FORUM DISCUSSION by Ronnie Brown

My work in Public Awareness arouse out of a set of fortunate invitations, decisions and meetings, but motivated by the slogan of `advanced mathematics from an elementary viewpoint'. I also recall a comment that a major problem in mathematical education is to convey the reality of mathematical objects. This is relevant to the theme of this conference, Mathematics and Art.

Of course not all advanced mathematics can be presented from an elementary viewpoint. Also, in undergraduate teaching, it seems essential that students should have a clear idea of why a topic is studied, so it is desirable to start, if possible, from a clear problem which the student can grasp. So I was led to abandon the teaching of homology, where the motivation is subtle, and the examples dealt with at the first level are rather dull.

Instead I developed a course on the theory of knots. The basic problems of how do we know a knot cannot be untied, and indeed what do we mean by this statement, are immediately accessible, and the mathematics which goes some way towards a partial solution is attractive, geometric, and allows for some nice algebra and computations. So when I was asked to give a lecture to the British Association for the Advancement of Science at Sussex in 1983, the subject of knots and the title `How algebra gets into knots' came to mind. In the teaching process, I had developed several tricks with physical knots and string, so these could be used. The audience, including professional mathematicians, liked the presentation and I was asked to repeat it on many occasions. I am still giving it!

The title became `How mathematics gets into knots', but even this was not accurate, since the aim was to use the theory of knots to explain some methods of mathematics to the general public. A key point of the lecture was the notion of prime knots and the analogy with prime numbers. After a lecture to children in 1985, a boy, aged maybe 14, asked if there are infinitely many prime knots! I had to say this was a very good question. After another lecture for schools, a teacher told me that nobody in his career had ever before used the word `analogy' in a mathematical context.

In the course of these lectures, I accumulated material to go in the foyer before the lecture, and we made the rash decision to consider putting this material into the form of a travelling exhibition. The naivety of this decision was of course that we had no design experience, and very little funds. In the end we accumulated £4,000 of funding from various sources, and were very fortunate in a series of designers who helped us enormously for little cost.

We also found that the exhibition format is one of the most difficult. Each board has to tell a story in itself, preferably largely through graphics, and each board has to be related to the other boards. It is not enough to say `This is a nice graphics, let's put it on' - you also have to be clear how the graphics contributes to the story you have to tell. There should be no `sugaring of the pill' ­ for this implies that the real mathematics is thought of as a `pill' to be disguised, rather than a delight to be revealed. Thus the form of the graphics has to contribute to rather than disguise the mathematics.

Further the whole exhibition has to have some clear message or impact - there has to be a decision as to what impression the viewer is supposed to gain from the exhibition as a whole, and this intention has to be implicit rather than explicit. As an example, there is no use in showing weird objects mathematicians study, unless you are trying to show that mathematicians are weird. This may in fact be true, but is not necessarily the impression you wish to convey.

The exhibition was designed to be a travelling exhibition, to be able to sent by carrier, and mounted easily. This militated against hands on material, which has problems of maintenance and security. The exhibition consisted of 16 A2 black and white boards, on bromide paper, mounted on polystyrene with an aluminium surround. A travelling case was designed. It was launched at the Pop Maths Roadshow at Leeds University in August 1989, and then toured the UK with the Roadshow

The process of design was very instructive to the design team (me, Tim Porter and Nick Gilbert). The necessity of thinking through the basic purpose of what we were hoping to convey has had an impact both on our teaching and our research.

Thus in our teaching we do think more of what should be the impact of the whole course on the student - what is supposed to be the impression of mathematics with which he or she is supposed to leave the course? What sort of qualities are we seeking to assess? There is a danger that courses are designed for an assessment, rather than the assessment is designed to assess the qualities which the course is designed to develop. Employers may want graduates who are good at planning their work, at assessing work done, at formulating problems as well as solving them, and finally at communicating what they have done - as was found by the assessment [5] of graduate mathematicians in employment!

In our research, we are inclined to question basic assumptions and to try to conceptualise the reasons for following a particular line of research. This perhaps forces us to think of more fundamental lines of enquiry.

An early aim of the exhibition was to include knots in history, practice and art. This was because of the feeling of presenting a wide context for mathematics - see the arguments in [2]. This is much more exciting to the viewer, who often sees mathematics as a subject isolated from general culture.

The difficulty in designing the mathematics board, and the sheer mass of potential material, made accomplishing this progressively more unreasonable. It so happened that in July, 1985, I passed the Freeland Gallery in Albermarle St., just off Piccadilly in central London. Having time to spare, and attracted by the sculptures of children in the window, I wandered in and was astonished to see sculptures of children happily at ease with strong and beautifully abstract bronze forms, many of a knotted nature. Both sets of sculptures were the work of John Robinson, and it seemed reasonable to me that his abstract work should be drawn to the attention of the mathematical community. In 1988, puzzling over my problems with presenting knots and art, I phoned John and asked him if he would like to contribute an exhibition to the Roadshow. He thought we could something really good in the time available, and I obtained permission from the organisers for such an exhibition to be included. In April 1989 I met John at his home. The Freeland Gallery had left Albermarle St in 1986, and the sculptures were transferred to his garden, where they looked magical. We chose thirteen substantial sculptures, and drove up to Leeds to decide where they should be placed. On the way up, I agreed to write `Conversations with John Robinson', giving titles and general mathematical background to the sculptures. When John obtained my text, he added material and 24 colour pictures, and we had a catalogue for the Roadshow! This did sterling service later as a way of introducing John's work to the academic and wider world.

In 1996, we agreed to try to put John's sculptures on the web. This was partly motivated by the greatly increased costs of colour printing, and was stimulated by the work of a student, Cara Quinton, who did an undergraduate project with us on putting mathematics on the web. She carried out the production work on the web site. This work was funded by John's Patrons, Edition Limitée, while her further work of putting the Knot Exhibition on the web was funded by the London Mathematical Society and The Philip Trust.

The web format proved very attractive to us, with its easy capacity for including pictures and animations, and for the use of hyperlinks, both internally and externally. Designing the web sites became an exciting adventure as the material and the complex inter-relationships grew. Thus the web format begins to allow the modelling of the complexity of mathematics.

There are still problems with the web format for wide use in schools because of problems of access to internet linked computers, download time, and cost of telephone calls. Thus the net should be supplemented by CDRoms, and we are producing an RPAMath CDRom based on the web sites. This also allows greatly improved animations, and a more integrated format.

What is the value of linking Art and Mathematics? Generally speaking, people feel that Mathematics is linked with the stifling of the spirit, and Art with freedom of the imagination. For the working research mathematician, it is mathematics which inspires the imagination, as totally new forms and almost unbelievable patterns and structures are revealed. These fruits of the imagination, verified through the tough testing of logic and calculation, have what has been described as an `unreasonable effectiveness in the physical sciences' (E. Wigner).

However, perhaps mathematicians can learn from the way education in art and design is carried out. A set of objectives for a course read: `The aims of this course in design are: 1. To encourage independence and creativity. 2. To teach the principles of good design. 3. To give a student a basis of skills to be able to apply the principles of good design in an employment situation.' I leave the reader to explain why it is, or is not, reasonable to replace the word `design' in the above by `mathematics' and whether the resulting aims are desirable, and are realised in current undergraduate courses!

The aims of Art and Mathematics are different. It has been said that Art has a foothold on emotion. This is not the intention of Mathematics, which, in order to find the essence of what is true, usually needs to strip off the inessential for the purpose at hand. Art and Mathematics have a common interest, in form and structure, in geometry and the way parts fit together. Also, it has been said that a major problem in Mathematics Education is to convey the reality of mathematical objects. Vivid realisations of such geometrical forms, with what an artist can give in craftsmanship, proportion, rhythm, and wider significance, can excite, enhance and entrance the imagination.

Mathematicians could also learn from the freedoms found in Art Education, and from the belief in discussion of what is `good art'. It is often held that discussion of `what is good mathematics' is inappropriate for an undergraduate education. I totally disagree with this. We have found in students a hunger for discussion on mathematics, and a placing of it in a cultural, technological and social context. It is wrong for students to graduate with a good degree in mathematics without acquiring and being able to conceptualise some professional values such as the notion of `good mathematics', the major achievements of the subject, and the place of mathematics in society. We have attempt to deal with these questions in several ways - I write `attempted' as we do not claim to have a final solution, and there are surely many modes of doing this.

We have run a course entitled `Mathematics in Context' - see [2]. After a few years we had a fair objection: `If Maths in Context is important, why are you running it as an optional third year course?' We have also run a first year course in this area, using [4] as a text. We also ask students to do evaluative projects or assignments of various sizes in other courses, for example `Write a brief account of the importance of fractals'. In this way students are forced to read books and search for material on the web, and put it together in some way. In a third year course on Groebner bases, students do a small assignment of considering a current paper using Groebner bases, and writing about it. Of course they do not have time to understand the paper properly, but they are asked to make an assessment of the use of these bases, and the practical relevance of the paper.

Two students on the Mathematics in Context course chose to do projects on Mathematics and Art. One wrote a year later that she was still haunted by the project! The other wrote in his project that having chosen this topic, it was about time he visited an art gallery!

It is not at all clear otherwise how we should bring the subjects of mathematics and art together in our curriculum, for those who wish it. Surely many more experiments need to be tried. I am sure that conferences of this kind will encourage such broader debates on mathematics, and on education.

REFERENCES

1. R.BROWN and T.PORTER, ``Making a mathematical exhibition'', in The popularization of mathematics, edited A.G.Howson and J.-P. Kahane, ICMI Study Series, Cambridge University Press, (1990) 51-64.

2. R.BROWN and T.PORTER, ``Mathematics in Context: a new course'', For the Learning of Mathematics, 10 (1990) 10-15.

3. R.BROWN and T.PORTER, ``The methodology of mathematics'', Math. Gazette, 79, July (1995) 321-334.

4. Philip J. Davis, Reuben Hersh , ``The mathematical experience / ; with an introduction by Gian-Carlo Rota", Harmondsworth : Penguin , 1983.

5. R.R.McLone, "The training of mathematicians", Social Science Research Council, 1974.

(The first three articles are available from the articles section of http://www.sees.bangor.ac.uk/public/cpm/)

(The first three articles are available from the articles section of http://www.sees.bangor.ac.uk/public/cpm/)