Why is this performance (for it is a performance), so valuable? I think there are two reasons. The first is the moral effect. Mathematics students faced with a new result have a natural tendency to believe that it is too hard for anybody to understand properly. If you copy out a proof on to the board or flash up the proof on a projector, the implicit message is that the proof is too hard for you to do anything but copy it out word for word. If you produce the proof without notes, the implicit message is that the proof is so easy that it is not worth making a fuss about.There is a second reason for this style of lecturing. Mathematics is not a collection of facts but of processes. A slide show (and what is a computer presentation but a slide show?) of theorems and their proofs is like walking through a museum full of stuffed animals. Only by watching you actually proving the results can your audience see the animals live in their native habitats.The non-mathematician may ask how a lecturer can possibly remember 50 minutes of mathematics. The answer is that she does not. Many proofs are entirely routine and can be constructed on the fly. Most of the remainder require one, or, at most, two, ideas and, once those are understood, the rest of the proof is again routine. A mathematics lecture is not like a classical symphony but like a jazz improvisation starting from a small number of themes.
I believe that the making of proofs (as opposed to the discovery of new mathematics) is like the making of shoes. It is better taught by watching someone make a shoe in front of you than by trying to figure it out from books.
“It is one of the first duties of a professor” writes Hardy “to exaggerate a little both the importance of his subject and his own importance in it.” Fortunately, beginning students automatically assume that you are one of the world’s greatest experts in your subject, so no lying is necessary. Since the mathematics taught to beginning students is necessarily elementary, it is highly unlikely that you will get lost but, if this happens, bluff (“I have given you the general idea, so go away and try and fill in the details. If you can’t, I will give them next time”) may be better than confession.
There is no doubt that weaker students value complete and accurate notes more than anything else. (The problem is that, having got complete and accurate notes, weaker students can do nothing with them.)Thosewho agree with theweaker students that the main purpose of lectures is to produce complete and accurate notes must answer the traditional question “Did Gutenberg live in vain?” Unless we believe that a university education may be summarised as “take notes, learn notes, pass exam, forget notes” we should include among our educational objectives that students should learn to use libraries and consult books (or failing that, that they should consult Wikipedia with its many excellent mathematics articles).
You cannot learn to ride a bicycle or play the violin from lectures. Instead you watch others riding bicycles or playing the violin and try to imitate them. You learn by long and painful practice (in the case of the bicycle painful to yourself, in the case of the violin painful to others). In the same way, you can only learn mathematics by doing exercises. It is possible (but quite hard) to learn mathematics without lectures by just reading books and doing exercises. It is possible (though, in my view, slightly unsatisfactory) to learn mathematics without books by just attending lectures and doing exercises. It is impossible to learn mathematics by just attending lectures and reading books.
Many good mathematicians are also good lecturers. (It is only youthful innocence that causes our students to believe that those who lecture badly do so because they are great thinkers.) After all, good research usually demands insight and clarity and these virtues provide a very good foundation for good lecturing. However, if it is necessary to make a choice, most mathematicians would prefer someone with something to say but who says it badly to some one with nothing to say who says it brilliantly. The better mathematician trumps the better lecturer.
I once asked Conway the best way to finish a lecture course. He replied “Early.”
All I remember is an elegant formula suddenly appearing and Gelfand saying that “A concrete formula like this reassures us that we must be on the right track.” Those who think that the only purpose of lectures is to communicate knowledge will not understand why Gelfand’s lecture is a treasured memory.
When I was young, I thought that the old sat in the front to demonstrate their eminence. Now that I am older I realise their choice has more to do with the state of their eyes and ears.