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Topic: Geometry Center Workshop, Summer '92
Replies: 1   Last Post: Sep 30, 1992 3:44 PM

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Laura Smith

Posts: 1
Registered: 12/6/04
Geometry Center Workshop, Summer '92
Posted: Sep 29, 1992 1:52 PM

Chaos at the Geometry Center

Laura Smith, U of MN, Summer 1992


I. Brief Introduction
A. What is the workshop on Chaos and Fractals?
B. Who did it?
C. What was the purpose of the workshop?

II. The Instructors
A. John Hubbard
B. Bodil Branner
C. Mette and Bjorn

III. The Teachers/Students
A. A typical day at the workshop.
B. Teacher presentations.
C. Teacher impressions.

IV. Incorporating the "new" mathematics into the High School Curriculum
A. Teacher teams in areas of Iterations, Modular Arithmetic, Farey Trees
and Fractions, Complex Plane. etc.

V. Conclusion
A. Overall impression and anticipation of next year.


I. Introduction

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

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.

V. Conclusion

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!


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