Notices of the American Mathematical Society
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Giving Talks for Undergraduates
Math talks for undergraduates pose a unique challenge. Only a portion of the audience is conversant on any given advanced math topic, and most are still formulating their career goals. Students are often promised pizza, cookies, coffee, or extra credit for attending. So, they arrive happily munching on snacks, but they may not (yet) care about whatever topic you chose to speak about. Crafting a presentation for this audience is a tall order, but with planning and creativity, undergraduate talks can be rewarding for everyone, including you.
Half the Battle: Find a Good Topic
It’s tempting to repurpose your research-centered talks for this crowd, but be cautious of this default option. Even if you can blaze a path from college-level math to your thesis results in one hour, it may not work well. Good topics typically have minimal prerequisites and also catch the interest of an unsuspecting math student. Don’t rush; finding a fun topic for students may take time.
Keep your eyes peeled for lesser-known mathematical gems or good storylines near your area of expertise. Spend time fleshing out the possibilities and be selective. Ask yourself: Why would this topic be fascinating to a student? If your answer is high-level (e.g., “This ends up being an important tool in studying open book decompositions of 3-manifolds.”), then keep looking!
Be honest with yourself about the prior knowledge that a topic requires. Don’t minimize a topic’s difficulty using the 20/20 vision you gained in graduate school. (“They just need to know what a faithful representation for a finite group is. I can explain that in a minute or two, right?”) Also, stay away from topics with heavy notation. If you charge ahead with abstract diagrams, subscripts, and the full Greek alphabet, you will leave a portion of your audience behind. The tradeoff isn’t worth it.
One of the best ways to develop a sense for good topics is to attend lots of undergraduate talks yourself. Go to the events at your own institution and check out great expository presentations at conferences and on YouTube. Reading expository math publications is also useful (such as Math Horizons or The Mathematical Intelligencer). Over time, you’ll see which types of topics work well and which fall short.
Craft a Storyline
After you choose a topic and before you make your slides, craft your talk’s storyline. See
Your goal in the introduction is to make the atmosphere lively and welcoming. It’s tempting to start your talk with definitions. We often begin research talks this way, right? For inexperienced students, nothing will kill interest faster. Instead, consider beginning with a hook that will shock or delight your audience. Give a fascinating example, tell a story, discuss a funny analogy, connect to real-life problems, or invite audience participation (and learn the names of the students who participate, if you can). If possible, you want students to feel a concept deep in their souls before it is formally defined. Or, you want the students to ask themselves a question about the topic before you ask the question.
Once the stage is set with your introduction, keep engaging the audience as much as possible. Remember, your audience is attending this talk voluntarily (plus or minus some free food), and they do not bring much prior knowledge to the table. So, enthusiasm and creativity will be appreciated. If you use slides, include plenty of color and relevant pictures. Avoid wordy slides at all costs. Don’t fall back on the standard beamer template unless you must. Insist on finding the best possible explanations of ideas.
Students are not accustomed to math marathons, so plan to “come up for air” frequently during the presentation. Include historical notes, humor, analogies, or personal ties that you have to the subject. Perhaps take a moment during your talk to have the audience members discuss a low-stakes question with their neighbor. Along the way, build several on-ramps back into the talk for students who get lost midway through. You might mention something like, “If you were confused during that discussion, then right now is the perfect time to jump back in! We are going to look at a beautiful example.”
People often say that the last fifteen minutes of a talk should be for experts only. Your audience includes zero experts, so steer clear of this advice. Craft a coherent conclusion that reiterates the presentation’s goals and extends an invitation to the students to keep thinking about the ideas (we’ll say more on this later).
A professor once told me, “No talk is so good that going over time will not make it a bad talk. And, no talk is so bad that ending on time will not make it better.” This is particularly true for undergraduate talks, so end your talk with a few minutes to spare.
A Few Ideas for Finishing Touches
Don’t hesitate to make your title and abstract inviting. Consider the following titles: “The second meanest problem of the 20th century” and “Ode to a blank sheet of graph paper.”Footnote1 These titles pique curiosity! The title “An introduction to the generalized Poincaré conjecture” is serviceable, but it does not exactly inspire curiosity.
These are two actual presentations given by my colleague Paul Zeitz, who is one of my role models for good talks.
When speaking to undergraduates, it is nice to “lift the curtain” on the mathematical process for them. Discuss the struggles and mistakes that were made along the way, if you can. This enables students to envision the development of math as it truly happens—with progress and struggle mixed together.
At the end of your talk, give students options to engage further. An easy way to do this is to mention a resource where students can read more about the topic. If possible, mention a resource that they can access immediately using the smart phone in their hands. You might also invite students to consider summer math programs or another relevant student-centered opportunity. Some students need to hear invitations such as these from lots of different people before they internalize that these options are for them.
Enjoy the Experience
Creating engaging presentations for undergraduates is fun, particularly since there is no pressure of assigning grades afterward. Enjoy this opportunity! Your speaking style may differ from some of the fine details we have discussed here. For example, maybe your thesis results make for a fantastic undergraduate talk? Regardless of specifics, the key is to find a fascinating topic and then be intentional (and creative!) about keeping your audience engaged from start to finish.
References
[ 1] - Satyan L. Devadoss, Giving Good Talks, Notices Amer. Math. Soc. 66 (2020), no. 10, 1647–1651.
[ 2] - Bianca Viray and Jonathan Voight, The Value of Mathematical Storytelling: Our Perspective on Giving Talks, Notices Amer. Math. Soc. 70 (2023), no. 6, 928–931.
Cornelia A. Van Cott is a professor of mathematics at the University of San Francisco. Her email address is cvancott@usfca.edu.
Article DOI: 10.1090/noti3058
Credits
Photo of Cornelia A. Van Cott is courtesy of the University of San Francisco.