M.I.T.’s communication-intensive offering of Real Analysis was created in the Fall of 2009 by Joel Lewis, Craig Desjardins, Susan Ruff, and Hans Christianson. It has been refined by subsequent recitation instructors: Todd Kemp, Mohammed Abouzaid, Peter Speh, and Kyle Ormsby.
At M.I.T., Real Analysis is typically taken by sophomores and provides many students’ first exposure to proof writing. Lectures on real analysis are accompanied by a weekly recitation that focuses on various topics of mathematical writing.
This end of term report from the first term describes the context and structure of the course and provides recommendations. In the report, the communication-intensive offering of Real Analysis is referred to by its course number: 18.100C. The concurrent version of Real Analysis that does not have an associated communication recitation is called 18.100B. The report ends with the semester schedule for the first term the course was offered. Here is a refined semester schedule from one of the more recent terms.
At the links below are additional details about the recitations, including lesson plans, handouts, and assignments, as well as additional comments from instructors. The structure of the semester varies each term; the structure shown below is loosely based on the first term the course was offered. All page numbers from Rudin refer to the required text for the course:
Rudin, Walter. Principles of Mathematical Analysis. 3rd ed. New York, NY: McGraw-Hill, Inc., 1976.
Recitation 1: Translating mathematical concepts into clear mathematical language
Recitation 2: Translating between mathematical concepts and notation
Recitation 3: Information order and connectivity
Recitation 4: Strategies for proof writing; guiding text
Recitation 5: Formulating precise questions; exam prep
Recitation 6: When to write conceptually vs formally
Recitation 8: Multiple quantifiers
Recitation 9: Proof-writing strategies and proof structure
Recitation 10: Communicating to different audiences
Recitation 11: Peer review
Students critiqued each other’s drafts of the writing assignment from Recitation 10. Peer review is described at Recitation 7.
Recitation 12 Communicating professionally
Recitation 13 Advanced LaTeX tools for mathematical communication
Recitation 13b: Feedback discussion
Instructor Observations
Comments on each recitation appear on the page for that recitation. Comments that appear here are about the semester as a whole.
Coordination between communication recitations and real analysis lectures
Students respond well to these recitations on the whole, but there is always a danger that the emphasis of the recitations will shift too far toward mathematical communication in general and too far away from real analysis. Ideally the recitation should help students to become better mathematical communicators while reinforcing the concepts of real analysis.
Kyle Ormsby writes, “From the planning stage onwards, the…recitation leader needs to work with the [Real Analysis] lecturer to coordinate the lecture and recitation components. Increasing the analysis content of the recitation will help the students with the material they’re learning and make the recitation seem like a more “legitimate” portion of their education.”
On each recitation page is listed the real analysis concepts that are likely to be difficult for students that week. Whenever possible, communication examples used in recitation should also serve the purpose of reinforcing these real analysis concepts.
Student/Teacher ratio Commenting on writing requires a great deal of time. In order for feedback to be given in a timely fashion, each recitation instructor should have only one 15-student recitation. If a recitation instructor is assigned to two recitations, a grader will be needed. The grader must be carefully chosen and supervised to ensure that the grader’s writing feedback is appropriate.
Require attendance Because so much of the learning happens in recitation, attendance must be required. An effective strategy has been to permit at most one unexcused absence.
Writing Math Peter Speh writes, “I think it would be very useful for students to have an introduction to how to write mathematics down in sentences. This could be done by requiring students to read Kevin Lee’s “A Guide to Writing Mathematics” for one of the first few recitations.”
LaTeX We find that it works well for students to teach themselves LaTeX from the provided resources. At the beginning of the term, a few minutes of each recitation should be devoted to student questions about LaTeX.
Students who are new to LaTeX sometimes spend countless hours trying to get LaTeX to produce documents that look like Microsoft Word documents.
- Tell students to use environments and reuse document skeletons.
- Students should accept defaults, not fight them. Explain that the point of LaTeX is to free mathematicians from having to worry about layout. Most of the time it does what it does for good reasons, such as readability.
- Monitor the amount of time that students spend tex-ing their assignments and intervene as needed.
Provide context: how is LaTeX used by mathematicians these days? E.g., mention journal article classes and style files (so students know we aren’t teaching them a dead tool).
Kyle Ormsby writes, “Provide an actual sample of accessible math writing with LaTeX code. Emphasize that it exhibits how references and theorem environments are used in math writing with LaTeX.”
Homework Problems Students were assigned traditional homework problems on real analysis in addition to the communication assignments described here for the recitations. One of the homework problems each week was written carefully and formatted with LaTeX. The recitation instructors commented on these LaTeXed homework problems as well as on the recitation assignments.
Mohammed Abouzaid writes,
- The first writeup of a homework problem and the first writing assignment were graded mostly on effort and ability to produce a compilable .tex file, with the instructor focusing on giving feedback on the students’ writing. Unfortunately, the students will have a natural tendency to pick the shortest possible problem to submit in .tex. They should be told ahead of time that it is to their benefit to submit a longer problem, so that they can get more comments that benefit them in future assignments. As things stood, the more advanced students, who wrote up long solutions, got the most and the most helpful comments.
- We made the homework assignment be due two days before the writing assignment was. Students, especially those encountering LaTeX for the first time, appreciated receiving comments about their homework assignment before the writing assignment was due.
Proof writing
Students sometimes comment that we don’t do enough proof writing. They have also said that the workload is light enough that an assignment could be added. Some ideas for additional proof-writing assignments:
- Students could revise a pset or exam proof, perhaps of their choice. To encourage students to choose well, the grade could be based at least in part on improvement rather than solely on absolute quality of the final.
- Mohammed Abouzaid writes, “If it fits the schedule, I would consider making an assignment just before epsilon-delta proofs are introduced in class, which consists of rewriting the proof that 2 has a square root in the reals in a way compatible with the expectations of the homework in class. We discussed it in recitation, and the students seemed to understand, but a few of them were just at the stage where having to do it by themselves would probably help.”
- Kyle Ormsby suggests writing timed proofs during recitation, perhaps as exam preparation. Further details about this suggestion are on the page for the exam review recitation.
Room seating for class discussions One day there were only four students, so we all sat around a table instead of using the usual arrangement for class discussions, in which all of the students face the front of the room. Kyle Ormsby writes, “The morning recitation now consists of four students, and it was very interesting to see the small group workshop version of this talk contrasted against the large group interactive lecture in the afternoon. The morning students were definitely more engaged, though I wonder if the afternoon discussion would have been better facilitated if the tables were rearranged so that the students spoke to the group instead of the front of the room.”
Scheduling assignments Be careful about scheduling difficult assignments to be due after a break, since the schedule for office hours may be disrupted.
