Mathematics education researchers frequently use task-based interviews to gain insight into mathematicians' practice. However, there are a number of factors that should prevent mathematics educators from extrapolating how individual mathematicians respond to researcher-generated tasks in laboratory conditions, to how mathematicians practice their…

In this article, nine mathematicians were interviewed about their why and how they presented proofs in their advanced mathematics courses. Key findings include that: (1) the participants in this study presented proofs not to convince students that theorems were true but for reasons such as conveying understanding and illustrating methods, (2)…

In this paper, we report a study in which nine research mathematicians were interviewed with regard to the goals guiding their reading of published proofs and the type of reasoning they use to reach these goals. Using the data from this study as well as data from a separate study (Weber, "Journal for Research in Mathematics Education" 39:431-459,…

In this article, we report two studies investigating what mathematicians value in a pedagogical proof. Study 1 is a qualitative study of how eight mathematicians revised two proofs that would be presented in a course for mathematics majors. These mathematicians thought that introductory and concluding sentences should be included in the proofs,…

Many mathematics educators have noted that mathematicians do not only read proofs to gain conviction but also to obtain insight. The goal of this article is to discuss what this insight is from mathematicians' perspective. Based on interviews with nine research-active mathematicians, two sources of insight are discussed. The first is reading a…

In this paper, 28 mathematics majors who completed a transition-to-proof course were given 10 mathematical arguments. For each argument, they were asked to judge how convincing they found the argument and whether they thought the argument constituted a mathematical proof. The key findings from this data were (a) most participants did not find the…

The purpose of this article is to investigate the mathematical practice of proof validation--that is, the act of determining whether an argument constitutes a valid proof. The results of a study with 8 mathematicians are reported. The mathematicians were observed as they read purported mathematical proofs and made judgments about their validity;…