Since I came to Princeton, I've taught six sections of Math 104 (second-semester calculus) and four of Math 204 (linear algebra), as well as some more advanced courses. On this page, I'll record my observations about various techniques I've tried out with (on?) my students.
I welcome any comments from others who have tried, or want to try, the techniques described below; I am especially interested in comments from students (mine or others) who have experienced these techniques from the other side. Please e-mail me!
Things that worked really well.
The students are supposed to work collaboratively on the problems. Then, at the end, each student chooses one problem to write up. I ask that the writer include not only a solution to the problem, but a brief description of the process by which their group arrived at the solution.
I've been extremely happy with the results of these exams; the students tend to write lengthy and thoughtful answers which often go beyond the boundaries of the specific questions asked. Many students have commented in evaluations that the group midterm was the most enjoyable and most mathematically intense part of the course. I intend to give a group midterm in every elementary course I teach. On the other hand, I had hoped that giving the group exam early in the course would establish a pattern of collaboration, making students more likely to do homework together, study for final exams together, and talk with each other about math outside the class hour. I don't think this has really happened. I'm still looking for ways to help it happen.
(Fall 2001) The group exam went poorly in my calculus course this semester; I got several complaints about dysfunctional groups during and after the test, and comments about the format on end-of-term evaluations were overwhelmingly negative. We made the exam much too hard, which is surely part of the problem. But it's possible this format works better in a more advanced class like linear algebra than it does in calculus, where depth is not as central a goal.
Here are some questions I imagine one might ask about the group midterm (because they are the questions I ask myself.)
(Fall 2001) See above: if the exam is too hard, many more complaints will arise.
There's no question, of course, that the group midterm means that I
assess students partly based on other students' work. To students who
find this inherently unfair, I can say only that mathematical
collaboration is one of the skills I hope to build in the course, and
that I don't know how to assess this skill on a purely individual
basis. Of course, a student who is concerned that his groupmates'
work will bring the grade down should be encouraged to read through
and check his partners' work.
Things that worked pretty well.
It seems to me that many students's response to a traditional graded homework is to look at the number on the front and then put the assignment away in a folder. In particular, I think it's not common for students to spend time thinking about the problems they got wrong, which is precisely where they stand to learn the most. The main point of the homework rewrite policy is to encourage students to return again to the more difficult problems, armed with the superior understanding coming from a week's reflection or conversations with classmates.
I also liked the idea of removing the "numerical judgment" facing the students each week on their graded assignments; these numbers have a ring of finality which seems to me inappropriate for homework. My expectation was, given the chance to rewrite problems, most students would end up with near-perfect homework grades.
Overall, I was glad I used the new policy. By and large, students really did turn in rewrites; I was worried many wouldn't bother. Many students were able to go back and correct problems they'd gotten wrong at first, and this seems to me to be a very productive task. But I don't think there was a lot of discussion with classmates, and many students turned in rewrites which were still incorrect. This led to further problems, because I hadn't adequately worked out a "rewrites of rewrites" policy. Next time I teach, I'll make the deadlines and rules much more clear.
To my surprise, the final numerical homework grades were about the same as they were when I used the traditional system.
You should be aware that this policy significantly increases the
work of grading, since one has to provide comments on the
questions requiring rewrites, and many questions will be graded more
than once. If your grader is someone other than you, you should think
about assigning fewer questions or arranging for the grader to be paid
extra.
Things I haven't dared try.
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