Some notes on my teaching at Princeton

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.

• Group midterm. In addition to a traditional midterm (either in-class or take-home), I give my students one group exam. The students work on the test in groups of three; the test is take-home and open-book, and calculators and computers are allowed. Usually I give the students about a week. Consequently, I can make the problems much longer, deeper, and more computation-intensive than typical exam problems. Here are two examples: the the Spring 2000 group midterm for Math 204, and the Spring 1999 group midterm for Math 104. (Note: the absence of a Xeroxed handout makes problem 1 on the 2000 exam somewhat hard to follow.) The exams have four questions, of which each group chooses three.

  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.)

  • Q: How do you assign the groups? Why don't you let them choose their own groups?
  • A: I try to put students who live in the same dorm together, and I try not to make groups in which all three students are struggling or all three are acing the course. If it's the second half of the term and I have a better sense of the students' personalities, I try to put people together who I think will get along.
    I choose the groups myself because I imagine the process of undirected group formation could be, let's say, socially awkward. My desire that no one feel left out trumps my general feeling that students should make decisions about the course for themselves.
    
    
  • Q: Are you worried that one student will do all the work?
  • A: Not so much. If student A tries do a problem and can't, and student B shows student A how, and student A writes it up, I think both students benefit from the question.
    
    
  • Q: But what if student B does all the questions, writes them up, and then gives the solutions to the other students to copy out?
  • A: That would be bad. It's easy to imagine student B pressuring student A to copy out a correct solution so that student B's grade won't suffer. One reason I ask student A to write the process description is that I think it encourages student A really to make student B explain the solution, so that student A can explain it in his or her own words. Of course, student B could also give student A a process description to copy; my feeling, or maybe my hope, is that student A would balk at this point.
    I should say that I don't think this situation has arisen for me. I should also add that Princeton operates on the honor code, which means that students sign a pledge on the exam that their work is their own; it would be interesting to know how the presence or absence of such a pledge affects the dynamics of a group exam.
    
    
  • Q: Do students complain that it's unfair for their grade to be dragged down by a groupmate who didn't contribute?
  • A: It's happened a few times. I've had one case in which two of the group members refused to start working on the exam until the night before; but this is one group out of 45 or so who have participated. By and large, my impression is that even people who loaf in other aspects of the class put in the work on this exam, to avoid letting their classmates down.

    (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.
    
    

  • Q: Do the students really work collaboratively, or does each student pick a problem, write it up, and not look at the others?
  • A: I don't know. From the process descriptions I can see that significant collaboration took place in some cases; but it's not clear that it's the rule. I can encourage collaboration, but I can't enforce it.
  • Q: What exactly do students write on the "process description"?
  • A: Sometimes students write very useful narratives that give me insight into the process of solving the problem. More often, the description isn't so interesting. Sometime they just don't write it. I'll try next time to be more precise about what I want. If that doesn't work, I'll drop this part of the exam.
  
  
  
• Breaking the flow of lecture. Common sense, cognitive psychology, and my personal experience agree that, no matter how interesting a lecture, students' attention is much keener in minutes 1-10 than in minutes 40-50. I've found it makes a big difference to break every lecture up with some kind of active or reflective activity. For example:
  • I write an assertion on the board, like "If a matrix is diagonalizable, it has distinct eigenvalues." I ask for a show of hands on whether the assertion is true or false. Then I ask each student to turn to the person sitting next to them and try to convince that person of the truth or falsity of the assertion. After one or two minutes, I bring the class back together for another show of hands, followed by a brief discussion. I think the class finds it gratifying that the process tends to converge on the right answer; it seems to me to emphasize the useful lesson that something is mathematically correct because it can be argued, not because I say so. (I first heard about this technique from Eric Mazur at Harvard; I'm not sure whether it's original to him.)
  • I write an assertion on the board, as above, but ask students to think quietly about it for a minute; then we discuss what people came up with.
  • I ask the students to turn to their neighbor and try to produce as many examples as they can of a mathematical object satisfying some properties; say, "think of as many matrices as you can which are both upper triangular and lower triangular."
  None of these activities takes more than five minutes out of the lecture, and I find they are very useful in keeping the students awake, alert, and involved in the course material.
  
  
• Tactics for learning students' names. This may seem trivial, but I think it's vitally important if you want to establish any sense of collegiality in the classroom. I have two ways of getting this task done quickly. Note that at Princeton we teach in sections of no more than twenty-five students; these techniques presumably won't work as well if you have a hundred kids in front of you, and in that event learning the names may not be so crucial.
  • For the first week or so, I ask that anyone who says anything in class start with "I, so-and-so, want to know..." It seems silly, but it does work.
  • On the first day, I pass out index cards on which I ask the students to write their names, e-mail addresses, phone numbers and hometowns, and to draw a picture of themselves with any distinguishing features emphasized. This doesn't take long and is an extremely helpful way for me to learn who the students are (and what features of themselves they consider most distinguishing.)

Things that worked pretty well.

• Homework rewrites. In Spring 2000, I tried out a new homework policy which I learned from David Carlton. Instead of returning homeworks with a numerical score, the grader simply marked each question correct or incorrect. If a question was incorrect, I asked for a rewritten solution by the following week.

  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.
  
  

• Groupwork. I've occasionally broken the class into groups and given them a problem to work on for 20 or 30 minutes while I rove around offering advice and support. I've never really taken the plunge and made groupwork a big part of my course; students seem to like it at the time, but seldom mention it either positively or negatively at the end of the year. If you're interested in groupwork, you should go straight to David Carlton's teaching page..

Things I haven't dared try.

• Self-grading. I thought it would be interesting to let the students tell me the grade they thought they deserved at the end of the course; that would then count for 20% or 25% of their grade. The rationale is that their self-assessment is, in fact, very relevant to my best estimation of how much they accomplished in the course. The reason I haven't dared try it is that I'm concerned it would create too much stress and dread for the students as they wondered, "Am I supposed to be honest or am I supposed to get the highest grade I can...?" If anyone's tried this, I'd love to hear about it.

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