Teaching statements for math jobs? What is the purpose of the "teaching statement" or "statement of teaching philosophy" when applying for jobs, specifically math postdocs?  I am applying for jobs, and I need to write one of these shortly.
Let us assume for the sake of argument that I have a teaching philosophy; I am not asking you to tell me what my teaching philosophy should be.  I would like to know how those responsible for hiring view teaching statements, especially in the case of new PhD's who don't necessarily have extensive teaching experience.  
(I believe this is appropriate for mathoverflow because it is of interest to "a person whose primary occupation is doing mathematics", as I am.)
 A: I think the statement part of the application is supposed to reveal something about your personality that does not appear in other places.
A: 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. 
A: If the position you are applying for has a teaching component, then the hiring committee wants to know whether you can fulfill that part in a competent manner.  It's not really about some abstract philosophy of teaching and learning, but rather the practical side: what do you do in a classroom and why?  How do you design your course and for what reason?  
It also really helps to include examples where you put that philosophy into practice, even if it's on a small scale (say, a seminar or a workshop, or even tutoring).  If not, did you assist a teacher who did?  Such examples help demonstrate that you are prepared to carry out your teaching duties.
The hiring committee is also going to look for indications that you are willing to learn and adapt.  In your statement, it helps to talk about how and from whom you have learned teaching techniques and philosophy, and whether you are interested in learning more.
In most universities and colleges, the undergraduate student tuition makes up the bulk of the institution's revenue, and they are not nearly as interested in your research as in your teaching ability.  The hiring committee and the dean may not have your teaching potential as their primary focus, but they will certainly take it into consideration.
A: A sensible department faculty and dean will want to hire you, only if you will help them carry out the duties and responsibilities of the department. Although your research will be important for the reputation of the department and school, teaching is by far the most important day-to-day responsibility you will have.
A teaching statement should demonstrate that you have given some serious thought on the challenges of teaching and how to address them. It should express your commitment to doing it well, even if you put research first.
A: I found this MO question while writing my own teaching statement. I'd like to point out that the Notices of the AMS published results of a survey which identified quite a few themes in successful teaching statements. I will paraphrase the list, beginning with the most-cited comments:


*

*Specific examples linking philosophy to practice.

*Dedication to teaching.

*Good writing.

*Thoughtful reflection on teaching.

*Student-centered.

*Match between individual and department.


Here's a handy rubric from the University of Michigan. 
A: Just to expand on Michael's last comment, I found that same advice article, and think it's worth posting a link. In particular, it was the only thing I used as a guide towards writing my own statement.
http://www.math.lsu.edu/gradfiles/TeachingStatement_Oxley.pdf
A: FYI, there are lots of examples of teaching statements out there on the web. Do a search for "mathematics teaching statement" and you should get plenty of hits. That said, it's probably better to write your own statement before taking a look at what else is out there... in my mind, it's too easy for teaching statements to sound the same.
I would also suggest adding in specifics wherever possible to illustrate your point. Talk about how your "philosophy" translates in the classroom.
