As reformers urge elementary and secondary school teachers to teach mathematics in new ways that highlight problem solving and engage students in important mathematical ideas, researchers have been pointing out that few public school teachers know mathematics in the ways that they would need to know it in order to teach in these new ways. These researchers point to deficiencies in teachers substantive knowledge (their understanding of the "stuff" of mathematics), in their syntactic knowledge (their understanding of what mathematicians do and of the nature of mathematical evidence), and in their attitudes towards the subject matter; they raise questions about the possibilities for addressing these difficulties through school-based staff development or university-based mathematics courses. The present study explores the possibility that changes in teachers' own teaching practices may provide opportunities for learning of and about mathematics. The study examines the cases of three primary teachers who, influenced by the National Council of Teachers of Mathematics (NCTM) "Standards," made significant changes in the way that they taught second and third grade mathematics and who also reported significant changes in their understandings of topics in elementary math, their attitudes toward the subject matter, and beliefs about what it means to do math. The conclusion looks at some of the reasons that teaching math in new ways may help elementary teachers to learn some of what reformers say they need to know of and about mathematics. (Contains 31 references.) (Author/ND)