DOCUMENT RESUME ED 035 544 SE 007 720 ATTTFOP McNabb, W.K. miTLF Geometry in the Secondary School, A Compendium of Papers Presented in Houston, Texas, January 29, 1967, at a Joint Session of The Mathematical Association of America and The National Council of Teachers of Mathematics. TNSTITUTION National Council of Teachers of Mathematics, Inc., Washington, D.C. PUB T)ATP Jan 67 myrp, 58n. AVATLABLE FROM National Council of Teachers of Mathematics, 1201 Sixteenth Street, N.W., Washington, D.C. 20036 (0.5C) EDRS PRICF BAPS Price MF-T0.50 HC Not Available from EARS. DESCR7PTOPS Algebra, *Conference Reports, *Geometry, *Mathematics, Mathematics Curriculum, Mathematics Education, *Secondary School Mathematics IDENTIFIEPS Mathematical Association of America, National Council of Teachers of Mathematics ABSTP ACT This booklet is a collec:Ilon of the papers presented at a joint session of the Mathematical Association of America and the National Council of Teachers of Mathematics during the fiftieth annual meeting of the MAA. These papers were presented under the headings of "Geometry and school Mathematics," "High School Geometry," and "School Geometry and the Future." Eight papers are presented in their entirety. An abstract of a ninth paper, "Geometry: The Cambridge Conference View" by Fdwin E. Moise is also included. Authors of the individual papers do not act as spokesmen for the MAA or NCTM but express personal viewpoints. (FL) DOCUMENT FESUME ED 035 544 SE 007 720 ATTTT-TOP McNabb, W.K. mTTLF Geometry in the Secondary School, A Compendium of Paners presented in Houston, Texas, January 29, 1967, at a Joint Session of The Mathematical Association of America and '"he National Council of Teachers of Mathematics. TNSTITUTTON National Council of Teachers of Mathematics, Inc., Washington, D.C. PUB nAmp Jan 67 mOTE 58n. AVATIABLE FROM National Council of Teachers of Mathematics, 1201 Sixteenth Street, N.w.1 Washington, D.C. 20036 (110.0) EDPS PPTCP 7DrS Pricc MF-0.50 HC Not Available from EDPS. DESCRTDTOPS Algebra, *Conference reports, *Geometry, *Mathematics, Mathematics Curriculum, Mathematics Education, *Secondary School Mathematics TDENTTPTERS Mathematical Association of America, National Council of Teachers of Mathematics ABSTRACT This booklet is a collection of the Papers presented at a joint session of the Mathematical Association of America and the National Council of Teachers of Mathematics during the fiftieth annual meeting of the MAA. These papers were presented under the headings of "Geometry and School Mathematics," "High School Geometry," and "School Geometry and the Future." Eight papers are presented in their entirety. An abstract of a ninth paper, "Geometry: The Cambridge Conference View" by rdwin E. Moise is also included. Authors of the individual papers do not act as spokesmen for the MAA or NCTM but express personal viewpoints. (FL) U.S. DEPARTMENT OF HEALTH, EDUCATION B WELFARE OFFICE OF EDUCATION THIS DOCUMENT HAS BEEN REPRODUCED EXACTLY AS RECEIVEDFROI' ',Hi PERSON OR ORGANIZATION ORIGINATING IT.POINTS OF VIEW OR OPINIONS STATED DO NOT NECESSARILY REPRESENT OFFICIAL OFFICE OF EDUCATION POSITION OR POLICY. A A o si N papers presented at a joint session of THE MATHEMATICAL ASSOCIATION OF AMERICA THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS I (\\1 :-. -4° Pr\ GEOMETRY CD IN THE SECONDARY SCHOOL LLI A compendium of papers presented in Houston, Texas January 29, 1967, at a joint session of THE MATHEMATICAL ASSOCIATION OF AMERICA THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS "PERMISSION TO REPRODUCE THIS COPYRIGHTED MATERIAL BY MICROFICHE ONLY 146S BEEN GRANTED BY Nat.COUI1C.Teaal.Math. TO ERIC AND ORGANIZATIONS OPERATING UNDER AGREEMENTS WITH THE U.S. OFFICE OF EDUCATION. FURTHER REPRODUCTION OUTSIDE THE ERIC SYSTEM REQUIRES PERMISSION OF THE COPYRIGHT OWNER." Additional copies of this publication may be purchased for 50c from The National Council of Teachers of Mathematics 1201 Sixteenth Street, N.W., Washington, D.C. 20036 FOREWORD The National Council of Teachers of Math- ematics is pleased to make the papers in this booklet available to those who were unable to attend the sessions at which they were origi- nally presented: a program presented by the Mathematical Association of America, at its fiftieth annual meeting, with the cosponsor- ship of the NCTM. The papers were presented at three ses- sions, under the headings of "Geometry and School Mathematics," "High School Geometry," and "School Geometry of the Future"; but they are, of course, interrelated. The authors speak as individuals, not as official spokes- men for either sponsoring organization, and the viewpoints expressed are even more stim- ulating and thought-provoking in that they are not always in agreement. The NCTM is grateful to the contributors and to the MAA for permission to publish the material presented here. W. K. McNabb Program Coordinator and Editor CONTENTS GEOMETRY IN THE UNITED STATES 1 Bruce E. Meserve, University of Vermont THE ONTARIO K-13 GEOMETRY REPORT 8 H. S. M. Coxeter, University of Toronto GEOMETRY IN GREAT BRITAIN 13 Andrew Elliott, Royal Military College of Canada AFFINE GEOMETRY 18 G. P. Johnson, Oakland University VECTOR GEOMETRY 24 Herbert E. Vaughan, UICSM, University of Illinois GEOMETRIC TRANSFORMATIONS 29 Seymour Schuster, University of Minnesota GEOMETRY: THE CAMBRIDGE CONFERENCE VIEW (Abstract) . 39 Edwin E. Moise, Harvard University TOPOLOGY 40 Gail S. Young, Tulane University of Louisiana AN APPROACH TO ANALYTIC PROJECTIVE GEOMETRY 45 C. E. Springer, University of Oklahoma vii GEOMETRY IN THE UNITED STATES ByBruce E. Meserve University of Vermont Burlington, Vermont The changing role of geometry is a source of confusion to many teachers and administrators, a challenge to all who are alert to the needs of their students. In the few min- utes available I shall try (a) to identify some of the changes that have taken, or are taking, place, and (b) to suggest some of the implications of these changes for the future of geometry in mathematics education in the United States. Since several of you could probably undertake these tasks more effectively than I can, it is with consid- erable trepidation and humility that I make this initial presentation at our joint MAA -NCTM meeting concerned with geometry. As a basis for discussion, I present my concept of the problem. Other speakers will speak to various as- pects of the problem. You will have opportunities to ques- tion us individually here and at future meetings. What Is Going On? Solid geometry as a separate course has almost disap- peared. However, paradoxically, we seem to feel that the geometry of space deserves greater emphasis than it has received in the past. The traditional plane geometry course has lost its "sacred cow" status and, from the point of view of some mathematicians, has been defiled by the injection of the use of algebra, as in Coordinate systems Distance and midpoint formulas Slope Equations of common curves and surfaces Use of inequalities to represent regions on a plane and in space The geometry course that is evolving from the traditional 1 plane geometry course has been overwhelmed with topics proposed for inclusion: Coordinate approaches to geometry Transformation groups Vector approaches to geometry Solid geometry Emphasis upon postulational systems Emphasis upon logical structure Study of other geometries such as spherical geometry, finite projective geometries, topology, and the non-Euclidean geometries Sometimes it seems that for each course in geometry at a university there is a corresponding suggestion for includ- ing related concepts in school geometry. It seems to me that the formal course in secondary school geometry is currently floundering as an over- encumbered giant circumscribed by drastically sharpened standards for definitions, postulational systems, uses of logical concepts, and the use of mathematical terminol- ogy. Consider the postulates for secondary school geometry. Except for the works of a very few individualists such as Swenson, Veblen, Birkhoff, and Beatley, teachers formerly had some security in a very stable (even though imperfect) system of postulates. Today the exceptions of the past provide guidelines for a wide variety of postulational sys- tems. Mathematicians, almost universally, take an inde- pendent "I'll define it as I please and be consistent with myself" attitude. I appreciate and, to a considerable ex- tent, share this attitude. However, we need to let others understand the reasons behind some of our proposed changes. We especially need to allow and encourage secondary school teachers to comprehend what we are trying to accomplish by our independent approaches to geometry. Only with this background can teachers enter into the true spirit of the teaching of geometry. Let us look briefly into the background for some of our "idealistic whims" and see why we are changing the ground rules for secondary school geometry. Each of our changes is intended to enhance the student's understanding and appreciation of geometry as a mathematical system and, in 2 tr particular, as a system concerned with points, lines, planes, relations among figures, and the representations of figures in the world around us. 1. Existence. --We used to assume the existence of any geometric figure that could be constructed. Even though Euclid's postulates were probably based upon Plato's pos- tulates for constructions, we were very casual about the possibility of making constructions. In our present em- phasis upon geometry as a postulational system independent Df physical representations (models), we now include pos- tulates of existence in our mathematical system. For ex- ample: There exists at least one line. If111is a line, then there exists at least one point C that is not a point of AB. If ABC is a plane, then there exists at least one point D that is not a point of ABC. 2. Equality versus congruence. --We used to say, "Base angles of an isosceles triangle are equal." Now we reserve the equality relation for different names given to the same number, figure, or other element.