Including participation credit in the final grade calculation is a matter of some concern to me. I have found it can foster competition regardless of the classroom climate. It may also disadvantage some students who are uncomfortable. For these reasons, I now use participation, including observable improvement over time, to adjust grades upwards if a student's grade is borderline.

Quizzes
Regular in-class quizzes are given. For the first two, students are given the option of submitting an error analysis. This error analysis provides students with an opportunity to improve not only their grade but also their comprehension of the course material. The first grading is fast; I simply indicate where the student has made a mistake and award a mark, with no explanation. Grading on the error analysis is much more time-consuming; the importance of having students do the analysis far outweighs its value in increasing their grade. An early failure in a course can be very dispiriting. But also, a test should be a vehicle for learning, not solely an instrument of evaluation. Many students told me they appreciate the opportunity to learn from their mistakes without being penalized for making them.

Student participation in setting the final examination
I believe in formal examinations--a final performance, an opportunity for students to pull the course together, to see it as a whole and to demonstrate what they have learned. Last year, I experimented with making students responsible for setting 50% of the final examination. Time was devoted in class, about two weeks before the end of term, to reflect in small groups on the course content and brainstorm areas for examination.

Conclusion

For the past two years, I have been developing my approach to participatory teaching in a small third year Algebra course for majors. However, my first attempt at implementing these ideas was in a second year Finite Mathematics course for non-majors. It was a multi-section course; mine had sixty students enrolled and no attrition. The two other sections were slightly smaller and both instructors employed an exclusively lecture approach. My main objective was to prove to my colleagues that I could teach sixty students without lecturing and without consequently compromising standards or disadvantaging the students. Measured by a common final examination, the three sections of the course had almost identical class averages. However, students in my section obtained more A grades overall (27% compared with 22% and 19%) and only 15% of students in my section failed the examination where 24% and 30% failed in the other two sections. Covering the course content presented no difficulty for me or, judging by their examination success, for the students. Over two-thirds of them participated actively in the course in one way or another and only one student complained of discomfort.

One cannot conclude that the lecture method came out second best judging by the examination criteria alone; but it could be claimed that my "coaching" method held up well against the traditional approach. It is likely, however, that the impact of participatory, democratic teaching methods is not readily evaluated in the short-term nor by quantitative means.