April 1, 2002 Ms. Verity Keane Local University USA Hi Verity, Sorry haven't been in touch for a while, but school is very hectic. In addition to my class in elementary mathematics methods, am taking a course on teaching elementary physical education. am really getting into shape now. Whew! But would believe it's geometry again in math? We always hated geometry, didn't we? For me, it was just gymnastics in the mind. Talk about pain, no gain. should be a geometrical Olympian by now, it hurts so much. Now we have a term paper to do on Geometry Begins, so after this letter it may be a while before write again. know what these professors are like. problem arose for me almost as soon as we started geometry. Don't forget, had spent the whole of my mathematical life trying to remember things didn't understand -- the sort of neurological equivalent of defying gravity. First off, the professor shook a colorful box spilled out the contents onto the floor. That led to a bit of excitement. would we be doing? Out came various blocks -- know, sort of like the ones preschool children play with. (See Figure 1.) Kid's stuff. We nearly died. Then he said to a student named Kyra, Pick up a shape, tell me about it. It's blue, she said. I think it's a cylinder. Good, he said. Now, to another student, you pick one up, tell us how it's similar to or different from hers. Mine's red, Paul said, and it's more square than hers. What would call it? A box? cube? When it was my turn, picked up one that was green. It sort of looked like a tent. (See Figure 2.) had a feeling this wasn't about the colors, so said, Well, mine is 'pointier' than Paul's Kyra's. But don't know what it's hated saying that, but it was true. Then get this: he didn't say what it was called. He didn't say! Instead, he displayed an overhead showing a lesson from a reform- based textbook. about that? It was from a section on shape, here is what it said: Give each student an attribute-block triangle, rectangle, square, hexagon (or geometric shapes from plasterboard). Ask the students to examine each shape, touching the sides corners. Have the students tell how many sides corners each shape has. For the rectangle, square, hexagon, have the students point out pairs of opposite sides. Taking an attribute out of his pocket, he said, This is part of a geometry kit used in elementary school classes, he handed it to Juan. (See Figure 3.) How many corners does it have? he asked. Juan looked it over. Three, he responded rather confidently. I see, our professor said. Tell me, Catherine, how many corners does yours have? Verity, was a bit unsettled by this whole scenario because wasn't at all confident that would get the answer right. took quite a bit of time looking my shape over touching each vertex. Six, answered somewhat hesitantly with a bit of fear trepidation. Isn't that silly? But math gets me like that. It just wasn't clear where all this was going. Anyway, he turns back to Juan says, Look at yours again. Why did say 'three'? Juan examined his block then took mine as well. Then the most amazing thing happened. Juan got this quizzical look on his face. You know, now that I've got both of them in my hands, they feel the same, he said. have the same number of edges, faces, corners. And know Catherine's shape is not a triangle because it's too long. But, in my mind, saw this thinner one as a triangle -- even though it is nearly half an inch thick. These two shapes are the same! They really were the same. believe it hit all of us at once that the attribute could not really be a triangle because the two shapes had identical properties. …