Learning to speak effectively is well worth our time, at all stages of our development. A lot of excellent material is available on this topic. Hopefully, the pointers below add something to the mix. I first assembled them for my 3rd-year undergraduate students in 2019 when I was coordinating a junior seminar (flipped classroom) in math at Princeton University.
Before your talk
Make a short list of the messages you want to convey.
Do this first. Only after you picked your messages should you start working on your presentation (slides / blackboard notes / speech outline). It’s much easier now: everything you say must contribute to conveying your messages (and yes, jokes and stories can help with that). Everything else must go.
Challenge that list.
Ask yourself: reasonably, what can I hope my audience will remember five minutes after my talk? The next day? A year later? How does that affect my choice of messages and their relative importance?
Open with a crisp message.
What are we learning today? What’s the point? This opening must be at a technical level your audience can grasp now, and it should entice their curiosity.
Don’t just answer “What?”—answer “So what?”
Ask this of every one of your main messages and supporting messages: so what? If you don’t, your audience will.
Introduce technical tools only as a means to an end.
Stated differently: align technical content with motivation. By default, your audience doesn’t want to make the effort to learn a random technical fact without motivation; and even if they indulge you, the concepts may not stick long enough to be useful later in the presentation. It’s hard to remember a purely technical fact. They need to have an anchor to the main narrative, and a heads up that it’s important.
Choose your medium.
Figure out which media will be available to you in the room (e.g., projector, blackboard). Choose one, or two, or none depending on what works best for your messages. It’s fine to switch back and forth if the room setup permits it with little friction.
Notice a strength of blackboards: they have permanence. Also, writing on a blackboard limits your speed. That’s good for long talks.
Talks are linear: carry the context.
Especially with slides, your audience cannot easily “go back” to check missed details. If you need them to commit something (reasonable) to memory, tell them that before you tell them the thing.
If you find yourself relying on minor points you made early on, or moving back and forth between slides (e.g., between a theorem statement and its proof), reconsider: Can you bring the required context closer to where it is first used? Can you repeat the useful part of that context aloud or visually?
During your talk
Show up early.
Take the room in. Setup your gear. Take a deep breath, great people, and remember to enjoy the moment.
The floor is yours: take control, respectfully.
The table is in your way? Move it. The computer is awkwardly far away from the screen, making it hard for you to point at things while controlling slides? Move it. There’s noise outside? Close the door or window. The lighting is wrong? Change it. You may want to ask for permission of course, but it’s not rude to ask.
And remember: the floor is yours for some time. Going overtime is rude.
Your duty is to deliver, not merely to try.
This means that if the projector doesn’t work, or there is no blackboard even though you thought / were told there would be one, or the internet is down and you can’t access your slides—it’s on you to adapt. People will help you fix the issue, and if the circumstances are beyond your control, your audience will sympathize, but they still expect to learn something from you: that’s why they showed up. Ask for a moment to re-assess, adapt, and they will be forgiving.
About math talks specifically
Anatomy of a theorem
Your audience needs to understand the claim, and its relevance (contrast helps). We must understand the characters (variables, sets, objects of various kinds) and the assumptions. Importantly, we want to understand what the theorem “is really saying” before we go into the proof. Ideally, we want to get some sense of why the assumptions are there. Only after we know what the statement says and why we care, can we embark into a proof.
Anatomy of a proof
Proofs are the execution of a strategy. That execution can be messy: some tedious bounds and integrals here and there, annoying corner cases of lesser importance. The details are important of course, but they are not interesting. What your audience may remember is the overall strategy: how could one come up with this proof? What’s the key mathematical insight? What were the obstructions? In short: what is the take-away that might serve us in another context? Why is the claim credible? Lead with this.
Flashing a proof achieves little.
Your audience cannot read a technical slide that appears for a few seconds, let alone comprehend it. The only legitimate use-case I can imagine would be to convince your audience that the proof is short, but even for this I’m not sold: you might as well just say out loud that the proof is “half a page.” This sentence and the flashed slide convey the same amount of information. Also, when your audience gets the sense that they didn’t follow you for a moment, you run the risk of them disconnecting completely.
Doing math on the board is hard. Rehearse it.
Furthermore, doing math on the board takes time, because you need to carry your audience with you. There is only one way around it: practice. Rehearse (and time) all of the math you intend to do live.
A few more thoughts
All public speaking is storytelling.
Stories are engaging: they can be great communication devices. They require just as great a delivery. Stories have characters, tension and a resolution. This requires emphasis and time. Refine and rehearse.
Ask a friend.
We all know the importance of honest feedback. Yet in practice, we often don’t ask because we do not want to impose, or we are afraid to hear what we need to hear, or we don’t think we can be ready to give a mock talk days before the real one. Nevermind all that: if the stakes are high, plan a mock talk, or ask a friend to plan it for you.
It’s not about you. It’s about your audience.
Who are they? What are they getting out of this?