On Wed, Jul 17, 2013 at 9:22 AM, Robert Hansen <email@example.com> wrote:
> > On Jul 17, 2013, at 12:21 PM, kirby urner <firstname.lastname@example.org> wrote: > > Does Euclidean geometry make more sense than non-Euclidean? > > > To fourth graders? > > Bob Hansen >
They've never seen any such thing as an infinitely flat infinitely thin plane. Might we have geometry that goes away from those?
Karl Menger suggested a "geometry of lumps" where we didn't go with the standard ideas of 0, 1, 2, 3 dimensions -- he was a dimension theorist, originally Vienna Circle, later worked at Illinois Institute of Technology.
Everything is a lump. Everything has an interior.
Makes about as much sense as saying a cube is made of 0D points stacked together -- to fourth graders.
Anyway, it's a spiral. The field is wide open to lesson planning so I'm not going to nail down exactly when to say what to whom at what age level. There's a natural shake-out that occurs when many people try many things.
Pouring water or dry grain from a unit tetrahedron to other shapes, appropriately sized, is something I've done successfully with Montessori kids (pre-K).
I've also presented this Lesson in different countries, e.g. Lesotho and Bhutan.
Computing volume relative to a unit cube is scarier, algebraically, which is probably why, in the older curricula, polyhedrons are at the back of the book and rarely discussed in much detail, beyond the obligatory topics of sphere, cone and cylinder.
Message was edited by: kirby urner (fixed Menger affiliation)