This article documents differences between novice and experienced undergraduate students' processes of reading mathematical proofs as revealed by moment-by-moment, think-aloud protocols. We found three key reading behaviors that describe how novices' reading differed from that of their experienced peers: alternative task models, accrual of premises, and warranting. "Alternative task models" refer to the types of goals that students set up for their reading of the text, which may differ from identifying and justifying inferences. "Accrual of premises" refers to the way novice readers did not distinguish propositions in the theorem statement as assumptions or conclusions and thus did not use them differently for interpreting the proof. Finally, we observed variation in the type and quality of "warrants," which we categorized as illustrate with examples, construct a miniproof, or state the warrant in general form.