The Design and Validation of a Group Theory Concept Inventory

algebra is a standard course required of mathematics and mathematics education majors across the United States. For many students, this course is the first time concepts are to be reasoned about formally based on their properties (Hazzan, 1999). Dubinsky, Dautermann, Leron and Zazkis (1994) noted, “mathematics faculty and students generally consider it to be one of the most troublesome undergraduate subjects” (p. 268). While this statement is largely accepted, student understanding in the course has not been empirically evaluated on a large scale. When the Force Concept Inventory (FCI) was introduced into the physics community, the state of students’ conceptual understanding was similarly not understood (Hestenes, Wells, & Swackhamer, 1992). The inventory was created to evaluate students’ conceptions of force via a validated and quick multiplechoice test. As a result, the test became widely used at the collegiate level. A concept inventory distills concepts from procedural and other knowledge to probe at a deeper level than many traditional assessments. Results from the FCI showed that students might do procedures correctly while still maintaining fundamentally flawed understandings of Newtonian force. Advanced undergraduate mathematics education finds itself where the physics community was decades ago when the FCI was created. There is a body of research related to student understanding about abstract algebra, but it remains small scale. In general, the research falls into the categories of student understanding and instructional innovation (Weber and Larsen, 2008). Currently 2 literature exists to describe mechanisms that could lead to concept acquisition (Dubinsky, 1997), student difficulties with proofs, abstraction and complexity (Dubinsky et al., 1994; Hazzan, 1999; Leron, Hazzan, & Zazkis, 1995; Selden & Selden, 1987; Weber & Alcock, 2004; Weber, 2001) and instructional innovations (Larsen, Johnson, & Weber, 2013; Dubinsky, 1997). However, these studies were primarily exploratory and often relied on in-depth interviews with small samples. Student conceptions in algebra have not been explored systematically or comprehensively. In fact, Larsen, Johnson, and Bartlo (2014) have recently called for the “creation of validated assessment instruments” (p. 709) in group theory for the purpose of evaluating instructional innovations. Recently, Weber (2013) similarly called for an increase of quantitative studies in the Research in Undergraduate Mathematics Education (RUME) community noting that qualitative studies dominate the undergraduate portion of the mathematics education community. Ideally, quantitative and qualitative methods could be used in a complementary fashion to establish both meaningful and generalizable results. The Group Concept Inventory (GCI) was created to address this call. The GCI is a quantitative measure created through both qualitative methods (such as interviewing students) and quantitative methods (such as item analysis and reliability see methods section.) Ultimately, the validated tool serves a powerful purpose to complement qualitative explorations and evaluate larger populations.