Mathematical Communication is a developing collection of resources for engaging students in writing and speaking about mathematics, whether for the purpose of learning mathematics or of learning to communicate as mathematicians.

Critiques–Steven Kleiman

[The following notes are about Steven Kleiman’s 2010 Undergraduate Seminar in Computational Commutative Algebra and Algebraic Geometry, in which the students give the lectures.]

This spring of 2010,  I tried something new: in-class critiques.  They worked out very well, far better than I  ever expected.   When the students arrived for class, I gave them each a standard sheet of blank paper.   During each student’s lecture, the others then wrote a critique, describing what was done well and what needed improvement,  so as to reinforce good practices and suggest opportunities for growth.  The critiques are marvelous — friendly and constructive, addressed directly to the lecturer.  Between classes, I or my TA made copies, with the writers’ names covered up,  to give to the lectures next class.  We kept a record of who wrote about whom, really just an attendance record, but we made no attempt to grade them or even comment on them.

The critiques provided each lecturer with valuable feedback,  which is their primary purpose, of course.  However, they had several other unexpected benefits.  First of all, they provided the students in the audience with an added reason to come to class.  As a result, for the first time in twenty years, attendance was nearly perfect; the exceptions were for medical reasons, for job interviews and grad school visits, and  for that occasional extra heavy work load in other subjects.  Furthermore, the audience paid greater attention to the lectures.   So there was more discussion of the subject matter during the lectures than ever before, with questions from the audience, answers from the lecturer, and comments from myself.  In addition, the class become more sensitive to the qualities of good lecturing,  as the students paid attention to the performance of their peers.    Consequently, this term, the lectures were the best ever.

Here is a fictitous review, made by Susan Ruff and myself, on the basis of actual reviews.  It can serve as a model to be handed out on the first day of class.


Most of your lecture was clear and easy to follow.  I like your calm,
collected style.  You projected a good dose of confidence and comfort,
both when explaining the material and when fielding questions.  It made
me feel that you were really comfortable with the material.  Also, you
opened up, faced the class, and made eye contact.  So it felt more like
a discussion than a lecture, and questions felt a lot more welcome.

You did a good job cutting the book’s material into a concise form
suitable for presentation.  You always said why you were presenting
information; so I never felt lost.  The first example was very helpful,
but should have been done before the general algorithm, which was hard
to follow.  Also, I had some trouble “believing” the theorems; so some
examples would have been useful to illustrate *why* they are true.

Sometimes there were pauses to think in the middle of an argument,
disrupting the flow of ideas.  Pauses aside, the pacing was good, but
maybe a bit slow.  Slow is ok, even perfect if there’s enough time.
But the final part was a bit rushed, so hard to follow.

My main complaint is about the board work.  It was all neat, clean, and
readable.  Also there wasn’t too much detail, just enough to help with
the explanation.  But the space could have been used better.  Sparce
writing all placed in the middle meant frequent erasing.  Once, a new
board was pulled down, but the other not pushed up; so the old writing
was hidden.  Also, it’s good to speak as you write; when you are in
front of an equation, saying it is helpful.

License: CC BY-NC-SA Page content licensed by Steven L Kleiman under the license:
CC BY-NC-SA (Attribution-NonCommercial-ShareAlike)