The Design and Validation of a Group Theory Concept Inventory

Abstract

Within undergraduate mathematics education, there are few validated instruments designed for large-scale usage. The Group Concept Inventory (GCI) was created as an instrument to evaluate student conceptions related to introductory group theory topics. The inventory was created in three phases: domain analysis, question creation, and field-testing. The domain analysis phase included using an expert protocol to arrive at the topics to be assessed, analyzing curriculum, and reviewing literature. From this analysis, items were created, evaluated, and field-tested. First, 383 students answered open-ended versions of the question set. The questions were converted to multiple-choice format from these responses and disseminated to an additional 476 students over two rounds.Through follow-up interviews intended for validation, and test analysis processes, the questions were refined to best target conceptions and strengthen validity measures. The GCI consists of seventeen questions, each targeting a different concept in introductory group theory. The results from this study are broken into three papers. The first paper reports on the methodology for creating the GCI with the goal of providing a model for building valid concept inventories. The second paper provides replication results and critiques of previous studies by leveraging three GCI questions (on cyclic groups, subgroups, and isomorphism) that have been adapted from prior studies. The final paper introduces the GCI for use by instructors and mathematics departments with emphasis on how it can be leveraged to investigate their students' understanding of group theory concepts.Through careful creation and extensive field-testing, the GCI has been shown to ii be a meaningful instrument with powerful ability to explore student understanding around group theory concepts at the large-scale.iii Dedication To my parents for teaching me to have a strong work ethic, to live life with kindness, and to pursue a career that truly fulfills you.iv Acknowledgements

Exploring Learning Difficulties in Abstract Algebra: The Case of Group Theory

“…Over the past 10 years, research into educational aspects of abstract algebra has gained increasingly more traction. With the works of Wasserman et al [1][2][3][4] and Melhuish [5][6][7], various studies have shown the benefits of learning abstract algebra and group theory in particular, as well as different pitfalls for students-an overview of the research results in this regard is presented in [8].…”

The role of affective learner characteristics for learning about abstract algebra: A multiple linear regression analysis

“…In order to facilitate quantitative research into educational aspects of group theory and enrich findings from qualitative research (cf. Baldinger, 2018;Cook, 2018;Even, 2011;Suominen, 2018), two concept inventories have been developed in recent time: the GTCA (Group theory concept assessment) by Melhuish (2015) and the CI²GT (Concept inventory for introductory group theory) by Veith et al (2022b), respectively. Here, we conceive that conceptual understanding "reflects knowledge of concepts and linking relationships that are directly connected to (or logically necessitated by) the definition of a concept or meaning of a statement" (Melhuish, 2019, p. 2).…”

“…In recent years a body of research has emerged, entirely dedicated towards exploring educational aspects of abstract algebra (Wasserman, 2014(Wasserman, , 2016(Wasserman, , 2017(Wasserman, , 2018 and group theory in particular (Melhuish, 2015(Melhuish, , 2019Melhuish & Fagan, 2018;Pramasdyahsari, 2020;Veith & Bitzenbauer, 2022;Veith et al, 2022aVeith et al, , 2022bVeith et al, , 2022c. Even though group theory is mostly taught on university level mathematics, numerous connections to primary and secondary school mathematics have been uncovered (cf.…”

The Impact of Computer-Assisted Abstract Algebra Instruction on Achievement and Attitudes Toward Mathematics: The Case of ISETL

“…Soyut olmanın yanı sıra soyut cebir kavramları, yüksek düzeyde karmaşık yani anlaşılması ve kullanılması için koordine edilmesi gereken birçok öğeyi (Melhuish, 2015) içinde barındıran kavramlardır. Melhuish'a (2015) göre bu durum APOS terminolojisinde şemalara karşılık gelmektedir.…”