Giving Mathematical Talks

Curtis Bennett and Frank Sottile

Over the course of the last several years, we have seen many talks of
varying quality.  Below is an article written after reflecting on the talks
we have heard. While most of the suggestions are obvious, we have seen many
speakers who could benefit from these suggestions.

When preparing a talk, you should first ascertain who you  will be speaking
to. Then you should decide what is the objective of your  talk. A guiding
principle is that while everyone in the audience has chosen to attend, it
is your job to make it worth their  while to listen.

There are three basics which go into the preparation of  a good talk:
research, materials, and organization. Research is what most of us are
familiar with. By this I mean knowing your topic well enough that you can
answer questions honestly, whatever your audience. Specifically you should
know some of the basic theory underlying the topic as well as some of the
consequences of the results discussed. Ideally you should also know some of
the history of the subject.  Depending on the setting, your talk need not
be original material. If not, then know the history well enough to give
proper credit and references.

The materials for the talk are the handouts, the overhead slides, the
chalkboard, or whatever else you need. You should have practice using the
materials. Typically you give handouts when conducting a series of seminar
talks on a topic. Handouts will help attendees  keep track of what you are
doing and give them something to refer to later. If you make handouts, be
sure that they are neat and clear.  Remember to make large headings for
topics so that people can use your handouts easily.

Exercise particular care when using overheads. Clear, concise slides  can
hold your audience's attention, but messy, disorganized slides may  be the
swiftest way to lose it. You should use 18 point (1/4") type  or larger if
typeset and half an inch if handwritten (neatly!). Consider your slides
carefully. When a slide goes on the screen, everyone will  start reading
it. Thus, don't put it up until you want it read.  As well, keep the slides
concise and to the point. You can fill in details  verbally.

Board technique is as important in talks as it is when teaching a class.
This means write neatly and think about what will be on the board  at all
times. If you will want something later in the talk, write it where  you
can save it without too much effort. Some of these touches can turn a  good
talk into a great talk. Similarly, bad board technique can take a good talk
and make it horrible. Steven Krantz's book How to Teach Mathematics has
several pages on blackboard technique. Every suggestion he makes is as
appropriate for a lecture or seminar as it is for a classroom.

When organizing a talk, remember that not all of your audience will be
paying attention at any given time. Try to make it easy for those who have
drifted to rejoin you. You will also hold more of your audience for a
longer time if you sufficiently motivate the subject early in the talk. Put
important comments in the beginning, as well. Every good talk I have ever
attended has been well-motivated from the start. Remember that  it is hard
work to watch a talk. As the speaker, you must continually convince your
audience that it is worth their while to pay attention.

Try to give basic examples illustrating the main points of your talk. This
can be very hard as basic examples sometimes don't exist. In that case, try
and come up with examples that motivate the ideas even if they are not
precisely examples of what you are speaking about. (In this case, DO warn
your audience what you have done). Remember, your job is to help your
audience understand the subject. They will be able to do so more easily if
they have examples to ponder.

Carefully consider which details to include. While details are frequently
at the crux of the matter in mathematics, they may also serve to confuse.
Those having little to do with the ideas of the subject are best left out.
We have been told that many details are best  done "in private, between
consenting adults." Always keep in mind the  goal of conveying the ideas
behind the subject, and let that be your guide.

Another organizational issue is one of time. Do not go over. Nobody enjoys
having a talk go overtime, and nobody minds when a talk is a little bit
short. Ideally, have something at the end that can be omitted if necessary
or lengthened if necessary. This will help you end your talk before the
allotted time.

If people ask questions during your talk, give them honest, thoughtful
answers. The audience usually asks questions to help them understand. If
you don't know an answer, say so. If you have a conjecture, explain why you
believe it. If, however, the issue brought up is subtle, then say so and
mention what makes it subtle without going into too much detail.  Little
else annoys people more than speakers who pretend to understand  more than
they really do.

There are several common formats for talks. Colloquium talks are given to
general audiences and should not do too much. In a colloquium talk,  give
some history of the field and motivate the problem. By all means state your
main theorem and discuss the proof, but be sparing with the unimportant
details. A colloquium audience is much more interested in understanding the
field and the importance of the result than the details of the proof.

A second common format for a talk (or series of talks) is a research
seminar. At your home institution, your audience is the (other) faculty,
and your purpose is to keep your colleagues abreast of your work. When
visiting another institution, your intended audience is the three or four
specialists in attendance. They invited you to see what you are doing and
to learn something of value. Make it worth their while. Having said that,
this is not a carte blanche to bamboozle the rest of the people in
attendance.

A third common forum for a talk is at a meeting of specialists in  your
area. Such talks are typically 10 to 20 minutes. These talks are often seen
as "advertisements" for your research. Again you should be clear about the
motivation of the question, but for an audience of experts, you can
motivate the problem while assuming the audience knows a lot about the
field. By their very nature, 20-minute talks skip details. As before, you
need to figure out what is important in your work and explain that.  The
statements of the theorems are far more important than the proofs. The
general idea behind the proof should be given if possible, but don't go
through messy calculations if you can avoid it. Probably the best comment
is that you should be willing to cut material.

A last format is the job talk. This is like a colloquium, but much harder
to give. In this case, you MUST know at least a week in advance who your
audience is and what they expect out of your talk. What makes this type of
talk much harder to give is that members of the audience will be looking
for different qualities. At a research school, many will want to see your
research and they will want to know what makes it good. Some will also be
interested in seeing how well you organize a talk. Others will be
interested in how well you answer questions, and some people will have
other interests still. At a smaller school, you might be giving a talk to
undergraduates. At some schools, you might even be teaching a class. It is
extremely important to do a good job at this talk. Many members of the
department will base their final decision on the talks given.

In all cases, you should practice your talk before giving it. The amount of
practice depends on the type of talk, and as a realist, I know that not
every talk gets practiced. The best practice comes before a live audience
(if you can find one). Grab some friends and ask them to watch you
practice. Then listen to their comments. Practice helps you fine tune your
timing, and it also helps you discover where the rough spots are. Lastly,
practice gets you used to the materials of the talk. I think a job talk
should be practiced a minimum of 3 times. That way, you avoid major gaffes,
know what you are going to say, and can be ready for the unexpected. For
other talks, one practice run is probably enough, although if you encounter
difficulties the first time, it is probably worth it to practice again.

This advice should make it sound like preparing a good talk is hard work.
While it may be hard work to prepare a good talk, there are many benefits.
The obvious benefits are communicating a subject you love to your peers,
and gaining a reputation for being a thoughtful and considerate speaker.
In addition,  preparing a talk is usually a boon to your research. By
seeking a way to express my ideas in a clear and concise manner, you are
forced to rethink the basics, and this often leads to new, simplifying
ideas. For this reason, talks (particularly research seminars) are good to
give when you are in the midst of some research, not after you have tied up
all of the loose ends. (Although in a job talk, it is better if the loose
ends are tied up.)

Perhaps the best way to improve your speaking is by observing others  and
thinking about their presentation. When you are paying rapt attention  to a
speaker, consider why your attention has been captured. If you lose
interest in a talk, think about what the speaker has done. If you drift
back into a talk, what enabled you to do so? By emulating the good and
discarding the poor, your speaking ability will surely improve.

It is also worthwhile to read about giving talks. One good reference is
Paul Halmos' article on the subject, "How to Talk Mathematics," in the AMS
Notices 21 (1974), pp. 155--158 (and presumably reprinted in the recent
book of his collected expository articles).