In Praise Of Lectures

In Praise of Lectures
T. W. K¨orner
October 31, 2004
The Ibis was a sacred bird to the Egyptians and worshippers acquired
merit by burying them with due ceremony. Unfortunately the number of
worshippers greatly exceeded the number of birds dying of natural causes
so the temples bred Ibises in order that they might be killed and and then
properly buried.
So far as many mathematics students are concerned university mathe-
matics lectures follow the same pattern. For these students attendance at
lectures has a magical rather than a real signiﬁcance. They attend lectures
regularly (religiously, as one might say) taking care to sit as far from the
lecturer as possible (it is not good to attract the attention of little under-
stood but powerful forces) and take complete notes. Some lecturers give out
the notes at such speed (often aided by the technological equivalent of the
Tibetan prayer wheel — an overhead projector) that the congregation is fully
occupied but most fail in this task. The gaps left empty are ﬁlled by the more
antisocial elements with surreptitious (or not so surreptitious) conversation1,
reading of newspapers and so on whilst the remainder doodle or daydream.
The notes of the lecture are then kept untouched until the holidays or, more
usually, the week before the exams when they are carefully highlighted with
day-glow yellow pens (a process known as revision). When more than 50% of
the notes have been highlighted, revision is said to be complete, the magical
power of the notes is exhausted and they are carefully placed in a ﬁle never
to be consulted again. (Sometimes the notes are ceremonially burnt at the
end of the student’s university career thereby giving a vivid demonstration
of the value placed on the academic side of ﬁfteen years of education.)
1A lecture is a public performance like a concert or a theatrical event. Television allows
channel hopping and conversation. At public performances, private conversation, however
interesting to the participants, distracts the rest of the audience from the matter in hand.
It must be added that just as good eaters make good cooks so good audiences make for
good lectures. A lecturer will give a better lecture to a quiet and attentive audience than
to a noisy and inattentive one.
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Many students would say that there is an element of caricature in my
description. They would agree that the lectures they attend are incompre-
hensible and boring but claim that they have to come to ﬁnd out what is
going to be examined. However, even if this was the case, they would still be
behaving irrationally. The invention of the Xerox machine means that only
one student need attend each lecture the remainder being freed for organised
games, social events and so on2. Nor would this student need to take very
extensive notes since everything done in the lecture is better done in the
textbooks.
Even the least experienced observer can see that the average lecturer
makes lots of little mistakes. Usually these are just ‘mis-speakings’ or mis-
prints sometimes spotted by the lecturer, sometimes vocally corrected by
a wide awake member of the audience, sometimes silently corrected by the
note taker but often passing unnoticed into students notes to puzzle or con-
fuse them later. The experienced observer will note that, though the general
outlines of proofs are reasonably well done, the ﬁne detail is often tackled
ineﬃciently or vaguely with, for example, a four line proof where one line
will do. A lecture takes place in real time, so to speak, with 50 minutes of
mathematics occupying 50 minutes of exposition whereas a chapter of a book
that takes ten minutes to read may have taken as many days to compose.
When the author of a book encounters a problem she can stop and think
about it; the lecturer must press on regardless. If the notation becomes too
complex or it becomes clear that some variation in an early deﬁnition would
be helpful the author can go back and change it; the lecturer is committed to
her earlier choices. When her book is ﬁnished the lecturer can reread it and
revise at leisure. She will get her friends to read the manuscript and they,
viewing it with fresh eyes, will be able to suggest corrections and improve-
ments. Finally, if she is wise, she will oﬀer a graduate student a suitable
monetary reward for each error found. Even with all these precautions, er-
rors will still slip through, but it is almost certain that the book will provide
a clearer, simpler and more accurate exposition than any lecture notes3.
Students may feel under some obligation to go to lectures; their teachers
2In the past some universities made lectures compulsory. In Cambridge during the early
19th century attendance at lectures was not compulsory but attendance at Chapel was.
‘The choice’ thundered supporters of compulsory chapel ‘is between compulsory 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 couple of
years producing a perfect set of lecture notes, in eﬀect a book. For the rest of their
professional lives their lectures consisted of reading these notes out at dictation speed.
Their exposition was then clear, simple and accurate but, in view the invention of printing
some centuries earlier, the same result could have been obtained more eﬃciently.
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are under no such compulsion. Yet mathematicians go to seminars, collo-
quium talks, graduate courses all of which are lectures under another name.
Why, if lectures have all the disadvantages that I have shown, do they persist
in going to them? The surprising answer is that many mathematicians ﬁnd
it easier to learn from lectures than from books. In my opinion there are
several interlinked reasons for this.
(1) A lecture presents the mathematics as a growing thing and not as a
timeless snapshot. We learn more by watching a house being built than by
inspecting it afterwards.
(2) As I said above, the mathematics of lecture is composed in real time.
If the mathematics is hard the lecturer and, therefore, her audience are com-
pelled to go slowly but they can speed past the easy parts. In a book the
mathematics, whether hard or easy, slips by at the the same 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 the audience. Most
lecturers can sense when an audience is puzzled and respond by giving a new
explanation or illustration. When a lecture is going well they can seize the
moment to push the audience just a little further than they could normally
expect to go. A book can not respond to our moods.
(4) The author of a book can seldom resist the temptation to add just
one extra point. (Why should she, when purchasers and publishers prefer to
deal in ‘proper’ books rather than slim pamphlets?) The lecturer is forced
by the lecture format to concentrate on the essentials.
(5) In a book the author is on her best behaviour; remarks which go
down well in lectures look ﬂat on the printed page. A lecturer can say ‘This
is boring but necessary’ or ‘It took me three days to work this out’ in a way
an author cannot.
There is another advantage of lectures which is of particular importance
to beginners. There is a slogan ‘We learn mathematics by doing mathematics’
which like many slogans conceals one truth behind another. We do not learn
to play the violin by playing the violin or rock climbing by climbing rocks.
We learn by watching experts doing these things and then imitating them.
Practice is an essential part of learning but unguided practice is generally
useless and often worse than useless. People who teach themselves to program
acquire a mass of bad programming habits which (unless they wish to remain
hackers all their lives) they then have to painfully unlearn. Mathematics
textbooks show us how mathematicians write mathematics (admittedly an
important skill to acquire) but lectures show us how mathematicians do
mathematics.
In his book Science Awakening Van Der Waerden makes the following
suggestive remarks about the decline of the ancient Greek mathematical tra-
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dition.
Reading a proof in Apollonius requires extended and concen-
trated study. Instead of a concise algebraic formula, one ﬁnds
a long sentence, in which each line segment is indicated by two
letters which have to be located in the ﬁgure. To understand the
line of thought one is compelled to transcribe these sentences in
modern concise formulas. The ancients did not have this tool;
instead they had the oral tradition.
An oral tradition makes it possible to indicate the line seg-
ments with the ﬁngers; one can emphasise essentials and point
out how the proof was found. All of this disappears in the written
formulation of the strictly classical style. The proofs are logically
sound, but they are not suggestive. One feels caught in a logical
mousetrap, but one fails to see the guiding line of thought.
As long as there was no interruption, as long as each gener-
ation could hand over its method to the next, everything went
well and the science ﬂourished. But as soon as some external
cause brought about an interruption in the oral tradition, and
only books remained it became extremely diﬃcult to assimilate
the work of the great precursors and next to impossible to pass
beyond it.
Many students simultaneously expect too little and too much from their
lectures4. If asked they might say ‘The purpose of lectures is to enable me
to understand the material’ or ‘The purpose of lectures is to enable me to
do the exercises’. Since the lectures do not achieve this end the students
assume either that the lecturer is incompetent or that they are. Often both
assumptions are false.
Suppose that that you visit a large town and you wish to learn how to
get around. One way of learning is to go by foot on a guided tour which
includes the main landmarks. At the end of the walk, even if you remember
everything your guide has shown you (that is ‘you have learnt the proofs by
heart’) you will not know the town in the way that your guide knows it. In
order to know the town ‘like a native’ you will need to explore for yourself.
Instead of using the main road to get from the market to the station you will
need to try other routes and see whether they work. (Naturally you will get
lost from time to time but because you have been shown routes between the
main landmarks you will be able to recover your bearings.) Your guide may
4I went to a lecture on the violin but when I tried playing one it sounded horrid. The
lecturer can’t have been any good.
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have explained that the road system runs the way it does because there are
only three bridges across the river but only by walking the roads themselves
will you be able to internalise this knowledge. However hard your guide may
have tried there are clear limits to how much you can learn on the ﬁrst walk.
But, without that ﬁrst tour given by a native, you would ﬁnd it very hard
to learn your way about town. Lectures by themselves can not give you a
full understanding of a piece of mathematics but, without lectures to get you
started, it is very hard to gain that full understanding.
In my view students should treat lectures not as a note taking exercise
but as a dialogue between themselves and the lecturer. They should try to
follow the argument as it emerges and not just take it down blindly. ‘But’
the reader will exclaim ‘this is an impossible and futile council of perfection’
and, after having thrown these notes into the nearest available wastepaper
basket, she may well resolve her indignation into a series of questions.
What about note taking? If you look at experienced mathematicians in
a lecture you will see that their note taking is an automatic process which
leaves them free to concentrate on the lecture. Most mathematics lecturers
follow two conventions which make automatic, or at least semi-automatic,
note taking possible
(a) Everything that is written on the blackboard is to be copied down
and nothing that is spoken need be taken down.
(b) It is the responsibility of the lecturer to ensure that what appears
on the board forms a decent set of notes without further editing.
Semi-automatic note taking is a skill that has to be learnt, but it seems to
be an easy one to acquire.
Would it better not to take notes? Some mathematicians never take notes
but most ﬁnd that note taking helps them concentrate on the job in hand.
(When the audience at a seminar stop taking notes the experienced seminar
speaker knows that they have lost interest and are now using her as a gently
babbling source of white noise whilst they think their own thoughts.) Further
even the largest blackboard will eventually be erased and notes allow you to
glance back to earlier parts of the lecture.
What should you do if you get lost? The ﬁrst and most important thing
is to remember that most mathematicians are lost most of the time during
lectures. (If you do not believe me, ask around.) Attending a mathematics
lecture is like walking through a thunderstorm at night. Most of the time you
are lost, wet and miserable but at rare intervals there is a ﬂash of lightening
and the whole countryside is lit up. Once you realise that your plight is
neither an infallible sign of your incurable stupidity nor a clear indication
of the lecturer’s total incompetence but simply a normal occurrence, it is
clear how you should act. You should continue taking notes watching all the
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time for a point where the lecturer changes the subject (or ﬁnishes a proof
or whatever) and you can rejoin her exposition as an active partner.
It is obvious that if you study your lecture notes after the lecture with
the object of understanding the point where the lecturer has got to you will
have a better chance of understanding the next lecture. If you are one of the
majority of the students who ﬁnd this a counsel of perfection then you could
at least use the ﬁve minutes before the next lecture rereading the last part
of your notes. (If you do not do even do this, at least ask yourself why you
do not do this.)
What should you do if you understand nothing at all of what is going
on? At an advanced level it is possible for an entire course of 24 lectures to
be devoted to the proof of a single theorem. If you get really lost in such a
course (and probably by the end everybody except, perhaps, the lecturer will
be really lost) you stay lost. However ﬁrst and second year undergraduate
lectures consist of a set of short topics chained together in some reasonable
order. Even if you completely fail to understand one topic there is no reason
why you should not understand the next (even if you do not understand
the proof of Cauchy’s theorem you can still use it). On the other hand if
incomprehensible topic succeeds incomprehensible topic then taking notes
in the hope that all will become clear when you revise is not an adequate
response. You should swallow your pride and consult your director of studies.
What about questions? There are three types of questions that an audi-
ence can ask.
(a) Questions of Correction If you think the lecturer has missed out
a minus sign or written α when she meant β then you should always ask.
No lecturer likes to spend a blackboard of calculations sinking further into
the mire because her audience has failed to point out an error on line one.
Sometimes very polite students wait until after a lecture to point out errors
with the result that the lecturer knows that she has made an error but that
she cannot correct it. So the rule is ask and ask at once.
(b) Questions of Incomprehension It takes considerable courage to admit
that you do not understand something in front of other people. However
if you do not understand something it is likely that many others in the
audience will be in the same boat and you will have their silent thanks. You
will usually also have the audible and honest thanks of the lecturer since,
as I have indicated above, most lecturers prefer to keep in touch with the
audience5. (There is a small and unfortunate minority who would prefer to
lecture to an empty room, but give your lecturer the beneﬁt of the doubt
5I have often thought that the technology of the TV game-show should be adapted to
the lecture theatre. Each seat would have a concealed button which the auditors could
press when they wanted the lecturer to slow down. The ‘votes’ could be added and the
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and ask.)
(c) Questions of Extension If you are in the happy position of under-
standing everything the lecturer says then you may wish her to go further
into a topic. Your modest request to hear more about the general case is
unlikely to go down well with the rest of the audience who are still struggling
with the particular case. Such questions should be left until after the lecture
when the lecturer will be happy to oblige (few mathematicians can resist an
invitation to talk more about their subject). If you ﬁnd yourself asking more
than one question per lecture, examine your motives.
It is noticeable that at seminars it is often the most distinguished mathe-
maticians who ask the simplest (if they were not so distinguished, one might
say naive) questions. It is, I suppose, possible that they only began to ask
such questions after they became distinguished, but I believe that a willing-
ness to ask when they do not know is a characteristic of many great minds6.
Mathematical sayings tend to have multiple attributions (perhaps because
mathematicians remember processes rather than isolated facts like names).
The ancient Greeks attributed the following saying to Euclid among others.
Ptolomey, King of Egypt, asked Euclid to teach him geometry. ‘O King’
replied Euclid ‘in Egypt there are royal roads and roads for the common
people, but there are no royal roads in geometry.’ Mathematics is hard,
there are no easy ways to understanding but the lecture, properly used, is
the easiest way that I know.
[Printed out October 31, 2004. These notes are written in LATEX2e and stored in and may
be accessed via my web home page
http://www.dpmms.cam.ac.uk/~twk/.
My home page includes other guides to things like writing essays and applying for Cam-
bridge fellowships.]
result shown on a dial visible only to the lecturer who would then be in the position of a
motorist trying to keep to the speed limit.
6Though there is no unique recipe for greatness. When the very great physicist Bohr
was visiting the great physicist Landau in Moscow he was invited to give a talk to the
graduate students with Landau translating. Bohr concluded his talk with the assertion ‘I
attribute my success to the fact that I have never been afraid to let my students tell me
what a fool I am’. The Russian translation ended ‘I attribute my success to the fact that
I have never been afraid to tell my students what fools they are’.
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