[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.]