The year has started well, but getting good discussions going is still hard. I find it so difficult to shut up and stay in the background. But yesterday one of my classes had a great discussion, and it was a real pleasure to see it happen. We had a rich problem to work with—a simple word problem that previewed most of the concepts of differential calculus—and students with differing answers, who cared about hashing out the differences and figuring out what was going on. I’m sure I still intruded too much, but they really did it mostly on their own. Now I just have to replicate that over and over.

I’ve also been enjoying decorating my classroom. I had not made a serious effort to enrich my teaching space(s) for a while, since I had been switching from room to room a lot. But this year I am always in the same room, and the only other teacher in that room is quite amenable, so we’ve got some cool math posters up (some from cafepress.com, some from the American Mathematical Society, plus a huge New Mexico map from Raven Maps & Images). I hung some Zome constructions up from the ceiling (so far: segment, square, cube, projected 4-cube, projected 5-cube; to be hung: icosahedron, dodecahedron, truncated icosahedron [soccer/Buckyball]), and put up a lot of printouts of Petrie projections of higher-dimensional polytopes. So the room has a pretty mathematical flavor to it, with a strong emphasis (right now) on polytopes. If I’m really ambitious I’ll redecorate every semester with a new theme—maybe next time it will be families of algebraic curves and surfaces, or topology, or something else.

I don’t have many fractal posters on the walls, but I’ve got one, and I’ll probably get another. But, while fractals are amazingly cool and beautiful, I have to say they’ve stolen a lot of press from many other equally cool and beautiful parts of mathematics. So I don’t mind not giving them pride of place.

I know that there are many teachers out there who have got me way beat on how cool and mathematical their classroom is. But I’m still proud of my new look, compared to what I had before (such as teaching in a history classroom, which has its own interest, I must say). One of my fellow teachers has got me beat hands down on number of posters, especially Escher posters—which I would get a few of, except that it would just be copying her 😦

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I’d love to hear more about the problem that you used for your discussion.

Regarding decoration, my best recommendation is to rotate the crops much more often than once a year. Definitely put up some amount of student work, but only if it’s “good.” Every kid does good at some point or another, but you gotta catch it when it happens.

The problem was from a very early section in the Stewart text. It posited an airplane taking off from one city and landing in another, and asked for a graph of the vertical velocity. This was way before we had even talked about slopes etc., much less derivatives, or second derivatives, or continuity. We got discontinuous graphs, graphs that were only positive, graphs that had sharp corners…and the students were able to see that all of those features were physically unrealistic. I wouldn’t say most of the students assimilated all that went on in the discussion, but that wasn’t necessary. It was just great to see them talking and getting close to the magic “level 3” discussion.