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Talking about Mathematics
Speaking in public is a daunting and nervejangling prospect for most of us, no matter how experienced. Nerves may be beyond our control, but every other aspect of the job is amenable to reason and, like all other skills, can be developed and improved with practice. The essential point to bear in mind is that the object of the exercise is to communicate a piece of mathematics in a comprehensible (even enjoyable) way. The following checklist may help to smooth the ride, particularly for a blackboardstyle presentation. 
What do I want to say? To be clearminded about this will increase your confidence, and also help to calm the nerves. One of our biggest fears is that we will "blank out" when the big moment arrives. At the very least take time to write out the mathematics you are going to present, and make sure you have your 'script' to hand, even if you end up surprising yourself and impressing everyone else by not using it!

How do I want to say it? Mathematics uses a very concise language, and just regurgitating a string of formulas will probably not go down too well with your audience. Indeed the essential detail of a mathematical argument is best conveyed in written form, which usually means on the blackboard. Nevertheless, you will have to supply a spoken commentary. Again, think about this in advance. You may want to annotate your notes to remind you of helpful remarks. For example, if you want to write down the inequality, n^{3}/3 > n^{3}/3  n^{2}/2 + n/6, your commentary might be "because the square of any nonzero integer is always greater than onethird of that integer", which, after a suitable pause (never underestimate the power of a pause!), should convince even the most sceptical members of your audience! And remember to speak up, particularly if you are naturally quietlyspoken. 
How will I use the 'props'? We have already mentioned the blackboard, which most mathematicians will find to be the ideal prop in presenting their work. The big advantage of blackboards over more sophisticated gadgets (like overhead projectors, with prepared transparencies, or data projetors, with PowerPoint) is that the necessity of writing on them encourages the gradual unfolding of complicated mathematical arguments, at a pace which gives everyone a fair chance to register what's going on, aided and abetted by your verbal pearls of wisdom. This is the ideal; but in practice the following glitches tend to creep in, and detract from the overall effect. 
It takes time to get used to writing on a blackboard. For visibility from the back of the room you will probably need to write larger than you naturally would, and with more pressure on the chalk. 
Unless you break it in half first, a new piece of chalk may tend to 'screech', causing extreme discomfort to everyone present. 
Getting the pace right can be difficult. If you are desperate to get the whole thing over and done with, the temptation will be to rush, and your writing will degenerate to scrawl. On the other hand, if writing on the blackboard slows you down too much, an element of tedium will creep in (yawning in the audience will alert you to this). 
Writing and speaking simultaneously can be a strange experience. The biggest temptation is to address your entire presentation to the blackboard! When writing, rather than facing the blackboard try to stand 'side on', and in a position which allows people to see what you are writing as you write it. Also, take the occasional opportunity to turn and speak directly to your audience. This can be scary, but will seem a lot easier if you focus on an individual in the room and imagine you are speaking to him/her. However, remember to change your point of focus regularly; otherwise you are in danger of turning your public presentation into a private conversation.

How did I do? Your first presentation might turn out to be a brilliant tour de force. If so, congratulations! However, the chances are that this will not be the case (and certainly nobody expects it to be). It is useful to get some feedback, so why not have a quiet word with the seminar leader just after the seminar to find out how it came across, and maybe pick up a few suggestions, for the next time!
Edited 4 Aug 2008  11:51 by admin
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