## 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.

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## Notes

- 1.
An abundant number is a natural number

*n*whose positive divisors add up to more than 2*n*. A perfect number is a natural number*n*whose positive divisors add up to exactly 2*n*. A deficient number is a natural number*n*whose positive divisors add up to less than 2*n*. - 2.
Of course, this methodology has its own weaknesses. For instance, participants may be unable to provide an accurate account of their actual mathematical behavior (Inglis & Alcock, 2012). Our position is that all methodologies have weaknesses, but that through their complementary use we can obtain a picture of mathematical practice that is more accurate and robust than the one we obtain when we restrict its study to any one of those methodologies.

- 3.
Having surveyed 291 mathematicians about their use of examples when exploring a new mathematical conjecture, Lockwood et al. explained that they had decided to conduct task-based interviews in order to get access to “what mathematicians do in practice”. Lockwood et al. claimed that data from the survey and the interview study were “meant to be complementary, and the interviews served to elaborate and refine the initial results from the survey” (p. 169).

- 4.
All survey data reported by Lockwood et al. (2016) comes from mathematicians’ response to a single prompt: “If you sometimes use examples when exploring a new mathematical conjecture, how do you choose the specific examples you select in order to test or explore the conjecture? What explicit strategies or example characteristics, if any, do you use or consider?” (p. 169). As the aim of Lockwood et al. (2016) was to make a qualitative contribution, Lockwood et al. did not report any quantitative analyses of these survey data. We believe such quantitative analyses would need to be scrutinized to determine how well these generalizations are supported.

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Mejía-Ramos, J.P., Weber, K. Using task-based interviews to generate hypotheses about mathematical practice: mathematics education research on mathematicians’ use of examples in proof-related activities.
*ZDM Mathematics Education* **52, **1099–1112 (2020). https://doi.org/10.1007/s11858-020-01170-w

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### Keywords

- Task-based interviews
- Mathematical practice
- Examples
- Conjecturing
- Proof