*In this section, Michel Goemans, Peter Shor, Lorenzo Orecchia, and Susan Ruff reflect on the use of recitations to address communicative aspects of mathematical content.*

Our goal was to help students communicate as mathematicians.

—Susan Ruff

Recitations, associated with this course, are weekly one-hour meetings, during which recitation leaders elaborate on subject matter introduced in the lectures and discuss effective communication of mathematics. Students typically complete pre-recitation assignments that prompt them to think about the content prior to the meeting. During the recitations, students and instructors examine the mathematical content of a writing sample and discuss how the content is communicated. Students then apply the concepts they’ve learned in their homework assignments.

The purpose of the recitations is to help students develop a critical appreciation of writing in mathematics. That is, we want students to understand that mathematics *is* communication. We want students to appreciate that the way we understand mathematical concepts is related to the way we communicate those concepts, and that mathematicians have different styles of communication, and thus, of understanding mathematical phenomena.

An appreciation of these subtleties is something mathematicians develop over the course of a career. We realize that it’s unlikely all students will come away from the course with a nuanced understanding of communication in mathematics; nonetheless, the recitations offer a critical arena in which students can begin to develop this important understanding.

18.310 did not originate as a communication-intensive course. It was taught for many years as a pure mathematics class. When MIT began to require that students take communication-intensive courses, we restructured the class to incorporate writing. Our goal was to help students communicate as mathematicians. We wanted students to see communication as an inherent part of doing mathematics.

The problem sets we assigned in the re-structured course included several writing prompts, and they were very time-consuming. We felt this format did not allow students enough time to sufficiently deepen their mathematical knowledge. We came up with the idea of having students attend recitations to help them develop their communication abilities while they delved more deeply into mathematical content.

The process of developing the recitations occurred over several years. In Fall 2011, Professor Peter Shor, our colleague in the Mathematics Department, and Susan Ruff submitted a proposal to the Subcommittee on the Communications Requirement to add recitations to 18.310. Susan proposed the communication topics, while Peter identified the math content students tended to find challenging at different points in the semester.

In Fall 2012, we offered the recitations for the first time: Michel was lead instructor, and Peter and Lorenzo were recitation leaders. We met each week, as a team, to plan, in more detail, the recitation agendas and assignments. Michel created writing samples for us to discuss in the recitations and wrote the corresponding pre-recitation and recitation assignments. Lorenzo's recitations were first, which meant he fleshed out the outlines of the recitation plans. Peter and Susan attended his recitations to learn from what he did.

During Fall 2013, we again taught with Lorenzo. We still met as a team (this time without Peter and with Richard Peng) to plan out what to do in the recitations, including revising the foci of some of the recitations. Again, Lorenzo's recitations were first. He made great improvements to the recitations based on his experience in Fall 2012, and we think the success of the current form of the recitations is largely due to him.

We have found that our most successful recitations are those that authentically integrate subject matter and communication. The “Incorrect Proof” assignment is an example of a recitation assignment that does this in a particularly productive way. This assignment involves a brief probability proof, in which probabilities are multiplied because independence is assumed without that assumption being explicitly stated. That assumption is incorrect. So the mathematical purpose of the assignment is to increase students’ awareness of the importance of independence when multiplying probabilities, while the communication purpose is to illustrate the importance of explaining assumptions in sufficient detail to ensure that proofs are rigorous. Students revise the proof to practice this skill.

During the recitation, students discuss their work and talk about the fact that it is difficult to “catch the bug” in the incorrect proof. They discuss the communicative aspects of the proof that contribute to this difficulty. Specifically, the recitation leaders help them attend to the use of the slightly ambiguous notation in the proof. The need to define notation properly becomes very clear. This assignment helps students develop precision and rigor in their writing.

In general, Lorenzo has found that it is particularly important that students feel that the communicative aspects of the recitations help them better understand the focal mathematics. This means that the mathematical content presented in the recitations has to be sufficiently challenging. If the content is too simplistic, students tend to tune out. They need to sense that attending to the communicative aspects of the mathematics, as in the “Incorrect Proof” assignment, will help them understand the mathematics more deeply.

It’s also helpful for students to have encountered mathematical concepts once or twice before focusing on communicative aspects of the content. Having familiarity with the content allows students to think about the nuances of the focal problem—and frees them to think about how communication and content intersect in important ways. This means that educators considering transforming a course into a communication-intensive one might consider placing it later in the sequence of required courses, as opposed to earlier in the sequence, so that students have an opportunity to develop familiarity with several fundamental concepts before attending to the subtleties of mathematical communication.

Finally, student engagement is key to successful recitations. Requiring that students complete pre-recitation assignments helps us foster engagement because it enables us to ask students questions about the material right away during the recitations. The assignments also prepare students to come to the recitations with their own questions. Lorenzo has found that creating time for students to work in groups also promotes engagement during the recitations.

Peer review is an important part of the writing process in mathematics. Serious peer review entails much effort and impacts published pieces in critical ways. Thus, as part of the recitation sessions, we ask students to review each other’s term papers. Students read each other’s writing outside of class, submit their comments as part of a homework assignment, and discus their work in groups during the recitation.

We have found that students tend to be a bit cautious about making comments that are critical of others’ work. Lorenzo has sensed that, in the past, students have not been as invested as they could have been in the success of other students. One way to incentivize students to feel more invested might be to make their peers’ term paper grades count for a portion of their own grades, but we’re unsure about how fair this approach would be.

Another idea for improving our peer review assignment might be to break the process down into smaller components. We can imagine, for example, asking students to review each other’s introductions during one peer review session, and other sections of the paper during different sessions. This would help students’ focus their comments and would reflect the true effort that is involved in critically reviewing a piece of writing prior to its publication.