Jason Starr, in comments, makes the excellent point that listing good things to do in talks helps society more than listing bad things. Here’s his list:
- Tell a joke.
- Answer good questions from the audience.
- Give a simple example before giving a difficult one.
- Explain some history.
- Explain why some famous problem is hard.
- Break the chalk so it doesn’t squeak.
- Repeat a soft-spoken audience member’s question / remark so everybody hears it.
Good stuff! In the same spirit, here’s a tip sheet I wrote I few years ago for grad students giving talks at the Graduate Student Conference in Number Theory. The first tip on the list is “Tell a story.” and I stand by that placement.
Contribute your own must-do’s in comments!
Quomodocumque 
A big one: if you are giving a laptop talk, if at all possible make sure that you can actually get your computer up and running with the projector before the talk, and have all the right adapters/etc.
And if it isn’t possible, make sure it is possible anyhow. Nothing worse than waiting 10-15 minutes while the A/V person is summoned to solve the insoluble.
(oops, saved too soon)
Carry around the presentation on a memory stick so that you can plug it into J. Random Organizer’s laptop when your laptop goes on the blink.
Make multiple copies of the presentation, including one you (and preferably other people) can download from the Net.
If you have slides with bullet points, make sure they are headlines (each one a complete sentence that tells a story) and not merely captions like “Extended applications”. Of course, the title of a slide can and should be a caption.
Don’t give talks about hypertwistoploppic pseudotheomorphisms. You are one of only six people in the world who understand them, and that just isn’t enough for a decent audience. Instead, give talks about the application of HTW-PTMs, treated as black boxes, to some well-known problem or issue.
If your comment is of the form “A good thing to do in talks is to avoid doing bad thing X,” it should be a comment on the previous post!
Put as little as possible on each slide.
If you are using slides, make them interesting to look at. Illustrate what you’re saying as vividly as possible.
Also, rehearse your talk in advance. Repeatedly. First, by yourself–that way you can time it–and then in front of a trusted non-expert who can tell you what parts are unclear. And then by yourself again. Timing it.
Every mathematician needs to keep “A River Runs Through It” in mind for inspiration.
My must-do would be to use the blackboard.
As an audience member, my brain automatically switches off whenever someone turns on the computer, and I inevitably get little if any value out of a computer talk.
In no particular order of importance:
Firstly, regardless of whether you use a blackboard or slides, these aids are just for the key points of your talk. They should not be the entire content of the talk. A good technique is to write a point, or display a slide, then talk about it, expanding the point, explaining the reasoning, and so forth. (Alternatively, depending on the material, you may prefer to explain / expound, then display a slide that sums up what you’ve said.) Conversely, a really bad technique is to just read your slides to the audience, or simply say aloud what you’re writing on the blackboard.
Secondly, remember the ‘5 plus or minus 2’ rule for the number of things that people can hold in their minds simultaneously. Try to break up long chains of reasoning into shorter chains (perhaps by grouping steps – ‘I can provide further details if anyone’s interested’). The same rule applies for the number of slides in a talk – if you follow the advice given in the first point, seven slides should be ample for a 50 minute talk (allowing time for questions).
Thirdly, try to convey at least some sense of enthusiasm. Academic detachment is fine, but you also need to engage the audience. If you behave as if you couldn’t care less about the subject of your talk, then you’re essentially talking only to yourself.
Prove a small result but don’t spend the whole talk proving a complicated result. That goes with Lenstra’s joke that a talk without a proof is like a movie without a love scene.
Make eye contact with the audience once in a while.
Lenstra’s joke describes some of my favorite movies, which I guess explains my talks.
Lenstra’s joke describes some of my favorite talks, which I guess explains my favorite movies.
• Start with a question.
• Use language that appeals to the physical senses.
• Use concrete language if possible.
• People are better at seeing generalisations for themselves than coming up with examples for themselves. (Just think about your own experience!)
• Give an example so easy that it’s insulting. Then give an example that’s slightly less insulting. Finally give an interesting example.
• State connections explicitly, even if it’s obvious if you think about it long enough. (Audience members don’t have enough time to think it through whilst listening to you.)
• Pause. Own the silence.
• Either say out-loud what you’re writing before or after you write it on the board, not during.
• Google public speaking advice, dramatic performance advice, and creative writing advice.