# Teaching and Learning Geometry.

Teaching and Learning Geometry Douglas French Published by Continuum International Publishing Group, 2004 ISBN 0826473628 169 pp., soft cover; 25.00 [pounds sterling]

This book is designed to offer practical advice and ideas for the classroom in the teaching and learning of school geometry and has presented a variety of ways of achieving this. However, to change what happens in practice in the classroom may be difficult!

In the book are twelve chapters:

* The Role of Geometry

* Learning Geometry

* Beginnings: Experimental Geometry

* Polygons: Symmetry and Angle Properties

* Construction and Congruence

* Perimeter, Area and Volumes

* Enlargement and Similarity

* The Theorem of Pythagoras

* The Circle

* Polyhedra

* Vectors.

Working through Chapters 3, 4 and 5 highlight important initial ideas and skills to do with recognising shapes and their properties, the importance of symmetry, tessellations and the transformations of reflection and rotation, angle properties and triangles, the properties of quadrilaterals and polygons. Also to be found is an introductory lesson plan on regular polygons using LOGO. The plan indicates lessons are timed for fifty five minutes. Constructions and congruence and Van Shooten's Theorem conclude this section.

In Chapter 6 the author suggests finding the area of a sector of a circle is a good example of a task that should be presented as a problem to be solved from first principles, rather than as a task to be worked out using yet another remembered standard procedure. At this stage in the book only the area of a circle has been met. Thus only first principles are possible.

The direct link of similarity to the transformation of enlargement and for the introduction of scale factors provides a more simple approach than using equal ratios. Such is the content of Chapter 7. This also includes both Varignon's theorem and the intercept theorem being proved using the midpoint theorem.

Pythagoras' Theorem has a multitude of different proofs but those considered in this book involve areas, congruence, transformations, similarity, trigonometry and algebra. The trigonometric identities and Apollonius' Theorem are also introduced.

The linked circle theorems of Chapter 9 are familiar to many students and Miquel's six-circle theorem also appears.

In the following chapter the use of a coordinate system allows both an algebraic and a geometric perspective.

It was a surprise to find a chapter on polyhedra included in a course on geometry, so often has it been considered a recreational topic in Australian courses.

The book closes with an interesting chapter on the use of vectors in solving geometry problems. For many this could be unfamiliar territory. A slight typing error appears on page 159:

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Throughout the book use has been made of dynamic computer programs and software as well as graphical calculators. Diagrams are many and include graphs and tables of values. The presentation of the book is pleasing to the eye. This book has presented a way of looking at geometry from various perspectives. Consequently each teacher of geometry should have access to a copy.

Margaret McDonald
COPYRIGHT 2005 The Australian Association of Mathematics Teachers, Inc.
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