The two were very much of a matched pair (serendipity, but I hope it gives an illusion of organization!) The first was based on what Jim King and I and others do in institutes and workshops for teachers. The second was based on what TERC is developing for teachers to do in the classroom. If the two had not proved closely related it would have been extremely embarrassing. A bit different as regards levels, though. For the first we brought in snap-together triangles--many, many of them. The activity of the day was snapping them together into polyhedra (entitled deltahedra because of the universally triangular faces). Had we been in a full-day workshop a considerable period of pure snapping and observing would have ensued, with participants pursuing questions they generated themselves, and also being enticed into pursuing questions like "How can you describe that multi-sided object you just produced? What are its salient characteristics? What distinguishes it from that one over there?" and (arising naturally from those) "Which of its characteristics can you attach a number to? How can you count the number of vertices?" (Once you get beyond ten either you line up a friend or else you've got to get organized.) With the right balance of freedom and provocation, the discussion can pretty safely be counted on to give rise to a definition of convexity and of valence of a vertex (= number of triangles around it)--along with sundry observations about what the valence can and can't be. Also to noticing that you just can't come up with a deltahedron with an odd number of faces, because you'll have an edge hanging out in space--literally (that's the virtue to the snap-togethers).

All of this we finessed, though Jim described it, and instead we dove straight into a more structured phase involving a direct challenge to produce as many different convex deltahedra as possible. Some of us started with a nice friendly tetrahedron and worked our way out from there, and others took a no-holds-barred approach to producing an icosahedron. Then we filled in between (there are some pretty elusive little characters in there, not to mention one impossibility.) And the discussion filled in as well, punctuated by cries of "I've found another one!" The underlying question is "So why do this?" and the answer ranges from the richness of the discussion and questioning and conjecturing and verifying it can generate to the fact that over the years the school curriculum has somehow managed to convince itself that we live in Flatland--the third dimension has been quietly shown the door.

The third dimension was, however, very much present when Andee Rubin from TERC took on the next Brown Bag. She was talking about TERC's Investigations curriculum, which is an NSF-supported series that has been working its way up from kindergarten over the past several years. Each Investigation is designed to take about a month to do, and to take some idea or topic and let students really submerge themselves in it. An aspect that I found particularly impressive is that they have noticed that children have parents: each Investigation has a booklet for the child to take home, explaining the mathematical objective and what projects the parents can expect the kid to come home with, and suggesting helpful and enjoyable ways for parents to join the activity.

For us the activity was box-making. Very specific boxes: using square grid paper, see how many ways you can make an open-top box with two squares on the bottom. Done with third graders it would have been preceded by a considerable time spent on open top boxes to hold a cube, and the directions would have been a gentle "Do you suppose you could make a box that would hold two cubes side by side?" For us it was more like "So cut loose, already!" and it was great. Our official answer is LOTS! Another thing there are lots of is questions it can generate. The most obvious (on which, I might say, we made remarkably little progress!) is "How can you organize your findings?" Once you get beyond a couple of families it is highly non-obvious. Another was "Do you suppose there is one of these nets that could be folded into the required shape in two different ways?" That one I can answer. About twenty minutes into this, Jack Lee exclaimed "Got it!" While we hung upon his every word, he proceeded to demonstrate. We couldn't all see, though, so he had to do it several times. Which is probably why he suddenly discovered that in fact there were three distinct ways to fold it.

We did have a good time!