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Interactions Between Teaching and Research: Developing Pedagogical Content Knowledge for Real Analysis

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Learning Through Teaching Mathematics

Part of the book series: Mathematics Teacher Education ((MTEN,volume 5))

Abstract

In this chapter, I describe five practices that I use in teaching undergraduate mathematics: regular testing on definitions, tasks that involve extending example spaces, tasks that involve constructing and understanding diagrams, use of resources for improving proof comprehension and tasks that involve mapping the structure of a whole course. I describe the rationale for each of these practices from my point of view as a teacher and relate each to results from mathematics education research. The discussion regularly returns to two overarching themes: the need for students to develop skills on multiple levels and the question of how best to use available lecture time. In a final section, I discuss these themes explicitly, focusing on their relevance to the work of teaching at all levels.

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Notes

  1. 1.

    Obviously this function cannot be sketched accurately, but the intention was for the students to notice this and to think about how to give some reasonable representation.

  2. 2.

    If a function f:[a, b] → R is continuous on [ a, b ] and differentiable on ( a, b) , then there exists c ∈ ( a, b) such that \(f'( c) = \frac{{f(b) - f(a)}}{{b - a}}\).

  3. 3.

    Under “Examples” at http://www.geogebra.org/cms/

  4. 4.

    For instance, in considering the Interior Extremum Theorem (if f is differentiable on (a, b) and attains a maximum or minimum at c ∈ ( a, b) , then f ' (c) = 0), I prove the maximum case and students are advised to write out a proof for the minimum case.

  5. 5.

    With the support of a Loughborough University Academic Practice Award.

  6. 6.

    Beyond collecting fairly basic, anonymous, opinion-based feedback.

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Authors and Affiliations

  1. Loughborough University, Loughborough, Leicestershire, UK

    Lara Alcock

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  1. Lara Alcock
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Correspondence to Lara Alcock .

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Editors and Affiliations

  1. Faculty of Education, University of Haifa, Haifa, 31905, Israel

    Roza Leikin

  2. Fac. Education, Simon Fraser University, University Drive 8888, Burnaby, V5A 1S6, British Columbia, Canada

    Rina Zazkis

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Alcock, L. (2010). Interactions Between Teaching and Research: Developing Pedagogical Content Knowledge for Real Analysis. In: Leikin, R., Zazkis, R. (eds) Learning Through Teaching Mathematics. Mathematics Teacher Education, vol 5. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3990-3_12

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