Title: Fractals for the Classroom: Strategic Activities Volume
Three
Springer-Verlag, New York, Inc. 1999 (ISBN 0-387-98420-8) 107p., 49,-DM
(in cooperation with the National Council of Teachers of Mathematics
(NCTM), New York)
This is the last volume of a series of five books on fractals, three of
which on Strategic classroom activities in teaching fractals,
(Springer, Volume One: 1991, and Volume Two: 1992)
intended to provide companion materials for the two books
Fractals for the Classroom. Part one: Introduction to Fractals and
Chaos, and
Fractals for the Classroom. Part 2: Complex Systems
and Mandelbrot Set, written by the first three authors, and also
published by Springer in 1992.
These Strategic activities have been developed from a sound
instructional base, stressing the connection to the contemporary
curriculum for school mathematics. They may change the teaching of
mathematics! The understanding of fractal objects provides a wonderful
setting for students to enjoy the amazing dialogue between numeric and
geometrical processes, and the fascinating interaction between
mathematics and computer science. Occasionally a topic from Mathematics
goes beyond the usual mathematical bondaries and captures the
imagination of the public at large: fractals and chaos theory achieved
such prominence. The authors of the book reviewed here contributed to
this popularity.
The volume offers a hands-on approach to gain deeper mathematival
insights into iterations, iterated function systems in two dimensions,
and iterated function systems and geometric genetic codes.
The general pattern and specific steps used to construct a fractal
image illustrated throughout this volume comprise an iterated function
system. The objective of this volume is to investigate the processes
and often-surprising results of applying such systems. By themselves
the fractal figures posses fascinating features, but the simple
mechanisms are also fascinating by which the fractals are formed.
The two units of this volume are developed from an intuitive
perspective that connects it to the familiar notion of linear maps,
symmetries, addressing, and draw a nice link to Pascals's triangle. The
concepts are introduced at the lowest possible level of mathematics. In
every unit the activities take advantage of the technological power of
the graphics calculator. (This possibility may compared with the BASIC
programs given in the textbooks [op.cit.].) Thus, this is again an
exercise and worksheed book. Everything is explained in great detail,
and some is probably school level. Counting, constructing, computing,
and visualizing: it is rewarding after a bit of concentration.
Answers are provided for all the exercises.
The book is very well written, and furthermore, I could not find any
misprint. However, the
organization of the material preceding the worksheets is a little
confusing, and occasional redundant. It is easier to go back to the
textbooks [op.cit.].
The first unit in this volume, unit 7 of the series, explains the
Sierpinski curve and the Sierpinski gasket, connects it with Pascals's
triangle via combinations, and leads over mappings and copying machines
to self-similarity. The last unit discusses symmetries and
compositions, algebraic mappings and fractal images, and chaos game
variations. The connection between geometric genetic codes and chaos
game developments is presented in what ought to be an exciting
experience for the reader who walks the path suggested in this volume.
I highly recommend this journey into chaos of the wonderful series of the five books on fractals, including this volume, for everyone interested in these topics. They are a necessary exercise for the serious student, and a fun game for the amateur with some background, interest and perseverance. Not much is actually needed to do many of the projects and exercises, and much can be figured out by doing. The volumes of the Strategic Activities can be used as the main part of a semester course on fractals. They help students at all levels to see mathematics as an exciting, dynamic discipline.
Wolfgang Quapp (Leipzig)