In this chapter we introduce concepts for analyzing proofs, and for analyzing undergraduate and beginning graduate mathematics students’ proving abilities. We discuss how coordination of these two analyses can be used to improve students’ ability to construct proofs. -/- For this purpose, we need a richer framework for keeping track of students’ progress than the everyday one used by mathematicians. We need to know more than that a particular student can, or cannot, prove theorems by induction or contradiction or can, (...) or cannot, prove certain theorems in beginning set theory or analysis. It is more useful to describe a student’s work in terms of a finer-grained framework that includes various smaller abilities that contribute to proving and that can be learned in differing ways and at differing periods of a student’s development. (shrink)