Student sensemaking of proofs at various distances: the role of epistemic, rhetorical, and ontological distance in the p
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Student sensemaking of proofs at various distances: the role of epistemic, rhetorical, and ontological distance in the peer review process Daniel L. Reinholz 1
& Mary E. Pilgrim
1
Accepted: 29 October 2020/ # Springer Nature B.V. 2020
Abstract
This manuscript focuses on how students make sense of proofs. Participants were students who engaged in peer-review conferences of each other’s attempted proofs in a graduate-level real analysis course for mathematics teachers. Building on the concept of distance from conversational analysis, we distinguish how three types of distance (epistemic, rhetorical, and ontological) between a student and a particular claim influence sensemaking. This article also explores the impact of students’ sensemaking on their perceptions of proof. Keywords Distance . Peer assessment . Peer review . Proof . Proof comprehension . Student perceptions
1 Introduction A wealth of research describes the teaching and learning of proof (e.g., Byrne, 2014; Moore, 1994). This article extends existing work to highlight social factors related to learning proof. Beyond proof construction, students can engage in a variety of proof sensemaking practices (Selden and Selden, 2017). By engaging with these practices, students experience the tentative nature of mathematics in the making, challenging the notion that mathematics comes prepackaged in a neatly finished form (Hersh, 1997). Our study focused on 13 students who engaged in peer-review conferences of each other’s proofs in a graduate-level real analysis course for mathematics teachers. To understand student
* Daniel L. Reinholz [email protected] Mary E. Pilgrim [email protected]
1
Department of Mathematics & Statistics, San Diego State University, San Diego, CA 92182-7720, USA
Reinholz D.L., Pilgrim M.E.
sensemaking, we explored how the relationship between a student and a particular claim (formally defined below as distance) impacted sensemaking. Our study also explored the resulting impact on student perceptions of proof. We made important theoretical contributions to the study of proof, by proposing a new construct—ontological distance—and by empirically documenting the role of various forms of distance in proof sensemaking.
2 Theoretical framing 2.1 Proof as a social practice From a situated perspective, a proof is a logical argument that meets negotiated social standards (Weber, 2008). We use proof to denote an argument that is deemed complete and correct, contrasting an attempted proof, which may not meet the negotiated standards (Byrne, 2014). Beyond constructing proofs, mathematical proficiency requires students to make sense of proofs. In this article, we define sensemaking as the set of practices typically associated with proof comprehension and validation (Selden and Selden, 2017). Validation focuses on determining the correctness of an attempted proof (Weber, 2008), whereas comprehension describes how one understands a proof (Mejia-Ramos, Fuller, Weber, Rhoads, and Samkoff, 2012). Below in our methods, we operationalize sens
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