Choose the topic with both the assignment and the audience in mind. If the talk is a course assignment, the audience is usually the students in the class. For senior research, it is other math majors and faculty. Remember that some faculty may not be familiar with your particular problem. Design the talk with other aspects of the assignment in mind: time limit, goal (inform, entertain, teach, ...), etc.
The broad outline of your talk should be Introduction, Major Topics, Conclusion, References.
The audience will absorb what you are saying more slowly than you think. Incorporate simple examples of all the important ideas, and illustrate with pictures or diagrams whenever possible. Don't forget "antiexamples," things that do not satisfy a definition. Summarize often what you have just said.
Use prepared overheads or Power Point. The math office can supply transparencies, and I can help you with Power Point. Planning the slides and planning the talk are mutually supporting activities – do them together. Your slides will help you to give a more effective presentation:
Write a rough draft of your talk, then design your slides on scratch paper. See below for guidelines on slides. Run through the talk once with the rough slides and time yourself. Most first run-throughs are too long for the time alotted, and you will have to decide what to cut. In general, cut details and computations first. If you have to cut out a topic decide which topics best fit in with the theme of the talk and be sure to keep them.
Now plan your slides in earnest. Put the first two or three that you make on the machine that will be projecting them to see how they look. The first slide should give the title of the talk and your name. The last should give your references. Other slides should summarize the content of the talk. Important features of the slides are
Practice the talk with the finished slides. Make sure you use the same technology you will use during the real talk, overhead projector, computer, whatever. Time yourself. Make changes as necessary, revise your slides.
Practice again. Look at the audience most of the time, don't talk to your slides. If you point, point on the overhead or use the computer mouse so that you keep your face to the audience. Power Point is useful for changing the highlighting of items as you progress through the talk. Speak deliberately, not too fast. Repeat important points in different words. Try to anticipate what questions you might be asked.
Practice again. Get me or a friend to listen.
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