Fun and learning with a computer; Turtle Geometry: The Computer as a Medium for Exploring Mathematics, by Harold Abelson and Andrea diSessa. Cambridge, Mass.: The MIT Press. $20.
This is a book for people of a wide range of ages who have access to a small (home or school) computer and who would like some insight into so-called "advanced" geometry.
In spite of its formidable title, you don't have to be a math "heavy" to enjoy and benefit from it. This is mathematics at the intuitive, rather than axiomatic, level. It is learning by discovery rather than by dull drill or confusing symbolic manipulations.
That is why the book can be aimed at the high school or college undergraduate level and recommended for an even wider general readership. Indeed, some of the materials that have gone into it grew out of work with preschool children by the Logo Group at the Artificial Intelligence Laboratory at the Massachusetts Institute of Technology.
With the help of this book and the simple computer programs involved, readers can explore the properties of space by following the wanderings of an imaginary turtle across the computer video screen. Simple commands direct the turtle to move. In this way, the authors help readers gain an intuitive feel for such abstractions as curvature -- how curved or pointy something is, a notion that seems obscure in mathematical formalism.
This way of coming at geometry emphasizes procedures rather than equations. And, since computer programs often produce surprises even for their programmers, studying geometry becomes a voyage of discovery.
The book is not for everyone with a home computer. It does aim to teach something of advanced geometry and thus demands a learning effort. But for those willing to exert that effort, there are rich rewards. With this approach, the authors succeed even in giving one a feeling for Einstein's theory of relativity without the struggle to learn the tensor calculus that usually is required.
The simple programming involved should be well within the skills of anyone with a home computer that has some graphics capability. Indeed, turtle geometry is already incorporated in one option offered with the popular Apple II computer , being included in the Apple Pascal (programming language) system.
As the authors note: "The computer will have a profound impact on our educational system, but whether or not it will enrich the lives of students will depend upon our insight and our imagination." Their book is an exce llent example of this potential for enrichment.