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Prospective secondary mathematics teachers are usually required to complete several university advanced mathematics courses before being certified to teach secondary mathematics. However, teachers usually do not find these courses to be valuable for their teaching. We designed an experimental real analysis course with the goal of making real analysis content useful and relevant to teaching. Our approach was to ground the real analysis content in pedagogical situations that problematized a secondary mathematics topic, where the nuances of teaching secondary mathematics could be informed by the real analysis that was covered. The experimental course was implemented in a graduate teacher education programme with 32 pre- and in-service teachers (PISTs). After the course, we conducted focus group interviews with 20 of these PISTs to get feedback on how the course was valuable to their teaching practice. Many PISTs found the course to be valuable for teaching secondary mathematics, as well as for their understanding of secondary mathematics and real analysis.

We argue that mathematics majors learn little from the proofs they read in their advanced mathematics courses because these students and their teachers have different perceptions about students' responsibilities when reading a mathematical proof. We used observations from a qualitative study where 28 undergraduates were observed evaluating mathematical arguments to hypothesize that mathematics majors hold four specific unproductive beliefs about proof reading. We then conducted a survey about these beliefs with 175 mathematics majors and 83 mathematicians. We found that mathematics majors were more likely to believe that when reading a good proof, they are not expected to construct justifications and diagrams, they can understand most proofs they read within 15 minutes, and understanding a proof is tantamount to being able to justify each step in the proof.