In 1995, the Mathematical Association of America (MAA) launched Project CLUME (Cooperative Learning in Undergraduate Mathematics Education) with an intensive 12-day workshop in cooperative learning. Following that summer's workshop, the participants implemented cooperative learning in their own classes at their home institutions in many different settings. They used their experiences from the summer workshop along with suggestions found in "A Practical Guide to Cooperative Learning in Collegiate Mathematics" (MAA Notes Volume #37). This "Companion" to that volume will contain reflections on the experiences of the participants in Project CLUME as they implemented cooperative learning in their undergraduate mathematics classes at a variety of institutions as well as practical tips and anecdotes about their experiences. Although this book is proposed as a kind of "sequel" to the "Practical Guide", it will be written as a stand-alone volume.

The title of this volume ("A Companion to the Practical Guide …") is the current working title; the authors simply refer to it as "The Companion". This is probably not the correct title for the book, and expect that the title will change as the project moves forward. We are open to suggestions (particularly from the MAA Notes editorial board) for a practical and marketable title. Similarly, the following chapter titles are current working titles. Some of these titles may change as this project evolves.

The authors hold the view that cooperative learning strategies when used effectively promote learning. They have used cooperative learning in their own classrooms, and have seen that students seem to have a deeper understanding and better retention of concepts when they have learned them with each other in cooperative groups. This chapter will describe what these authors mean by cooperative learning, and will include a rationale for using cooperative learning in undergraduate mathematics classes. It will contain a brief overview of the book.

In this chapter, we will present examples of activities and strategies that we have used in our classes, and we will discuss how they worked. We plan to discuss practical concerns that an individual instructor may have in beginning to introduce cooperative learning activities into the undergraduate mathematics classroom. In particular, we will present some activities that were particularly successful, and discuss why we think they worked. We will also discuss some activities that didn't work so well, and what we think went wrong.

In this chapter we plan to address both concerns about curricular issues and concerns about creating a cooperative climate in the classroom. We will address the following issues: setting the climate, getting started, simple group structures and strategies, more complex structures, the nuts and bolts, and success stories.

In our classes, particularly in classes with undergraduate students, we use "tests" and "grades" both to motivate our students, and to assess their progress. (It is unfortunate, but true, that some of our students need to think that the activity "counts" in their grade in order to give it their best effort.)

When we ask our students to work in groups in the classroom and laboratory, and when we ask them to get together to do their homework together, how do we test them over the course content? Having used cooperative learning in our classrooms, we have found that we can use all the "traditional" forms of testing with which we are all familiar. And in addition, we have the possibility of group projects, group quizzes, and group exams. This chapter will include some to the ways that we have used to test and assess our students. We will discuss what seemed to work for us, and why we think it worked. We will also include some discussion of things that seemed not to work well, and what we think went wrong.

What are the practical nuts and bolts of implementing cooperative learning in undergraduate courses? What is the impact on the instructor? the students? the department? In this chapter we address practical concerns related to implementing curricular change and its impact on the institution. We discuss our experiences in implementing (or attempting to implement) cooperative learning in our own institutions, and the impact that this has had beyond our own individual classrooms.

Section 1: This section contains an introduction to the chapter. The primary purpose of this section is to establish that learning theory is an important consideration in developing activities that will be effective in a cooperative learning environment. Both our beliefs about how people learn and our beliefs about the nature of mathematics influence decisions that we make as instructors in our classrooms. In this section we explore what some of these beliefs might be, and in the following sections of this chapter we show how we have constructed cooperative learning activities for our students which keep these beliefs in mind.

Section 2: In this section we will attempt to illustrate the beliefs about learning mathematics and about the nature of mathematics which are built into various cooperative learning activities. We will look at some activities which have been used in our own classrooms, and show how these reflect -- consciously or unconsciously -- beliefs about learning and about the nature of mathematics.

Section 3: We have argued above that the activities one chooses to use reflect, consciously or unconsciously, one's beliefs about teaching and learning and the nature of mathematics. We have also argued that an effective way to begin to change the way one teaches would be to adopt a theory of learning mathematics and then design teaching strategies as a consequence of that theory. For example, suppose that an instructor has adopted a constructivist theory of learning based on applying the ideas of Piaget concerning reflective abstraction, and reconstructed them in the context of a particular undergraduate mathematics concept. Now this instructor wishes to construct activities based on that theory. How does one go about doing this? In this section we shall answer this question by describing the development of some specific activities based on particular learning theories.

How can we structure our curriculum in undergraduate mathematics so that our students engage in active inquiry? Is it really possible to expect undergraduates to "discover" any mathematical concepts? And what is the value of having them do this? What is the overall impact on the curriculum when we do this?

The authors will offer four perspectives on the role of guided discovers in a structured, well-developed mathematics curriculum.

We have developed a set of outlines for workshops and presentations on cooperative learning of various lengths. These workshops are intended for collegiate mathematics faculty, and the outlines contain suggestions for getting participants at these workshops involved in cooperative activities. Our underlying philosophy here is that we learn best by being involved in the learning experiences ourselves, and that we tend to teach the way we have been taught.

We will include a bibliography of resources on cooperative learning which are particularly useful for collegiate mathematics.