The Teaching of Geometry at the Pre-College Level: Proceedings of the Second CSMP International Conference Co-Sponsored by Southern Illinois University and Central Midwestern Regional Educational Laboratory PDF
Preview The Teaching of Geometry at the Pre-College Level: Proceedings of the Second CSMP International Conference Co-Sponsored by Southern Illinois University and Central Midwestern Regional Educational Laboratory
THE TEACHING OF GEOMETRY AT THE PRE-COLLEGE LEVEL THE TEACHING OF GEOMETRY AT THE PRE-COLLEGE LEVEL PROCEEDINGS OF THE SECOND CSMP INTERNATIONAL CONFERENCE CO-SPONSORED BY SOUTHERN ILLINOIS UNIVERSITY AND CENTRAL MIDWESTERN REGIONAL EDUCATIONAL LABORATORY Edited by HANS-GEORG STEINER University of Erlangen-Niirnberg, PH Bayreuth, West Germany First published in EDUCATIONAL STUDIES IN MATHEMATICS, Vol. 3, No. 3/4; Vol. 4, No.1 ISBN 978-94-017-5638-9 ISBN 978-94-017-5896-3 (eBook) DOI 10.1007/978-94-017-5896-3 All Rights Reserved © Springer Science+Business Media Dordrecht 1971 Originally published by D. Reidel Publishing Company, Dordrecht, Holland in 1971 No part of this book may be reproduced in any form, by print, photoprint, microfilm, or any other means, without written permission from the publisher EDITOR'S PREFACE It has been a privilege for me to collaborate with the Comprehensive School Mathematics Program (CSMP), Carbondale, Illinois (U.S.A.), as European co-director, mathematician-in-residence, and consultant since 1965. It was during this time that what was once a dream became a reality: children, teachers, mathematics educators, mathematicians, educational measurement and evaluation specialists, media and systems experts are working together under one roof for the purpose of finding new ways of viewing and orga nizing, of learning and teaching mathematics. Significant features of CSMP are its strong emphasis on mathematical content, the active involvement of mathematicians at every stage of the development of a highly individualized mathematics curriculum, and the collaboration with other mathematics projects in the U.S.A. and abroad. The CSMP International Conferences on the Teaching of Mathematics at the Pre-college Level bring together pioneers from other projects and mathe maticians from various countries who are specialists in their fields and who are deeply concerned with the teaching of mathematics and the mathematical learning process. Each conference consists of a variety of activities. Like the first conference on the teaching of probability and statistics (Carbondale, March 18-27, 1969) *, this second CSMP international conference was not restricted to giving lectures. Films and learning materials were presented, discussions and workshops among the participants and the CSMP staff were held, classes in all phases of the CSMP curriculum development observed, and demonstration lessons arranged. This publication is a collection of the papers which were presented or prepared by the participants. The spectrum ranges from basic reflections on the goals of geometry teaching, historical perspective and psychological investigations on the learning of mathematics, to didactical analysis of particular areas of geometry and proposals for the incorporation of more advanced and recent mathematical ideas in the curriculum. The Conference * Proceedings of this conference have been published: see The Teaching of Probability and Statistics. Proceedings a/the First CSMP International Conference, (edited by Lennart Rade), Almqvist and Wiksell, Stockholm, 1970; John Wiley and Sons, Inc., New York, London, Sydney, 1970. v VI PREFACE Recommendations express a consensus which grew out of extensive and lively discussions. The conference chairman and editor of this book wishes to express his gratitude to Southern Illinois University Extension Services, especially to Chancellor R. W. MacV icar and Mr. Andrew Marcec for their cooperation in providing conference rooms and in arranging for various social activities, to the CSMP staff for their great help in organizing the conference and preparing the papers in their final form by translating, editing and typing, and to Professor Freudenthal for his support in making this publication possible. HANS-GEORG STEINER Bayreuth, West Germany CONTENTS EDITOR'S PREFACE v BURT KAUFMAN I Background THE CSMP STAFF I The CSMP Development of Geometry 5 GEOMETRY CONFERENCE RECOMMENDATIONS 10 FRIEDRICH BACHMANN/ n-Gons 12 H. s. M. COXETER I Inversive Geometry 34 ROBERT B. DAVIS I The Problem of Relating Mathematics to the Possibilities and Needs of Schools and Children 46 ZOLTAN P. DIENES I An Example of the Passage from the Concrete to the Manipulation of Formal Systems 61 ARTHUR ENGEL I Geometrical Activities for the Upper Elementary School 77 T. J. FLETCHER I The Teaching of Geometry-Present Problems and Future Aims 119 HANS FREUDENTHAL/ Geometry Between the Devil and the Deep Sea 137 PETER HILTON I Topology in the High School 160 M.A. JEEVES I Some Experiments on Structured Learning 178 PAUL J. KELLY I Topology and Transformations in High School Geometry 201 VICTOR KLEE 1 The Use of Research Problems in High School Geometry 206 HOWARD LEVI I Geometric Algebra for the High School Program 214 KARL MENGER I The Geometry Relevant to Modern Education 225 G. P APY I A First Introduction to the Notion of Topological Space 242 GUNTER PICKERT I The Introduction of Metric by the Use of Conics 255 VII VIII CONTENTS ANDRE REVUZ I The Position of Geometry in Mathematical Education 272 THOMAS G. ROOM I The Geometry and Algebra of Reflections (and of 2 x 2 Matrices) 277 SEYMOUR SCHUSTER I On the Teaching of Geometry -A Potpourri 300 HANS-GEORG STEINER I A Foundation of Euclidean Geometry by Means of Congruence Mappings 311 MARSHALL STONE I Learning and Teaching Axiomatic Geometry 315 HERBERT E. VAUGHAN I The Development of Euclidean Geometry in Terms of Translations 328 MARIO VILLA I A Logical Approach to the Teaching of Geometry at the Secondary Level 335 JOHN WILLIAMS I Problems and Possibilities in the Assessment and Investigation of Mathematics Learning 359 HANS ZASSENHAUSI Genetic DevelopmentoftheCongruenceAxioms 374 BIBLIOGRAPHY AND LIST OF FILMS 377 BURT KAUFMAN BACKGROUND Mathematics teaching is in a state of transition in all parts of the world. Numerous experiments aimed at modernizing the mathematics curriculum have been undertaken. The Comprehensive School Mathematics Program (CSMP) was envi sioned in 1963 as a response to a research impetus in mathematics education manifested in the reports of several prominent committees of mathematicians and mathematics educators from the United States and abroad. CSMP, located in Carbondale, Illinois, U.S.A., is a major effort to mo dernize both the content of the pre-college mathematics curriculum and the methods of teaching. In 1966, CSMP was established and became affiliated with Southern Illinois University. A proposal for a long-range curriculum development project was written during this year. The proposal was presented to the Central Midwestern Regional Educational Laboratory (CEMREL), and CSMP was incorporated as one of its major programs in the spring oi 1967. The Central Midwestern Regional Educational Laboratory was established under Title IV of the Elementary and Secondary Education Act of 1965, and began operation in June, 1966. One of 15 educational laboratories in the United States, whose purpose is to improve education for the children of the nation, CEMREL's mission is the development, CEMREL has set as its mission the development of individualized curricula in mathematics and the arts and humanities for all students in kindergarten through twelfth grade; it is also concerned with the development of effective instructional techniques and materials for children who have learning disabilities. Foremost author ities from across the nation compose the advisory committees to these programs. CEMREL is also concerned with developing instructional man agement systems, making use of the latest technology, to be used in conjunc tion with the curriculum development programs. Various related projects as well are a part of CEMREL's activities. CEMREL is governed by a Board of Directors made up of outstanding educators, civic leaders, businessmen and labor officials from Illinois, Ken tucky, Missouri and Tennessee. Headquarters for the laboratory are located in St. Ann, Missouri at 10646 St. Charles Rock Road, and programs offices are maintained in Carbondale, Illinois, Bowling Green, Kentucky, and in 2 BURT KAUFMAN Chattanooga and Nashville, Tennessee. Wade M. Robinson is the executive director. CEMREL works cooperatively with the State Departments of Education in the four states, as well as with colleges, universities and, of course, the elementary and secondary schools. The primary goal of CSMP is the development of individualized mathe matics curricula for students of ages 5-18 which provide for each student a program sound in content, enjoyable, and most appropriate for his needs and abilities. Each activity package will provide a teaching program which may involve individual study, teacher instruction, small group interaction, reading a text, watching a demonstration film, listening to a tape, playing mathematical games, or a combination of these and other procedures. In considering strategies to achieve this ultimate goal, namely the develop ment of mathematics curricula that are sound in content and appropriate for the needs of future adults in a changing society, the developers of CSMP decided the program must be discipline-oriented. By this is meant that, while all pedagogical aspects of mathematics education are of deep concern, priority is given to the selection of mathematical content that is sound, rele vant, and enjoyable. The implications of this decision are that the mathe matical community must be deeply involved in the program, that there be mathematicians physically in residence, and that mathematicians must guide the program. To date, this has been the case, and it is a principle of CSMP procedure that every phase of future development of the program will continue to enjoy the strong involvement of mathematicians. The major role of mathematicians in CSMP is the selection and analysis of content. The CSMP attitude is that content selection should, as far as is possible, be unfettered by traditional notions of "what children can do" or "what teachers can teach"; it should instead be guided by what is important in mathematics, by what the outcomes of mathematics instruction should be, and by what students actually demonstrate they can do. For these reasons, content selection within CSMP is based on empirical data gathered from probes made with students in the trying-out of ideas generated by mathe maticians. Serious content suggestions are accepted or rejected only after adequate trials in the classroom. To have a truly individualized curriculum, one must take into account the different views of mathematics held by various people. The creative research mathematicians view it in various ways, depending on their fields of research. There are various users of mathematics (scientists, engineers, social scientists, businessmen, etc.) who view mathematics in other ways; mathematics educators are concerned with curriculum and methods, while school administrators, parents, students, and the man in the street view it in
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