The mathematics that teachers at any level need to teach well is not identical to the mathematics that students are expected to learn. This mathematics, which we call "teachers' mathematics", includes at least the following: alternate definitions of ideas and their consequences, multiple explanations and representations of the ideas, a wide range of applications of the ideas, connections with ideas studied in other courses, why concepts arose and how they have changed over time, responses to questions students might have about the mathematics, alternate ways of approaching problems - with and without technology, and how problems and theorems can be extended and generalized. Because the mathematics emanates from the classroom, we view this as a subfield of applied mathematics. The mathematics is quite rich, touching on many parts of pure and applied mathematics. A typical high school teacher rarely takes a course in this field but learns the content haphazardly through reading and attending conferences. A recent report of the Conference Board of the Mathematical Sciences recommended a capstone (i.e., senior level) mathematics course for prospective high school mathematics teachers, and we believe teachers' mathematics is appropriate content not only for one course, but for a sequence of upper division and graduate courses. With the aid of a grant from the Stuart Foundation to the mathematics department of the University of California at Berkeley, four of us have been working for the last few years on materials that could be used in such a sequence. In this work we are working to conceptualize the field. Examples of teachers' mathematics from these materials will be described