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.

Peer Critique

Context: This lesson plan is from a weekly communication recitation that accompanies M.I.T.’s Real Analysis (18.100C). This week students learn about continuity and compactness (Rudin pp. 85-93). The material is relatively easy, but students may have trouble with writing epsilon-delta proofs. This recitation occurs after students have written drafts for this proof-writing assignment.

Authors: This recitation was suggested by Susan Ruff with refinements by Kyle Ormsby. The skit was suggested by Katrin Wehrheim, and was developed by Katrin, Susan Ruff, and Joel Lewis. The skit was converted to a handout by Kyle Ormsby.

Communication objectives: Give and receive collegial peer critique.

Recitation Logistics

Assign peers to each other so students working together have written on different topics.

Split the class time evenly and give students time warnings so each student has the same amount of time to give and receive critique (but don’t require students to stick rigidly to the schedule: some papers will need less time than others).

Should critique be given in class or outside of class?
The exact logistics of this recitation vary from term to term depending on scheduling: sometimes students read their peers’ writing before class and write a formal critique to discuss in class, sometimes students read before class and simply jot notes to themselves to guide in-class peer critique, and sometimes students begin reading and critiquing each other’s writing in class and complete the critiques outside of class. For the advantages and disadvantages of each, see the logistics section of the peer critique page.

If you have students begin reading each other’s papers in class. Warn them that they won’t have time to finish reading, so they should read and comment on only the introduction (or other suitable chunk of text).

What peer group sizes should be used?
In the past, students have been paired so that each student gives a critique to and receives a critique from one peer; however recently Kyle Ormsby moved this class toward a writing workshop model, in which each student gives a critique to and receives a critique from multiple peers. Kyle used group sizes of 3-4, and this group size worked quite well. An advantage of using groups instead of pairs is that the effects of individuals’ idiosyncrasies are moderated: authors can get a sense of the degree of consensus among the critiques.

Should the instructor provide comments as well?
When we’ve provided instructor comments on the same draft that peers reviewed, the comments from the staff diluted the value of receiving comments from peers. On the other hand, when we haven’t given instructor comments, peers have missed important mathematical issues. A good middle ground has been for the peer critique to be preceded by the instructor doing a brief read for content issues only and to then requiring students to do quick content revisions only if needed. To avoid adding too much time to the revision schedule, the turnaround for the instructor’s quick read and the content revisions is fast: both occur in the week that precedes the peer critique.

What logistical problems are common?
Students may be sensitive about their own writing and can take critique personally. Monitor the tone of groups and intervene if needed.

Tell students to bring a printout of their writing to class.

Be prepared for how to handle students who don’t bring printouts, don’t finish their preparation on time for class, or don’t come to class. For peer critique to work equitably, everyone must do the critique and everyone must attend, so consider setting a grade penalty for attendance or late homework for this class if you don’t have one for the semester as a whole.

Also decide how to handle peer critique if the chosen group size doesn’t evenly divide the class size (e.g., if a class has groups of 3 and 4, will some students read for two peers and some for three, or will one student in each group of 4 be sitting idle while the other two peers give comments on the paper for the fourth?).

Skit providing guidance for collegial peer critique

Before students give each other face-to-face peer critique, we provide guidance for doing so collegially by using a skit to model common mistakes and how to avoid them. The following skit is from the Fall 2009 term, with Joel B. Lewis, Craig Desjardins, and Susan Ruff. In the skit, Craig critiques  Joel’s writing and Susan provides commentary. The writing sample was constructed for the skit.


Scene 1: Brief superficial comments (e.g. spelling) followed by “looks good”
Commentary: Goal is to help peer improve writing. To do so, you must be willing to criticize. Author: you can help commenter by encouraging him/her to point out ways in which you could improve the paper.

Scene 2: Comments are critical, general, not constructive
Commentary: Goal is to help peer improve, so be constructive: comments should be specific enough so author can identify how to address the issue. Give suggestions for improvement if necessary to clarify point. (But it’s up to the author to decide how to solve problems.)

Scene 3: Good comments, but author responds defensively by explaining why he did what he did.
Commentary: Author: your job is not to defend your work but rather to understand the problems identified by your peer. You don’t have to follow your peer’s suggestions, but to make an informed decision you must be sure you understand your peer’s concerns/suggestions. Practice active listening.

Scene 4: Good comments with good active listening response.

(optional scene 3.5: author and peer try to fix the problems together during the critique. Commentary: not enough time for that, and it’s the author’s job to revise, not the peer’s)

Commentary and alternative to skit

Kyle Ormsby writes: “This recitation was devoted to small group peer review.  Eschewing previous years’ skits, a small handout was prepared illustrating (in)effective peer review through sample dialogues.  Students were asked to read through the dialogues on their own as they entered the class, and a quick (5 minute) discussion was held to determine what was good and bad about each scenario.  I think this worked quite well and got the students thinking about how to interact with each other during the reviews.

“Peer review went smoothly, though even with groups of four with two reviews per author, students finished before the end of the recitation.  It’s tempting to add ten minutes of material into this recitation, but I think it would distract from the primary objective:  effect peer reviews.  I used the extra time to answer student questions.”


Peer critique assignment Whenever scheduling permits, students are assigned to write a formal critique outside of class in addition to giving some face-to-face critique in class. Here is one such assignment, briefly stated, and here is another version of the assignment, with more guidance for students.

When a formal written critique hasn’t been assigned, students have commented that the peer critique they received was not very helpful because the critique wasn’t being graded so it wasn’t taken very seriously. On the other hand, when a formal written critique is assigned and graded, students have said that giving and receiving peer critique is helpful.

Revision assignment After students have received their peer critiques, they are assigned to revise their writing.

We briefly gave guidance for how to handle conflicting comments &/or an overwhelming number of comments.

  • Guidance for handling conflicting comments:
    Often academic peer reviewers give conflicting comments too. You are the author: be sure you understand the reasons behind the conflicting comments. In this class you have the luxury of asking for clarification if necessary. Then choose the solution that you think best addresses the underlying causes-to your satisfaction.
  • Guidance for handling an overwhelming number of comments:
    Go through the comments with a highlighter and mark the most important comments that require the greatest changes. Address those first. As you address a comment, but a checkmark next to it.


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