Giving up lectures: Lecturing is a very satisfying activity for us. We can organize the materials, display the logical structure, and introduce just the right examples. When the period is over, we feel that we have given a good account of ourselves, a good performance. It is hard to accept that this may not be the best way for students to learn. However, respectable educational research suggests that prolonged lecturing is the classroom activity least likely to result in learning.

Giving up control: When you move to a classroom mode that is more student-directed, you will feel as if you are giving up control of the class. You cannot necessarily tell what students will want to discuss, what suggestions they will make. It is likely that students will ask questions for which you do not know the answer. This is painful the first couple of times it happens. However, it is very enlightening to the students. When you say, "I don't know what happens if we let b=10,000; let's try it and see," they realize that it is not a personal failing when they have to admit that they don't know the answer. To become good learners, they need a good learner as role model.

Listening to students: One of the things you will learn as you read student papers and hear them discuss topics in their groups is that they are not thinking what we thought they were thinking. They have some surprising ideas fixed in their minds; a new, more useful concept cannot take the place of a faulty one until the old notion has been dealt with.

This course has taught us to work harder at listening to what the students are telling us, not assume that we are going to hear or read one of the most "five common errors." One of the instructors told us that he always thought his students learned from an organized, interesting, informative lecture. He had always asked a few questions to be sure that the students were understanding. When he started to use cooperative learning in the classroom (after giving one of his thoughtful, informative lectures), he got a chance to listen in while students told each other what they thought they understood. He was amazed at how little of his careful, clear lecture they understood.

Dealing with writing: Many mathematicians admit that they do not like to deal with student writing. One of the reasons they went into mathematics is that they did not like to write a lot of papers! The type of writing we are expecting of the student, expository writing, is not particularly hard as writing goes. Previous instructors have found themselves to have the ability to improve their students' writing level dramatically simply by stressing the importance of carefully written solutions.

When you are working with student writing it helps to be very explicit about what you want students to do -what you expect an assignment to look like. You will find that the more writing the students do the better they will get. This is especially true if you ask various students to read what they have written out loud to each other in small groups and to the large group.

Working too hard: It is hard teaching a new course for the first time; this course is newer than most. Instructors commonly report that teaching this way requires more emotional energy than teaching traditional math classes. If you actually know whether or not students are learning, you tend to maintain a higher level of personal engagement - you worry about them.

There is a tendency in dealing with things we are unsure of (grading student writing, for example) to compensate by being too conscientious, spending too much time on the work. At the beginning, it is helpful to allow a limited amount of time for dealing with each paper. Nothing terrible happens if you do not make a comment that you might have made if you had more time. The students understand time pressure.