By Daniel Erman
There is a wealth of information on the internet about how to give a good Mathematics talk, including:
(*) Gian-Carlo Rota’s essay “Ten lessons I wish I had been taught”
(*) John Baez’s “Advice for the young scientist”
(*) Terence Tao’s “Talks are not the same as papers”
In this post, I’d like to consider an aspect of lecturing which is not emphasized in the above sources, but which I think is extremely important. Lecturing necessarily involves choosing to emphasize some aspects of the material over others. A common pitfall I’ve seen among speakers—especially student speakers—is to apologize during the talk for such choices, or to make self-deprecating jokes. This is nearly always a bad idea, as it distracts from the point of your talk.
Let’s take an example. Imagine you are giving an expository talk on a topic which involves a theorem with a complicated proof, and you don’t understand the proof very well. However, you have a clear view of why this theorem is important, and you know how to apply the theorem. So you decide that, in your talk, you will avoid the proof of the theorem and focus instead on an application which illustrates the big picture. You prepare your talk along these lines.
As the date for the talk approaches, you start getting nervous about the fact that you don’t understand some of the details of the proof. A common reaction to such nervousness is, when skipping the proof, to apologize for this choice; e.g. “Now I’m going to skip the proof, since I don’t understand it.” Offering up this personal detail doesn’t help the audience understand your talk any better, and it generally makes the speaker look uncomfortable. After presenting the statement of theorem, it would be better to state: “We will focus on applications of the theorem rather than the proof.”
This is not to say that you should try to fool the audience into thinking you know more than you do. However, you should only include such details when relevant. For instance, if an audience member asks, “Could you tell us about the proof?”, then this is the time to say, “Honestly, I didn’t understand the proof well enough to give a sketch.” However, preempting such questions by discussing your own struggles tends to distract from the message of your lecture.
Your lecture should focus on the material you have chosen to present. Though you may feel the desire to apologize for/joke about some aspect you are omitting, such apologies/jokes are usually distracting.