Towards a viable structure for our undergraduate course
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There is at present a complete lack of honesty or clarity of thought in
our undergraduate teaching. Together with our students, we are trapped
in a treadmill that has grown up around us without anyone planning it,
where genuine issues of learning, teaching, science and student welfare
are all sacrificed to ridiculous rituals of exams and preparing for
them, a system having no detectable purpose other than upholding a
self-justifying class division of graduates into firsts, seconds,
thirds and fails. We make ourselves the laughing stock of the
university and of the country by taking in students with higher A level
grades than other departments, and passing out a higher proportion of
them with thirds, pass degree and fails. Each of us puts a big fraction
of our professional lives into teaching, knowing that much of it is
wasted, and that a significant number of our students understand less
about math on graduating than when they entered. At the other end of
the scale, we have many excellent students; but our system also
misdirects much of their time, effort and competitiveness into a quest
for exam marks that is frankly contemptible, when they could be
studying serious math or acquiring other cultural accomplishments.
Instead of offering them a challenge and rewarding original thought, we
put a ceiling on their aspirations and reward poverty of ambition.

A couple of weeks ago at the University of Madison, I noticed a little
plaque on a building saying that in the 1880s some famous crank
introduced Revolutionary Teaching Methods: students were not examined
at all and could study whatever they liked (with some slant towards
dance and other forms of physical self-expression). But can anyone
seriously maintain that this would be worse than the disaster we
currently have on our hands?

I propose an analysis from first principles along the following lines:
 (1) What are we good at?
 (2) What is our student intake?
 (3) Can we replace our exams by checks and tests designed to enhance
learning, teaching, science and student welfare?
 (4) What kind of education can we give to students not destined to
become research mathematicians?

We all know the answer to (1). We're good at doing math research, at
training research mathematicians, at giving courses at MSc or doctoral
level. Along with that, we also know exactly what we want to teach in
3rd or 4th year MMath courses; the quality and variety of our MMath
courses speaks for itself. The inevitable conclusion is that one
emphasis of our undergraduate teaching must be to recruit and to
prepare students for our MMath, MSc and PhD programs.

Question (2). We are in mass education. We take in 200 or 250 kids,
selected from the brightest school-leavers in the country, with top A
level grades. We don't have any control or even serious influence over
how or what the schools teach, and we have absolutely no way of
distinguishing the students that are growing into research
mathematicians from those whose A level grades will be the peak of
their intellectual career. (Cambridge does no better: apart from the
top 2 or 3, the quality of their students is exactly the same as ours.)
Of this intake, perhaps one or two will fail at the end of their first
year and another half-a-dozen at the end of their second year, perhaps
two dozen will transfer to other Warwick departments such as business
studies, and each and every one of the remainder will win a Warwick
math degree in 3 or 4 years time. The inevitable conclusion is that it
is our moral and political duty to provide each of these young human
beings with an appropriate course of study.

It is at this point that an honest analysis would benefit us, and lack
thereof is extremely damaging. As we all know, whereas our top students
are capable of becoming research mathematicians, and the majority can
do undergraduate math perfectly well, a significant minority are never
happy with first year math, and go from bad to worse in later years. In
fact our course structure, coupled with misplaced perseverance,
unrealistic expectations and lack of imagination on the part of the
students means that many of them will end up doing as many as 8 or 9
specialist math courses in their 3rd year of which they understand not
one word. The guy who has demonstrated that he can't do quadratic forms
in MA242 Algebra I or least common multiple of integers in MA246 Number
Theory may well end up offering MA359 Measure Theory and MA408
Algebraic Topology. There is no point in arguing about numbers here --
whether this third category is 10\% or 40\% of our students does not in
any way affect the argument that we have the duty to cater for them.

Question (3). Our current elaborate system of rigorous exams is
incredibly destructive, and it is hard to understand why on earth we
have slavishly accepted it as necessary for so long. A scientific test
should take away a tiny sample for analysis, without interfering with
the working of the organism as a whole. Instead of this, our exams cut
off the entire body and soul of our undergraduate teaching. It is
completely grotesque that instead of seeking self-fulfilment in math or
other subjects, our students are forced to spend 2 1/2 terms of every
year preparing for exams. Incidentally, the obsession with examinations
that blights our teaching is a close reflection of the childish fallacy
that progress can be achieved in any area by measuring results, close
monitoring of figures, compiling league tables and naming-and-shaming
the under-performing.

Setting and marking exams involves a huge and unpleasant effort on the
part of the lecturers, and sitting them take up 6 entire weeks of Term
3 (Academic Office is currently working out plans to devote another
week to them). What on earth is the higher purpose served by this
rigorous classification of students that makes this expenditure of
time and effort worthwhile? Although the exam results are expensive to
produce, at no point do we subject them to serious statistical or
epidemiological analysis to help us improve the health of our course or
the lot of our victims. In fact the main use of the exams is to enforce
the British class system, and to provide an easy criterion for financial
houses to recruit our better graduates.

Why should all the course have the same style of exam? Even if we
accept the necessity of classifying our victims rigorously, we don't
need all the main courses in Years 1--2 to have 2 hour exams with 4
questions, all at the same level of difficulty. In view of the stated
aim of establishing the Honours/Pass and the I/II.1 borderlines, some
of the courses with extensive assessed components could be one hour,
with just one hard and one easy question. Surely it would be worthwhile
to experiment around with other models to find something that works. We
should think out the aims of each assessment and exam, and limit its
scope to what is necessary to achieve it.

We must start heading towards some kind of recovery from the current
disaster over the next few years. For practical reasons, it is
imperative that any change to our teaching or assessment from now on
should reduce, not increase, the effort of examination. At present,
most first year courses have innumerable in-term assessments (whose
marking is stretching our supervision system to breaking point),
followed by exams in Week 7 and september resits. When these
assessments were introduced, we should thought of ways of reducing the
weight of the exams.

Let me suggest another principle: of the taught material, about one
third should be assessed on the spot, only about one third examined at
the end of the year, and the rest should be studied by ambitious
students as background for future courses, and for its interest. (That
is an ideal aim -- I don't know how we persuade students to take the
third component seriously.)

Question (4). The main beneficiaries of our course are the students
leaving with a low I or II.1. This is the fraction who have no ambition
to do math research, but who can study undergraduate math perfectly
well for 3 or 4 years before starting the personally fulfilling,
intellectually challenging and well remunerated career in a financial
house they thoroughly deserve.

Here I'm concerned with the tail; we must find an effective and cheap
solution to this problem. Whereas TQA told us that we shouldn't fail a
single student, for 3 years up to the present year, our BSc lists
included something like 20 x II.2s, 15 x III, together with a good
number of passes and fails. It is dishonest of us to deny a share of
the responsibility for this dreadful waste of young people's lives.
Just because this year's bunch happens to be high scoring is no cause
for complacency -- the current first and second year lists give every
indication that next year BSc list will be as bad as usual. I am
convinced we can and should find a system that is simple, effective and
cheap, and allows us to eliminate the pass/fail tail-enders in our BSc
lists, and reduce the number of II.2 and III down to a handful. Our
aims should be:
 (a) To detect possible tail-enders in good time.
 (b) To find out how their interests can best be served inside the
department or outside.
 (c) To get them into some rewarding math courses (modules).
 (d) To prevent them from doing an overload of math courses they will
not benefit from.
 (e) To persuade them to take outside options where appropriate.
And most important, to do all this effectively and cheaply.

As models for what can be achieved, take the interviews with first year
tail-enders at Jan of Y1, and MA397 Consolidations; each of these is a
life-saver for a significant number of students. But much more is
needed: year after year, both Y1 and Y2 boards pass through a couple of
dozen students who we known perfectly well are potential problems. The
Y1 board rarely fails anyone, and does almost nothing to give
tail-enders a shock, despite holding trump cards in the shape of
september resits. Every year, the Y2 board fails half a dozen students
and puts another half dozen on the pass degree. Next year we will have
11 pass degree students in Y3, and I'm jolly glad that I won't be exam
secretary when they come up before the board.

I propose that we should award ourselves additional powers to impose
conditions on these tail-enders. We should introduce a "pre-pass"
category (or "proceed to honours"), drawing the line at 55\% at the end
of Y1 and Y2 (with the usual bordeline arguments about people just
below the line). Students in this group should be subject to
restrictions on what courses they are allowed to register for:
 (i) they are treated as BSc in Y2 even if on the MMath;
 (ii) they are prevented (or strongly discouraged) from taking the 2/3
core courses in Y2;
 (iii) they are capped at 90 Cats of Math courses (thereby obliging them
to look for outside options);
 (iv) they are put on MA397 Consolidations in Y3.
We don't need any change to regulations to introduce this, just to
make creative use of the existing exam boards and registration
arrangements. My scheme does not involve very much in the way of new
resources (except for teaching MA397 Consolidations, which is money well
spent). We tell the students in PYDC that the department operates
such-and-such rules. They can argue about any of the rules, for
example, getting back onto the 2/3 core courses by showing that they
can handle the material in assessments. (In any case, this is a real
mistake of strategy for them, since they get better marks and more
credit by taking the exam one year later.) We enforce the rules where
appropriate by using the deparmental signature on registration
paperwork (that is, these named students are not allowed to use the
rubber stamp on their registration forms, that must be signed by the
named tutor or year coordinator). In practice half of these students
will pull their socks up in any case and get back to reasonable scores
in later years (and some from the higher classes may come down), but the
aim is to ensure that they all get the best chance to do so. (These
rules are not in course regulations. But the rules that are solemnly
written out in course regulations are not actually enforceable if the
student is determined to get away with it. If we tell the students
firmly that the department will not allow them to register unless
they conform, they will mostly obey.)

This is my final diatribe as exam secretary.

Miles Reid, Mon 2nd Jul 2001