Often, the best way to teach a proof is by drawing an image that illustrates the ideas. While mathematics is based on abstract concepts, humans are creatures whose intuition is largely based on images. This book describes many ways in which concepts can be visually represented and these images truly are worth a thousand symbols. Part I consists of twenty chapters, which are called:



*) Representing numbers by graphical elements

*) Representing numbers by lengths of segments

*) Representing numbers by areas of plane figures

*) Representing numbers by volumes of objects

*) Identifying key elements

*) Employing isometry

*) Employing similarity

*) Area-preserving transformations

*) Escaping from the plane

*) Overlaying tiles

*) Playing with several copies

*) Sequential frames

*) Geometric dissections

*) Moving frames

*) Iterative procedures

*) Introducing colors

*) Visualization by inclusion

*) Ingenuity in 3D

*) Using 3D models

*) Combining techniques



Each chapter has several sections where a specific type of problem is examined in each section. For example, the three sections of chapter 1 are:



*) Sums of odd integers

*) Sums of integers

*) Alternating sums of squares



Each chapter terminates with a section called "challenges", which is a set of problems that are to be solved by creating images similar to those demonstrated in the chapter. Hints and solutions to these problems are provided in an appendix. Part II of the book is called "Visualization in the classroom" and is a short history of visualization and a demonstration of how physical objects can be used to illustrate the concepts.

This is an outstanding math book, demonstrating some of the very best strategies that can be used to develop and explain a proof. It is not possible to use a diagram to present every mathematical proof that you may explain to your students. However, it can be done more often than you may realize, so I recommend that all teachers of mathematics examine this book, you most certainly will find a gem that you can use.



Published in Journal of Recreational Mathematics, reprinted with permission.