But certain tools are actually more than tools, and provide rich environments for exploration and learning, once known as

*microworlds.*The microworld concept originated in the Logo movement in the 1980s. Logo was a programming environment where children could explore programming and geometry, but it was also a place for teachers to do interesting math and create lessons for their students. A microworld is accessible ("low threshold"), but at the same time it offers near-limitless possibilities ("high ceiling".) Contemporary descendants of Logo include MIT's Scratch and UC Berkeley's Snap. Interactive geometry (GeoGebra, for example) is another type of microworld.

I maintain that the geoboard, as well, is more than a learning tool: it is a microworld. Yes, it can be used for excellent lessons on slope, area, distance, the Pythagorean theorem, and simplifying radicals (as you can see in my

*Geometry Labs*.) But it is also a wonderful arena for mathematical explorations, including some fun puzzles and some unsolved problems. I will focus on such teacher-level questions as I co-lead a workshop on

My co-presenter will be Paul Zeitz, of the University of San Francisco, the co-founder of Proof School, the Bay Area Math Olympiad, and the SF MTC.Connecting the Dots

Saturday, March 4

San Francisco Math Teachers Circle.

9:30-12:00

Proof School, 555 Post St.

Free. (Lunch included.)

RSVP here.

And yes, most of the questions we will explore

*can*be adapted for use in the classroom.