I have given bad talks. I bet you have too. My guess is the people who think they haven’t given a bad talk give the worst ones.

What follows refers to colloquia at schools like mine, where the main purpose is to get the undergraduates excited about some sort of math that they’ve never seen before, and maybe show us faculty something cool too. And by the way, this stuff also works for job talks, at least here at Hood, a small, primarily undergraduate liberal arts college. If your environment is completely different, please leave a comment with your own tips.

I didn’t know colloquia like this existed when I first started giving talks. I’d only seen them at R1s, which were obviously another animal entirely. And even many of those violated a lot of the excellent advice in John E. McCarthy’s article How to Give A Good Colloquium: they confuse colloquia with seminars and overshoot their audience, they don’t give enough examples, and prove too much. And since those are the kinds grad students see, those are the kinds we give, and the cycle of violence continues.

I thought if you got invited somewhere, you had to talk about your research. And when they said to “make it mostly accessible to undergraduates, and the first part accessible to undergraduates that have only had calculus,” that just meant include more definitions and talk slower.

And I can never talk slower.

But once I got to this job, and started seeing the talks people would give here, and started hearing about the talks my colleagues would give at other schools, I finally realized that there were other options. And I think I’m finally getting better. So I’ll list a couple things I’ve learned the hard way.

1. Your colloquium is not your defense

I mean this both in terms of content and purpose. Your job is not to prove how smart you are, or how much math you know. The people who’ve invited you know you’re smart and know math. We just want you to show our students (and us) some cool stuff that you’re passionate about. If your slides are largely recycled from your defense with some extra definitions added, you’re about to give a lousy talk.

And don’t worry if somebody in the audience thinks you’re not a hotshot researcher because you don’t act like every result is trivial and every definition well-known. There aren’t as many of those people as I used to think, and impressing them isn’t worth losing almost everyone else in the room.

2. You don’t need to talk about your research

If you’re asked to give a talk aimed largely at undergraduates, and most undergraduates don’t have any of the prerequisite knowledge to understand your research, don’t think it’s your job to slam a whole graduate education into 50 minutes. I don’t care how good of a teacher you are. It’s just not going to work.

Find something else to talk about! Maybe your favorite theorem? A neat application of your research area? Is there some problem you ran across somewhere that you found interesting? I can’t speak for everyone at every school, or even every small liberal arts college, but I’d rather my students see you explore some fun old Martin Gardner puzzles in depth than sit through an unintelligible research talk.

3. Don’t make us wait to the end to see the point

Most of the research talks we see at conferences are ones where the first 20-30% are definitions and setup, the bulk is results and maybe their proofs, and the last bit has an application or some ideas for future work. And that’s fine for talks amongst ourselves – we know the drill, we can see the big picture, and we’re patient. Undergraduates don’t, can’t, and shouldn’t have to be. Tell us at the beginning why your talk is going to be interesting, and spend significant time putting your result and/or broader research area in context, either within mathematics or with respect to applications. If there are any bits where you can get away with just arguing by analogy, or using informal language instead of slogging through slide after slide of definitions, do it. If anybody wants details they can look them up.

The reason cold opens are used so often in movies and tv is that they’re compelling: they show us something interesting that happened, and make us wonder how that came to be. Whereas movies that start with a lot of clumsy exposition are bad movies. Colloquia are no different.

4. In your slides, less is more (unless it’s pictures or intermediate steps)

Take out as many words as possible, and then take out more. Your slides aren’t a script for you to read out loud. They’re a place to display the stuff the audience won’t be able to keep in their heads while you’re talking.

But if you’re running through a calculation of something, don’t (necessarily) skip all the algebra because it’s boring and a pain to typeset. That might be the one part of your talk that one particular undergrad in the audience can follow. Maybe it’s not thrilling for you or I, but giving the student audience a few little easily-digestible crumbs might be just enough to get them to stick with you for a few more minutes. Talks to a non-technical audience are as much about making people feel like they know what’s going on as making them actually know what’s going on. It’s possible to do the former even when you’re not doing a great job at the latter. Don’t speed through the easy stuff if you can help it.

Also, find more pictures. They don’t even have to be super relevant. Everybody likes pictures. Citing somebody’s famous result? Throw a picture of them in there. Why not.

5. If you don’t know exactly how to pitch your talk, ask

This goes double for job talks. Not everybody has the same idea in mind when they say “accessible to undergraduates.” And not every group of undergraduates is the same. Maybe the usual attendees happen to have all taken a few advanced courses, or maybe they’re all a bunch of calc 1 students who get extra credit for attending. Maybe the school culture means the attendees are almost all faculty and they’d like you to aim the level a little higher. I can’t speak for everyone, but I’d rather you ask me too many questions about the audience than just guess and guess wrong. And don’t just go by what the school calls their talk series. We often call ours a seminar, but please don’t come here thinking it’s your weekly research seminar in grad school all over again.

Even if you’re giving a job talk, they might not care if you talk about your actual research, or just any topic you find interesting. If it’s not clear from the communication with the committee, ask. In retrospect, I think some of my interviews would have been a lot better if I’d come up with a talk that wasn’t about my research instead.

I’ve given as many caveats as possible here because I know this advice won’t apply to every talk at every school. If your environment is completely different and you have different, or even contradictory advice, please leave a comment. Together, maybe we can all sit through (and give) slightly fewer bad colloquia.