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A  Supervisors Primer\\
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{\bf Being a collection of hints for new supervisors compiled by\\
T. W. K\"{o}rner\\
and reflecting nobody's opinions but his own.\\}
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Those that can, do. Those that can't, teach.\\
And those who can't teach, teach teaching.\\
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The Faculty Board also issues a guide to supervisors\\ which
has the advantages of being shorter,\\more official and
considerably less opinionated.
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There are many ways of supervising well. The purpose of
these notes is not to provide a set of unalterable
rules but to provide a model for you to modify and to
react against. My own practice differs in many respects
from that set down here, sometimes because I fall short
of my own ideals (I am too disorganised to run any sort
of record system) and sometimes because I have made a
deliberate decision to do things differently (as a
Director of Studies I use a
mixture of linked supervisions and classes for my
first two years).
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During their first two years at Cambridge mathematics students
are usually supervised in pairs twice a week. The supervisions
will be in a group of subjects and each supervisor will
expect to see their pairs once a week for eight weeks.
The supervisors are chosen by the Director of Studies
of the particular College the students belong to. 
In the third year students have supervisions in
specific subjects (generally 4 supervisions for a 24
hour course and 3 supervisions for a 16 hour course).
Most Colleges belong to one of three groups (or `circuses')
which organise supervisions on a joint basis.

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\noindent
{\em Why Supervise?}\/ Here are some possible reasons
with my comments.

(1) {\sf The money.} This is not a bad reason but you would do
well to calculate how much you expect to earn before you
start. You will find that the sum involved is useful but
no more than that.

(2) {\sf It will look well on a CV.} This is not usually
a good reason. The number of openings for which having
supervised at Cambridge is a strong positive recommendation
is non-zero but small.

(3) {\sf `My College or my Research Supervisor expects me to.'}
This is a bad reason. If you are a research student your
College and your Research Supervisor want you to do good
research. Everything else is secondary.

(4) {\sf `It will help me decide if I like teaching.'} This
is an excellent reason. If you like supervising you may
like teaching in general --- if you do not like supervising,
you certainly will not.

(5) {\sf It is a good introduction to teaching.} Again an
excellent reason. Like most skills teaching is mainly
learned by doing and where better to start than with
a small group of able and motivated students? In supervisions
you can actually see the effects of your teaching and learn
both what teaching can achieve and what it can not.

(6) {\sf The best way of learning something is to teach it.}
Undoubtedly true, but you must remember that you will be
improving your general mathematical culture rather than
learning things which are directly useful in your research.

(7) {\sf It is a change.} Research is a lonely occupation
in which long periods may pass without apparent progress.
Supervising provides social contact and immediate rewards
as you see your teaching having its effect.

(8) {\sf It provides a way of putting back into the system
something of what you have gained from it.} Not a bad reason,
particularly
if you were a Cambridge undergraduate.

Whatever your reason for starting supervising you should not
continue with it if you find that you do not
enjoy it or that it takes up too much time. In the first
case you are cheating your students, in the second you are
cheating yourself.

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\noindent
{\em Why Let You Supervise?}\/ Stephen Leacock described
the Oxford method of supervision as follows. (His description
dates from 1922 but things change slowly in that university.)

\begin{quotation}
I understand that the key to this mystery [the Oxford
method of education] is found in the operations of a person
called the tutor [i.e. supervisor]. It is from him or
rather with him, that the students learn all they know:
one and all are agreed on that. Yet it is a little odd
to know just how it does it. `We go over to his rooms,'
said one student, `and he just lights his pipe and talks to
us.' `We sit round him with him,' said another, `and he
simply smokes and goes over our exercises with us.'
From this and other evidence I gather that what an Oxford
tutor does is to get a little group of students together
and smoke at them. Men who have been systematically
smoked at for four years turn into ripe
scholars\footnote{Leacock goes on to say that `Of all
the various reforms that are talked of at Oxford,
and of all the imitations of American methods that
are suggested, the only one worthwhile to my way of thinking,
is to capture a few millionaires, give them honorary
degrees at a million pounds sterling apiece, and tell
them to imagine they are Henry VIII.'}.
\hfill{[From {\em My Discovery of England}]}
\end{quotation}

Many `arts' subjects rely on the pipe-smoking method
of supervision
and, since only mature scholars can successfully smoke
at students, they tend to look down on subjects like
mathematics which use graduate students as supervisors.
There are however many positive reasons for using young
supervisors. Here are some.

(1) Younger supervisors still remember vividly the
problems they had with the subject and are
therefore (with the exception of a few arrogant bastards
whom no Director of Studies in their right mind would employ)
more sympathetic to the needs of students.

(2) Even the youngest supervisor appears to students
as incredibly old, but, with
an effort of imagination, students can identify with a graduate
student in a way they can not with a more senior figure.
It is important to convey to them that mathematicians are
endlessly building and rebuilding their subject and that
mathematics is not a fixed permanent structure
designed and inhabited by men wearing frock-coats and
long white beards. It is therefore your duty to
make remarks like  `I use this result in my research'
or `As Wiles said in his seminar'. Hardy held that
it was the job of lecturers\footnote{He actually wrote
professors, but we live in a democratic age.} to exaggerate both
the importance of their subject and of their place within
that subject.

(3) Although mathematicians learn more about their own
particular subjects as they get older, their approach
to the remainder of mathematics tends to stick in the
groove formed when they were undergraduates. The use of
younger supervisors counteracts the natural conservatism
of the system.

(4) Older supervisors can not help feeling that the have been
teaching students the same thing for twenty years {\em `and
they still don't understand it!'}. There is a genuine gain
in expertise with age but, unfortunately, it is often more than
balanced by a loss of hope and enthusiasm. As a Director of
Studies I prefer hope and enthusiasm to expertise and
so do most of my colleagues.

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\noindent
{\em How Many Hours?}\/  Three hours of supervision use up
one afternoon a week. Most people will find that their
main work does not suffer, and may, indeed, benefit from
one afternoon a week doing something different.
Six hours of supervision use up two afternoon a week.
If you are badly organised, or if afternoon seminars
and morning lectures already eat deeply into your time,
this may be too much but many people can manage this
amount of supervision without their main work suffering.
If you find yourself doing more than six hours of supervision
a week on a regular basis you are almost certainly doing
too much for the health of your research. (In addition,
many scholarships and awards specifically
limit the amount of teaching you may do to six hours
a week. The same restriction applies to all students
studying for a Cambridge PhD\footnote{See paragraph
12 of the BOGS memorandum to Graduate Students}.)

There is another way of approaching the matter. Observe
that the amount of preparation required is much the same
whether you give one or three supervisions. There are
thus substantial economies of scale in giving several supervisions
on the the same exercises but, as the number of supervisions
increase, you will become aware of the diseconomies of
boredom and staleness (`Have I already told them that, or was
it the previous pair?'). Most supervisors find that if they
give such a series of supervisions they are happiest with
the second or third supervision and that their satisfaction
with their own performance then tails off.


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\noindent
{\em Finding Supervision Work}\/  Supervisions are arranged
very early in the term so, if you wish to supervise, you
should start looking for supervision work a few days before
lectures start. (There is rarely any point in starting earlier
than this since Directors of Studies have better things to
do with the vacations than worry about the next term.)
If you are a member of a College which takes undergraduates
it is considered courteous to offer your services first
to the Director of Studies at your own College. Even
if he or she can not offer you anything directly they may
pass your name on to an organiser of a Part II circus.
(Even if you belong to a Graduate College it may be
best to start with with your own Director of Studies,
if such a person exists, since he or she may well have good
connections with an Undergraduate College.) If this does
not work, your Research Supervisor may be able to help
(though such assistance is not part of their duties
towards you). Both Departments maintain Notice Boards
for advertising demands for, and offers of, supervision
work\footnote{Directors of Studies in subjects like Natural
Sciences and Economics may also be on the lookout for
supervisors. Such supervisions are what
is called {\em service teaching}\/ and you may find that
quite a number of your supervisees are less competent
and less enthusiastic than you would, ideally, wish.
If you are a good teacher and you have the backing of
a good Director of Studies such teaching can be very rewarding.
Under other circumstances it may come to resemble breaking
stones.}. In addition there is now a World Wide Web page
(accessible from the mathematics faculty page) for advertising
demands and offers. I suspect that, despite the Luddite tendencies of 
people like myself, this will rapidly become the main forum
for this kind of thing. It is generally considered that Part III students
should be so busy
following this gruelling and testing course that they should
have no time to supervise others.

Directors of Studies prefer, if possible, to use experienced
supervisors for the general first and second year supervisions
where bad supervising will do most harm. Frequently
the Director of Studies will do mainly first and second year
supervisions in order to get to know their undergraduates.
Whenever possible, also, they use members of their own
College who will, in theory, and usually in practice,
feel personally involved in the success of their
College's candidates\footnote{According to Cornford in his
{\em Microcosmographia Academica}\/ `An appeal should be made,
wherever it is possible, to {\em College Feeling}.\/ This, like
other species of patriotism, consists in the sincere belief
that the institution to which you belong is better
than an institution to which other people belong.
The corresponding belief ought to be encouraged in others
by frequent confession of this article of faith in their presence.
In this way a healthy spirit of rivalry will be promoted.
It is this feeling which makes the College system so valuable;
and differentiates , more than anything else a College from
a boarding-house; for in a boarding-house hatred is concentrated
not upon rival establishments, but upon the other members
of the same establishments.}. On the other hand, few, if
any, Colleges can fill their Third Year supervision requirements
from their own resources and many specialist subjects have
only a limited pool of potential supervisors. Thus the majority
of supervisors start by taking on specialist Part II or Part III
supervising.
Finally you should remember the following points.

(1) Unless you are a Teaching Fellow of a College
you have {\em no duty}\/ to supervise for your College or for
anybody else. If you do not wish to supervise just
say so. Nobody will hold it against you.

(2) Directors of Studies may not be able to supply the
exact number of hours that you want. If you ask for three hours
they may offer two or four. It helps to be flexible.

(3) Directors of Studies are in the business of
providing the best possible
teaching for the undergraduates in their charge --- not
of providing pin money and teaching experience to post-graduates
however deserving.

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\noindent
{\em College Post}\/ The College Post is free but is
sufficiently slow and erratic to
allow students endless scope for excuses. If you have anything
urgent (and most things involving supervisions turn out
to be urgent) deliver it by hand to the Porter's Lodge
of the College involved (or use a trusted messenger, or
first class post).
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\noindent
{\em The First Meeting}\/ It is possible to make all
your supervision arrangements by post, but most supervisors
find it easier to have a preliminary meeting to fix
supervision times, give arrangements for handing in work,
and to set supervision work. Here are some hints concerning
these matters.

\noindent
\underline{Supervision Times.} If you have, say, four supervision
pairs a week the simplest procedure is to name four times
and leave it to the students to choose which pair goes
to which supervision. (Remember that students are here to
do mathematics, not to play cricket, act or row and that,
outside lectures, supervisions have first call on their time.)

\noindent
\underline{Arrangements For Handing in Work.} Almost all supervisors
prepare for supervisions by marking students' work in advance.
In this case you must set a deadline for handing in work.
(If you have an official deadline and a later unofficial
deadline you will find, within two or three weeks that all
the work is handed in five minutes before the later
deadline. You are therefore well advised to have a single
deadline and stick to it.) It is unfair to students to
ask them to hand in work more than 24 hours, say, before
the supervision\footnote{Do you really need longer to
mark it?} and, if you do demand a longer period the students
may forget what they were thinking when they wrote their work.
Make sure that the students know where to hand in the work
(`My pigeonhole at College', `The pigeonhole marked $K$
in DPMMS' [remember that the departments are closed over
the weekend] etc).

\noindent
\underline{First Week's Work.} If you intend to use the
lecturer's example sheets, remember that many lecturers
do not issue the first example sheet until after a week
or so has passed. You may be able to get round this by
asking the lecturer for a copy of his or her first
example sheet and Xeroxing it or by getting a copy of the
first example sheet for the previous year. The main problem
(and the reason for the lecturers' delay) is that
most courses start slowly and students will not
have covered much ground in the first week. You might
therefore prefer to set questions which revise relevant
previous work or questions which take them slowly through
some exemplary case. (For example, the motion of a slightly
perturbed particle in a bowl with equation $z=x^{2}+2hxy+y^{2}$
provides an excellent and easy introduction to normal modes.)

\noindent
\underline{Lecturer's Handouts.} It is a good idea to ask
one of your students to take an extra copy of anything
handed out by the lecturer and to give it to you.
Example sheets for Applied Courses and Core Courses are
placed on racks in DAMTP and example sheets for Pure
Courses are kept in a small filing cabinet close to the
mail pigeon holes in DPMMS. (You can also sign a list in
DPMMS asking for Pure Examples Sheets to be sent to you.)
Example Sheets for Statistics Laboratory courses are held in
pigeon holes by the Statistics Laboratory enquiry office.

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\noindent
{\em Pairings}\/ Usually your students will have been put in
pairs by their Director of Studies or by the circus organiser.
Although an attempt is made to pair according to ability,
other factors affect the success of a pairing.
It is, for example, much more important that a pair be friends
than that they be intellectually matched and, at Part II
level, circus organisers try to pair people from the same
college. If you feel that some other pairing might be more
successful feel free to try it but my own experience
suggests that unless two students are obviously
ill matched it makes little difference.

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\noindent
{\em Number in Supervision} If you are supervising 1A or 1B
(non-O) courses no student should drop out and the number you teach per
hour is a matter for the Director of Studies. If you are supervising
Part II or a 1B O course then students may drop out. Unless a College
has specifically told you otherwise you should assume that it would prefer
paying for supervisions in pairs to paying for a singleton.
You should do your best, by merging and rearrangement to fall in with this
assumed wish --- but if it can not be done then it can not be done
and you should not worry.


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\noindent
{\em What Does The Course Contain?}\/ The contents of each
undergraduate mathematics course are closely specified
in schedules which bear the lapidary phrase
`These schedules are minimal for lecturing and
maximal for examining'. Copies of the schedules are
given to each undergraduate and further copies are
available from the Faculty Office in DAMTP.
Once a course has been running for a couple of years
the Tripos questions will help flesh out the schedules
and should also be consulted. (Copies of the last three
or four year's Tripos papers, or `Specimen Papers' where
appropriate may be obtained from DAMTP enquiries. Copies
of the last 30 years Tripos papers are held in both
Departmental Libraries.) If you are in doubt as to how the
lecturer intends to cover part of the course ask the
lecturer or consult the lecture notes of one of your
better students\footnote{It is useless to ask a student directly
whether a lecturer has done such and such in the lectures.
You will meet either with puzzlement or stout denial.}.

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\noindent
{\em What Work Should You Set?}\/ If you are teaching a first or
`general' second year course then you should discuss what
to set with the Director of Studies. In many Colleges the
Director of Studies may specify the questions exactly.
(For example the third term work may consist in going
through the questions of last year's examination on the subject.)

If you are teaching a `specialist' second year or third year
course then it is up to you to choose what to set. The
traditional system was to set five or six old tripos questions
per supervision. The modern method is to use the example
sheets provided by the lecturer. The lecturer should make
it clear how many example sheets he or she intends to use.
(Frequently the lecturer for a 24 hour course
issues 4 example sheets, each one
intended to supply enough work for a complete supervision
for an able student. This is not, however, a universal
model.)

The majority of supervisors use the modern method.
(If the course is new, there is no other possibility.
Even if the course is an old one, syllabus changes may
mean that the old Tripos questions no longer reflect
the course content accurately. Using the example sheets
keeps the lectures and supervisions in parallel.)
There are however some arguments
for the old system.

(1) If you choose the questions your supervisions will better
reflect your style of teaching and, perhaps, the needs
of your students. There is a general belief that it should be
possible (if only the lecturer worked a little harder)
to produce an example sheet suited to all students and
all supervisors. I do not share this belief.

(2) The occasional question not yet covered in lectures
will encourage students to consult the College Library.
(But the first few times you will need to give pretty
explicit instructions --- the idea of doing mathematics
by consulting books rather than lecture notes is
will seem strange to most first years.)

(3) Since the students will have to do Tripos questions
rather than example sheets in the exam it makes sense
to give them some practice at doing Tripos questions.
In my experience students' confidence in themselves
rises substantially after they have conquered a few
genuine Tripos questions. (It is usually possible to
find a few Tripos questions which, although genuine,
are rather easy.)

In spite of these arguments, beginning supervisors will
probably prefer to stick to the lecturer's example sheets.
(Though even here you will have to decide what it is
reasonable to ask your weaker students to do.)
As you gain in experience and confidence you ought,
at least, to consider replacing some of the lecturers
questions by Tripos questions or other questions
of your own.

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\noindent
{\em Rearranging Supervisions}\/ You may well need to
rearrange your supervisions once or twice during the
term. In return you should be sympathetic to students
who need to change their supervisions for good cause.
However a small percentage of students do trade on the
good nature of their supervisors so you should make sure
the cause is good (being ill is a respectable reason,
college football is not). Whether it is you or the student who
wishes to rearrange, as much notice as possible should
be given.

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\noindent
{\em Failure to Attend Supervisions}\/ My comments here
depend on whether the supervision is a Part I or a Part II
supervision.

\noindent
\underline{Part I (non O-courses)} It is very unusual for a first or second
year student to fail to turn up to a supervision
without notice . You should ask his or her supervision
partner to check what has happened. It usually turns
out that the student is ill and all you have to do is
send a get well soon message. If the student is not ill
(or if the illness persists through to a second supervision)
you should contact the Director of Studies and tell
him or her what is happening. (Do not expect to be effusively
thanked, you have just added another worry to the Director
of Study's load.)

\noindent
\underline{Part II (and O-courses)} Unfortunately failure
to turn up to a Part II supervision is a less rare event
and often only reflects the thoughtlessness of the
student. Reasonably enough, students normally start
rather more courses than they expect to take seriously
for the exam. More unreasonably, they sometimes forget
to tell the supervisor when they do drop out. You
should make it clear that if a student fails to turn
up at a supervision without telling you in advance
you will

(a) charge the College for the supervision, and

(b) tell the College why you are charging.

\vspace{.5\baselineskip}
\noindent
{\em Other Bad Signs}\/ Very good students produce uniformly
good work even when suffering from flu and a rocky love
life. Good students may produce the occasionally poorly
prepared supervision and you should not worry if a
first or second year student
has one bad supervision out of eight. If however he or
she has more than two such supervisions in a term, or
if the general standard of the work is such as to make
it clear that the student can not cope, you should not
wait for the end of term but discuss the matter as
soon as possible with the Director of Studies
\footnote{You will
probably only see a third year student four times
in all and
you do not know if your course is one that he or
she is taking seriously. If the student is doing
badly you should say so in the supervision report
but you need not take any other action.}.
Once you have done this, your responsibility ends.
The Director of Studies is paid to worry about such things.
You are not.

More generally, if Directors of Studies wished to
hire amateur psychiatrists
or social workers in place of a mathematicians, they would do
so. If a student is going through a bad patch, he or she
will probably derive greater security from a supervisor
who demands good work and no excuses
than from supervisor who pries,
however sympathetically, into their private lives.
Although Mathematics students exhibit
(in common with their elders) a range of behaviour which
might cause comment in the outside world, they are
in Cambridge parlance `low tutorial risks' and you are very
unlikely to see anybody with really serious problems.

\vspace{.5\baselineskip}
\noindent
{\em Copying}\/ From time to time you will become aware that
two students are producing identical stupid mistakes.
This is not as easy to handle as it looks for two reasons.
The first is that students often react like the prisoner
who complains that that he is the victim of a miscarriage
of justice because he has been convicted of robbing a bank in
Bristol at a time when he was actually robbing a bank in London.
If you accuse A of copying from B when in fact B copied from A
or cite as an example of copying the one question in which copying
did not occur, the students may use indignation to obscure
the issues not only from you but from themselves.

The second is more fundamental. Mathematicians spend much more
of their time co-operating than competing whereas clever
schoolchildren spend more of their time competing than
co-operating. Many Directors of Studies actively encourage
students to discuss questions and problems together.
Your best plan is, perhaps, to talk generally about the virtue
of collaboration (where ideas pass through the brain before
reaching the paper) and the vice of copying (where the brain
is by-passed). I tell my students that if, when
they finally write out their supervision work, they are alone
in a room with no-one else's answers, 
they will not have overstepped the boundary of acceptable
co-operation.


\vspace{.5\baselineskip}
\noindent
{\em Pie-Crust Promises}\footnote{Easily made, easily broken}\/ One
of the supervisors who commented on the first draft of these notes
remarked that novices should be warned against aceding to requests
to defer supervisions to the start of the next term `since \ldots
experience suggests that never quite as much or as good work
is executed in vacations as students predict'. More generally
supervisors  have problems with what  I would call `pie-crust
promises' --- that is promises which 10\% of students actually keep
but which 90\% believe when they make them. All that I can suggest 
is that you try to modify the conditions to make it easier for
the student to stick to their resolution --- for example by
deferring the supervision to a week after the end of term rather
than the beginning of the next term. (However, I must admit that
even this ploy fails more often than not.) If a student uses the
line `Well I tried really hard on lots of the questions but I wasn't
able to write down anything' tell him or her that they should
write down everything that they think is relevant  (definitions,
statements of theorems etc) and that you expect at least ten
lines of writing for each question. Sometimes students suffer
from `mathematician's block' and this demand actually produces
a cure. Sometimes a student is coasting and will decide that it easier to 
do the work than feign incompetence.
In the remaining cases,
though the knowledge may not be of any practical use,
you will at least be able to distinguish
whether the trouble stems from laziness or genuine inability to
cope.


\vspace{.5\baselineskip}
\noindent
{\em What Should a Supervisor Teach?}\/ The supervisor's
main duty is not to the Departments or even the College
but to the student and, so far as the student is concerned,
your main duty is to teach them to do as well as possible
in the examinations. Since the way to do well in the
examinations is to know lots of mathematics, to be
able to write it down coherently and apply it appropriately
it will be seen that your duty runs almost in parallel with
the higher ends of education.

If asked what a supervisor actually does, the student
will probably reply that supervisors answer questions
about the the courses, mark supervision work and show
how to do the exercises the student can not do.
The student thus sees the supervisor as teaching
specific items of knowledge. However a good supervisor
teaches more general things as well.

Since students are not taught how to write mathematics
in school one of your main tasks, particularly in the
first year, will be to teach them how to organise
a mathematical argument. One of the hardest parts
of supervising is to convince students that a particular
argument is inadequate and genuinely needs to be replaced.
(The lines `He who is convinced against his will is of
the same opinion still' come to mind.)

Most students come from school with the impression that
mathematics is taught and learned in a dull monotone.
(witness the set of notes in which every second
sentence is highlighted in Day-Glo yellow). They need to learn
that each course has a few central ideas and that once
these ideas are identified and understood the rest will
fall into place. In the same way they have to learn
that the gigantic dinosaurs of formal proof are frequently
controlled by a rather small brain of an idea.

Students, in general, arrive with no mathematical conscience.
If they can get to the end of a question they breath a sigh
of relief and move on to the next one. They do not
ask `Does it generalise?', `Is the answer plausible?',
`Where have I used each hypothesis?', `Which general
theorems did I use?' and they would not dream of subjecting
their own arguments to a hostile examination. Initially
you must do all this for them, acting as an external conscience,
but over two years students must learn to do this for
themselves --- internalising their supervisor, as it
were --- and you must help them do this.

In the long run, too, students must learn what it means
to think like, say, a numerical analyst or an algebraist.
You can help their development by taking time off from
{\em marking}\/ questions and using it to {\em discuss}\/
questions --- what they test, where they come from, where
they might lead and so on. 

\vspace{.5\baselineskip}
\noindent
{\em Be Specific}\/ I was once told by a Swimming Trainer
that it took 24 hours of training to modify a swimmer's
stroke. It is very hard for students to modify their
behaviour and it is helpful to give advice in the form of simple
rules. For example the advice `Write out your questions more neatly'
might be usefully replaced by `Write on alternate lines'.
If a student fails to give the statement of the theorems
he or she uses tell him or her to write out the statement 
of the main theorem used in {\em every}\/ question.
This produces a close approximation to correct behaviour
and once the student has achieved a little extra mathematical
maturity the advice can be allowed to lapse.

\vspace{.5\baselineskip}
\noindent
{\em Timing}\/ An hour is a long time and being supervised is
a stressful experience for a student. Towards the end of the
supervision the supervisee's only thought may well run along
the lines `Please
teacher, can I be excused? My brain is full.'  
You should therefore deal with the most important things
(perhaps the questions that the students could not do,
or perhaps the questions which are most central to the
course) first leaving less essential things to last
or even ommiting them. It is not how much you
tell your students which is important, it is how much they
remember.
It will help your students if you vary the pace. After half an
hour stop for a few minutes and ask them about the courses,
or offer them a cup of tea or talk about your aunt who once
killed a tiger with her bare hands, so converting a long one hour
supervision into two short half hour ones.

You should occasionally  check how much
time you are spending on each member of a supervision
pair. It is not necessary that you should divide your time
equally between them, indeed students would expect you to
spend more  time on the weaker member of a pair, but if you
do not divide your time equally you should be aware that
you are doing so. There are certain kinds of supervisees
such as `Black Holes' (who sit poker-faced through an entire
supervision), `Barrack Room Lawyers' (who spend their time
arguing about the exact meaning of your questions and their answers)
and `Gee Whizzers' (who talk endlessly and enthusiastically just
off the point) 
whom it is easy to unconsciously neglect.

\vspace{.5\baselineskip}
\noindent
{\em Preparation}\/ In my experience as Director of Studies
the only time students complain about supervisors is when
the supervisors can not do the exercises they have set.
This kind of failure is much less frequent than it was
twenty or thirty years ago when supervisors used to claim
that `It does the young men good to see a mathematician at
work'. Do not deceive yourself by such specious phrases.
As your attempt to solve the problem goes round and round in circles
the weaker students will become more and more confused
and the stronger students will simply switch off.
Both weak and strong students will resent the fact that you
could not be bothered to think about what you set them.

If you mark your students' work beforehand this will tell
you what they can not do and what you have to prepare.
(If you do not mark your students' work beforehand you
will have to go through the all the questions making
sure you can do them.) If you can not do a question
the simplest thing to do is to ask around at coffee
or tea time in the department. If this fails and the question
comes off an example sheet contact the lecturer. Answers
to the last five years of Tripos Examination are held
by the librarians of the two departments\footnote{It is
Faculty policy that the answers are not available to students.
Most, though not all, Directors of Studies feel that
if answers were available they would represent too much
of a temptation to the lazier students.}.

The questions that students raise or which arise naturally
in the course of the supervision will give you ample
opportunity to think on your feet and students do not
object if in such cases you say that you can not give an immediate
answer but will try to have one by the next supervision.

\vspace{.5\baselineskip}
\noindent
{\em First Year Supervising}\/ At school students are used
to being closely supervised and having their work marked
in great detail. They find the transition to the more
open atmosphere of University life rather disorientating.
In addition mathematics students are often desperately unsure
of their ability to cope with the intense courses here.
(It is much more common for our students to underestimate their
ability than to overestimate it, though the reverse problem
does occur.) For these reasons it is very important to
comment on first year student work in greater detail
than is necessary (or, indeed, desirable) in later years.
You should also explicitly tell first year students if they are
doing well or badly.

\vspace{.5\baselineskip}
\noindent
{\em Third Term Supervising}\/ At the beginning of the third
term students have less than six weeks to go until the exam
and the amount of new material they can absorb is limited.
It makes sense to concentrate on making sure that what
they can do they can do well. If one more push will enable
a student to master a topic then help him or her to make
that push; if a substantial effort is required it may
make sense to abandon the topic for this term. Since it is
important to keep the candidate's morale up you may choose
to let certain things which you would criticise in another
term pass without comment. If you think students are going
to do well do not leave it to the final supervision to tell
them so. Be as optimistic as possible about the prospects
of all your students.

It is traditional
for the supervisions in this term to centre round the previous
year's examination questions. If you decide to do something
else you should explain to the students why you are breaking
with tradition. Although some Colleges give mock examinations
in the second term (when they may well spur students to
work harder) I think that little purpose is served by such
things in the third term. What, after all, does a student learn by
rushing through a Tripos question that he or she will not
learn better by thinking about it at leisure?

\vspace{.5\baselineskip}
\noindent
{\em What is an Alpha Question?}\/ With the exception of the
Essay Paper in Part IIB Tripos questions are either standard
or short. A standard question should take a well prepared
candidate between 20 and 30 minutes. A standard question
carries a possible mark of 20 and a candidate who scores
15 or more is given an `alpha'. A candidate who scores
between 10 and 14 marks inclusive is given a `beta'. A short
questions should take half as long and carries 10 marks.
A candidate scoring 8 or more obtains a `half alpha'
and a candidate scoring between 5 and 7 marks obtains a
`half beta'\footnote{Inevitably the younger supervisors are
beginning to think in terms of `new alphas' whilst
the older generation still think in terms of `old alphas'
with a conversion rate `two new alphas to to one old alpha'.
These notes are written in `old alphas'.}.

Initially the examiners order the candidates by total
marks and draw the class border lines using this list.
They then look at the number of alphas of candidates
near the border lines and promote and demote candidates
using the number of alphas as a (perhaps the) major criterion.
At the second/third border line they may also look
at the `beta count'. (The third/fail decision may
involve detailed discussion of how particular candidates obtained
their marks.) The border line marks for the last year
are not secret (though they should not be shouted from the
rooftops) and may be obtained from your Director of Studies
\footnote{There is a persistent but false rumour among the
undergraduates that there is a magic formula used by the
examiners in deciding the border line. What is true is
that the computer program used by the chairman of the
examiners has a formula by which it tries to {\em guess}\/ what
the examiners will decide. The computer is rarely far
wrong but it is also rarely exactly right.}. Note, however
that the borderlines change from year to year according
to how the marks pan out.

\vspace{.5\baselineskip}
\noindent
{\em Examination Technique}\/ All students seek the holy
grail of an exam technique so powerful that its owner
can dispense with any actual knowledge. At a more
rational level, exam technique does make a small
difference but the great majority of your students
have been competing vigorously in examinations
for many years and know at least as much about the
subject as you do. In the unlikely event that
you have a student who needs help in this direction
the following simple maxims will probably be sufficient.

(1) Don't panic.

(2) A standard 1A or 1B question is supposed to take the average
student half an hour. It can not therefore be very hard.

(3) Questions which take a long time to state (and so
contain lots of clues) are often easier than short ones.

(4) Help the examiner by underlining answers. The less time
the examiner spends on a question the less time he or
she has to find errors.

(5) Read all the questions on the paper. Candidates tend
to start at Question 1 and work forward but the easiest
questions may be at the end.

(6) It is hard to get full marks on multipart and essay
questions but it is also easy to get some marks. If things
are going well avoid them; if things are going badly try
them.

(7) Keep a check on the time. If you have been working
on a question for 20 minutes without result, move on.
Inspiration may strike while you are doing another question.

(8) If you can not do anything rotate your efforts round
a limited number (three or four at most) questions
rather than rushing around attacking them all.

(9) Never ever leave an exam early. While there is life
there is hope.

(10) Don't panic.

\vspace{.25\baselineskip}
The various pieces of advice just given may not be very useful
but at least are fairly harmless. Students also ask,
directly or indirectly, what is
`the least that will satisfy the examiners'.
Do not indulge them. Explain that {\em time}\/ is not a
problem for the average examinee. If a candidate can
answer four standard questions well in a three hour exam
he or she is doing very well
\footnote{What about the best candidates?
In my view there are few more futile occupations than
trying to train students to come top in Cambridge exams.
If they are good enough to have a chance of coming top, they
should be good enough to have higher ambitions.}.
Remind them also of the importance the examiners attach 
to the number of Alpha Questions and that
it is therefore wise for the average student not to throw
away marks by skimping on his or her good questions.


\vspace{.5\baselineskip}
\noindent
{\em Very Strong Students}\/ It is said that there are fewer
luscious blonds\footnote{Of either sex.}
and large companies in Sweden than is
generally supposed and that most of them are in private hands.
In the same way, you may find fewer really able students
among your supervisees than you expected. If you do have
a very good student remember that even bright students
need reassurance that their work is going well.
Remember too, that bright students can acquire
bad mathematical habits like explaining too little
(after all, it is obvious) or too much (`I write at length
because I am too lazy to be brief') so you must
read their work critically. With such students the
main business of the supervision will only take
part of the hour, so you can discuss matters arising
out of the questions (can the hypotheses be weakened,
what happens if the field is non-constant, etc) or
the course (how is Galois theory applied to differential
equations, why did Markov invent his chains, etc).
If you can think of nothing further to say nobody,
least of the student, will mind if you end the supervision
early.

No meeting to discuss teaching is complete without someone
talking about the need for `challenging problems for
the very able'. In my view this is mistaken. Many
of the undergraduates whom we see as coasting effortlessly
along see themselves, more correctly,
as engaged in a desperate and
continuous struggle to keep their heads above water.
In any case, it seems to me presumptuous to think that
we have anything extra to teach the exceptionally
gifted. Surely the best thing we can do is leave them time
to think, read and learn by themselves.


\vspace{.5\baselineskip}
\noindent
{\em Professional Courtesy}\/  Inevitably a student
will come and ask you to explain some point in his or her
notes which, it seems to you, the lecturer has got wrong.
If you give way to your initial impulse and tell the
student that the lecturer has made a complete foul up
then the main lesson the student will have learned is not
that mathematicians sometimes make mistakes or even that
you are cleverer that the lecturer but that the
subject is so hard that even the world's greatest
experts can not get it right\footnote{Many students are
under the impression that their lecturers are
the world's greatest experts. Remember the placebo
effect and do not disillusion them.} and so the
poor student might as well give up. Instead you should
consider the the following models.

\vspace{.2\baselineskip}
\noindent
{\bf Wrong} The lecturer has got the sign wrong.

\noindent
{\bf Right} You seem to have copied the sign wrong.

\vspace{.2\baselineskip}
\noindent
{\bf Wrong} The lecturer's method of proof went out with
the Penny-Farthing.

\noindent
{\bf Right} The lecturer's proof emphasises the concrete
aspects but you may prefer the following more abstract
version.

\vspace{.2\baselineskip}
\noindent
{\bf Wrong} This syllabus misses out everything that is important.

\noindent
{\bf Right} Now that we have finished this weeks work, let
me talk a little about where the subject goes.

\vspace{.2\baselineskip}
\noindent
If you think there is a real problem with the syllabus
you should discuss it with the lecturer or other senior
staff. (Remember that syllabuses are constantly revised.)
If you think that there are problems with the example
sheets write to the lecturer; on the whole lecturers
complain that they get too little feedback on their
examples sheets.

It is your duty to stress the importance
and interest of your subject. (If you tell students that
a subject is dull they will believe you.) You should,
however not do this by denigrating other
subjects --- if you refer to statisticians as failed
bookmakers make sure your students know you are joking.

\vspace{.5\baselineskip}
\noindent
{\em Records}\/  You will find it easier to fill out claim
forms and supervision forms at the end of term if you keep
some notes on the supervisions as you give them. One simple
form might be as follows:-
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}\hline
Athos&$\alpha$&$\alpha$&$\beta$&$\alpha$&$\alpha$&$\alpha$&$\alpha$&
$\alpha$\\ \hline
Porthos&$\beta$&$\alpha$&$\beta$&$\gamma$&$\alpha$&$\alpha$&$\beta$&
$\beta$\\ \hline
Aramis&$\beta$&$\beta$&$\gamma$&X&$\beta$&X&$\gamma$&
$\beta$\\ \hline
\end{tabular}
\end{center}
where X marks a supervision missed and the Greek letters represent
the quality of work produced.

\vspace{.5\baselineskip}
\noindent
{\em Claim Forms}\/ Colleges should send you supervision
forms and claim forms towards the end of term. You
will not be paid until the College has received
your completed form back. Given the informal nature
of the supervision system it is inevitable that
mistakes get made in sending out claim forms. If
you receive a claim form from a College for which you
have not supervised just ignore it. If do not
receive a claim form from a College for which you
have supervised walk round to its tutorial office
and ask for one. Colleges do not wish to loose
your goodwill or your supervision reports and the
office will be pleased to supply supervision
and claim forms.

In my opinion, if you have given 7 out of 8 or 3 out of 4
of a course of supervisions it is reasonable to claim for
the whole course. In any case, remember that Colleges
want the supervision forms returned quickly and,
if you insist on only charging for work done, make
a separate claim the next term rather than delaying
the forms until you have done all the supervisions.

\vspace{.5\baselineskip}
\noindent
{\em Filling Out Supervision Reports}\/ Supervision reports are
used by Colleges in several different ways.

(1) At the end of each term the Director of Studies discusses
with each individual student how the term has gone and what
aspects of his or her work need improvement. Supervision
reports usually provide the basis for the discussion and
this fact explains why you are urged to return supervision
reports early. A  long supervision report which arrives after
the end of term is worth much less than a shorter one which
arrives on time.

(2) At the end of a student's time at Cambridge,
the Director of Studies will use supervision reports
in compiling references. Such reports are particularly
useful in adding a bit of convincing colour.
(`All her supervisors comment on how hard she works',
`Although his overall examination results have not been good,
his statistics supervisor speaks highly of his statistical
intuition.')

(3) When students are unable to take their end of year
exams because of illness or for some other reason
a University committee has to decide whether they can
proceed to the next stage of the course (or, if
they have reached the end of the course, whether they
should be given a degree). Supervision reports usually
represent the only hard evidence available. Supervision
reports are also used if a student fails an exam and
the College must decide what to do about it.

(4) Supervision reports may also give early warning of
impending problems.

\vspace{.5\baselineskip}
Many Colleges use a standard supervision form which
asks you to fill in

(i) {\sf Name of Student}

(ii) {\sf Subject} Fill this in by writing `Maths' followed
by the part of the Tripos (1A, 1B, II or III) and then
the specific subject. For example you might write
`Maths 1A, Prob \& Dif Eqns'.

(iii) {\sf Number of Supervisions on which report is based}
You should note in your report
any supervisions the student has failed to attend
without reasonable excuse.

(iv) {\sf Total number of supervisions you intend to give}

(v) {\sf Term and Year} In Cambridge the Autumn term is called
Michaelmas, the Winter term is called Lent and the Spring
term is called Easter. (In Oxford they are all called
something else again.) Since the academic year does not
coincide with the calendar year I give the academic year
so that I might write `Lent term 92/93'.

(vi) {\sf Is the report confidential?} In my view the report
is for the College and not the student so I mark my reports
confidential. You may have other views.

(vii) {\sf Summary of Performance} In this section you are asked
to estimate the ability, industry and probable performance
of your students. (The question asking you to estimate the
spot price of oil in Rotterdam in a year's time seems to have
been missed off.) It may comfort you to know that the
Director of Studies who will read your report finds it
almost as hard to answer such questions as you do and will
not attach an exaggerated degree of importance to your replies.
Here are some vague guidelines which you should feel free to ignore.

\  \ {\bf Ability} Put `excellent' if the student has star
quality, `excellent/good' if you have no real criticisms
(you are always free to mark two boxes), `good' if the student
has problems but can overcome them (most students should be `good'
or better),
`fair' if the student can just about cope now but may
run into real problems later. If you put `poor' this will
ring alarm bells.

\  \ {\bf Industry} Put `good' if the student has
worked hard  for all supervisions (most students should be
`good'). Put `adequate' if one or
two supervisions (out of eight, say) have been poorly
prepared. Anything worse than this should be marked
`unsatisfactory' and, again, such a notation will ring
the alarm.

\ \ {\bf Expected Grade} Use your liberty to mark more than one box
to the full. Since the weather tomorrow is likely to
resemble that today, the safest prediction in the second
or third year is that the candidate will repeat the
previous year's performance. In any case (unless you mark
`3rd/fail' or `fail') nobody will worry about your prediction.

(viii) {\sf Detailed Report} As a Director of Studies,
I find this is often the most useful part of the report.
\samepage{
However if you have nothing in particular to say
do not worry --- everybody has students who make no
impression on one whatsoever.}

\vspace{.5\baselineskip}
\noindent
{\em Taxes}\/ You are only liable to taxes if your total
earnings from all sources exceeds a certain amount.
First time supervisors are thus unlikely to need
to worry about taxes. However, if you have substantial
other income (for example if you are a research fellow)
then you will need to make an annual return to the Inland
Revenue detailing all your income including supervision
fees. Supervision fees are not taxed at source (so the
Inland Revenue are particularly concerned to know about them)
but Colleges send a list of their supervisors for the
year to the tax offices. All this means is that you must
keep a record of your supervision earnings by, for example,
photocopying the cheques and placing the photocopies
in a suitably marked file.

\vspace{2\baselineskip}
If these notes read like the clucking
of a mother hen I apologise.
If you have suggestions for improvement please
contact me by e-mail (\verb+twk@pmms.cam.ac.uk+) or
otherwise. In any case let me end with my best wishes
to you and to your future supervisees.

\vspace{2\baselineskip}

\begin{footnotesize}
\noindent
[Last revised 1 August 1996. Printed out \today. These notes are
written in \LaTeX2e and stored in
directory \verb+~twk/FTP+ on moa
in (I hope) read permitted form in a file labelled
\verb+SUP.tex+. It may be accessed via my web home page
\begin{center}
{\tt http://www.pmms.cam.ac.uk/home/emu/twk/.my-home-page.html}.
\end{center}
Also available:-

`Dr K\"{o}rner's Helpful Guide For Mathematicians Seeking A
Cambridge Research Fellowship',

`In Praise of Lectures' (how to listen to a mathematics lecture),

`An Unofficial Guide To Part~III',

`How to Write a Part~III Essay',]

\end{footnotesize}


\end{document}

