Creative Geometry: Introduction

As a Mathematics teacher who loves Geometry, I have always been disappointed to find that many students say they do not like Geometry. I believe they feel this way because it has been presented to them as complex theorems that they must memorize, long lists of properties, and difficult tests.

It is my goal in teaching to introduce students to the creativity and beauty in Mathematics, and to show them the connections between Mathematics and nature, art, science, and all other aspects of their lives. I believe that creative projects can teach students the concepts of mathematics, help them to understand the properties of geometric figures, remember the definitions and theorems in the Geometry curriculum and instill in them an interest in, and perhaps even a passion for the subject.

My students learn the properties of a parallelogram, rhombus, rectangle and square by creating paper models of the figures, which they fold, rotate, and compare. They see and remember volume formulas by creating 3-dimensional solids that dramatically illustrate these relationships. When a student holds a paper model in his or her own hand, and folds three pyramids together to magically form a perfect cube, how can that student forget that the formula for the volume of a pyramid as 1/3 the volume of the cube and therefore 1/3 base area times height?

Students can apply the concept of locus to a cleverly drawn treasure map, and use their understanding of reflection, rotation and other transformations to create Tangrams with the help of interactive Geometry software (The Geometer's Sketchpad). They can practice writing mathematics in creating Origami instructions written in mathematical terminology, and come to a concrete understanding of square roots by constructing a spiral of Pythagorean triangles.

Hands-on Geometry projects can be used to introduce geometry concepts, to help students remember theorems, and to assist them in seeing the connections between mathematics and other subject areas. Students find that mathematics is a fascinating and creative subject and that it has real applications in their lives. When students work together using indirect measurement to find the height of a tall tree on campus, and draw a creative poster explaining their methods, they have authentic and concrete experiences in real mathematics. They are pleased with their accomplishments, and proud of the product.

Writing something down on paper helps students to remember that which they have written. When a student measures, calculates, discusses, writes, compares, draws, cuts, folds and illustrates mathematics, he or she internalizes the concepts and "owns" them. Research has shown that when we see it, write it, say it, and even draw it" we are more likely to remember it. How much more, then, will concepts be remembered when the knowledge has been "created" by the students themselves!

I would like to dedicate these pages to my wonderful students, over 5,000 of you in my 36 years of teaching. You have inspired and taught me more than you will ever know. I hope this book will inspire other teachers to give their own students opportunities to demonstrate their talents and their creativity.

Cathi Sanders
"As students progress through the educational system their interest in mathematics diminishes. Yet there is an ever increasing need within the workforce for individuals who possess talent in mathematics. The literature suggests that mathematical talent is most often measured by speed and accuracy of a student's computation with little emphasis on problem solving and pattern finding and no opportunities for students to work on rich mathematical tasks that require divergent thinking. Such an approach limits the use of creativity in the classroom and reduces mathematics to a set of skills to master and rules to memorize. Doing so causes many children's natural curiosity and enthusiasm for mathematics to fade away as they get older. Keeping students interested and engaged in mathematics by recognizing and valuing their mathematical creativity may reverse this tendency." Eric Louis Mann, Ph.D.
All human knowledge thus begins with intuitions, proceeds thence to concepts, and ends with ideas. Hilbert, David (1862-1943)