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Group theory students’ perceptions of binary operation

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Abstract

Binary operations are one of the fundamental structures underlying our number and algebraic systems. Yet, researchers have often left their role implicit as they model student understanding of abstract structures. In this paper, we directly analyze students’ perceptions of the general binary operation via a two-phase study consisting of task-based surveys and interviews. We document what attributes of binary operation group theory students perceive as critical and what types of metaphors students use to convey these attributes. We found that many students treat superficial features as critical (such as element-operator-element formatting) and do not always perceive critical features as essential (such as the binary attribute). Further, these attributes were communicated across three metaphor categories: arithmetic-related, function-related, and organization-related.

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Notes

  1. We include closure in our binary operation definition, but acknowledge the inclusion of closure is not universal.

  2. A small proportion of students addressed the isomorphic copy of Z3 in Z6. This group is not included in the “incorrect” category. See Melhuish (2018) for a more nuanced treatment of this idea.

  3. A binary operation on set G is a function that assigns each ordered pair of elements of G an element of G. In both classes, notation typically used a generalized arithmetic notation (multiplication or addition).

  4. This is particularly the case for the sameness prompts, as the construct of sameness is not well-defined. Mathematically normative approaches to sameness include identical inputs producing identical outputs (potentially with restricted domains such as with subgroups having the same operation as the parent group) or as inducing isomorphic structures (magmas). See Melhuish and Czocher (2019) for a detailed discussion of the multitude of reasonable approaches to sameness of operations.

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Authors and Affiliations

  1. Texas State University, 601 University Dr., San Marcos, TX, 78666, USA

    Kathleen Melhuish & Michael D. Hicks

  2. Portland State University, 1825 SW Boadway, Portland, OR, 97201, USA

    Brittney Ellis

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  1. Kathleen Melhuish
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  2. Brittney Ellis
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  3. Michael D. Hicks
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Correspondence to Kathleen Melhuish.

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Melhuish, K., Ellis, B. & Hicks, M.D. Group theory students’ perceptions of binary operation. Educ Stud Math 103, 63–81 (2020). https://doi.org/10.1007/s10649-019-09925-3

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