Put together 40 energetic high school mathematics teachers, two prominent mathematics researchers, a progressive team of two high school teachers from Denmark, and the Geometry Center at the University of Minnesota. What do you get? Chaos! That is an intensive and imaginative course focusing on the mathematics involving Chaos and Fractals. For two weeks during the month of July, 1992, mathematician John Hubbard led a workshop for high school teachers in which the teachers became very active in the study of modern mathematics that incorporated dynamical systems, number theory, complex analysis, algebra, trigonometry, calculus, and various interesting sets such as the Mandelbrot Set. In this workshop the mathematics researchers became the teachers, and the teachers were mathematics researchers. As stated by Mette Veldesby, one of the instructors of the workshop, "[I hope the teachers gain] a feeling that math is not something old, but math is alive." This is only one of the many ways the workshop was successful.
II. The Instructors
Involved in the development and instruction of the workshop on "Chaos and Fractals" were John Hubbard from Cornell University, Bodil Branner from the Technical University in Denmark, and Mette Vedelsby and Bjorn Felsager, high school mathematics teachers from Denmark. These four made up a highly professional team.
John Hubbard is well known as a co-discover of the mathematical theory of the Mandelbrot set. In addition, he and Bodil Branner have been researching the dynamics of complex polynomials together for many years. With much enthusiasm for the teachers, they opened the door to their research. Bodil Branner remarks, "When [we] do research and work, whenever we answer one question there are ten more that pop up that we don't know the answer to. I think it's necessary to try to build a course like this that [the teachers] will see how we work in mathematics, not just present finished results. . . .A mathematician working is like being a part of creating something." Not only did they share their mathematics, they shared their thoughts and feelings during the research process. To the teachers' enjoyment during one session, Hubbard expressed how he thought he was in control of the making of the Mandelbrot set until, quit unexpectedly, this set appeared in other places as well.
As Hubbard and Branner brought in their research expertise, Mette Vedelsby and Bjorn Felsager contributed their knowledge and experience in high school teaching. They had also worked with Bodil Branner for many years and as a consequence, took sabbatical leaves to work with Hubbard at Cornell University last year. There they worked on software development for mathematics teaching and became involved in the development of the Geometry Center's Chaos and Fractal workshop. Having already done several workshops for teachers in Denmark on Chaos and Fractals, Vedelsby and Felsager had suggested this topic for the Geometry Center's summer workshop. In addition, they had also used many of the mathematical topics they presented at the workshop in their own geometry classes in Denmark. "Our primary role," says Felsager, "was to make sure that there were materials for the high school teachers that they could use when they teach. Also, we made sure there were exercises at a level appropriate for the high school teachers and to assist in the planning of the course always with the high school teachers in mind. Bodil and Hubbard are leading experts in the field [Chaos and Fractals], but neither has experience teaching high school. So we are from the education side."
So, combining research and education with mathematics, the essence of the workshop involved the dynamics of quadratic polynomials. Topics developed naturally from iterations with Newton's Method which eventually led to phenomena such as the Mandelbrot and Julia sets in the complex plane. Felsager is enthusiastic,"We certainly hope that [the teachers] get a feeling for mathematics. That they feel it and learn some modern mathematics that's interesting and has been an inspiration. [We hope] we've sort of succeeded in giving a unified treatment of mathematics. Everything was integrated, so if you understood one part, then that would throw light on another part. I hope they go back and feel inspired to try to integrate their curriculum more, so they don't teach the separate objects, but simply teach it as mathematics."
III. The High School Teachers
A typical morning of the workshop consisted of two one-hour long lectures, each followed by 30 to 40 minutes of "exercise" time where the teachers would work together on problems related to the previous lecture. Mette Vedelsby reports, "[The teachers] are really interested. They are working very hard. They want to understand all that is available to them. They are asking, asking, asking." As the teachers worked together on their exercises they began to visualize the mathematical relationships in their minds as if they were "imagining movies." They talked through their solutions, while sharing ideas and distributing duties. The instructors were always ready to help.
A long lunch break was scheduled, but many teachers came back early to continue working on exercises from the lecture or on exercises requiring use of the computer. "We're not just studying fractals." says Stanley Goldady from Breckinridge High School. "There is so much we're doing and learning about graphing calculators and computer software. We're combining algebra, geometry, and calculus." The teachers used the computers to analyze the dynamics of various functions, create various fractal sets, and explore on their own.
Another very important aspect of the workshop was the series of one and one-half hour afternoon sessions where some of the High School teachers gave presentations on mathematical activities they were doing in their geometry classes. One teacher shared his own investigation of the sphere packing problem. Another demonstrated computer simulations of fractals and chaotic motion. A creative invention of the use of various shaped protractors was presented as well as the mathematics behind the robotic arm. At times the presentations became very active with people representing lockers in the locker room problem, an "Algebra" walk, and the construction of a three dimensional fractal!
Although each day of the workshop officially ended at 5 p.m., many of the teachers studied together at night as well. Some of them even admitted to working at the Geometry Center until 11 pm. The facilities at the Center were available to them whenever they wished to use them. Spending days and nights studying mathematics, a couple of teachers mentioned they were even beginning to dream about fractals and functions!
During the second week some of the teachers admitted fatigue, but that did not seem to stop them in their motivation to explore various mathematical situations. The teachers were excited to be mathematics students again. One teacher announced, "It's been 24 years since I've studied Calculus!" Another teacher said, "I knew absolutely nothing about this kind of math (fractals) before. I wish I could understand more. There is just a ton of stuff in a short period of time." Some of the teachers admitted that the concepts they were learning were "way up here," but they were persistent in learning and wanted to work together. As one teacher shared, "It takes awhile to see how everything is brought together. Today (Tuesday of the second week), we just started working in the complex plane."
And towards the end of the week many things were brought together in the form of the Mandelbrot Set: the iterations, the functions, the complex plane, the patterns of fractions in the Farey Tree, and much more. Joan Jurichko from Mahtomedi High School was amazed, "I had no idea what dynamical systems were before this. You can keep zooming in on the Mandelbrot Set and something more beautiful shows up."
IV. Incorporating the "new" mathematics into the High School curriculum
The teachers expressed excitement about the "new" mathematics they could bring into the classroom. Even though students may not be able to go in so much depth as the teachers did, they could be shown what some of the mathematics they are learning, in particular functions, are used for. For a real "hands on" experience in the classroom, a three dimensional fractal of Serpinski's Triangle could be built, as was done in the workshop.
Vicki DeVoss, a teacher at Work Opportunity Center in Minneapolis, was enthusiastic about bringing in some new mathematics into her classroom, "Alot of kids use math in different ways. It's not just the gifted kids who can explore this kind of math. This is more visual, attracting kids from the "art line." In a visual manner, we can explore fractions with Farey Trees by shading [in colors] and observing symmetries."
The teachers are excited yet also concerned about incorporating some of the mathematics they studied into their own high school curriculum. Many teachers were concerned about the computer availability in their schools, as well as computer illiteracy among teachers and students. It was suggested by some that a "pre-done" computer disk would be very helpful. One teacher mentioned that there was too much preliminary work that is necessary to be able to plop it right into the curriculum, however, there are also many patterns in functions, for example, from some of these preliminaries that students could experience.
In order to develop appropriate mathematical activities for high school students, several teams of teachers continued working at the Center for a few weeks following the workshop. These teachers developed ten sets of activities: Ordering and the Farey Tree, Modular Arithmetic, Basic Iteration, Angle Doubling Basics, Geometric Iteration, Fractal Complexity, Graphical Iteration, Geometric Transformations in the Complex Plane, Angle Doubling and Binary Functions, Angle Doubling and Chaos. The activities are designed to supplement a variety of mathematics courses such as Pre-Algebra, Algebra, Geometry, and Calculus.
The workshop on "Chaos and Fractals" may have been a little chaotic, but it was very successful. The teachers accomplished a vast amount in only a few weeks. Jurichko summed up her experience in three words, "Amazement, confusion, insight!"
What will the workshop be next summer? Geometry and Art!