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Fractals, Graphics, & Mathematics Education

Teaching First: A Guide for New Mathematicians

Benoit Mandelbrot & Michael Frame, Editors

Catalog Code: NTE-58
Print ISBN: 978-0-88385-169-2
204 pp., Paperbound, 2002
List Price: $49.95
MAA Member: $37.50
Series: MAA Notes

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Fractal Geometry is the latest addition to the collection of mathematical tools for describing nature and the works of Man. It made possible for the first time a rational study of roughness. Fractals are encountered in mathematics and many natural sciences, but also in finance and in art, music, and literature, most often without being consciously included by anyone. Therefore, fractals interconnect the arts and the natural and social sciences in many intrinsic ways. Rarely, if ever, are students exposed to anything like this in mathematics and science classes.

This book collects essays about fractals’ role in mathematics and science education. In the first four chapters the editors address general issues. The next twelve chapters are invited case studies authored by educators who began years ago to use fractal geometry in classes that range from second grade elementary school, through public and private high schools, to state universities and private colleges. Some contributors survey literature and software they have used, others present fairly detailed sample lessons. The chapter from Florida Atlantic University reports on a program training teachers throughout Broward and Dade counties.

Many teachers developed fractals courses on their own in isolation from one another. This book is a token of how widespread such courses have become. The common themes that appear throughout mark the coming of age of this subject.

Table of Contents

Foreword, by Lynn Arthur Steen

Introductory Essays
1. Some Reasons for the Effectiveness of Fractals in mathematics Education, by Benoit B. Mandelbrot & Michael Frame
2. Unsolved Problems and Still-Emerging Concepts in Fractal Geometry, by Benoit B. Mandelbrot
3. Fractals, Graphics, and Mathematics Educations, by Benoit B. Mandelbrot
4. Mathematics and Society in the 20th Century, by Benoit B. Mandelbrot

Classroom Experiences
5. Teaching Fractals and Dynamical Systems at The Hotchkiss School, by Melkana Brakalova & David Coughlin
6. Reflecting on Wada Basins: Some Fractals with a Twist, by Dane Camp
7. Learning and Teaching about Fractals, by Donald M. Davis
8. The Fractal Geometry of the Mandelbrot Set: Periods of the Bulbs, by Robert L. Devaney
9. Fractals—Energizing the Mathematics Classroom, by Vicki Fegers & Mary Beth Johnson
10. Other Chaos Games, by Sandy Fillebrown
11. Creating and Teaching Undergraduate Courses and Seminars in Fractal Geometry: a Personal Experience, by Michael Lapidus
12. Exploring Fractal Dimensions by Experiment, by Ron Lewis
13. Fractal Themes at Every Level, by Kenneth G. Monks
14. Art and Fractals: Artistic Explorations of Natural Self-Similarity, by Brianna Murratti & Michael Frame
15. Order and Chaos, Art and Magic: a First College Course in Quantitative Reasoning Based on Fractals and Chaos, by David Peak & Michael Frame
16. A Software Driven Undergraduate Fractals Course, by Douglas C. Ravenel

A Final Word
17. The Fractal Ring from Art to Art through Mathematics, Finance, and the Sciences, by Benoit B. Mandelbrot

18. Panorama of Fractals and Their Uses. An Alphabetic Wordbook-Index, by Michael Frame & Benoit B. Mandelbrot
19. Reports of Some Field Experiences
Guide to Topics

About the Editors and Other Contributors

About the Editors

Benoit B. Mandelbrot, best known as the author of The Fractal Geometry of Nature, 1982, is Sterling Professor of Mathematical Sciences at Yale University and IBM Fellow Emeritus (Physics) at the T. J. Watson Research Center. His life’s work seeks a measure of order in physical, mathematical, or social phenomena that are characterized by extreme variability or roughness. The Wolf Prize cited Mandelbrot for having “changed our view of nature.” Recently, his interests have broadened to include science education.

A fellow of the American Academy of Arts and Sciences and a member of the U.S. National Academy of Sciences, his awards include the 1993 Wolf Prize for Physics, the Barnard, Franklin, Steinmetz and Richardson Medals, the Science for Art, Harvey, Humboldt, Nevada, Honda, and Proctor Prizes, the Caltech Distinguished Service and Scott Awards, and a number of honorary doctorates.

Trained as a topologist at Tulane University, Michael Frame became interested in fractal geometry in response to a question from a student in 1983. He has taught at several schools, most recently Union College, before moving to Yale. With Physicist David Peak, he is the author of Chaos Under Control: the Art and Science of Complexity, a text from humanities students. Recently he has begun investigating using the web in teaching.

MAA Review

Reviewed by George Ashline

...In this text, Michael Frame and Benoit Mandelbrot have written general essays describing the impact of fractal geometry on mathematics education, and included case studies of teachers using fractals in a variety of curricular approaches. As a mathematics professor at a small liberal arts college, I am very interested in ways to help mathematics "come alive" for my students. In this review, I will describe the features of this text and the plethora of innovative ideas found within it for enlivening and enriching mathematics education.

The foreword and first four chapters of Fractals, Graphics, and Mathematics Education contain general remarks about the role that fractal geometry has played in mathematics education. In the foreword, as reiterated in other parts of the text, L. A. Steen describes the unique opportunity fractals provide to re-emphasize the experimental side of mathematics and help to balance the theory and abstraction which dominated twentieth century mathematics and mathematics education. There are multiple and varied reasons behind the popularity of this area of mathematics. Steen, Frame, and Mandelbrot, and other contributors to the text detail the many characteristics of fractal geometry that can enrich mathematics education, including

  • simplicity of many of its underlying ideas, with a short distance to fascinating complexity and unsolved problems.
  • accessibility to students with various mathematical comfort levels, and ability of those students to take ownership of the material.
  • beauty of the graphics which connect to the visible world, stimulate student curiosity, and motivate further experimentation.
  • centrality of the computer in creating these graphics and allowing students to explore, conjecture, and gather new insights.
  • fun, excitement, and spontaneity created by new topics and problems that arise from often unexpected directions.
  • emphasis on the human aspect of science and the middle ground offered between quantitative and qualitative fields.
  • widespread applications and a "common language" for various disciplines, including architecture, art, business, dance, economics, literature, music, psychology, sociology, and technology.
  • suitability of this new language for the study of "roughness", in many contexts and situations.



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