I would like to thank John Conway for providing an example of meaningful classification, where there is real mathematical content. The rough idea is that the more "regularity" there is in a shape, the more symmetries it has, e.g. the square has more symmetries than the rectangle. "What has more symmetries?" then gives rise to a meaningful hierarchy which can extend in many directions (e.g. where's the circle in such a hierarchy?). This is exactly the sort of thing that is missing from much of our teacher education materials. We sell our future teachers short by asking too little real cognition of them.
==================================== Judy Roitman, Mathematics Department Univ. of Kansas, Lawrence, KS 66049 firstname.lastname@example.org =====================================