School of Mathematics

The University of Wales, Bangor

In this article we describe a new style of course entitled "Mathematics in Context". This course was tried out with a group of eight 3rd year honours students in the first two terms of the year 1988/89. The aim of the course was to encourage students to examine global views of mathematics and current issues for a professional mathematician.

Our new course turns upside down many of the basic methods of teaching in undergraduate Mathematics. It is "top down" rather than "bottom up". It starts off with general discussions as to the nature and extent of Mathematics. It proceeds through a decision by students as to what they want to write about. Its end result is an extended essay in which the student is expected to communicate a viewpoint, and to discuss issues. We start with the following basic assumption: An education of undergraduates should teach not only how to do something but should also give some training and consideration to the following:

- The context of particular techniques.
- Relationships to other disciplines.
- The role of the subject in history and in society.
- The notion of value in the subject.
- Communication skills.
- The methodology of research.
- The extent and quality of current research.
- Current issues.

There seemed to be a danger that none of these aspects were covered in any part of the current courses. It seemed to us that many students at both the undergraduate and postgraduate levels were starved of a global viewpoint. So in the session 1987/8 a small working group was set up to look at this problem. It recommended a new course of 1/6th of the degree (replacing a traditional three hour examination paper) to consist of a half course on the History of Mathematics, and the other half on Mathematics in Context. Each part was to be assessed by two long essays, to be handed in at the beginning of the second week of the second and third terms. Here we will not describe the History of Mathematics course, but are entirely concerned with the course on Mathematics in Context, since this breaks new ground.

The Aims of the course were stated as:

- To present some global picture of mathematical activity, rather than an individual topic.
- To encourage the student to assess, to evaluate, and to write and communicate.
- To encourage participation in a joint project.

To these ends, the course was intended to discuss some or all of the following Themes:

- The impact of Mathematics on science and society.
- Mathematics and technological advance.
- The extent and size of the Mathematical research activity.
- The employment of mathematicians.
- The teaching of Mathematics.
- The communication of Mathematics.
- The funding of Mathematics.

Our initial investigations suggested that a considerable amount of source material was available, for example the following:

- American Mathematical Society publications and articles on the uses of Mathematics.
- Reports of Government commissions in various countries. (for example, the ACARD report on Software).
- Department of Education and Science publications on the teaching of Mathematics.
- Local sources on the preparation of exhibitions and the popularization of Mathematics.
- University Grant Committee Statistics, and reports from Committee of Heads of University Departments of Mathematics, the London Mathematical Society, and the Institute of Mathematics and Its Applications.
- Reports of IMA conferences on the applications of Mathematics.
- The McLone report on "The Training of Mathematicians".
- Articles in popular journals such as the Mathematical Intelligencer and Scientific American.

The proposal was agreed by the Board of Studies. In view of the radical nature of the course, the External Examiners were informed of the proposal at our Examiners Meeting. They both expressed enthusiasm for the idea, and looked forward very much to seeing how it worked in practice.

So the new session started with eight bold students and two bold staff coming to the first session and wondering what was going to happen.

Teaching has been by one discussion period of 1-1.5 hours per week taken by both of us. The advantage of having two people, is that they can put views which complement, or even disagree with, each other. This conveys to the students the fact that the arguments in this area are not cut and dried, and it also improves the atmosphere for discussion. The venue was a seminar room with comfortable chairs; this arrangement encouraged easy and informal discussion.

Our first session explained the aims of the course and why we had set it up. First we pointed out some characteristics of standard Mathematics courses.

(i) Each course is given through lectures which cover fully the material of the course. Very often, this material is written fully on the blackboard and the students copy this into their notebooks.

(ii) The courses are aimed at giving a thorough grounding in a particular topic, and in acquiring a particular range of techniques.

(iii) The students are marked on their ability to reproduce these techniques with rigour and accuracy under examination conditions.

(iv) While the course may contain some motivation, and background, the examination reflects the context either not at all or to a very small extent.

(v) There is little room for the evaluation either of techniques or of particular results.

We also pointed out some of the main conclusions of the McLone Report [2], particularly the employers' rather jaundiced view of the Mathematics graduate (see below). We explained that we were concerned that students were not given a professional's view of the place and role of Mathematics, and that they were bound to be unaware of the nature and extent of current mathematical research. The students agreed that though they had deduced that mathematical research probably went on in Mathematics Departments, since they were aware that research was a general requirement for Universities, they had no idea what the research consisted of, nor how much Mathematics research went on generally.

So the first task that was set was for students to go to the Library, to find Mathematical Reviews, and using this as a source, to come to some conclusions on the quantity of Mathematical research, its changes over time, and the balance of subjects. Of course, it would be quite impossible to read the whole material, and so some kind of sampling, and some splitting up of the analysis among the group, would be desirable.

The response to this exercise was excellent. The next week it was reported by the students that Mathematics was clearly expanding, judging by the increase in the number of pages of Mathematical Reviews over the years. One student had done a survey of numbers of papers in Ballistics over the years, and had come to some interesting conclusions. One student asked why we could not have just given them the information on the numbers of pages in Mathematical Reviews over the years. This perhaps illustrates the way in which students have been conditioned to expect information to be given to them in handy parcels. We explained that there was no substitute for picking the volumes off the shelf: Never mind the quality, feel the weight! All of the class were astonished at the extent of current Mathematical research.

The initial handouts for the course included the following:

- The transparencies for a talk by R.Brown on "The Public Image of Mathematics" which was given to the British Mathematical Colloquium at St.Andrews in 1986.
- Copies of articles from the Notices of the American Mathematical Society on current progress in Mathematics.
- A detailed statement of Themes for the whole course (see Appendix 1). This list was not intended as "definitive", but rather as a working document that would evolve with the course and with the students. It was intended to indicate some of the issues and topics that required discussion for a full view of Mathematics in Context, and some of the Themes which could form the general area for projects.

The aims of the sessions were to discuss these themes, with the intention of helping students to formulate their decisions as to specific projects to be undertaken for the essays. Unfortunately, notes were not taken of the discussions, and so we do not have a proper record of the progress of the course. It would have been a good idea if each week, a student was required to take notes and prepare a written record of the topics and of the discussion.

Both of us were nervous about the whole course; about our preparation, or lack of it; and as to how the students would react to such an unstructured and open-ended course. What was actually going to happen?

We need not have worried! The course gave people an opportunity to discuss questions which had been bothering many for some time about the nature of Mathematics and the purpose of the courses as a whole. The discussion was vigorous and stimulating. Our aim was not to give final answers on these questions, but to give a professional background and methodology to the discussion of these questions. Is Mathematics important? If so, for what and in what way? Should there be more, less, or the same amount of Mathematics taught? On what basis does one begin to formulate answers to these questions?

Discussion of these questions is clearly not the same as a traditional Mathematics course. Further, a proper discussion of these issues and questions brings up issues and methodologies of politics, sociology, psychology, education, history, and so on, for which the previous education of the students gives no pointers and no expertise. Thus there is a severe problem of giving the discussion and the resulting projects an appropriate standard of rigour. On the other hand, many of these issues with regard to Mathematics can be discussed properly only by someone with a good background in Mathematics. So if they are not discussed in a Mathematics course, they will be discussed nowhere. It seemed better to attempt the study, to make students aware of the problems, to help them to assess these problems themselves, and for us to be aware in making the final assessments of the limitations of the training of the students. It was also helpful to make it clear to the students that, in the real world, issues and even questions are not as clear cut as they are in a Mathematics proof, and that even in Mathematics the question of what is interesting and important is not as susceptible of a final answer as is the question of correctness.

It is relevant that the McLone Report [2] recorded the following summary:

A description of the employers' view of the average Mathematics graduate might be summarized thus: Good at solving problems, not so good at formulating them, the graduate has a reasonable knowledge of mathematical literature and technique; he has some ingenuity and is capable of seeking out further knowledge. On the other hand the graduate is not particularly good at planning his work, nor at making a critical evaluation of it when completed; and in any event he has to keep his work to himself as he has apparently little idea of how to communicate it to others.

Essentially each session was a discussion lead by us, and with a view to focusing on and formulating issues in mathematics with a view to directing students to areas where they might carry out specific projects. Because of the broad nature of the course and the issues, there is a danger that the course could become completely shallow, and reminiscent of the apocryphal essay question: "Write a brief history of human thought, and compare it with some other kind of thought." The intention was to avoid this danger by leading the discussion to the suggestions for projects which had a base of source knowledge, or where something could be attempted with a reasonable standard of rigour.

Part of our discussion was on the nature of the projects, and what was expected. The qualities to be looked for were:

- Understanding of the overall aims of the area of study.
- Assessment of materials and facilities.
- Formulation of specific objectives for the essay or project.
- Clarity of thought and exposition.
- Evaluation of the difficulties in carrying out the project as formulated, given the limited time and the possible difficulties in finding the material and documentation, and evaluation of the extent of achievement of the original objectives.
- Quality of presentation.
- Possibilities for further work.

The aim has been to concentrate on issues, and to raise questions on methodology, judgment, and values. The standpoint is the opposite of authoritarian, and is intended to encourage the exchange and development of viewpoints. Some of the topics require some methodology and terminology from the Social Sciences, such as the difference between "normative" and "positive" statements.

One of the interesting features of the course is that the matters raised are appropriate for controversy. One of the sad features of Mathematics even at the research level is the lack of controversy, or of discussion of aims and methodology. The concentration is always on what has been done, rather than on how to "bring new concepts out of the dark" [1].

In thinking about the design of this course, we were influenced by two useful courses at Southampton, one on "Curriculum Studies", and the other on "Problem Solving". However, their Curriculum Studies course is more related to education, whereas the aim of our course is to consider the variety of issues and areas which should be part of the proper consideration of a professionally trained mathematician.

As one example of an issue, one could mention the contrast in Mathematics between problem solving, and the development of concepts and language. Mathematicians tend to rate the former the most highly, and the solution of some classical problem tends to get the most prizes. Indeed, those who attempt to develop new language and concepts are often treated with derision, until the concepts are proved the correct ones. That is, the training of mathematicians has its emphasis on rigour, technique and achievement, and has little emphasis on problem formulation, or concept formulation. By contrast, a study of the history of Mathematics shows that in the applications of Mathematics it is the concepts and language which are often more important than the particular theorems.

We believe that a number of students feel they are starved of a global point of view, and of any discussion which will help them come to terms with the rather peculiar nature of the activity called Mathematics. So we want to contrast the presentation of Mathematics as a normal activity, related to discovery and our exploration of the world and the over cynical view that the purpose of a course is simply to find some justifiable way of arranging students in a linear order.

For source material, we regard books like Devlin "Mathematics: The Golden Age", books by Ian Stewart, Imre Lakatos, Martin Gardner, Davis and Hersch, Morris Kline, articles from the Mathematical Intelligencer, Notices of the American Mathematical Society, London Mathematical Society Newsletter, Gazette de la Societe Mathematique de France, etc., as all fair game.

The qualities we look for in a project are a clear statement of aims, some kind of conclusion, and some assessment of methodology and the extent to which it has been possible to achieve the aims within the time scale and other restrictions necessarily imposed. In this respect, a properly carried out analysis of failure is as interesting in its own way as success. As we explained to students, part of the prescription for success is a proper analysis of failure (on the grounds that if you never fail at anything, then you have probably not set yourself sufficiently hard tasks). We are interested in processes involved in the development of Mathematics both generally, and in the progress of an individual.

We have been surprised and delighted by the quality of the work that has been produced, and the way in which students have responded with energy, enthusiasm, and independence. We particularly liked the way in which two students not only wrote a booklet on a specific task, but also wrote a separate booklet analysing the methodology underlying their approach.

It is, we believe, a severe criticism of Mathematics as a profession that there has been so little interest in the understanding and the teaching of methodology.

It is also important that we are looking to see ways in which students have exercised judgment, and made some assessment of values and context. Once again, we are looking for the human side of mathematical activity (of course some may take the view that that is a contradiction in terms!).

Recall that the difference between a professional and amateur has been analysed as that an amateur can do things, but a professional not only can do things, but also knows how to do things. That is, the professional approach is based on a fund of knowledge, i.e. example, analysis, history, evaluation, and communication. It is some discussion of this distinction that we regard as contributing to methodology (see also the definition in the Fontana Dictionary of Modern Thought). Of course, top flight students may not need this kind of analysis, or even welcome it, because they may naturally have such an intuitive basis of what is going on.

However, our experience is that such kind of metadiscussion is welcomed by the majority, particularly those with some kind of initiative and thought, though not necessarily the most technically competent. In the long run, those who not only do something but think about what they are doing will win out. We owe it to our students to explain to them what it is we think we are trying to teach them. It may also be that a proper analysis of aims, together with some attempt at an evaluation of whether our methods are the correct ones for achieving these aims, will lead to some improvements in the presentation of Mathematics, and its enjoyment by students.

We like the view that one of the aims of education in Mathematics is to prove to pupils and students that they are more clever than they thought they were. To our surprise and delight, the course has in some cases released independence and creativity which was unexpected and unseen.

The real problem for the management of such a course is to relate specific projects in the course to the discussion of general issues. For example, if the topic under discussion is the applicability of Mathematics, it is clearly not enough just to give an example of an application of Mathematics. Some kind of conclusion and judgment has got to be drawn, there has to be some analysis of why this particular example was chosen, and the discussion should consider the example in its role as evidence for, or lack of evidence for, the social importance of Mathematics.

A further advantage of the style of the course is that it gave us an opportunity to tell anecdotes and jokes which we felt every student of Mathematics should know, and to convey attitudes towards the subject which we hoped would enliven and encourage.

It might be thought that it is necessary to have a year to prepare the course, with the collection of duplicated articles on the various aspects that are to be considered. On the other hand, students should have some experience of the fact that the way in which Mathematics is taught has nothing to do with the real difficulties of gathering information in real life, when you want to do something specific. Thus one of the considerations for a choice of topic is the availability of information. Also, there is for this course no established body of knowledge (whoopee!), much of the material involves primary sources, and too much preparation could well rigidify the course, and make the tutors less responsive to the students. We knew in some sense what we were aiming at, but were quite unclear whether these aims were realizable. The starting point was that there seemed to us to be a gap in the students' professional training. We were well aware that there was a large element of risk in the success of the course, and this led us to concentrate on conveying the overall aims.

The course has to be run by iconoclasts, who are prepared to listen, and to examine every point of view for its value. The attitude has to be not "You are wrong.", but "How would you give that point of view some rigorous and justifiable basis?", or "Have you considered the following evidence?". What the leaders have to give to the course is the benefit of their knowledge, and the fact that they have considered and discussed most of the points at issue at some time or another. Thus a lot of the ground work had been unconsciously prepared over a period of years. The excellent reactions of the students has encouraged us to pursue these ideas.

The course had for us a large element of risk, and we have not been at all sure how it was going to go. We shall certainly ask the students for an evaluation at the end of the course.

The fact that we can now regard the course as successful and worth developing is due to the responsiveness of the students. They have surprised us in their clear decisions as to what projects they wanted to do, and their success in taking into consideration the guidelines and criteria which were discussed in the course. Much of our detailed advice was concerned with cutting projects down to size, and with helping in focusing aims.

At the Examiners Meeting in June, 1989, the External examiners commented that the course was very successful. We have so far had written reaction from two students, both very favourable, apart from comments in the projects which made it clear how much the preparation of the projects had been enjoyed.

- [1] A.Grothendieck, Private communication (1983).
- [2] R.R.McLone, "The training of mathematicians", Social Science Research Council, 1974.

Click here for Appendix 1, **Themes for Discussion**.

Project Titles: 1988/9 (The following are listed in pairs since each student writes two projects.)

DJA (i) The importance of Mathematics for commerce and finance (ii) The usefulness and aim of an education in Mathematics

SLB (i) Changes in primary Mathematics- its effects (ii)Applications of Statistics (or manipulation of Statistics)

KC (i) Worksheets bases on Patterns and Colourings Report based on Worksheets (ii) Maths and Art

DCH (i) Popularization with puzzles (ii) Thoughts on the funding of Mathematics in British Universities

SK (i) Math's in a Topic (ii) The contents of Mathematics Degree Courses around the World

CWP (i) An Introduction to Fractals and Fractal Geometry; To examine and aid the popularisation of Mathematics; Exhibition on Fractals (ii) A study of basic epidemiology

FR (i) Mathematics across the curriculum (ii) The employment of mathematicians

AS (i) A lesson in Mathematics should be a voyage of discovery (ii) Game theory: concepts and applications

File translated from T