# [plt-scheme] The Philosophy of DrScheme

On 02/12/2008, at 5:26 AM, Greg Woodhouse wrote:
>* A minor nit: There is no reason why mathematics cannot be taught as
*>* an active process of discovery. The problem (well, one problem) is
*>* that the only way to really learn mathematics is by doing, and that
*>* means calculating. Still, there is no reason it can't be
*>* interesting. I'll give you an example: one thing that always
*>* intrigued me, even as a child, is that there are only 5 regular
*>* polyhedra (the tetrahedron, octahedron, cube, dodecahedron and
*>* icosohedron), but I didn't realize until much later how accessible a
*>* result it really is. You could almost make it a homework exercise!
*>* Start with Euler's famous formula V - E + F = 2 (for a topological
*>* sphere) and then suppose you have refgular polyhedron the faces of
*>* which are n-gons. It all comes down to counting: If there are m of
*>* them, how many times will you count each vertex in m times n
*>* vertices per face? How many times will you count each edge? What
*>* happens if you plug these numbers in Euler's formula? Even if youer
*>* students take euler's formula on faith, the result is still
*>* impressive.
*>*
*
An aside:
Greg's example of Euler's formula is used to good effect in a
wonderful book by Lakatos, "Proofs and Refutations", that reads almost
like a play about what an idealised mathematical classroom might look
like. [If you "look inside" on Amazon, you can read the first few
pages, which gives the flavor of the book.]