Abstract
Undergraduate concepts are often first introduced in a single-dimensional setting and then extended to multiple dimensions. For instance, many undergraduate real analysis students will first learn of the metric topology on \({\mathbb{R}}\) before being exposed to more general metric spaces. I conducted a paired teaching experiment (Steffe & Thompson, 2000) with introductory real analysis students that explored their learning of the general metric function. In this experiment, I was able to observe the students’ mathematical activity as they made sense of analytic ideas in increasingly general settings. I present a construct, called component-wise reasoning, that offers an explanatory mechanism for the ways that the students leveraged their understandings of phenomena on \({\mathbb{R}}\) to construct new schemes for similar phenomena in other metric spaces. I discuss how component-wise reasoning can offer explanatory power for students’ sense-making in abstract spaces across the undergraduate curricula.
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Notes
Univalence can refer to one-to-one functions and can also refer to the property of functions that domain values do not map to multiple range values. Dorko (2017) used the latter interpretation of univalence.
Such development is beyond the scope of this paper.
Note that this account of learning is a direct consequence of a radical constructivist theoretical perspective.
While prompts themselves are not causal in the constructivist tradition, researchers must consider themselves as agents of interaction in teaching episodes and must consider the students to be independent and self-regulating (Tallman & Weber, 2015). An important theoretical consideration that researchers must make in analysis includes accounting for the students’ possible interpretations of given tasks as part of their independent contributions (Steffe & Thompson, 2000, p. 288).
That sequential convergence can motivate distance measurement in various spaces.
We did not use the “\({\mathcal{l}}_{p}\)” notation or language in the teaching experiment.
They eventually refined this characterization to be \(L_n=|v_{x,n}-v_x|+|v_{y,n}-v_y|\).
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Acknowledgements
Much thanks to Elise Lockwood, Michael Tallman, and Michael Oehrtman for their counsel during the writing of this paper.
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This project was funded by NSF Grant #1419973.
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Reed, Z. Component-Wise Reasoning as a Mechanism of Sense-Making in Real Analysis. Int. J. Res. Undergrad. Math. Ed. 9, 217–242 (2023). https://doi.org/10.1007/s40753-022-00198-5
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DOI: https://doi.org/10.1007/s40753-022-00198-5