Many researchers and curriculum frameworks recommend that the concept of "proof" and the corresponding activity of "proving" become part of students' mathematical experiences throughout the grades. Yet it is still unclear what "proof" means in school mathematics, especially in the elementary grades, and what role the teacher has in cultivating proof and proving among their students. In this article, I propose a conceptualization of the meaning of proof in school mathematics and use classroom episodes from third grade to elaborate elements of this conceptualization and to illustrate its applicability even in the early elementary grades. Furthermore, I use the conceptualization to develop a tool to analyze the classroom episodes and to examine aspects of the teachers' role in managing their students' proving activity. This analysis supports the development of a framework about instructional practices for cultivating proof and proving in school mathematics. (Contains 1 table, 6 figures and 7 footnotes.)