It is widely recognized that developing a successful teaching practice, one that is grounded in the principles that guide the current effort to reform mathematics education, requires a qualitatively different and significantly richer understanding of mathematics that most teachers in the 1990s possess. However, it is not as clear how teachers' mathematical understandings develop and how those understandings affect instruction. This paper explores two avenues for K-6 teachers' mathematical investigations--inquiry into mathematics itself and inquiry into children's mathematical thinking--and illustrates how they arise in elementary teaching situations and how they can be explored in a professional development setting. Contains 43 references. (Author)