Table Of ContentTHE 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
