Using task-based interviews to generate hypotheses about mathematical practice: mathematics education research on mathem
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ORIGINAL ARTICLE
Using task‑based interviews to generate hypotheses about mathematical practice: mathematics education research on mathematicians’ use of examples in proof‑related activities Juan Pablo Mejía‑Ramos1 · Keith Weber1 Accepted: 24 May 2020 © FIZ Karlsruhe 2020
Abstract 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 craft in authentic settings. In this paper we critically analyze the rationality of using task-based interviews to investigate mathematical practice, focusing on how task-based interview studies have been used to inform our understanding of mathematicians’ use of examples in mathematical practice. We discuss four specific generalizations about mathematical practice drawn from these studies, and suggest other types of studies that can be used to corroborate or challenge those generalizations. Keywords Task-based interviews · Mathematical practice · Examples · Conjecturing · Proof
1 Introduction 1.1 The use of task‑based interviews to investigate mathematical practice Mathematics educators sometimes conduct research to understand how mathematicians practice their craft (Weber, Dawkins, & Mejía-Ramos, 2020). In some cases, mathematics educators would like to know how mathematicians successfully complete challenging tasks to gain insight into how students can be taught to complete these tasks. In other cases, mathematics educators would like to understand how mathematicians engage in activities of their professional practice to inform the design of classrooms in which students engage in similar activities. A common methodology that mathematics educators use to understand how mathematicians practice their craft is the semi-structured, task-based interview (Maher & Sigley, 2014), in which mathematicians
* Juan Pablo Mejía‑Ramos [email protected] 1
Graduate School of Education, Rutgers-The State University of New Jersey, 10 Seminary Place, New Brunswick, NJ, USA
are asked to complete mathematical tasks in a laboratory setting. Using this methodology, mathematics educators have investigated how mathematicians engaged in a wide variety of mathematical activities, including solving challenging problems (Carlson & Bloom, 2005; deFranco, 1996; Schoenfeld, 1985), using diagrams (Samkoff et al., 2012; Stylianou, 2002; Stylianou & Silver, 2004), learning new mathematics (Wilkerson-Jerde & Wilensky, 2011), creating and evaluating conjectures (Alcock & Inglis, 2008; Inglis et al., 2007; Lockwood, Ellis, & Lynch, 2016; Lynch & Lockwood, 2019), writing proofs (e.g., Kidron & Dreyfus, 2014; Weber, 2001; Weber & Alcock, 2004), and evaluating proofs for correctness (e.g., Weber, 2008), among other topics.
1.2 The problem from generalizing from task‑based interviews Mathematics educators conduct t
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