These geometric shapes are formed by repetition: Take a shape, do something to it, then do it again for as long as you want. Fractals generally start simple enough, but they can get hideously complicated.

But it turns out that fractals are an excellent tool for math teachers. At least that's the opinion of fractal fan Cynthia Lanius, a math teacher at Milby High School, Houston, Texas.

For one thing, she says, fractals make the shapes seen in nature. "You don't see a lot of triangles, squares or rectangles in the woods," but fractals show up in everything from cumulus clouds to fern fronds.

For another, they are a field of modern math research that students can grasp. "Kids think that all mathematics was learned 500 years ago, that math is a dead study like Latin is a 'dead' language. I want them to understand that there are people living, breathing, and making a living as math researchers."

A geometric basis for math lessons
Each fractal becomes the basis for lessons on geometry, exponents, patterns and fractions, Lanius says. "I didn't want teachers to see this as just fluff. I wanted to incorporate some math so they could use fractals as a way to teach what they are already teaching." That helps counter an argument that Lanius knows all too well: "I don't have time to teach this stuff." With fractals, she says, "they are doing neat stuff and still meeting the curriculum."

She developed the fractals unit, and other math web sites, in work funded by the RGK Foundation for the Center for Research on Parallel Computation, with headquarters at Rice University in Houston.

Lanius says she designed the fractals unit to be as flexible as possible. Obviously, it's best to have Internet access for every pair of students, so they can discuss what's on the screen and follow links, but she's duplicated the pages in a "print-only" format (with links stripped out) for teachers whose classes don't have web access.