In the past two decades since Benoit Mandelbrot wrote Fractal Geometry of Nature (Freeman, 1982), fractals have grown in popularity and arisen in many contexts and forms in the mathematical, natural, and even social sciences. Some of the foundational ideas for this field can be traced back to the research of Pierre Fatou and Gaston Julia, who wrote important papers on the iteration theory of rational functions in the early twentieth century. Mandelbrot's pioneering work has brought fractal geometry to the fore and energized a new area of mathematics education. Creating a language for the study of "roughness" and developing the associated computer graphics, he provided the general theory for fractals, which offer new insights into understanding many aspects of Nature and the world around us. This is evidenced in such diverse examples as an island coastline or paper folding patterns or even a cauliflower head!
Fractal geometry provides an accessible, interesting setting not only for mathematical research, but also for mathematics education, and there have been many curricular initiatives in this area. In addition to its impact on middle and secondary school mathematics, fractal geometry has influenced undergraduate mathematics, and has been the source for many undergraduate research efforts. For example, consider the programs for the Hudson River Undergraduate Mathematics Conference (HRUMC) over its nine-year history. According to my count, there have been sixteen sessions on Fractals and Chaos over that time span, with typically three student speakers per session. That represents many student research projects, and indicates some of the student enthusiasm that fractals have engendered. Also, it is noteworthy that the keynote speaker at the 1997 HRUMC was Mandelbrot himself presenting The Beauty and Usefulness of Fractals and at the 2002 HRUMC was Robert Devaney presenting Chaos Games and Fractal Images. Another illustration of the interest in this area was the special issue of the College Mathematics Journal in January of 1991 devoted to chaos and fractals. (Those who have access to the "Arts and Sciences II" collection in JSTOR can browse the issue online.) According to Robert Devaney in the foreword to that issue, "I feel that these ideas will have their biggest impact in mathematics education. The mathematics of chaos and fractals is at once accessible, alluring, and exciting. Fractal geometry offers a wonderful arena for combining computer experimentation and geometric insight." Those words are certainly echoed throughout the essays and case studies contained in Fractals, Graphics, and Mathematics Education.
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.
The authors have incorporated these ideas into a general introductory course on fractal geometry at Yale University, and the other contributors to this text have used these alluring properties of fractals to engage students in many different ways.
A variety of teachers who have successfully integrated fractals into the curriculum describe their classroom experiences and offer practical advice in the twelve case studies found in Fractals, Graphics, and Mathematics Education. These articles provide the nuts and bolts information which can be tailored to meet the needs of the distinctive student population at one's own institution and in one's own courses, either those of an introductory flavor or those intended for students with more background in mathematics and science. The authors of these chapters represent a wide range of schools and colleges/universities into which they have successfully implemented fractal geometry curricular enhancements. The case studies describe initiatives at such diverse institutions as Boston Technical High School (MA), Broward County middle and high schools (FL), the Hotchkiss School (CT), Lehigh University (PA), New Trier High School (IL), Nickel District Secondary School and various York Region Schools (Ontario, CAN), St. Joseph's University (PA), Union College (NY), the University of California at Riverside, the University of Rochester (NY), the University of Scranton (PA), and Yale University (CT).
Rather than highlighting all of these interesting case studies, I will instead focus on a couple of specific ones. For example, Dane Camp of New Trier High School describes a lesson that he has incorporated into his AB Calculus class to experiment with a special class of fractals called Wada Basins. In his lesson, he relates such fractals to an attractive pendulum, Newton's method for complex valued functions, and light reflected in four mirrored spheres stacked in a tetrahedral shape. Quoting the end of his article, "the fact that connections can be found between the seemingly dissimilar areas of magnetism, function iteration, and reflecting light reveals that mathematics is a field that will always contain wonder for those willing to explore." Camp's lesson helps students appreciate how fractal insights can shed light and understanding on their surrounding world, which is a common theme throughout the case studies contained in this text. The Maryland Chaos Group provides a Chaos Gallery with nice pictures of Wada Basins, the related four mirrored spheres, and other interesting graphics, animations, and links to pertinent mathematics.
Vicki Fegers and Mary Beth Johnson of the Broward County School Board (BCSB) wrote another one of the case studies, describing their fractal enrichment efforts for middle and high school mathematics instruction in their region. Some of these efforts were supported by a three year NSF grant attained in the mid 1990's by H.-O. Peitgen in a collaborative endeavor between Florida Atlantic University and the BCSB. A rich collection of fractal geometry teacher enhancement materials was created as a result of these efforts. Summer institutes were offered for middle and high school mathematics and science teachers interested in chaos and fractal curricular enrichment. The Lead Teachers participating in the first institute created the initial drafts of a number of Curriculum and Textbooks Enhancements (or CATEs). Involving hands-on, cooperative activities and explorations in fractal geometry, these CATEs were subsequently revised and rewritten in future years. In their article, Fegers and Johnson include two CATEs (Fractal Wallhanging and Lessons in Paperfolding, respectively) that they have written for the project. They provide detailed classroom implementation instructions for these activities, student and instructor notes, and appendices containing sample transparencies, activity sheets, solutions to those sheets, and relevant graphing calculator programs. (As an aside, it should be noted that Figure 25 of the Fractal Wallhanging article is incomplete, but the other 26 Figures and 17 Appendices contained in their article nicely complement their lesson descriptions!) These useful, classroom-tested materials offer instructors some specific ways to incorporate fractal concepts into courses and to discuss some significant mathematics at the same time.
The other case study authors also provide useful fractal enrichment ideas. For example, some contributors include actual syllabi for fractal courses they have designed, others specify procedures (in a laboratory or on a computer/graphing calculator) for running fractal experiments, and others offer details (including relevant figures and descriptions) of fractal projects and units they have created. At the end of the text, biographical descriptions of each contributor are given, as well as a bibliographical listing of relevant texts and other materials. In the center of the text, another of its features is a color graphics section to enhance the two case studies described above, as well as another case study by Brianna Murrati and Michael Frame on "Art and Fractals: Artistic Explorations of Natural Self-Similarity".
I will conclude this review by specifying several fractal enrichment websites, in addition to the previous ones that I have included. These Internet resources contain ideas for creating fractal geometry courses, or enhancing already existing courses. Some of these sites are also mentioned in the case study chapters and the Reference section of the text.
- As described in the last chapter of the text, Frame and Mandelbrot have created a Panorama of Fractals and Their Uses. This is intended to be a "detailed wordbook or lexicon that orders alphabetically a wide mixture…" fractal uses for the classroom. Those wishing to submit additional applications are encouraged to contact the authors at email@example.com.
- Kenneth Monks at the University of Scranton provides a couple of versions of his chapter Fractal Themes at Every Level, as well as his fractal course software and some of his pertinent student papers.
- Robert Devaney at Boston University offers an interactive version of his chapter on The Fractal Geometry of the Mandelbrot Set: Periods of the Bulbs. Also, see his NSF-sponsored Dynamical Systems and Technology Project for ideas on effectively using technology to integrate fractals and chaos into the classroom.
- Cynthia Lanius at Rice University details a fractals unit for elementary and middle school students that "adults are free to enjoy". The unit describes various fractal properties and involves creating and interpreting such figures as the Sierpinski triangle, Jurassic Park fractal, and Koch snowflake.
- Eric Weisstein's World of Mathematics (a web resource provided by Wolfram Research, the maker of Mathematica) contains information about fractals and links to pertinent topics and resources. Also, Alexander Walz provides information on fractals and their creation using the Maple V.
- Mathematics Archives and the Math Forum both provide a variety of fractal information and links to other online resources.
- The journal FRACTALS publishes peer-reviewed work in the form of full-length research papers, short communications, reviews of both technical and pedagogical nature, and popular articles. The table of contents and paper abstracts for previous issues are accessible.
- As suggested by a number of the contributors to the text, one particularly effective means to generate fractal graphics is through the freeware fractal generator Fractint, created for IBM PCs and compatible computers. The authors provide more information on other fractal software.
George Ashline (firstname.lastname@example.org) is an associate professor of mathematics at St. Michael's College in Colchester, VT. He is a member of Project NExT, a professional development program for new or recent Ph.D.s in the mathematical sciences who are interested in improving the teaching and learning of undergraduate mathematics.