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
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
(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?
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
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
Q: What exactly do students write on the "process
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
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
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
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
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
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.