Table Of ContentFUNDAMENTALS
TEACHING
H EICUMS
AT U NIVERSITY
LEVEL
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1
FUNDAMENTALS
TEACHING.
OF
ifltraENMIO
UNIVERSITY
AT
LEVEL
B BAUMSLA6
Sweden
Imperial College Press
4mu
Published by
Imperial College Press
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Covent Garden
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A catalogue record for this book is available from the British Library.
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,
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ISBN I-86094-214-8
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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Preface
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.
vii
Fundamentals of Teaching Mathematics at University Level
Vlll
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.
Preface IX
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 their teaching and teaching methods, and it has not been
possible to present their ideas in this book. To do so, even briefly, would
have required a much larger book than this one. I hope that the bibliography
at any rate will help the reader explore some of those ideas.
