career - Teaching statements for math jobs?

Having been on both sides of the issue, I might say that having considered it for some time, I really don't know! But in reality if you are looking for a position at a research university, the Dean will want to have evidence (or the non-research faculty will want to have evidence) that you care about teaching. More precisely, some subset of your peers might have a very specific teaching philosophy although they may not be able to articulate it. Those peers want to know if your teaching philosophy coincides with theirs.

A few years back everyone was "hot" on the use of technology in the classroom. I don't know what that means, but suppose that it means using TI calculators, power point (the horror, the horror) or a course blog. If you have a point of view on the positive value of these things then you should say so.

The problem is that each department has its own mix of bozos. I am pretty much a chalk on slate kind of guy, and when someone tells me they like clickers in large classes, I wonder do they turn around to look at their students faces. So in an ideal world you would tailor your teaching statement to the place you want to go, or to the place that you are applying. Of course, you don't want to write 200 teaching statements, so that won't work.

So I am back to the original premise. They want to know that you have thought about teaching.