H EICUMSAT U NIVERSITY
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FUNDAMENTALS1 OF TEACHING.
ifltraENMIO AT UNIVERSITY
B B A U M S L A 6 Sweden
4mu Imperial College Press
Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE
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FUNDAMENTALS OF TEACHING MATHEMATICS AT UNIVERSITY LEVEL
Copyright 2000 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented without written permission from the Publisher.
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Printed in Singapore by Regal Press (S) Pte. Ltd.
In memory of my parents, Braine and Kalman,
my first and best teachers
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Mathematics is an important subject in the universities of the world, and it is therefore unfortunate that there are so few books and courses on the teaching of it at university level. As a larger percentage of the young are coming to study in the universities, teaching is becoming harder not easier. In my view this provides the justification for this book.
This book is concerned with the basics. Beginners and those who want an overview will benefit most. The book would also suit a variety of other readers: legislators and educational administrators, heads of departments, directors of undergraduate studies, even established lecturers. I feel this is natural because education cannot be duscussed piece-meal, instead the global picture must be kept in mind. Lecturers have considerable freedom, but are bound by decisions made by governments, states, their universities and departments. They are also strongly dependent on the interests, attainments, abilities and determination of the students, and thus can not afford to go ahead without keeping all these matters in mind.
The book is divided into four parts as follows:
PART I Education in General: Chapters 1-4. PART II General Theory of Teaching: Chapters 5-7. PART III Departmental Matters: Chapters 8-10. PART IV The Individual Lecturer: Chapters 11-13.
Thus the topics are discussed down the chain of command, from the more powerful down to the less powerful, from society's over-riding demands, through the university and then the department, to finally the part that the lecturer can control.
This book can be used in a course in mathematics teaching or else for individual reading. Although I believe the order of the chapters is optimal, it is not necessary to read them in that order, since there are appropriate references. One can restrict one's reading to Chapters 6-7, and 11-13 for practical steps to improve teaching.
Vl l l Fundamentals of Teaching Mathematics at University Level
I believe that academics must take a much greater interest in education as a whole, because they are well suited to help formulate education policies, and that on the whole they have not taken this responsibility seriously. I think it is sensible to have a global view of the education system in one's own country, and to have some idea of at least one other country. Thus some examples of how this can be done are given in Chapter 1 and Appendix A.
The rise in both the number of universities and the number studying at university throughout the western world is striking. This poses opportunities, but also brings dangers. It is possible that the best students will be neglected due to the obvious importance of helping the weaker ones. And obviously there should be a greater variety of courses to match the greater variety of student ability, attainments and interests. But there should always be deep and demanding courses for the best students, otherwise mathematics will gradually die.
Since in many countries schools attended before university have ceased to regard one of their aims as matching their final courses to the university's requirements, I feel it is important for lecturers to obtain as clear a knowledge of the final years of school mathematics as possible.
Teaching in my opinion will be improved if the university and the department take a greater interest in organisation and how it can be used to improve teaching.
We mathematicians should make an effort to formulate and teach study skills, and I have several suggestions. There is a series of steps for studying a book (briefly described as SQERPSR2), a technique for linking ideas together (City Explorer), and what I call the Jigsaw Puzzle method, which aids progress even when the overall picture is confused. I emphasise the value of a summary book, the value of tables and diagrams, and the role of memory and mnemonics. Students should be encouraged not only to learn a theorem but also to learn an example to illustrate it; the two should always appear together in the student's mind.
As a practical means of analysing teaching I propose ten fundamental rules of teaching, and also suggest a systematic but simple system of recording results from one year to the other.
Elegant proofs and arguments are all very well, but even more important are the ways used to help construct these ideas, which often are omitted from the final version. There are some useful techniques for lecturing, like writing an example on one side of the blackboard, and the general proof on the other side, and matching them step for step as far as possible.
Comprehensive and intelligible previews are important. Thus for instance I introduce point set topology with a fable of the mathematician who had his ruler stolen while out to lunch. As the course progresses and the student learns more ideas and concepts, a more precise and detailed overview should be given.
There are new methods of teaching now coming into use, but I remain convinced that much can be done by simple means, that attention to even small details can make a useful difference. Unfortunately there are many small details, and it is hard to keep them all in mind simultaneously. Having them listed is a useful first step.
I believe one should spend considerable time trying to deal with topics that one knows from one year to the next will turn out to be tricky. There is for instance a method of making the epsilon delta definition of limit more concrete, continuity can be illuminated by betting on the horses, and an analogy with spices can help convey the idea of linear independence.
Our object as mathematicians is deep and genuine knowledge. Analogies and rough ideas should appear often and freely, but one must ensure the students know what is an intuitive explanation and what is a correct and precise mathematical formulation. This can be done for instance with marking the rough ideas by a special symbol.
It is important to relate the teaching to the students' interests, so at least mentioning the applications which appeal to the groups being taught is helpful, and likewise reference to the history of mathematics, and striking quotations from the great mathematicians of the past do help to increase motivation. On the whole simple techniques and methods which do not require special equipment are best.
Thus this book is a practical how-to-do-it book, and also a theoretical book. It raises issues which need to be discussed and gives a suitable framework for this discussion.
My comments will all basically be about mathematics at university. I am talking about my own experiences and thoughts from 1960 till 1998.
Most of my lecturing career was spent at Imperial College, London, where I was a lecturer for 25 years. I had begun my teaching career in the University of Witwatersrand in South Africa, where I had been a student. I then proceeded as a Ph.D. student under Professors Hirsch and Gruenberg at Queen Mary College, London. After my Ph.D. I taught for a year at the University of South Wales at Cardiff. I then spent two years at the Courant Institute in New York, returning to Imperial College. After leaving Imperial
X Fundamentals of Teaching Mathematics at University Level
College I spent six months at the University of Lund, and since then five years at Malardalen University, one of Sweden's newest universities. These represent a large variety of different types of universities, but even so, because of the large variation in universities, it is not possible to say something definitive for all universities. Yet much of what I say would have some application to most.
My purpose in this book is to state, underline and remind all concerned with the teaching of mathematics, of the fundamental measures which are needed. Many of the remarks I make may strike some readers as being obvious or just plain common sense1, but in practice it is difficult for teachers to effectively achieve even those things that most would agree are obviously sensible. There must always be a great deal of compromise.
There are many mathematicians who put an enormous amount of energy and passion into th