The mediocre teacher tells. The good teacher explains. The superior teacher demonstrates. The great teacher inspires. (William Ward)

It is difficult to give good talks, especially when one is just starting out one’s career.

One should avoid the common error of treating a talk like a paper, with all the attendant details, technicalities, and formalism. (In particular, one should never give a talk which consists solely of transparencies of one’s research paper!) Such talks are almost impossible for anyone not intimately familiar with your work to be able to follow, especially since (unlike when reading a paper) it is difficult for an audience member to refer back to notation that had been defined, or comments that had been made, four slides or five blackboards ago.

Instead, a talk should complement a paper by providing a high-level and more informal overview of the same material, especially for the more standard or routine components of the argument; this allows one to channel more of the audience’s attention onto the most interesting or important components, which can be described in more detail.

[Another aspect that a talk can cover that usually not covered in papers is how the final proofs were obtained.  See this Abstruse Goose comic for an illustration of this.]

A good talk should also be “friendly” to non-experts by devoting at least the first few minutes going over basic examples or background, so that they are not completely lost even from the beginning. Actually, even the experts will appreciate a review of the background material; even if none of this material is new, sometimes you will have a new perspective on the old material which is of interest. Also, if you organize your presentation of background material correctly, your treatment of the new material should flow more naturally and be more readily appreciated by the audience.

One particularly effective method is to present a proof of New Theorem Y by first reviewing a proof of Standard Theorem X in the style of the proof of Y, and then later in the lecture, when the time comes to prove Y, just note that one simply repeats all the steps used to prove X with only a few key changes, which one then highlights. (Of course, it would be a good idea to keep the proof of X on the blackboard or on screen during all of this, if possible.) This often works better, and can even be a little bit faster, than if one skipped the proof of X “to save time” and started directly on the proof of Y.

For “typical” conference or seminar talks aimed primarily at specialists, it is acceptable to focus primarily on the new results after the introductory portion of the talk, as much of the audience will already appreciate the motivation for these results.  But for talks aimed at a broader audience (e.g., colloquia) one ought to put some effort into motivation and on the structure of the talk.  One analogy that I have found helpful is to treat mathematical talks as similar to popular performing arts such as film, television, theater, or music, in that the narrative devices such as themes, motifs, plot, etc. play an important role in making the performance memorable and in communicating the key points. Some typical narrative framings of good talks include things like:

  • Posing one or more open problems at the start of the talk, and devoting the rest of the talk to showing progress on those problems.
  • Describing a mysterious mathematical phenomenon (or mysterious analogy or connection between different fields), and devoting the rest of the talk to revealing some partial or complete explanations of that phenomenon or analogy.
  • Introducing a new result in a way that makes it surprising or striking, and devoting the rest of the talk to explaining how it can actually be established despite its unusual nature.
  • Stating some slogans near the start of the talk which may initially seem cryptic or unmotivated (perhaps because it involves terms that are not familiar to most of the audience), but over the course of the talk the slogan is elucidated and makes perfect sense by the end of the talk. (Or one can replace the slogans with some initially cryptic but ultimately informative table or graphic; a famous example is Mumford’s sketch of Spec Z[x].)
  • A three act structure in which a problem is set up, difficulties are encountered, and solutions are presented in the third act.  One can also try to implement more advanced narrative devices such as a surprising “plot twist” in the third act, nonlinear storytelling or building up a feeling of suspense, but these are harder to pull off well, especially if the content of the talk does not naturally suggest such a device.
  • A small number of judicious jokes (e.g., somewhat exaggerated analogies with real-life situations, illustrated perhaps with a humorous image or two) can help make the subject of the talk less intimidating, though if the joke does not actually assist in clarifying the rest of the talk it could safely be removed.
  • In addition to just objectively stating results, one is permitted to also provide some opinion and speculation, for instance concerning future directions, or in drawing some insights or morals from the results just established.  (I recommend Minhyong Kim’s analogy that definite mathematical results are like “money in the bank”; once you earn enough of this currency of credibility, you can “spend” it by philosophizing, as long as you don’t run your bank account too low because of this.)  It is particularly common (though not mandatory) to pose some open problems at the end of one’ s talk.
  • Tangential material (or expanded details of arguments) that would disrupt the narrative flow of the talk can be prepared as “deleted scenes” that one can show in response to an appropriate audience question, or if one somehow has additional time at the end of the talk.  Being able to anticipate an audience question with such backup material demonstrates preparation on the part of the speaker (as well as good taste on the part of the questioner), and can make the talk more effective and memorable.

There are three main formats in which one gives mathematical talks: blackboard (or whiteboard), transparencies, and computer presentations. They all have their strengths and weaknesses:

  1. Blackboard talks are very flexible, allowing for rather nonlinear and adaptable presentations. A good lecture hall with plenty of blackboard space allows for the audience to see a large part of the talk at any given time, making it easier to follow and to refer back to previous parts of the talk.
  2. Transparencies can convey detailed information, such as tables, computations, or graphics, efficiently and rapidly (sometimes too rapidly!). If two projectors are available, make full use of both; in particular, it can be invaluable to have a key transparency with some crucial definitions or theorems on one of the projectors during the main part of your talk.
  3. Computer presentations (slides) are of course excellent for animations, graphics, and other “eye candy”, although one should not let the style of the presentation obscure the substance. They also have the advantage of being easily made available on-line. One can also use “hypertext” features, such as popup windows, to good effect, although this requires some careful thought and planning to be effective.

One should try to keep these various attributes in mind when designing the format and content of one’s talks. Sometimes a hybrid approach works well (e.g. transparencies for some key details, blackboard for the intuitive “big picture”, and/or computer for illustrative examples).  Note also that some conferences (particularly those held in conference centers or hotels) may not have blackboards or overhead projectors available.

For slide talks, one way to try to design a narrative structure (as recommended above) is to first create a summary slide that conveys the key points one wants the audience to take away from your talk, and then organise the rest of your talk to be a logical progression towards that summary slide.

It takes a bit of practice to figure out how much material one can fit into a given time frame (e.g. a 50 minute lecture). Cramming in too much mathematics, or running hopelessly over time, is generally not a good thing, unless your work is really, really exciting (and this, honestly, only occurs very rarely). It therefore is a good idea to move the more “optional” part of the talk to the end (or to be separated off into “deleted scenes”, as suggested above), so that it can be easily dropped or abridged if necessary. After a while, you will get a sense of how many of your slides or how many pages of your handwritten notes can typically be presented effectively in any given time frame. I of course can’t tell you what these numbers will be for you, since each person’s style in writing slides or notes is so different; you’ll have to find out for yourself.

If you have to give the very first talk of your career, it may help to practice it, even to an empty room, to get a rough idea of how much time it will take and whether anything should be put in, taken out, moved, or modified to make the talk flow better.

See also John McCarthy’s “How to give a good colloquium“, Jordan Ellenberg’s “Tips for giving talks“, or Bryna Kra’s “Giving talks“.