There are many potential goals undergraduate research mentors have for their students when leading a mathematics REU project. Students might learn new math, prove new theorems, and write and publish a paper. Less tangible, but equally important, are the transferable skills that students develop. Students can improve their mathematical writing (see Allison’s post on this), their collaborative and independent learning, and their presentation skills. One of the stated goals of Summer Undergraduate Math Research at Yale (SUMRY) is to help students improve their oral presentation of mathematics, and we will explore how we work to achieve this.

## We hope to improve the presentation skills of all students, regardless of what skills and anxieties they bring with them, and we believe that early and frequent presentations of their mathematical progress and results help accomplish this goal.

We know students enter an REU with a wide array of mathematical backgrounds, but they also enter with diverse writing, learning, and oral presentation skills. While most students have the shared experience of presenting a homework problem in class, some students might have been star members of a debate team while others fear public speaking. We hope to improve the presentation skills of all students, regardless of what skills and anxieties they bring with them, and we believe that early and frequent presentations of their mathematical progress and results help accomplish this goal.

Starting in the third week of SUMRY, we have each group give weekly twenty-minute presentations. These usually start as chalk talks and switch to projected PowerPoint or \(\LaTeX\)-ed Beamer talks later in the summer. At first, these presentations are rough, with students maybe defining relevant objects and stating open questions; these presentations are polished throughout the summer and become worthy of professional conferences. This transformation does not occur by magic, but instead through constant feedback from mentors and other audience members. Mentors understand that even if they themselves do not enjoy presenting, their experience listening to and evaluating presentations is valuable to their students. In addition to advising on presentation content, mentors help students develop and incorporate best practices, such as erasing the board completely, addressing the audience, and using more images and fewer words on slides. Peers and other audience members are also excellent resources for feedback. They help speakers learn which ideas have been clearly explained and which explanations need improvement. It is through a constant conversation around presenting that students improve throughout the summer.

### Students Presenting at SUMRY

Of course with so many presentations, we need many audiences. In previous summers, students felt as though they were attending so many talks that they did not have time for mathematics. We now have bilateral presentations, where groups are paired each week and present only to each other. The smaller audience allows for more questions and engagement, and this arrangement facilitates feedback for the presenters. Students present to larger audiences when they travel to REU conferences such as the Young Mathematicians Conference (YMC), and also at workshops hosted by Yale. During the last week of the program, we have final presentations where students present to the entire cohort; this serves as a wonderful venue to celebrate our students, their work, and the summer.

This past SUMRY, I mentored a group of students exploring closed geodesics on translation surfaces. By the middle of the summer, they had honed a presentation of their results and traveled to present at their first conference. Coincidentally, the colloquium speaker who visited Yale the day after the students returned from the conference was an expert in this same field. After her talk, the students successfully engaged the speaker in an impromptu presentation of their results. This prompted a detailed discussion of their project with the speaker that led to new research questions and new ideas for how to approach them. By improving their ability to present mathematics verbally, these students experienced firsthand that communicating mathematics is an integral part of the research process.

Ian Adelstein is a Lecturer in the Department of Mathematics at Yale University. He has mentored several undergraduate research projects in geometry, mostly through Summer Undergraduate Mathematics Research at Yale (SUMRY), an REU program that he directs. In addition to mathematics, he believes that REUs should foster community, and always budgets plenty of money for pizza.