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The road not taken—Investigating affordances of infinitesimal calculus for enriching the repertoire of secondary mathematics teachers

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Abstract

There is a growing body of evidence suggesting that secondary mathematics teachers may not be drawing significant benefit from the tertiary mathematics courses they take in preparation programs or in-service for professional development. While there are some theory-driven attempts to re-design or complement such courses for teachers, empirical evidence on how teaching might actually draw on these courses in practice is scarce. The M-Cubed project utilizes an innovative approach to generate empirical data for exploring this issue – a “lab” where mathematicians and experienced secondary mathematics teachers examine authentic mathematics lessons and discuss teaching alternatives – roads not taken – in specific situations. This article demonstrates this approach in an investigation of the affordances of Infinitesimal Calculus (IC) for secondary mathematics teaching. We draw on the theoretical framework of Commognition in conceptualizing affordances of IC for teaching, and show nuanced ways in which the content and practices of IC can support pedagogical practices and enrich the repertoire of secondary mathematics teachers in contingent situations, even when IC appears unrelated to the taught content. Our findings suggest that interactions of mathematicians and teachers may be a necessary ingredient for learning how IC could be brought to bear on secondary mathematics teaching. We conclude by discussing possible implications for IC courses for teachers.

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Notes

  1. Acknowledging that the term ‘tertiary calculus course’ may mean different things in different contexts, we use ‘infinitesimal calculus’ inclusively to refer to the study of continuous change in the context of real numbers as taught in tertiary courses such as single- and multi-variable calculus and real analysis.

  2. Some turns were omitted to maintain coherency, as evident in the turn numbering.

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Israel Science Foundation (IL), 2391/20, Alon Pinto

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Correspondence to Jason Cooper.

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Pinto, A., Cooper, J. The road not taken—Investigating affordances of infinitesimal calculus for enriching the repertoire of secondary mathematics teachers. Int. J. Res. Undergrad. Math. Ed. 8, 318–338 (2022). https://doi.org/10.1007/s40753-021-00161-w

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