In university mathematics courses, the activity of proof construction can be viewed as a problem-solving task in which the prover is asked to form a logical justification demonstrating that a given statement must be true. The purposes of this paper are to describe some of the different types of reasoning and problem-solving processes used by undergraduates to construct proofs in their university mathematics courses, and to consider the relationship between the reasoning that students use when constructing a proof and what they have the opportunity to learn from their proving experience. (Contains 1 figure.)