*In this section, Prof. Haynes Miller and Susan Ruff describe the criteria for presentations and the components of the presentation workshop.*

The course fulfills the CI-M requirement at MIT, and as such, offers instruction in mathematical communication. As part of that instruction, each student team is required to give a fifty-minute presentation on one of its three projects, and these presentations are intended to parallel the seminar lectures that form an important mode of communication in the discipline.

Effective presentations provide motivation, communicate intuition, and stimulate interest, all while being mathematically accurate and informative. As is true with their experience with mathematical writing, many students do not enter the course in possession of the tools to do much more than present the facts. For example, students often come to practice presentations with the mistaken belief that a mathematical presentation must be extremely formal throughout, every term must be rigorously defined, all facts must be proven, and pictures are too infantile for this level of presentation. We try to counter these preconceptions and urge flexibility and a sense of appropriateness: sometimes things need to be presented rigorously and formally, but sometimes a picture, conceptual explanation, or example is much more effective.

Characteristics of an Effective Undergraduate Research Talk (PDF)

For the presentation workshop, which typically lasts 50 to 80 minutes, we begin by having the two co-instructors each give a short mock presentation. These presentations are designed to address common student misconceptions about mathematics presentations. For example, to help students realize that presentations should not be relentlessly formal, the first presentation might be good in every way except that it is dull and difficult to follow because it is unnecessarily formal throughout. In contrast, the second presentation might cover the same material but use examples and figures to introduce some concepts informally, while reserving rigorous formality as a strategy for clarifying and solidifying the most subtle or important concepts.

To help students recognize the value of the second presentation relative to the first, after each presentation we ask the students a question designed to check their understanding of the content. The goal is to allow students to discover their natural tendency to overlook weaknesses in presentations. When they try to answer questions about it, they may discover that they got less from it than they had thought. The second presentation is then intended to offer a more understandable approach to the same material. Of course it's the second time students will have heard this material, so they will naturally understand it better. But this serves a pedagogical purpose too, as it reinforces our point.

We follow the presentations with a class discussion on how to give a good presentation. Carefully designing two mock presentations has the virtue of drawing attention to key learning objectives, but doing so is challenging. In Spring 2013, each mock presentation was delivered by a different instructor and so had different advantages and disadvantages, as is stressed by Haynes' comments on the workshop (PDF). In the past we have reduced accidental differences between the presentations by having a single instructor present both, and we may return to that approach in the future.

After the mock presentations, the class discusses the characteristics of a good presentation. Questions we discuss often include the following:

- What are the reasons to include a proof in a presentation?
- What other strategies are available for achieving these goals?
- What strategies can be used to make a math presentation engaging for the target audience of math majors?

In Spring 2013, the mock presentations ran long, and the class session was shorter than we had originally planned because of scheduling disruptions at MIT. Thus, the subsequent discussion was rushed. The presentation workshop works best when there is ample time for discussion.

We hope that students come away from this workshop with an appreciation for some of the complexities in designing a good presentation. Pretty much every choice involved has both pros and cons.

This video features the presentation workshop from Spring 2013. The co-instructors deliver mock presentations, which are followed by a brief class discussion comparing the two presentations.

Different instructors have set different expectations for the presentations. Some have insisted on slide presentations. More typically, students are encouraged to use media suited to the demands of the presentation.

When discussing slide presentations in mathematics, we usually make the following points:

- When slides contain large amounts of text (or equations), the audience cannot read and listen at the same time, so strategies are needed either to reduce the content on the slides or to guide the audience through the content.
- The audience needs time to absorb math concepts, but it is very easy to click through slides too quickly, especially when the presenter is nervous, so strategies are needed to give the audience time to think.
- The audience cannot refer to past slides to remind themselves of the meaning of new notation or of the purpose of details being presented, so strategies are needed to help the audience remember important points.

In the course, roughly one group presents each week. Experience has shown that the first team to present sets the bar for the rest of the semester. It is important that the first team be chosen carefully and be guided well so that they give a strong presentation.