Mathematics   >>   Critical Thinking   >>   Secondary

Web Resource—Chaos in the Classroom     Robert Devaney

Critical Thinking
Often discussion of chaos and fractals leads to simple “pretty picture shows” without mathematical content. This site focuses on the mathematics behind the pictures. The chaos game allows students to deepen their understanding of algorithmic thinking, geometric transformations, probability and randomness. Critical thinking is highlighted by asking students to study what happens when the rules of the game are changed and to determine the algorithms used to create different fractals. Creative thinking is also encouraged by asking students to make “fractal movies.”
This site provides mathematical exploration of fractals appropriate for secondary students. The chaos game can be played with or without the use of technology. This game affords students the opportunity to see how the Sierpinski Triangle is generated. Searching for the winning strategy, they discover the logic imbedded within apparent randomness. The site is actually written for teachers with good suggestions for classroom implementation of the game. After playing the game, students can directly access sections on self-similarity, fractal dimension, changing the rules of the game, and rotations and animations. The last section provides extensions and high-level critical thinking.
Appeal and User Friendliness
This site is written for high school math teachers. It explains the mathematics behind the Sierpinski Triangle and how to use the Chaos Game to encourage students to discover this mathematics. Although it is not in lesson plan format per se, it is well written with clear directions and excellent illustrations. Teachers will need to generate discussion questions. It is subdivided into sections that make it well organized and easy to follow. The follow-up sections to the chaos game are appropriate for students to work on without teacher guidance and encourage critical and creative thinking.
Sample Problem
Students are provided with three different “fractal movies” and asked to determine what algorithms were used to create them.