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Monster-Barring as a Catalyst for Bridging Secondary Algebra to Abstract Algebra

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Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers

Part of the book series: Research in Mathematics Education ((RME))

Abstract

This chapter details a teaching experiment in which a pair of prospective secondary mathematics teachers leverages their knowledge of secondary algebra in order to develop effective understandings of the concepts of zero-divisors and the zero-product property (ZPP) in abstract algebra. A critical step in the learning trajectory involved the outright rejection of the legitimacy of zero-divisors as counterexamples to the ZPP, an activity known as monster-barring (Lakatos, Proofs and refutations, 1976; Larsen and Zandieh, Educational Studies in Mathematics 67:205–216, 2008). This monster-barring activity was then productively repurposed as a meaningful way for the students to distinguish between types of abstract algebraic structures (namely, rings that are integral domains vs. rings that are not). This chapter makes two primary contributions. First, it illustrates how students might be able to develop abstract algebraic concepts using their knowledge of secondary algebra as a starting point, thus addressing the issue of perceived irrelevance by forging a direct cognitive connection between ideas that are central amongst the two subjects. Second, and more generally, this chapter emphasizes the importance of identifying, attempting to understand, and leveraging student thinking, even when it initially appears to be counterproductive. The chapter concludes with a discussion of the implications for secondary teacher preparation.

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Notes

  1. 1.

    Informally, an equation has a solution in the ring (R,  + , ⋅) if (1) its coefficients are all elements of the set R, and (2) x is an element of R. For example, 4(x − 5) = 0 has a unique solution (x = 5) in (ℝ,  + , ⋅) but four solutions (x = 2, 5, 8, 11) in (ℤ12, +12,·12).

  2. 2.

    Although a comprehensive account of these analysis techniques is beyond the scope of this chapter, the interested reader may consult Cook (2018) for a detailed account from a very similar teaching experiment.

  3. 3.

    Henceforth, for brevity, I suppress the notation for a ring’s binary operations and simply denote the set.

  4. 4.

    The interested reader can consult Cook (2014) for more information about how ℤ12 can be introduced in an experientially real way.

  5. 5.

    It is not completely inconceivable that Brian viewed ℤ12 as a contrivance that I created purely for the purposes of this teaching experiment. There would be no such concerns with M2(ℝ).

  6. 6.

    Unfortunately, Julie was unable to complete the final session of the teaching experiment.

  7. 7.

    I did not include an example in which the additive identity could be interpreted as a nonzero number (for example, viewing ℤ12 as the set {1, …12}). Exploring how such a representation of the additive identity might shape a student’s conceptions of zero-divisors is an interesting question that I leave for future research.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Oklahoma State University, Stillwater, OK, USA

    John Paul Cook

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  1. John Paul Cook
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Correspondence to John Paul Cook .

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Editors and Affiliations

  1. Department of Mathematics, Science and Technology, Teachers College, Columbia University, New York, NY, USA

    Nicholas H. Wasserman

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Cook, J.P. (2018). Monster-Barring as a Catalyst for Bridging Secondary Algebra to Abstract Algebra. In: Wasserman, N. (eds) Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-319-99214-3_3

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  • DOI: https://doi.org/10.1007/978-3-319-99214-3_3

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