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WU MATH 249 - Study Guide in Math

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IN PRAISE OF LECTUREST. W. K¨ORNERThe Ibis was a sacred bird to the Egyptians and worshippers acquiredmerit by burying them with due ceremony. Unfortunately the numberof worshippers greatly exceeded the number of birds dying of naturalcauses so the temples bred Ibises in order that they might be killed andand then properly buried.So far as many mathematics students are concerned university math-ematics lectures follow the same pattern. For these students attendanceat lectures has a magical rather than a real significance. They attendlectures regularly (religiously, as one might say) taking care to sit asfar from the lecturer as possible (it is not good to attract the atten-tion of little understood but powerful forces) and take complete notes.Some lecturers give out the notes at such speed (often aided by thetechnological equivalent of the Tibetan prayer wheel — an overheadprojector) that the congregation is fully occupied but most fail in thistask. The gaps left empty are filled by the more antisocial elementswith surreptitious (or not so surreptitious) conversation1, reading ofnewspapers and so on whilst the remainder doodle or daydream. Thenotes of the lecture are then kept untouched until the holidays or, moreusually, the week before the exams when they are carefully highlightedwith day-glow yellow pens (a process known as revision). When morethan 50% of the notes have been highlighted, revision is said to becomplete, the magical power of the notes is exhausted and they arecarefully placed in a file never to be consulted again. (Sometimes thenotes are ceremonially burnt at the end of the student’s university ca-reer thereby giving a vivid demonstration of the value placed on theacademic side of fifteen years of education.)Many students would say that there is an element of caricature inmy description. They would agree that the lectures they attend areincomprehensible and boring but claim that they have to come to findout what is going to be examined. However, even if this was the case,they would still be behaving irrationally. The invention of the Xerox1A lecture is a public performance like a concert or a theatrical event. Televisionallows channel hopping and conversation. At public performances, private conver-sation, however interesting to the participants, distracts the rest of the audiencefrom the matter in hand. It must be added that just as good eaters make goodcooks so good audiences make for good lectures. A lecturer will give a better lectureto a quiet and attentive audience than to a noisy and inattentive one.12 T. W. K¨ORNERmachine means that only one student need attend each lecture the re-mainder being freed for organised games, social events and so on2. Norwould this student need to take very extensive notes since everythingdone in the lecture is better done in the textbooks.Even the least experienced observer can see that the average lecturermakes lots of little mistakes. Us ually these are just ‘mis-speakings’ ormisprints sometimes spotted by the lecturer, sometimes vocally cor-rected by a wide awake member of the audience, sometimes silentlycorrected by the note taker but often passing unnoticed into studentsnotes to puzzle or confuse them later. The experienced observer willnote that, though the general outlines of proofs are reasonably welldone, the fine detail is often tackled inefficiently or vaguely with, forexample, a four line proof where one line will do. A lecture takes placein real time, so to speak, with 50 minutes of mathematics occupying50 minutes of exposition whereas a chapter of a book that takes tenminutes to read may have taken as many days to compose. When theauthor of a book encounters a problem she can stop and think aboutit; the lecturer must press on regardless. If the notation becomes tocomplex or it becomes clear that some variation in an early definitionwould be helpful the author can go back and change it; the lecturer iscommitted to her earlier choices. When her book is finished the lec-turer can reread it and revise at leisure. She will get her friends toread the manuscript and they, viewing it with fresh eyes, will be ableto suggest corrections and improvements. Finally, if she is wise, shewill offer a graduate student a suitable monetary reward for each errorfound. Even with all these precautions, errors will still slip through,but it is almost certain that the book will provide a clearer, simplerand more accurate exposition than any lecture notes3.Students may feel under some obligation to go to lectures; theirteachers are under no such compulsion. Yet mathematicians go toseminars, colloquium talks, graduate courses all of which are lecturesunder another name. Why, if lectures have all the disadvantages thatI have shown, do they persist in going to them? The surprising answeris that many mathematicians find it e asier to learn from lectures than2In the past some universities made lectures com pulsory. In Cambridge duringthe e arly 19th century attendance at lectures was not compulsory but attendanceat Chapel was. ‘The choice’ thundered supporters of compulsory chapel ‘is betweencompulsory religion and no religion at all’. ‘The difference’ replied one opponent‘. . . is too subtle for my grasp’.3At one time it was the custom for beginning lecturers to spend their first coupleof years producing a perfect set of lecture notes, in effect a book. For the restof their professional lives their lectures consisted of reading these notes out atdictation speed. Their exposition was then clear, simple and accurate but, in viewthe invention of printing some centuries earlier, the same result could have beenobtained more efficiently.IN PRAISE OF LECTURES 3from books. In my opinion there are several interlinked reasons forthis.(1) A lecture presents the mathematics as a growing thing and notas a timeless snapshot. We learn more by watching a house being builtthan by inspecting it afterwards.(2) As I said above, the mathematics of lecture is composed in realtime. If the mathematics is hard the lecturer and, therefore, her audi-ence are compelled to go slowly but they can speed past the easy parts.In a book the mathematics, whether hard or easy, slips by at the thesame steady pace.(3) Some lecturers are too shy, some too panic stricken and a few(but very few) too vain or too lazy to respond to the mood of theaudience. Most lecturers can sense when an audience is puzzled andrespond by giving a new explanation or


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