Fair warning: if you are not into outreach, you probably don't want to read this one. The entire content deals with the efforts of our Creating a Community of Mathematics Learners project. Not the cup of tea for all of you, but very much that for some, so I am going ahead with it.

This is old news, but is aimed at answering a question that I have heard from several tongues and seen on other eyebrows: "What is it you CCML guys do when you get a whole mob of middle school teachers together for a week-long summer institute?" No generic response here, but we had an exceptionally fine time doing the mathematics this past summer, so that seems worth the recounting.

The tale gets off to a rocky beginning, because we set a record for false starts. We of the math planning committee kept brightly presenting yet another batch of lessons and activities, and the rest of the group kept saying, in extremely polite phrases, "Ja, so?" The breakthrough came at the meeting when Jack Beal asked "So why is it that we are doing geometry, anyway?" Instant answer was that that's what the teachers had requested, because the wave of the K-12 future includes more geometry, earlier. But that didn't account for the enthusiasm of the planners, and it certainly didn't provide any kind of coherence for the whole set-up. Ramesh Gangolli rose to the occasion with a pair of characteristics:

- Geometry is a field in which a small collection of elementary results can lead directly to universal and powerful conclusions.
- Geometry encourages developing the ability to relate two different cognitive functions. For example, interaction between "visual experimentation" and verbal description of shapes and relations is valuable and needs developing. Likewise interaction between the visual and the computational, and between the visual and the algebraic.

These we promptly dubbed the Glorious Features of Geometry (GFG's for short) and we were off and running. As we designed and as we taught we kept checking in on the GFG's, and they provided exactly the glue we needed.

Day One was dedicated to Pythagoras. For the first session we presented the participants with diagrams providing cues to proofs of the Pythagorean Theorem by President Garfield, Leonardo da Vinci and a couple of other guys, and turned them loose on providing proofs (defined as arguments clear enough to convince the Friendly Skeptic. We got to be the Friendly Skeptic at first, but worked on turning this responsibility over to them as the week went along.) Some groups came up with one proof, some with four, but no thumbs were twiddled.

In the afternoon they tackled a batch of applications of the Pythagorean Theorem, ranging from railroads in the sun to maypoles. Also included was a real doozy of a problem from a 12th century Chinese manuscript. A couple of groups got it, which wowed me considerably.

On Day Two the morning was on similarity. Michael Keynes and I put that one together. Actually, I'd say we had probably gotten a bit carried away. We not only had a whole sequence of activities on perspective, some involving drawing on acrylic panels held at arms length, and others involving distances in photographs (our pride and joy was a picture of Michael apparently holding my 6-foot son in the palm of his hand), but also a set using flashlights and plastic figures to reproduce the thinking of Thales. By the end the participants were a bit wild-eyed, though definitely still game.

Then in the afternoon we launched another topic: three dimensional coordinate geometry. For this we used the magnificent coordinate systems produced by Caspar Curjel's inspired raids on Eagle Hardware's plumbing section, plus some little ones the participants made for themselves with CD jewel cases. Very tactile, and very helpful in establishing just what those planes and lines and points could be doing as they wander around off the page.

Day Three was something of a mathematical rescue job: tesselations have become a very popular teaching topic, for some excellent reasons. Unfortunately, in the process the mathematics has tended to submerge beneath the charm and the razzle- dazzle. So we decided to dig it back out again. We took them through basics up through semi-regular tilings with a lot of "How do you know that will tesselate?" and "Why can that be a vertex?" type questions, and watched their respect for the whole topic soar. In the afternoon we pushed the same logic up a dimension and worked some with Platonic solids. A very jazzy day.

Day Four was on symmetries and transformations, working our way up to "So how could you produce the rotation that would move the box we just threw into that corner over there to coincide with the one we threw over to that side of the room?" I treasure the mental image of several members of my class deep in consultation as they stood on tables for a clearer view of the boxes.

On Day Five they spent the morning presenting to each
other lessons they had been working on in other parts
of their days. The afternoon was then the Tetrahedron
Fest. Using straws and tissue paper,
they produced tetrahedra and figured out their
surface area and volume. Then they combined forces
and produced tetrahedra made up of four of their
original tetrahedra, and figured out the new
surface area and volume, and the shape of the
empty space in the middle. That's as much as could
be carried through the door, so at that point we
headed for the school playfield. At the June
session, we went on combining and stacking and
connecting until
we had an incredibly gorgeous object made up of 256
of the smallest size tetrahedron. Kodak loved us that
day. The July session had fewer participants, so
instead we flew them as kites. Small snag: no wind.
But the runners kept them in the air for a minute or
so, and the guys that put theirs out the back
window of their station wagon got quite a lot of
loft on it. In either format,
the tetrahedra provided a nicely festive conclusion
to a rigorous, but highly enjoyable week.