Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving

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The Mathematical Nature of Reasoning-and-Proving Opportunities in Geometry Textbooks (2014, background): “…deductive reasoning does not imply that inductive reasoning has no place in mathematical reasoning, but rather that deductive reasoning also has a place, especially with regard to the justification of claims about infinite classes as opposed to single or finite classes (Stylianides & Ball, 2008).”
Proof constructions and evaluations (2009, background): “Given that teachers’ ability to teach mathematics depends on the quality of their subjectmatter knowledge (e.g., Hill, Rowan, & Ball, 2005), a necessary condition for the realization of the aforementioned goal is that teachers of all levels have good understanding of proof (see Stylianides & Ball, ”
Effects of Video Club Participation on Teachers' Professional Vision (2009, background): “…programs have been designed with the goal of developing teachers’ mathematical knowledge for teaching (Ball & Bass, 2003), an understanding of mathematics that is “useful for, and usable in, the work that teachers do as they teach mathematics to their students” (Stylianides & Ball, 2008, p. 308).”