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The Group Theory Concept Assessment: a Tool for Measuring Conceptual Understanding in Introductory Group Theory

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Abstract

Introductory group theory provides the foundational course on algebraic structures. Yet, we know little about students’ underlying conceptual understandings. In this paper, I introduce the Group Theory Concept Assessment (GTCA), a measure created for the purpose of conducting large-scale studies of student conceptual understanding in group theory. The GTCA measures this understanding divorced from proof construction ability. I outline the design process and share results from various stages of analysis and refinement. I then present an item-level analysis of the GTCA including a detailed look at four items. These items focus on subgroups, inverses, quotient groups, and kernels, respectively. I conclude by reflecting on the GTCA’s potential usage as a research tool and practitioner tool.

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Notes

  1. A more thorough treatment of this framework can be found in Dubinsky et al. (1994) and Arnon et al. (2014).

  2. The 18 topics ranked were expanded to 19 as cosets and quotient groups were separated to be consistent with topic treatment in curriculum. However, modular groups were classified as an example group and therefore not targeted as an independent concept.

  3. In this context, college refers to the common United States usage: undergraduate universities including colleges and universities.

  4. As in any random sample, there are limitations due to response bias. Particularly, institutions that did not have their textbooks available via their website were likely underrepresented as not all instructors responded to email requests for current textbook.

  5. Note: Hungerford (2012) begins with rings, but provides an alternate route starting with the group chapters.

  6. This analysis occurred in 2014, and thus does not include more recent articles.

  7. This student was using multiply as a generalized operator for addition.

  8. In earlier rounds, the focal element was a.

  9. Access to the website can be requested via http://groupconcepts.wp.txstate.edu/

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  1. Department of Mathematics, Texas State University, 601 University Drive, San Marcos, TX, 78666, USA

    Kathleen Melhuish

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Melhuish, K. The Group Theory Concept Assessment: a Tool for Measuring Conceptual Understanding in Introductory Group Theory. Int. J. Res. Undergrad. Math. Ed. 5, 359–393 (2019). https://doi.org/10.1007/s40753-019-00093-6

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