This article reviews some key ideas about the nature of proof, kinds of proofs, and the way of reasoning called proving. This provides background for a discussion of the teaching of proof, in which some important aspects of teaching that support students' proving are described, and the ideas of a tool-box of accepted premises and local organization of knowledge are explored as ways to structure teaching. Finally, lessons learned from approaches to teaching proof and problem solving in the past are connected with the earlier discussions of understanding and teaching proof to suggest a transformation of proof teaching into proof based teaching, in which proof is a process of coming to understanding rather than a topic to be taught and learned. [For the complete proceedings, see ED585874.]