1. Introductory comments
2. Brief workshop outlines
3. A summary of our discussions at the June CLUME meeting that fills in some details on the outline.

We provide in this document a brief outline of what a half-day, a one-hour, and a two-hour workshop on cooperative learning in mathematics might contain. You will find a brief outline in Section 2, and comments in Section 3 that flesh out some of the sections.

Our intent was to draw up an outline that would give possible designers/presenters a place to start. It is by no means the last word on the subject, and in fact, we expect to revise it as we gain experience giving workshops.

If you are planning to give a workshop, you might be interested to know that Neil Davidson has offered to act as a coach/mentor and you should feel free to bounce your ideas off him. Do you need a co-presenter? Both Neil and Barbara Reynolds have indicated an interest in co-presenting -- contact either one for further details.

The following outline is intended for a half-day workshop. This is a brief outline, and a more detailed discussion of several of the components is attached. There are also some comments below about possible plans for 1 and 2 hours presentations about cooperative learning.

• Brief introduction to answer these questions:

• What is cooperative learning, generally?
• What will happen during the workshop?

• Ice-breakers (when done, Broni says participants will be "ice-broken"), such as:

• Handshakes all around
• Line up by height, or birthday, or number of years taught, or number of years you have used CL
• See discussion summary for other possibilities.

• Group formation: leader will use method of choice.

• If possible, try to get information before the workshop about whether any of the participants have previous experience using CL, and in which courses they think they would like to use CL.
• If you have some participants that have previous experience with CL, you could use the ice-breaker where they line up according to the number of years they have used it. Use this to make sure that each group has a variety of levels of experience represented.
• You might sort them into groups according to the content area they would be most likely to use CL in.
• Leader might mention some other methods for forming groups.

• Team-building for workshop

• 3-step interview structure, with attention-getting questions
• Uncommon commonalities
• Team name

• Easy strategies (this should form the bulk of the workshop)

• Think/Pair/Share, Roundtable, etc.
• Choice of content depends on audience
• Include an opportunity for participants to reflect on their experiences with these activities, and then report out to the whole group (one new observation per group).

• Leader describes some other strategies (perhaps supported with a handout for the participants), since it is not possible to do all of them. This is followed up by an activity in which each group tries to generate example activities for each of the strategies mentioned. You could use Jigsaw, where expert groups focus on one of the easy strategies and try to generate examples, and then report back to their home group.

• Dealing with concerns

• Ask participants to express those concerns that they would most like to have resolved in order to be able to use or try to use CL. One way to do this is to have groups discuss concerns, with one member recording them, and follow up with a large-group discussion.
• The leader could set up a situation (perhaps spinning off participants expressed concerns), and then have focused brainstorming for solutions
• What might come up here?
• Assessment concerns (formative and summative)
• Problems with group dynamics
• The "coverage" problem
• Special problems like difficult seating arrangements, size of class, size of groups to use

• Rationale for using cooperative learning
• Summing up

ONE-HOUR PRESENTATION: The only feasible goal of such a presentation is to bring the participants some level of awareness about cooperative learning. It is practically impossible to build skills in only one hour. Such a presentation could be titled "A taste of cooperative learning". A possible basic structure follows:

• Introduction: success stories, pictures of cooperative settings on the overhead, stories detailing classroom examples
• Do two to four simple activities involving the easier strategies, such as Think/Pair/Share and Roundtable. Some commentary by the leader on what is happening would be appropriate here.

Summing up: Include a brief reference to the issues that concern many who are reluctant to try cooperative learning (group forming, problem students, summative assessment). Reassure participants that it is okay to learn to use cooperative strategies a bit at a time. The purpose of such a presentation would be to inspire people to begin to use CL.

TWO-HOUR PRESENTATION: Such a presentation could include some skill-building. The suggested breakdown of time is 1.25 hours of activities and .75 hours of overview via lecture. Use the same basic format as the one-hour presentation, but augment with some of the components from the longer workshop outline.

HINTS FOR WORKSHOP PRESENTATION

• Practice what you preach! That is, don't preach!
• Story-telling is a good idea: both in terms of the leader sharing stories, and getting the participants who have some experience with cooperative learning to share stories.
• Be sure to include activities in the workshop where participants reflect on what they are learning and how they might apply it in their own classroom.
• Give handouts: e.g. bibliography, principles of designing activities.
• Move from the simplest activities to the more complex. This is valid advice on many levels:

• for students: from narrowly focused questions to open-ended questions
• for students: from simple CL techniques to more complex techniques
• for teachers: from simple implementations to the more complex, according to the growth in one's skill in designing and managing activities
• for leaders: the basic workshop to the more complex workshop

Below are our ideas for fleshing out each of the components of the workshop outline. This summary is not intended to be exhaustive, but should give prospective workshop leaders a place to start in designing their own workshop. Referring to the literature (including the handouts from the 1995 CLUME workshop and the Practical Guide) would be the logical next step.

Overall, our philosophy is that we should model in the workshop what the participants might do in their classrooms. This is not content specific. The main areas are :

• Creating a cooperative climate
• Group formation
• Easy strategies for getting the novice started
• More complex strategies for the initiated
• Rationale: why is cooperative learning worth the trouble?
• Developing your own plan for getting started
• Concerns (especially assessment, group dynamics)
• Some principles for designing activities

There are two parts to this -- community building and team building. While the participants will likely be predisposed to cooperating, remind them that their students will not necessarily come into their classes knowing how to work effectively in groups. Some effort will be needed to help students develop the appropriate skills.

First, we would recommend some ice-breakers to begin the community building. Some examples are:

• Getting to know you (see handout in Neil's packet from 1995 workshop)
• Handshake problem (an elementary mathematics problem is how many handshakes would there be if each person shook everyone else's hand once. This can lead into some more complex mathematical questions, if desired. )
• You might want to adapt a strategy that Martha uses in her statistics classes. Each chair in the room has a number pre-assigned to it. When her students enter the room, each chooses a number at random from a bag and they sit in the corresponding chair. She then gives them some ice-breaking questions -- for a workshop, you might have them introduce themselves, and tell what courses they teach.
• Knots (a group of about 6-8 people stand in a circle. Each person joins hands with two people who are not standing next to them. The goal is to unravel the knots without letting go and without talking. Warning: we had fun with this at CLUME, but when Anne described the technique to a colleague, he said he would be afraid to use it in class because of the physical contact and the question of sexual harassment -YIKES!)
• Corners (people with a specified common interests or characteristics cluster in particular corners of the room)
• Circles (see handout in Neil's packet)
• Lining up according to various characteristics, Human Box Plots

After groups are formed, use at least one team-building activity. Some examples are:

• Three-step interviews: use a question that would catch interest, and would help participants remember each other
• Uncommon commonalties
• If the workshop is long, consider having teams choose a name. At this point, mention this useful guideline for the invention of names -- they must be positive, short, legal, and not obscene. We'd like to avoid things like "Three D's and an S", which was used in one class to stand for "dumb, dumber, dumbest, and stupid".

For the workshop, a primary goal is to use a quick method for putting participants in groups. However, the various methods of group formation in the classroom should be discussed. One possible approach is to have the leader present a range of alternatives, followed by a group activity in which the participants think about a class in which they might try to use group work, and then evaluate the list of alternatives according to which ones they might be most comfortable using and why, which they might try first, advantages and disadvantages in various situations, etc.

Here are some of the issues that the leader should be sure are brought out at some point in the discussion:

• permanent versus non-permanent groups
• student-selected, or student input solicited, or chosen by teacher alone
• homogeneous versus heterogeneous (with respect to background, ability)
• the use of "social engineering" (i.e. collecting some information about class members, and using your intuition to put people in groups that you think will work well together)

Use some easier strategies in the early activities done in the workshop so that participants see that it is not difficult to get started, and also begin to think about how they might use these strategies in their own classes (see also the comments under DEVELOPING YOUR OWN PLAN). At the same time, emphasize the importance of starting students with narrowly focused problems, using the easier strategies at first, with a progressive development during the course to more open-ended problems and more complex strategies. In each case, the actual activity chosen by the workshop leader for the groups to engage in would depend on the audience.

Here are some of the easier strategies to use in a classroom:

• Interviews

• Brainstorming (or Lotsa Thoughtsa)

• Think-Pair-Share OR Think-Pair-Square-Share

• Roundtable

• Triptych

• Numbered heads together

If your participants are up to it, you might try one of the following strategies to give them a taste of what is possible.

• Round-flocks (invented by Clare Hemenway, a play on round-robins)

• Jigsaw

• Data sharing

Given that becoming an expert in using CL takes considerable time and effort, and that when you give up some of the control in your classroom, some bad things can happen, one might well ask whether learning to use CL is worth the trouble. Some of the participants in your workshop might have this question, and it would be worth exploring this issue to help them resolve it while they still have the support of their group.

One effective way to get at the rationale question is to have the groups engage in brainstorming on the following question: Why might you be interested in trying to use cooperative learning for at least some of the time, in some of your classes? In brainstorming, ideas are generated, but cannot be criticized. Groups keep a list of the ideas that come up in their group.

To prepare to report out the results, the following might happen:

• Participants rank the list, and cross out anything bizarre that they do not want to make public
• List their best reasons
• Pick a spokesperson
• Spokespersons line up in front of room, and each give one item from their list, rotating until everyone has given all their ideas (no repeats allowed).

Here are a few reasons that students like to be in classes in which cooperative learning is used:

• Feelings of isolation are eased. Students learn that they are not the only one in the class who is struggling with difficult ideas. This helps develop self-confidence in dealing with mathematical ideas.
• Students tend to enjoy the work more if it is in a social setting.
• Students find it less intimidating to ask a question of a classmate. Often they will understand a response from a classmate better than that offered by the instructor.
• Cooperative learning prepares students for team work on the job.

If you ask your own students what they do and don't like about cooperative learning, you might get some quotable comments that could be used in this portion of the workshop. At this point, one might give a one-minute description of the research results on cooperative learning (Neil Davidson and Barbara Reynolds will put a bibliography out on the clume-list.)

In summing up, you might make these points (but remember, don't oversell or preach):

• There are many positive research results concerning cooperative learning. For instance, giving explanations during group work has been shown to be positively related to achievement. There are several reviews of the research available, and you can consult the bibliography that Neil and Barbara will post on the clume-list.
• There is a broad spectrum of mathematical opportunities for using CL. Any topic is fair game, and there is a wide variety of possible strategies to consider.
• There is more student accountability and responsibility in cooperative learning, contrary to what some might believe. In the traditional lecture courses, many individuals may not do much work, and simply take the tests. Our business is student learning and cooperative learning fosters, if not forces, students to do their share of the work.
• "Cooperative learning keeps the kids awake; if they are awake, they just might learn something". This is illustrated by Neil's story about visiting a high school and walking down a hall, looking into a succession of classrooms, and seeing teachers lecturing and students with their heads down on the desk, or flopped back, studying the ceiling. In contrast, when he reached the room where CL was happening in a mathematics classroom, everyone was energized, talking, working, enthusiastic, etc.

It would be good to have participants leave the workshop with at least a rudimentary action plan for how they might try cooperative learning. There are several things that can happen in this portion of the workshop that might help them do this.

• Using one of the easy strategies, do a cooperative review of the workshop so far. The goal would be to construct a list of two or three things that they might actually do in class. In some cases, participants will want to pick one or two strategies to use at first, until they feel comfortable.
• Have participants who have used cooperative learning tell how they got started.
• Neil's packet from the 1995 CLUME workshop has a handout entitled "Multiple Uses for Cooperative Learning in Mathematics" which shows that there is a broad spectrum of mathematical opportunities for using cooperative learning.

• Group homework check
• Presentation: simple structures
• Group discovery or problem solving
• Processing group interaction
• Cooperative review of mathematical learning
• Group homework preparation

• Be sure to mention to participants one easy way to start using cooperative learning -- the See-Saw approach. The idea is to make a balance between group work and whole class discussion/lecture. You give smaller tasks, engendering the growth of confidence (both in the groups and in the instructor, perhaps). It helps with pacing (assuming that you can regain the attention of the students once the group work is to be wrapped up). It helps keep the groups on task, because the task is smaller and has a single focus.

• Have the participant groups brainstorm ways to persevere.
• Here are some things we mentioned during our discussion:

• Set realistic and manageable personal goals (e.g. to use one cooperative activity in each class each week)
• Keep a journal (What did I do? What happened? How can I make it better next time?)
• Form a support group. (Anne described an interdisciplinary faculty group on CL that she belongs to. There are five members that meet every two weeks to talk about their efforts to use CL -- and now, other innovations --in the classroom. Not only has it been helpful in brainstorming solutions to problems, but it kept the members working on their teaching: no one wants to have to say that they haven't tried anything new in the intervening 2 weeks!)

When this section is polished, it might turn into a workshop hand-out. It seems that it would be difficult for workshop participants to have any real ideas about this more advanced topic, even by the end of the workshop. Perhaps they could work from the hand-out, brainstorming applications of some of the principles mentioned here.

• Make sure that your activities have very clear, step-by-step instructions, and check that students understand the instructions.
• In many courses (notably the courses for elementary education majors), the form and sequencing of group tasks is critical. Any course in which some or most of the students tend to be math phobes, you should make a special effort to "meet them where they are". Start out with easy strategies to get them to work on short problems with definite, confirmable answers. Over the semester, gradually loosen up the structure to include more complex strategies and more open-ended problems. Use a progressive development from very structured exercises to the more open ended.
• Aimless exploration is often unproductive, both in group tasks and in individual work, but guided discovery can be very successful. In this format, students generate data and respond to direct questions. The idea here is that if there is a particular objective that you want them to reach, explicitly pointing them in the right direction will make it highly likely that they will reach it. This does not necessarily make the exercise easy, it just makes it more likely to be productive.
• Along the same lines as the previous principle, a way to approach proof in group activities is to have students engage in concrete activities, generate conjectures, and then prove or disprove conjectures. Large tasks (such as proving Lagrange's Theorem in a group theory course) should be subdivided into more manageable sub-tasks. The design of these tasks might be informed by learning theory. For lower level courses, appropriate justifications can also be approached in this way.
• It helps to have students make predictions and then investigate them. For example, if a problem involved calculating a probability, you might have them estimate it before calculating it. There are many problems in that area which seem easy at first but are actually quite complicated.
• Find a way to pilot your activities. You might "bribe" a group of students (e.g. with extra credit) to try out the exercise before you do it in class. Another possibility is to require that one group meets with you every week. This is more plausible if there are four or more groups, and you are using permanent groups. Benefits include

• getting a birds-eye view of the dynamics of each group
• being able to demonstrate to your students that you care care about your teaching and that your activities are carefully planned

• In designing activities, first think carefully about the goals of the activity: Is it intended to

• develop skill,
• foster conceptual growth,
• encourage assimilation and review, or
• consider examples and counterexamples? The goal of the activity should have an effect on the choice of strategy.

• You might want to work on one course at a time, or part of one course at a time. Pick your opportunities carefully, there is no need to try to reform the entire course in one shot. This is the principle of progressive refinement, allowing the possibility for further development over time. (This again emphasizes the need to keep track of what you did, how it went, and how you might improve it in the future.)