\documentclass[12pt,a4paper]{article}
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\title{In Praise of Lectures}
\author{T.~W.~K\"{o}rner}
\begin{document}
\maketitle
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 mathematics 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 understood 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) conversation\footnote{A 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.}, 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.)

Many students would say that there is an element of
caricature in my description. They would agree that the lectures
they attend are incomprehensible 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 on\footnote{In 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 `\ldots is too
subtle for my grasp'.}. 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 misprints 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 confuse 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 improvements. Finally, if she is wise, she will offer
a graduate student a suitable monetary reward for each error
found. Even with all these precautions, errors will still slip
through, but it is almost certain that the book will provide
a clearer, simpler and more accurate exposition than any
lecture notes\footnote{At 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.}.

Students may feel under some obligation to go to lectures;
their teachers are under no such compulsion. Yet mathematicians
go to seminars, colloquium 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 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 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 \emph{Science Awakening} Van Der Waerden
makes the following suggestive remarks about the decline of
the ancient Greek mathematical tradition.
\begin{quotation}
Reading a proof in Apollonius requires extended and
concentrated 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 segments
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 generation 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.
\end{quotation}

Many students simultaneously expect too little and too
much from their lectures\footnote{I went to a lecture on
the violin but when I tried playing one it sounded horrid.
The lecturer can't have been any good.}. 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 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.

\emph{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.

\noindent
Semi-automatic note taking is a skill that has to be learnt,
but it seems to be an easy one to acquire.

\emph{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.

\emph{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 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 \emph{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.)

\emph{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.

\emph{What about questions?}\/ There are three types of questions
that an audience can ask.

\ \ (a) \emph{Questions of Correction}\/ If you think the lecturer
has missed out a minus sign or written $\alpha$ when she meant
$\beta$ 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) \emph{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 audience\footnote
{I 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 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.}.
(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 and ask.)

\ \ (c) \emph{Questions of Extension}\/ If you are in the happy
position of understanding 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 mathematicians 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 willingness to ask
when they do not know is a characteristic of many
great minds\footnote{Though 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'.}.


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.


\vspace{2\baselineskip}

\begin{footnotesize}
\noindent
[Printed out \today. These notes are
written in \LaTeX2e and stored in
and may be accessed via my web home page
\begin{center}
{\tt http://www.dpmms.cam.ac.uk/\~{}twk/}.
\end{center}
My home page includes other guides
to things like writing essays
and applying for Cambridge fellowships.]
\end{footnotesize}


\end{document}
