Nicole Engelke Infante's research while affiliated with University of Nebraska at Omaha and other places

We report on a series of task-based interviews in which nine mathematicians were asked to evaluate a series of six mathematical arguments, purportedly produced either by fellow mathematicians or undergraduate students. In this paper, we attend to the role of context in mathematicians’ responses, leading to four themes in expectations when evaluating the proofs that they read. First, mathematicians’ evaluations of identical arguments were sensitive to researchers’ manipulation of authorship, with most accepting arguments purportedly produced by a colleague while taking a more critical view of that same argument if produced by an undergraduate student. Our thematic analysis of interview responses led to three context-based factors influencing mathematicians’ responses when evaluating student-produced texts: course goals, instructors’ expectations, and assessment type. In the final section, we consider implications for researchers focused on understanding common practice amongst mathematicians as well as the pedagogic consequences of our findings for practice in the classroom.

Visual representations, such as diagrams, are known to be valuable tools in problem solving and proof construction. However, previous studies have shown that simply having access to a diagram is not sufficient to improve students’ performance on mathematical tasks. Rather, students must actively extract information about the problem scenario from their diagrams for them to be useful. Furthermore, several studies have described the behaviors of mathematicians and students when solving problems and writing proofs, but few have discussed students’ behaviors in the context of proof writing in introductory point-set topology. We present a case study of an undergraduate, Stacey, enrolled in a general topology course. Throughout a semester, we presented Stacey with several proof-related tasks and examined how and why Stacey used diagrams when working on these tasks. Based on our analysis, we concluded that Stacey’s diagram creation and subsequent use during the construction of a given proof was an effort to identify the key idea of the proof. We describe Stacey’s overall proving behaviors through the lens of a problem-solving framework and present Stacey’s use of diagrams as an aid to discovering the key ideas of proofs in topology.