In this paper, we distinguish between two ways that an individual can construct a formal proof. We define a syntactic proof production to occur when the prover draws inferences by manipulating symbolic formulae in a logically permissible way. We define a semantic proof production to occur when the prover uses instantiations of mathematical concepts to guide the formal inferences that he or she draws. We present two independent exploratory case studies from group theory and real analysis that illustrate both types of proofs. We conclude by discussing what types of concept understanding are required for each type of proof production and by illustrating the weaknesses of syntactic proof productions.