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.

Exam review

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 completeness and sequences and series (Rudin pp. 42-43, 47-69, 71-75). There’s an exam next week on the first third of the course. Likely trouble spots for students at this point in the term include infinite series and preparing for the exam.

Authors: This recitation was suggested by Susan Ruff. The description of the recitation below is by Kyle Ormsby.

Communication objectives: Formulating precise questions; informal oral communication (communicating to learn)


“We held a question discussion during this recitation.  Students were required to submit two precise questions on Stellar by the night before, I compiled these into a single document, and I then asked students to work in groups to determine which questions were easy/hard/important/irrelevant and discuss ones that the group was interested in.  Susan and I bounced between tables providing help when necessary; we also made sure students remained focused, but they mainly managed this on their own.

“I think this is a useful form of review, and I saw several examples of excellent peer-to-peer instruction.  Before the next exam I plan to repeat this type of question discussion, though I’d like to use something like Google Documents in order to create the question list.  This should accomplish several tasks, such as (a) making sure that at least some of the students read other questions in advance and (b) not requiring the instructor to compile questions manually.”

After holding a similar review before the second exam, Kyle wrote, “Another question discussion was held.  This time, questions were collected via a shared Google Document.  Response was good although the questions were still not great and early submitters did not go back and review later submissions.  Overall, there may be a more effective way to use this class time as exam review, though I’m not sure what to do.”

(In other semesters, only the first half of the recitation has been devoted to small-group discussion of questions. In the second half of the recitation, the class discusses as a whole those questions that groups were not able to answer on their own, with the recitation instructor providing explanations as needed. –Ed.)


The recitation is preceded by an assignment that has students write precise questions about analysis for discussion.

Depending on when during the following week the exam is held, there is often no recitation homework for the following week so students can focus on studying for the exam.

Instructor observations

Kyle Ormsby suggests including some timed proof writing in the recitations. The timed proof writing could be used for exam prep as follows (this idea has not yet been tested):

  • Begin by briefly giving some suggestions for timed proof writing.
  • Give students a timed sample exam problem to answer in class. I recommend at least two problems, one “easy” in the sense that its solution only depends on a thorough understanding of the concepts involved and one requiring some amount of insight. This will allow students to choose a problem they can actually make progress / cut their teeth on depending on skill / experience level.
  • When time is up discuss the solution to ensure students understand it.
  • Have students swap papers–students are now graders and should grade each others’ solutions, with an eye to what about the writing makes the solution easy or difficult to grade.
  • Students briefly give each other feedback.
  • Have a class discussion based on students’ observations.
  • Use remaining time to discuss content for the exam.
  • Additionally, revisions of these in-class writing assignments could be given as extra credit assignments post-exam.


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