with each student writing two questions and answers. In class, the teams decide on the correct answers and select two questions for to pass to another team. The new group considers the questions, adds their comments to the answers, and then passes the questions to a third team which also evaluates the questions and answers. Then the questions are returned to the original teams which consider all responses, and prepare and present the conclusions to the whole class. This activity is an assessment that students do on their own reasoning.
Teams-Games-Tournaments (TGT) is a Johnson & Johnson  cooperative learning activity which consists of teaching, team study, and tournament games. We use this at the conclusion of each chapter. The usual heterogeneous groups are split up temporarily. Students are put into homogeneous ability groups of three or four students for a competition, using the list of questions at the end of the chapter. Students randomly select a numbered card corresponding to the question they are to answer. Their answers can be challenged by the other students and winner keeps the card. Students earn points (one point for each card won) to bring back to their regular teams, a team average is taken, and the teams' averages are announced and all congratulated. On occasion, I follow up with a quiz for a grade. The quiz takes a random selection of three or four of the questions just reviewed; students write the quiz individually.
Jigsaw is a cooperative learning activity  where each member on a team becomes an "expert" on a topic. After the teacher introduces the material, each team separates, with the members joining different groups who study one particular aspect of the topic. In effect, they become "experts" on that topic. The teacher's role is to move among the student groups, listening, advising, probing, and assuring that the groups make progress and correctly understand the concepts. When asked, the teacher should not try to "teach" the material, but rather pose questions which lead the students to form their own correct conclusions. When the students return to their original teams, they teach the other members what they have learned. The teacher is responsible for choosing topics and for monitoring the groups to assist and verify that what is being learned is accurate. Ultimately, all students are responsible for knowing all the information. I use this method to reinforce the introductory discussion of non-Euclidean geometry. Each expert team is responsible for the material on a subtopic, such as Bolyai, Gauss, Lobachevski, and similar triangles. When the students are divided into "expert" groups and each group is assigned a section to learn, I listen to, and comment on, what they are saying.
Because each team gives an oral analysis of the definitions it wrote, the whole class is able to share in the decisions and reasoning. Since students' understanding of the need for clear definitions is essential to their understanding and using an axiomatic system, this activity provides them with needed practice. This assessment provided me with immediate feedback verifying that this foundation was indeed in place.
One of the goals of TGT is to give weaker students an opportunity to shine and carry back the most points for their team. After one tournament, a student was ecstatic at her top score! She had never before been best at mathematics.
Once I tried using team TGT average grades in an attempt to promote accountability. Students howled, "It's not fair." So, I stopped counting these grades, and yet they still worked diligently at the game. (I then replaced this group quiz with an individual one.) Several students commented that they liked the game format for review.
During Jigsaw, the students seemed highly competent discussing their topics. During such activities, the classroom was vibrantly alive, as groups questioned and answered each other about the meaning of what they were reading. When the students finished teaching their teammates, one group assured me that the exercise was a great way to learn the material, that it had really helped their team. Students teach the content and assess their knowledge through questioning and answering each other.
One day, before class, a student told me that his teacher-wife was pleased to hear I was teaching using cooperative learning; it is where the schools are heading.
Use of Findings
Teacher self-assessment of effectiveness of the classroom assignment was done informally through the questions heard when traveling through the room. It let me know how the students were learning long before test time, while I could still effect improvements. Depending on what was learned, the lesson was revisited, augmented, or considered completed. For instance, students had a very difficult time "letting go" of the Euclidean parallel postulate. When they would incorrectly use a statement that was a consequence of the Euclidean parallel postulate in a proof, I would remind them of our first lesson on the three parallel postulates.
Some group assessment for student grades was done (one was a team project to write proofs of some theorems in Euclidean geometry), but most of the formal assessment of students for grades was done by conventional individual examination.
Daily and semester reflection on the results of my evaluations allowed me to refine my expectations, activities, and performance for the improvement of my teaching. The students were learning, and so was I.
I would offer some advice and caution to making the change to a cooperative learning format in your classroom. The cooperative classroom "feels" different. Some college students may feel anxious because the familiar framework is missing. The "visible" work of teaching is not done in the classroom. Good activities require time and considerable thought to develop. Consider working with a colleague for inspiration and support. Don't feel that it is an all or nothing situation; start slowly if that feels better. But do start.
I continued because of my desire to help all students learn and enjoy mathematics. But more, it was a joy to observe students deeply involved in their mathematics during class. There was an excitement, an energy, that was missing during lecture. It was addictive. Try it, you may like it, too.
 Greenberg, M.J. Euclidean and Non-Euclidean Geometries (3rd ed.). W. H. Freeman, New York, 1993.
 Hagelgans, N., et al., A Practical Guide to Cooperative Learning in Collegiate Mathematics, MAA Notes Number 37, Mathematical Association of America, 1995.
 Johnson, D. and R. Cooperative Learning Series Facilitators Manual. ASCD.
 National Council of Teachers of
Mathematics. Professional Standards for Teaching
Mathematics, National Council of Teachers of Mathematics,
Reston, VA, 1991.