I actually didn't know that. I remember "Proffs and Refutations" being optional reading for a course I took in mathematical logic, but (contrary to my usual practice) I never got around to reading it. Sounds interesting.<br> <br> <div class="gmail_quote">On Tue, Dec 2, 2008 at 1:27 AM, Daniel Prager <span dir="ltr"><<a href="mailto:danprager@optusnet.com.au">danprager@optusnet.com.au</a>></span> wrote:<br> <blockquote class="gmail_quote" style="PADDING-LEFT: 1ex; MARGIN: 0px 0px 0px 0.8ex; BORDER-LEFT: #ccc 1px solid"> <div class="Ih2E3d"><br>On 02/12/2008, at 5:26 AM, Greg Woodhouse wrote:<br><br> <blockquote class="gmail_quote" style="PADDING-LEFT: 1ex; MARGIN: 0px 0px 0px 0.8ex; BORDER-LEFT: #ccc 1px solid">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.<br> <br></blockquote><br></div>An aside:<br><br>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.] <div> <div></div> <div class="Wj3C7c"><br>_________________________________________________<br> For list-related administrative tasks:<br> <a href="http://list.cs.brown.edu/mailman/listinfo/plt-scheme" target="_blank">http://list.cs.brown.edu/mailman/listinfo/plt-scheme</a><br> </div></div></blockquote></div><br>