Polyominoes, Polygons, and Polyhedra: Discovering Geometry through Exploration

This book is designed to engage students in the art and excitement of doing mathematics: of being curious, exploring possibilities, observing patterns, making conjectures, and justifying claims. Active play and discovery has been an important part of my own mathematical journey and continues to be an essential part of the field of mathematics.

Creative problem solving should not be limited to honors or upper-level courses; given the opportunity all students can grow and thrive through mathematical exploration. Problems can and should be chosen to be accessible yet intriguing. I visited kindergarten classes where students were enthusiastic about patterns and numbers, eagerly wanting to share their thoughts and try new things. As a university professor, I wanted to know more about why this enthusiasm and confidence dissipates by the time they reach my classes. For most students, mathematics becomes a set of rules to follow, a collection of techniques to master, and a jumble of examples to mimick. Despite the fact that some of our most treasured mathematical results took years— to be established, our society has emphasized speed and completion over contemplation. Most students (and adults) spend very little time thinking about problems and instead rely on other resources to provide information and demonstrate methods. They search for examples to mimic instead of creating personal strategies. Without the opportunity to think and explore, there is no excitement and sense of accomplishment. As a teacher and author, my mission is to share the joy of mathematical exploration with students, especially those considering a career in education.

Many resources have mathematical tasks and problems available for students to explore. Noncurricular puzzles, games, applications, and fun topics are great ways to encourage teamwork, creativity, and reasoning, but a mathematics course needs to have purpose and cohesion as well. In developing my course, I challenged myself to choose, sequence, and tailor activities so that students would develop and build upon foundational knowledge. As students work on activities in my class, I respond to their questions, celebrate their discoveries, address misconceptions, and probe deeper into interesting ideas. As a class, we discuss the key concepts that arise from the activities. As I write this textbook, I find that I need to anticipate how others might use the book and include more detail than I need in my own classroom. Some activities have more scaffolding to ensure that essential material is uncovered and introductory and summary reading has been added to focus and support learning. I hope that this additional detail and structure still allows for differing strategies, rich discussion, and the thrill of discovery.

Geometry is over 2500 years old, but continues to grow as a field of study. We see geometry, touch geometry, and create with geometry. There is geometry in the physical world around us, the universe beyond us, and in our imagination. Geometry invites us not only to explore, but to question, conjecture, and justify. It is a natural place to build logic and reasoning. In this textbook, my approach to logic is invitational, not rigid. Students are asked to explain their reasoning to a friend, sometimes in writing. They are asked to be clear and thorough, but there is no set format required. The goal is not to become mathematical proof writers, but to solidify their own understanding and to share it with others. The ability to write with detail and clarity benefits all students whether they aim to become teachers, lawyers, business professionals, or scientists.

The framework of this book is designed to meet most of our state’s cerfication requirements for elementary and early childhood educators, using exploratory activities to lead them to a deeper understanding of the geometry that they will be teaching. Despite the fact that students have been taught most of the formulas and facts before arriving at college, I have found that most students do not fully understand them. Approaching the material from an exploratory perspective with an emphasis on answering the question ‘why’ gives a fresh new look at familiar material. Before redesigning my course, I dreaded teaching the topic measurement as it seemed so dry and repetitive. Now area of a rectangle forms the foundation upon which much of this book is based. In particular, dissection and construction of polygons are used to observe and justify results like the Pythagorean Theorem. The list of formulas is kept short and intuitive, especially with volume and surface area. Instead, students are encouraged to use the definitions to compute these quantities.

The textbook includes a mix of interactive activities and open-ended explorations. Many of these are embedded GeoGebra and PolyPad applets, most created by the author. Other activities are tactile, using manipulatives, geometrical tools, and inexpensive office supplies. Often an option of completing tasks either digitally or tactilely is provided.

My hope is to continue improving the book, releasing a new edition each spring in time for fall adoptions. These new editions will incorporate additions and corrections suggested by instructors and students who use the text the previous semesters. Thus I encourage you to send along any corrections, suggestions, and comments as you have them.