HOUYHNHMS UNIVERSITY STAFF COLLEGE

MATHEMATICS DIVISION

FINAL EXAMINATION, 1987

TIME: 3 hours: answer at least one question from each section

Section A: Undergraduate

1. Discuss evidence relating to:

(i) the public image of mathematics;

(ii) the relationship between the public image of mathematics and the nature of mathematics;

(iii) the relationship between the public image of mathematics and its presentation by university staff.

2. "The reasonable requirements of an employer for education of a graduate are training in: (i) oral communication; (ii) written communication; (iii) use of sources of information; (iv) problem formulation; (v) problem solving." Discuss the relevance of these criteria to, and their actual importance in, the current training of undergraduate mathematicians.

3. "The difference between technical training and education for graduates is that a technical education teaches skills, teaches people how to do things, whereas an education for graduates also teaches judgement, assessment of context, relationships to other disciplines and sciences, awareness, curiosity, and the notion of value." Discuss in relation to the undergraduate curriculum in mathematics.

4. Choose a topic from one of the final year courses in your department and explain how you would present this to a class of intelligent 13 year olds. Give reasons for your decision to choose this topic.

5. Most first year analysis courses give deductions of the basic results of the calculus from the axioms for the real numbers. On the other hand, the idea of an axiomatic treatment is a complete novelty to almost all the students taking the course, and there seems to be no analysis text which makes any attempt to explain what is the purpose of such a treatment, or to relate this treatment to any other part of mathematics or science. Discuss the likely effects on the students.

6. What is the difference between a "proof" and an "explanation"? What is the difference between a "theorem" and a "fact"? Give examples of appropriate uses of all these terms in the teaching of undergraduate mathematicians.

7. In the future, there will be students entering mathematics courses at University who have attended Mathematics Masterclasses, and who have been involved in "investigations" in GCSE courses. Discuss the likely reactions of such students to the current range of courses and methods of teaching, and explain if any changes in the latter are desirable.

Section B: Postgraduate

8. " I am grateful to Professor X whose wrong conjectures and fallacious proofs led me to the theorems he had overlooked." Discuss this apochryphal Ph.D. dedication in relation to the problems of supervising research students.

9. Discuss the problems in training research students in the choice of area and specific problems in relation to the constraints of time, finance, and available expertise.

10. Describe your usual structure for the layout of a research proposal for a Research Council, and discuss the problems involved in the preparation of such a document.

11. Are there paradigms in Mathematics? Discuss in relation to the work of Thomas Kuhn on the Structure of Scientific Revolutions.

12. Discuss the problems involved in allocating research funding to pioneering work, with illustrative examples.

Background Information

Letters:

To a newspaper:

*The ‘one of us’ syndrome*

Sir-Mr A E Hare’s experience of the usefulness of mathematics education ("Maths mystique", July 17) reminds me of my own.

Towards the end of my BSc in "Pure" maths, I enquired of the departmental staff as to the degree course’s role and purpose. Their reply was that it "prepared students for further research in the subject".

Having avoided this option, I have since spent 14 quite successful years developing computer software, mostly in military or other engineering applications. I have found next to no need for any of the maths which I studied beyond early secondary school.

What I have come to realise is that employers (and others tasked with selecting personnel) take great comfort from an easily understood criterion, almost regardless of its efficacy. Whether this criterion is degree, GCE’s, old school or whatever, it removes or reduces the selector’s need to think out his or her decision.

No doubt selecting maths graduates is a fairly good way of finding the individuals best able to perform such tasks as programming computers. So too might be a crossword competition. It is certainly not necessary to 18 to 21 year olds through intensive education in irrelevancies in order to establish their relative abilities in essentially unrelated fields.

Yours,

DAVID DEMPSEY, Camberley, Surrey

To the Notices of the American Math. Society

SIR,

As I examine the cause of my disillusionment with academics, however, I find that they have much more to do with the academic system itself than with the lack of funding.

In the mathematical community today, there is no common view of what is good mathematics. There is no common definition of our purpose as mathematicians or our role in the world. There is no global vision that would allow us to see how our individual research contributes to the progress of man. Instead, each group of specialists seems to trudge forward with its had down, not even bothering to interpret its work for the rest of us, to say nothing of the general public.

Many believe that this kind of chaos is inevitable in a field as large as mathematics, but I think it is the result of an academic system that actively discourages the search for a global view, especially among younger mathematicians.

Of course more funding would be welcome. But if more money simply reinforces the notion that mathematics can sprawl endlessly without priorities or values, then the mathematical community may be better off without it.

Douglas J. Muder, MITRE Corporation

QUOTATIONS FROM `MATHEMATICS COUNTS' Report of the Committee of Inquiry into the Teaching of Mathematics in Schools, Great Britain Stationery Office, (1982) (The Cocckcroft Report)

Mathematical content should include elements which are intrinsically interesting and important

It is hoped that all that is proposed in this paper will make mathematics a more worthwhile experience for all pupils.

Mathematics as an essential element of communication

Mathematics from 5 to 16 – It is now almost 10 years since McLone published his research report. The training of Mathematicians, which covered the pure and applied branches and included statistics and operational research. In response to enquiry, employers revealed what they saw as the weaknesses of contemporary training, the most notable of which was the deficiency in communication skills found among the mathematicians in their employment. Inadequacies were found in both their oral skills and report writing.

The quality of pupils’ mathematical thinking as well as their ability to express themselves are considerably enhanced by discussion.

There is a danger that mathematics might be made to appear to pupils to consist
mainly of answering set questions, often of a trivial nature, to which the
answers are already known and printed in the answer book! But *pupils will
have developed well mathematically when they are asking and answering their
own questions…..why?…..how?….what does it mean?…..is
there a better way?…..what would happen if I changed that?…..does
the order matter?….*

Making, testing and modifying hypotheses are parts of the thinking processes of everyone at different levels within mathematics, within the whole curriculum and in everyday life.

The main reason for teaching mathematics is its importance in the analysis and communication of information and ideas. The mere manipulation of numerical or algebraic symbols is of secondary importance.

Successful exposition may take many different forms but the following are some of the qualities which should be present ; it challenges and provokes the pupils to think, it is reactive to pupils’ needs and so it exploits questioning techniques and discussion; it is used at different points in the process of learning and so, for example, it may take the form of pulling together a variety of activities in which the pupils have been engaged; and it uses a variety of stimuli.

There is much to discuss in mathematics: the nature of the problem in order to comprehend what is intended; the relevance of the data; the strategies which might be used to produce solutions; and the concepts which need to be clarified and extended. The correctness of results needs to be discussed; where mistakes have been made these should not be ignored or summarily dismissed as they are often profitable points for discussion of handled sensitively.

Show initiative and flexibility in their approach. The aim should be to show
*mathematics* as a *process*, as a *creative activity* in
which pupils can be fully involved, and not as an imposed body of knowledge
immune to any change or development.

Young children learn to understand and use the spoken word before learning to write. In many situations in life oral communication predominates. It is therefore necessary from the beginning that pupils should talk about mathematical ideas.

But oral skills in mathematics among older pupils are often neglected in schools as pupils are expected to play a more passive, listening role. Writing about mathematics is not generally developed. There is a lot of ‘written work’ connected with ‘exercises’ but there is little communication of mathematical ideas in writing.

But a classroom where a range of activities is taking place and in which pupils express interest and ask questions can also provide on-the-spot problems. Teachers need to exploit these situations because there is greater motivation to solve problems which have been posed by the pupils themselves.

It is worth stressing to pupils that, in real life, mathematical solutions to problems have often to be judged by criteria of a non-mathematical nature, some of which may be political, moral or social. For example, the most direct route for a proposed new stretch of motorway might be unacceptable as it would cut across a heavily built up area.

"Mathematics lessons in secondary schools are often not about anything. You collect like terms, or learn the laws of indices, with no perception of why anyone needs to do such things. There is excessive preoccupation with a sequence of skills and quite inadequate opportunity to see the skills emerging from the solution of problems."

"During every mathematics lesson a child is not only learning, or failing to learn, mathematics as a result of the work he is doing but is also developing his attitude towards mathematics. In every mathematics lesson his teacher is conveying, even if unconsciously, a message about mathematics which will influence this attitude. Once attitudes have been formed, they can be very persistent and difficult to change. Positive attitudes assist the learning of mathematics; negative attitudes not only inhibit learning but….very often persist into adult life and affect choice of job."

QUOTATION FROM THE 1974 McLONE REPORT:

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.

CONTEXT and PRETEXT

When I took part in a debate at College I was rebuked by a friend with the old phrase: `Text without context is only pretext.' What should one say about `Maths without context'?

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