Conjecturing, generalizing and justifying: Building theory around teacher knowledge of proving
The purpose of this study was to detail teachers' proving activity and contribute to a framework of Mathematical Knowledge for Teaching Proof (MKT for Proof). While working to justify claims about sums of consecutive numbers, teachers searched for key ideas and productively used examples to make, test and refine conjectures. Analysis of teachers' mathematical activity revealed knowledge of the proving process that would be useful for and useable in the teaching of proof. This includes knowledge of the interconnections among empirical exploration, conjecturing, generalizing, and justifying as well as an understanding of the characteristics of examples and conjectures that could support the proving process. The central premise of this paper is that delineating aspects of teacher knowledge is a first step to supporting teachers' efforts to engage all students in fundamental mathematical practices of conjecturing, generalizing and justifying.