There is much written about the achievement of quality in Higher Education, and the Government has set up quality assessments of teaching and research.

There are many points to be made about `quality'. One is that the use of the word is curious: there is a good English word `qualities', which implies that these come in many varieties and forms. The replacement of `qualities' by `quality' with a view that this is one something (one quality?) which can be judged in a purely rank order, or even linear scale, is an aspect of the degradation and perversion of language noted by Orwell, which, in his fable, is part of a Government mode of control.

Governments like to set in motion procedures to, as they say, `improve quality'. Usually, they do this by setting up a `Quality Assurance Unit', and asking for information, which requires systems and paperwork. Also the QAU requires staff, equipment and systems. These staff are a kind of guardians of quality. But then one is reminded of Horace's quip: `Quis custodiet custodiens?' How will we ensure that these guardians are of the appropriate quality? How will we ensure that the principles they use in judging quality are well attested and good? What do they mean by `quality'? (Welsh teaching assessments were patterned and graded differently from those in the rest of the UK.)

Here is a link to an article by a guru
of quality in Business and Commerce, which first appeared in a magazine for
A-level Business Studies students (thanks to Margaret Brown for pointing
it out). I think we can all agree that for this audience, **the points made
must be very elementary, and basic**. The overall theme can be summarised:

**Quality is about investment, training, people
and attitudes:
it is not about systems, it is not about paperwork**.

A friend with army experience called this last statement `trite'! Indeed, it is only common sense.

How then does it come about that we get exactly the opposite in the imposition of quality assurance regimes in Higher Education? For example, when the Welsh Assembly agrees to spend say £5 million on Quality Assurance in Higher Education, it does not at all mean that all maths lecturers gets a new laptop (unlike the `laptops for teachers' scheme). These monies do NOT go to investment and training for the people directly involved in teaching and research. It goes instead to the `Quality Assurance Unit'. Who ensures the quality of that unit? Experience shows their judgement can be bizarre, and in any case they do not publish their underlying principles for discussion, though they do say what paperwork they require, and what systems they are examining. They can spring a new principle on you in their final assessment, with no room then for argument!

How does it come about that very large sums are spent on **pursuing a path
directly opposed to the presumed aims**? It is like flying on a course
of 18 degrees when told the correct path for home is 180 degrees. (This happened
at a University Air Training Squadron, and the student involved, who luckily
did no damage to himself or the plane, was immediately discharged. But then
it was quickly obvious that his landing point was quite inappropriate!
Fortunately, I was not the student.)

All I can presume is that Governments want to reassure people that these aims of `quality assurance' are being pursued diligently!

There are now questions on: `Where will the next generation of UK mathematicians come from?'. This is related to questions on an examination paper for staff, the thinking behind which was based on the quotation from Julius Caesar:

`The fault, Brutus, lies not in the stars, but in ourselves, that we are underlings.'

Let us look, in the first instance, at what we do and on what we have control.

Is the word `underlings' appropriate in relation to mathematicians? There seems some justice, in that it seems top people will not analyse and present to the public or Government why maths is good, except in so far as it is applied. Am I wrong? An EPSRC document once had the comment: `If you are a sad and lonely mathematician, but wish to apply yourself to biology, we will throw money at you!'

EPSRC Panels described proposals we had put in for new methods in algebraic computation as asking for software which was `too expensive'. There were no suggestions for other software which could do the job, nor comparisons with software costs in engineering, chemistry, or physics. Does `too expensive' mean `too expensive for mathematicians'?

Yet history shows, and John O'Reilly has publicly asserted, that abstract mathematics is part of the necessary basis of high technology! John's hero for this point is Evariste Galois, with applications to modern coding theory!

Some mathematicians seem more interested in saying that an area, or even a particular mathematical structure (!), is `rubbish', than in defending the subject as a whole. See for example the views on category theory quoted in my article `Higher symmetry of graphs' (preprint 93.19), and compare those with the views in Yuri Manin's `Georg Cantor and His Heritage': ArXiv AG/0209244.

This is the theme of a meeting to be held in Manchester, in March 2005. Where indeed will the maths students come from? What will they learn? What is mathematics?

There is much to be said on these issues. See my article on `What should be the output of mathematics education?', which refers mainly to the question for schools, since that was the theme of the conference for the Proceedings of which the paper was written. Have Universities taken account of the findings of the McLone Report (1974) and a subsequent EPSRC funded study by Griffiths and McLone? What impression do Universities give to students of their possible highest achievement? Perhaps they get the impression that the main aim of University mathematics education is to teach students to write neat answers to examination questions?

If this last aim is the appropriate one, then indeed a department which for exactly the two consecutive years considered gets so called `poor results' (despite the results in previous years and prospects in succeeding ones) has perhaps failed.

One aim of education is to show people how they are cleverer than they thought
they were. If the aim is `educate', with its derivation from `educare'
(education) and `educere' (lead forth), then showing a range of people the
achievements, methodology and joy of mathematics may indeed be regarded as
a great success, even if they cannot show it well in an examination paper.
Being a mathematician is about :

developing, writing and explaining mathematics:

success in these activities, as for most, requires what a novelist
once described as `the five Ds'.

A student who obtained a pass degree in maths at Bangor in 1971 recently revisited the department. She was very proud of herself. She got involved in Cogs Management Services Ltd, designing management software for business and the Civil Service! Could she have done so well without the training of a mathematics degree? In the 1970s, we took on a self supported MSc student with a poor degree from a major civic University. After passing his continuous assessment, he decided to do a project on Mumford's Invariant Theory, and did a grand job! This is anecdotal evidence which suggests the possibility, confirmed by our experience with research students, that top class undergraduate examination technique and knowledge are neither necessary nor sufficient for a career in developing and using mathematical modes of thought, and presenting these to others. Curiosity and determination are not qualities easily measured by examinations!

Our experience is that students who are asked to reflect on their relation
to mathematics by writing about mathematics, can improve enormously. So at
Bangor we set store on students writing and presenting mathematics, as a
*compulsory part *of our degrees. My own research work was given a new
direction and impetus by the writing of a book on topology, which, by a set
of fortunate chances, set me on the line to investigate groupoids in
homotopy theory.

The best advertisement for a maths degree should be the enthusiasm of the graduates: with this, they will encourage others to join in! Is that enthusiasm there? Experience at Bangor is that a wide range of students are delighted to learn words, the language, the concepts, the context, to understand and justify a subject they love, and they wish to acquire and appreciate professional attitudes. Will these commonly be part of an undergraduate education in mathematics? You do not have to get a first class degree by performance in an examination, to understand and discuss such questions! Does a degree in mathematics shows an exposure to modes of thought more valuable than, or at least distinct from, many other degrees?

There is now pressure for a 4 year supported PhD programme in mathematics, after presumably a 4 year MMath degree. How much is this needed because the undergraduate programme is about knowledge and technique, rather than process, ideas, context, and imagination, and includes no discussion or experience of research and research methodology? In the social sciences, courses on research methodology are often part of the second year curriculum. It is an interesting sociological fact that this is considered impossible in mathematics, or indeed that such a subject is considered not to exist. I have even heard it said that it is possible to teach people to teach mathematics, but not to research in it. By contrast, we believe research students need such teaching in research methodology!

Is it fair to say that the undergraduate teaching culture is inimical to the research process, so that habits formed there have to be unlearned at the research level? In the old book `Men of Mathematics' there is famously asked the question: `How much do you need to know to do research in mathematics?' The answer was: `Everything, or nothing!' What is especially worrying is the paucity of debate, even of outlets for debate.

The future of mathematics depends on young people, and on their understanding of how mathematics works, how it has contributed to society, and their ability to communicate these ideas. See also the comments of Einstein. I have found excellent teachers who do not seem to have the language to communicate to pupils why maths is a good thing - their University courses have not educated them in this aspect. There are however no simple or conclusive answers as to how to make such discussion part of a mathematics degree couse. The old rule applies: if it is important, it should be part of the assessment.

It is claimed that mathematics is so advanced that it is impossible for students to read research papers in the subject. However, in a Bangor course on Groebner Bases, one quarter of the continuous assessment was the following: use a bibiliographic database to search for papers on Groebner bases, choose on, and write an account of its use of Groebner bases to the best of your ability and knowledge in the time available (which did not allow complete understanding). Papers generally chosen were papers in the previous 5 years in IEEE journals, and on coding theory. The message intended by this exercise was the potential importance in technology of abstract ideas, in this case of rings and ideals, particularly when combined with computational methods. It is also good (essential?) for students to appreciate that there are new and emerging mathematical areas, many of which currently are in relation to computer science.

This seems a good opportunity to record a favourite quotation from the philosopher G. Spencer-Brown:

`Unfortunately, we find systems of education today which have departed so far from the plain truth, that they now teach us to be proud of what we know and ashamed of ignorance. This is doubly corrupt. '

(Follow the link for more, which has reminded me of the cryptic proverb: `If a fool will but persist in his folly, he will become wise.' )

Teaching assessments are about students, and research assessments are about
the advancement of knowledge. It is thus `Logical, but absurd', as Samuel
Butler would have said!, that there seems no place for rewarding Public Awareness
work in either teaching or research assessments, despite the lip service
paid to the idea by Governments. Our work in this area has had its
*major* funding outside UK mathematics and science, with three major
proposals refused by UK research funding bodies. We have been very fortunate
in other support, particularly the European Social Fund, Edition Limitée,
and the EC, and 21 years of support by Anglesey Aluminium for the Masterclasses.

However, work in this area has been for us stimulating and rewarding. It has contributed to interdisciplinary contacts and presentations. Presenting `advanced mathematics from an elementary viewpoint' to school children and the public, has helped us to present these ideas to other scientists, and has forced consideration of the nature and structure of mathematics. All this has helped our presentation of the subject to mathematics students!

I hope this paper will stimulate discussion. The future of UK mathematics is at stake. Should one despair? Please email me with any comments.

See also the LMS Paper in Response to the Consultation Paper on the Future of Higher Education. For other documents from the LMS, see their education web page .

See also a Letter on Science Funding to `Science and Public Affairs', August 1999.

More later.

Ronnie Brown

February 10, 2005

updated March 19, 2005

November 21, 2005 Removed comments on the RAE to a separate page

January 19, 2009 revised

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