In traditional advanced mathematics lectures the instructor usually provides definitions, theorems, and proofs on the board rather quickly. The students often cannot make sense of these during the lecture as they are busy writing. In order to gain an understanding of the content, an intensive post-class processing on the basis of their notes would therefore be necessary for many students. However, students often do not post-class process lectures carefully. Furthermore, they often do not record the lecturer's oral explanations that are essential to make sense of the formal lecture content. One approach to address these problems is a flipped classroom, in which students come into contact with new content before class, while the in-class time is used for activities that help students make sense of it. We implemented an adaptation of this approach in a proof-oriented analysis course, in which the instructor used the in-class time to illustrate how experts in mathematics make sense of new definitions, theorems, and proofs, while keeping the lecture as the mode of teaching. In this article we describe this adaptation, its possible benefits for addressing students' problems in learning from traditional mathematics lectures, and its actual effects, which we investigated in an empirical study.