The formal expression of "straightness" is a part of differential geometry and is a very difficult formal area of mathematics. However, the concept of "straight," an often used part of ordinary language, is generally used and experienced by humans starting at a very early age. This book will lead the reader on an exploration of the notion of straightness and the closely related notion of parallel on the plane and on the sphere.
This book is based on a junior/senior level course I have been teaching for twenty years at Cornell for mathematics majors, high school teachers, future high school teachers, and others. Most of the chapters start intuitively so that they are accessible to a general reader with no particular mathematics background except imagination and a willingness to struggle with ideas. However, the discussions in the book were written for mathematics majors and mathematics teachers and thus assume of the reader a corresponding level of interest and mathematical sophistication.
The course emphasizes learning geometry using reason, intuitive understanding, and insightful personal experiences of meanings in geometry. To accomplish this the students are given a series of inviting and challenging problems, are encourage to write and speak their reasonings and understandings; and then I listen to and critique their thinking and use it to stimulate the whole class discussions.
Most of the problems are approached both in the context of the plane and in the context of a sphere (and sometimes a cylinder and cone). I find that by exploring the geometry of a sphere my students gain a deeper understanding of the geometry of the plane. For example, the question of whether or not Side-Angle-Side holds on a sphere leads one to pursue the question of what is it about Side-Angle-Side that makes it true on the plane. I also introduce the modern notion of "parallel transport along a geodesic" which is a notion of parallelism that makes sense on both the plane and on a sphere (in fact, on any surface). While exploring parallel transport on a sphere the students are able to more fully appreciate that the similarities and differences between the Euclidean geometry of the plane and the non-Euclidean geometries of a sphere and other spaces are not adequately described by the usual Parallel Postulate. I find that the early interplay between the plane and a sphere en! riches all the later topics whether on the plane or on a sphere.
In my course the distinction between learning activities and assessment activities is blurred. I present a sequence of problems (together with motivation, discussion of contexts, and connections of the problems with other areas of mathematics and life). I tell the students:
Write out your thinking to each problem. We will return your papers with comments about your solutions. Respond to our comments
The students then work on the problems either individually or in small groups and report their thinking back to me and the class. This cycle of writing, comments, discussion continues on each problem until both the students and I are satisfied, unless external constraints of time and resources intervene.
What I have discovered is that in this process not only have the students learned from the course, but also I have learned much about geometry from them. At first I was surprised; how could I, the teacher, learn mathematics from the students? But this learning has continued for 20 years and I now expect its occurrence. In fact, as I expect it more and more and learn to listen more effectively to them, I find that a greater portion of my students show something new to me about geometry. I have also discovered that I am learning more (percentage-wise) from those students who differ from me in terms of gender and race. For more discussion of this, see the "Message to the Reader" on pages xx-xxiv.
Useful Supplements
Instructors may obtain from the publisher an Instructor's Manual which contains for each problem a full discussion of possible solutions, examples of students' original work, and suggestions for class discussion.
For exploring properties on a sphere it is important that you have a model of a sphere that you can use. Some people find it helpful to purchase plastic sphere sets which include a transparent sphere, a spherical compass, and a spherical "straight edge" which doubles as a protractor. These sets should be available in your bookstore or from Key Curriculum Press, Berkeley, CA. They work well for small group explorations in the classroom. The instructor should also have a large "black board" sphere that can be written on with chalk these spheres are often common in chemistry classrooms. However, a beach ball or basketball will also work, particularly if used with rubber bands large enough to form great circles on the ball. Students often find it convenient to use worn tennis balls ("worn" because the fuzz can get in the way) because they can be written on and are the right size for ordinary rubber bands to represent great circles.
Acknowledgments
I acknowledge my debt to all the students and teachers who have attended my geometry courses. Most of these people have been students at Cornell or teachers in the surrounding area of upstate New York, but they also include students at Birzeit University in Palestine and teachers in the new South Africa. Without them this book would have been an impossibility.
Starting in 1986, Avery Solomon and I organized and taught a program of inservice courses for high school teachers under the financial support of Title IIA Grants administered by the New York State Department of Education. This is now called the Cornell/Schools Mathematics Resource Program (CSMRP). As a part of CSMRP we started recording classes and writing notes on the material. Some of the material in this book had its origins in those notes, but they never threatened to become a textbook. I thank Avery for his modeling of enthusiastic teaching, his sharp insights, and his insistence on preserving the teaching materials. In addition to Avery, my friends, Marwan Awartani, a professor at Birzeit University, and John Volmink, the director of the Centre for the Advancement of Science and Mathematics Education in Durban, South Africa, have for a long period of time consistently encouraged me to write this book.
A few years ago my colleague Maria Terrell suggested that five of us at Cornell who have been teaching non-traditional geometry courses (Avery Solomon, Bob Connelly, Tom Rishel, Maria, and I) submit a proposal to the National Science Foundation for a grant to write up materials on our courses. The fact that we were awarded the grant (in 1992) is largely due to Maria's persistence, clear thinking, and encouragement. It is this grant which gave me the necessary support to start the writing of this book. I thank the NSF's Program on Course and Curriculum Development for its support.
The major portions of this book were written during the 1992-93 academic year in which I taught the course both semesters. Eduarda Moura was my teaching assistant for these courses. She was supported by the NSF grant to assist me by describing the classroom discussion and the student homework upon which the content of this book is based. Much of this book (and especially the instructor's manual) are derived from her efforts. In addition to Eduarda, Kelly Gaddis, Beth Porter, Hal Schnee, and Justin Collins were also supported by the grant and made significant contributions to the writing of this book. I thank them all for their excellent contributions, their support of my work, and their friendship. The final writing and the decisions as to what to include and what not to include have all been mine, but they have been based on the foundation that was started with Avery and the CSMRP materials and was continued wtih Eduarda, Justin, Kelly, Beth, and Hal during 1992-93.
Since the spring of 1992, the early drafts of the book have been used by me and others at Cornell and 13 other institutions. Various other individuals have worked through the book outside a classroom setting. From these students, instructors and others I have received encouragement and much valuable feedback that has resulted in what I consider to be a better book. In particular, I want to thank the following persons for giving me feedback and ideas which I have used in this final version: David Bray, Douglas Cashing, Helen Doerr, Jay Graening, Christine Kinsey, István Lènárt, Julie Lubell, Richard Pryor, Amanda Cramer and her students, Erica Flapan and her students, Linda Hill and her students, Tim Kurtz and his students, Judy Roitman and her students, Bob Strichartz and his students, and Walter Whitely and his students. Susan Henderson, my wife, spent many hours proof-reading and refining the text and was always my consultant on matters of aesthetics! .
The entire production of the manuscript (typing, formatting, drawings, and final layout) has been accomplished using an integrated word processing software under my direction. Finally, I wish to thank George Lobell, Senior Editor at Prentice-Hall, for the vision and enthusiasm with which he shepherded this book through the publication process.
Ithaca, NY, May 1995