Big Ideas in Modern Geometry

This book was written to prepare future teachers, especially in the middle and secondary grades, for the mathematical work of teaching. The focus of this module is geometry, and mathematics doesn’t get any better than that. Geometry allows us to think spatially, to see structure in art and form, and to create and visualize new “worlds” with different rules.

Does the word “geometry” call to mind the two-column-proof of your high school days? Long ago mathematics educators decided that geometry class would be a good place to showcase the importance of definitions, reasoning, and proof in mathematical thinking – really, these things are vital in all areas of mathematics – not just geometry – and if you use any of our other modules, you’ll see that this is so. However, if the two-column proof has ruined geometry for you, then forget about it. You don’t need to do any here. You are free to reason in any form you see fit, as long as you can communicate your argument to others. After all, we mathematicians rarely write a proof in such a form. We’d hate to be constrained in that way.

As mathematicians, we want to convey to you the beauty of our subject. Mathematicians view mathematics as the study of patterns and structures. We want to show you how to reason like a mathematician – and we want you to show this to your students too. This way of reasoning is just as important as any content you teach. When you stand before your class, you are a representative of the mathematical community; we will help you to be a good one.

No one can do this thinking for you. Mathematics isn’t a subject you can memorize; it is about ways of thinking and knowing. You need to do examples, gather data, look for patterns, experiment, draw pictures, think, try again, make arguments, and think some more. The big ideas of geometry are not always easy.

Each section of this book begins with a Class Activity. The activity is designed for small-group work in class. Some activities may take your class as little as 30 minutes to complete and discuss. Others may take you two or more class periods. No solutions are provided to activities – you will have to solve them yourselves. The Read and Study, Connections to School, and Homework sections are presented within the context of the activity ideas.