Review

The review below is reproduced from THE MATHEMATICS TEACHER, 93(3), March 2000, p. 254. published by NCTM.

**Rethinking Proof with The Geometer's Sketchpad with CD-ROM**, by Michael de Villiers, Key Curriculum Press. To order, please contact: dynamiclearn@mweb.co.za

This book of thirty-one activities begins with a strong statement about the role of proofs as an integral part of mathematical knowledge. The author emphasizes that powerful computer tools, such as The Geometer's Sketchpad, do not make proofs obsolete but rather makes possible the visualization and measurement techniques that allow students to develop conjectures that can help them develop deductive proofs. His introduction of proofs uses a sequence of explanation, discovery, verification, intellectual challenge, and systematization in a kind of spiral approach that allows earlier reasons for proof to be revisited and expanded.

The activities are designed to be used with The Geometer's Sketchpad and are divided into five chapters that correspond to the sequence previously mentioned. The book contains a CD-ROM that contains programs to use with each activity. The programs are add-on modules for The Geometer's Sketchpad.

The first chapter addresses proof as explanation. These activities encourage students to develop a conjecture after investigating a given geometric problem, such as locating the placement of a water reservoir so that it is equidistant from four villages. The students are next asked to explain what they have observed and then present their explanations as an argument either in paragraph form or as a two-column proof.

The next chapter presents activities related to proof as discovery. Again students begin with a figure and are prompted to develop a conjecture and an explanation on the basis of their observations. However, in this chapter, students are led to discover new geometric properties through logical reasoning and then to follow it up with construction and measurement to test for false assumptions or conclusions.

Chapter 3 focuses on proof as verification. Students again begin by developing a conjecture and explanation. They are then led through developing a proof of the conjecture. They are encouraged to include a demonstration sketch to support and explain the proof.

Chapter 4 presents activities related to proof as challenge in a fashion similar to that used in the previous chapter. However, these activities are more intellectually challenging. One activity challenges students to locate an airport so that the sum of its distances to three cities is minimized and then to develop a proof of their conjecture.

The final chapter, "Proof as Systematization," guides students in developing proofs in a clear, systematic way, giving reasons for each step. For example, students are asked to systematize the properties of rhombuses and the properties of isosceles trapezoids.

The teacher notes will be very useful as teachers use these activities with their students. The notes describe the desired outcomes of the outcomes and suggest ways to maximize the experience for the students. The answers to questions are also included. This set of activities is an outstanding compilation that would enrich the geometry curriculum. The activities are easy to follow yet challenging enough to keep students interested. The five-step sequential model of developing the functions of proof is worthy of consideration and investigation. This approach will make proof development much more understandable to students. This resource is one of the best that I have seen!

*- Martha Cantrell, Tallulah Falls School, Tallulah Falls, GA 30573, USA*.