"Best practice" innovative teaching in mathematics Our department is currently revamping our first-year courses in mathematics, which are huge classes (about 500+ students) that are mostly students who will continue on to Engineering. 
The existing teaching methods (largely, "lemma, proof, corollary, application, lemma, proof, corollary, application, rinse and repeat") do not properly accommodate the widely varying students.
I am interested in finding out about alternative, innovative or just interesting ideas for teaching mathematics - no matter how way out they may seem. Preferably something backed up by educational research, but any ideas are welcome.
Edit: a colleague pointed out that I should mention that currently we lecture to the 500 students in groups of around 200-240, not all 500 at once.

“...no matter how way out they may seem.”
In that case you might want to consider the public-domain student exercises for mathematics that I have created. The address is: http://www.public-domain-materials.com/folder-student-exercise-tasks-for-mathematics-language-arts-etc---autocorrected.html
