We have been teaching an annual 8 lecture series mathematics course on topology of surfaces, Euler characteristic and loops on surfaces. It is more of discussion session than a lecture. Although it is primarily intended for a certain group of female undergraduates who are part of some program, I believe the topics could be introduced to any group of undergraduates who are interested enough! Here's the link for those interested :
http://www.math.sunysb.edu/~basu/courses/WSE187-spring11/WSE187.html
We assume no basic background (of calculus or linear algebra) on their part. It is very hands on and they try to develop a concept of what a surface is and what numerical invariants help distinguish between them. Among other things, we encourage them to use play-doh to model filled-in surfaces and help them visualize deformations. We also have fun cutting Mobius strips along various curves and single slice bagels into links! From my experience (and I have only taught it twice) some of these hands-on events always excite even the ones who aren't showing much interest. After all, it's fun for them to learn that the soccer ball has only so many pentagons and hexagons essentially due to Euler! And They have to give a presentation at the end figuring out and explaining (from basic principles) how some problem (assigned to them) can be solved via topology - the solution could be in terms of pictures, play-doh or any other material if need be. For the interested, have a look at the presentations from last year :
http://www.math.sunysb.edu/~basu/courses/WSE187-spring10/Lectures_WSE187.html
Lastly, although we haven't done it the course referred to above, including something leading up to the four colour theorem should be plenty of fun too, both for the teachers and the students!
