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.

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 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).

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, 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“.