TEACHING UNDERGRADUATE MATHEMATICS
This page is intentionally left blank
TEACHING UNDERGRADUATE Editors Bob Burn MATHEMATICS Reader in Mathematics Education, Agder College, Norway John Appleby Senior Lecturer in Engineering Mathematics, Newcastle University, UK Philip Maher Senior Lecturer in Mathematics, Middlesex University, UK Imperial College Press
Published by Imperial College Press 203 Electrical Engineering Building Imperial College London SW7 2BT Distributed by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Teaching undergraduate mathematics / editors, Bob Burn, John Appleby, Philip Maher. p. cm. Includes bibliographical references and index. ISBN 1-86094-115-X (acid-free paper) 1. Mathematics ~ Study and teaching (Higher) - Great Britain. I. Burn, Bob. II. Appleby, John (John C.) III. Maher, Philip. QA14.G7T43 1998 510\7ri41"dc21 98-24247 CIP British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. The copyright of all material reproduced from the Proceedings of the Undergraduate Mathematics Teaching Conferences, which forms the bulk of this book, is held by the Shell Centre for Mathematical Education, University of Nottingham, and is reproduced with permission. Material from the (named) plenary lectures at those Conferences is reproduced with the additional permission of the lecturer concerned. Copyright 1998 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 permissionfromthe Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. This book is printed on acid-free paper. Printed in Singapore by Uto-Print
Foreword Education is a continuous process leading from primary school right through to Higher Education and beyond. Nowhere is it more essential to maintain coherence in the process than in Mathematics where its hierarchical nature and multi-faceted applications require careful attention. Over the past twenty years, education at all levels has been in a constant state of flux and changes at one level have impacted on other levels. In particular, changes in the final school years have forced universities to reassess their courses with the aim of making them more suitable for the incoming students. This book documents the ideas that have been discussed in this context at the Undergraduate Mathematics Teaching Conferences. Since the problems of mathematics education are still very much alive, the material presented here may help those who are currently grappling with the issues. Michael Atiyah May 1997
This page is intentionally left blank
Foreword Sir Michael Atiyah Contents V Introduction List of Themes Acknowledgements Glossary of UK Educational Terms 1. The Undergraduate Mathematics Teaching Conferences - the Source of the Material 2. The Process of Teaching Mathematics 2.1 2.2 2.3 2.4 2.5 2.6 The New Student The New Lecturer Teaching Undergraduates 2.3.1 Some Inbuilt Problems 2.3.2 Modes of Teaching 2.3.3 Good Practice 2.3.3.1 Effective Lecturing 2.3.3.2 Tutorials 2.3.3.3 Self-paced Learning 2.3.3.4 Projects Communication 2.4.1 Communicating What? 2.4.2 Books 2.4.3 Microcomputers 2.4.4 Video Appraisal The Future 1 2 4 4 5 9 9 11 16 16 16 21 21 22 28 32 37 37 46 54 68 69 80 3. Content of A-level and Undergraduate Mathematics 3.1 Rigour at A-level 3.2 First Year Undergraduate Courses 3.3 Geometry 87 87 102 111 vii
viii Teaching Undergraduate Mathematics 4. Detailed Expositions 4.1 Analysis 4.2 Number Theory 4.3 Quadratic Equations 4.4 Friction 123 123 136 137 138 5. History of Mathematics 5.1 History of Mathematics in the Undergraduate Curriculum 5.1.1 Why History of Mathematics? 5.1.2 Construction of Courses in the History of Mathematics 5.1.3 The Cameo Approach 5.1.4 Bibliography and Other Materials 5.2 Education Imitating History 6. Needs of Society and the Professions 6.1 6.2 6.3 6.4 Mathematics for the Needs of Society 6.1.1 Adapting the Undergraduate Curriculum 6.1.2 Courses on 'Mathematics in Society' Mathematics for Employment Mathematics for the Teaching Profession 6.3.1 University Mathematics and the Teaching of Mathematics in Secondary Schools 6.3.2 Routes into the Teaching Profession Service Teaching of Mathematics 6.4.1 Traditional 6.4.2 Service Teaching to Biologists 6.4.3 Mathematics for Computer Science 6.4.4 Service Teaching for Liberal Arts 141 141 142 143 146 148 151 161 161 161 164 169 177 177 179 181 181 182 185 186 7. Applications and Modelling 7.1 Applications of Mathematics 7.2 Why isn't Modelling Taught More Widely? 7.3 Classical Applications vs Modelling 7.4 The Experience of Mathematical Modelling 187 187 196 199 203
Contents ix 8. Learning Mathematics 8.1 Enjoyment 8.2 Learning (or not learning) Mathematical Concepts 8.3 Self-assessment and Peer Support 8.4 Intuition and Experiment 8.5 What Balance of Learning Activities do Students Need? 223 223 226 233 234 236 9. Assessment 9.1 Assessment by Lecturers 9.2 Peer Assessment Index 247 247 259 265