A comparison of students’ models of knowledge to be learned in an introductory linear algebra course with results from prior research on such models in college calculus courses

Abstract

"Research done from an institutional perspective has found students to develop non-mathematical practices in college calculus courses that emphasize routinization of knowledge. The knowledge students are expected to learn, as indicated by tasks determining their grade in the course, enables students to routinize techniques and use non-mathematical considerations, such as didactic and social norms from their course, to justify their techniques. Such research has mostly been done in the calculus context. To calibrate the study of the effects of institutionalized routinization of knowledge, I investigated these in the context of a course in a different domain of mathematics and regulated by institutional mechanisms similar to those regulating college calculus courses. To this end, I adapted, to an introductory college linear algebra course at a large urban North American university, the framework and methodology from a body of research that qualifies students’ activity by attending to institutional mechanisms that regulate it. The framework appends to the Anthropological Theory of the Didactic (ATD) (Chevallard, 1985, 1999) notions from the Institutional Analysis and Development framework (IAD) (Ostrom, 2005). The ATD provides tools through which to model activities that occur in institutions and the IAD elaborates institutional mechanisms that regulate activity that occurs in institutions. I analyzed curricular documents to develop task-based interviews (TBI) that could draw out the nature of the knowledge students mobilize. I conducted interviews with ten students shortly after they had completed the course. The qualitative approach I used included an analysis of curricular documents to model knowledge to be learned in the course that relates to each TBI task, as well as an analysis to model the knowledge students mobilized in response to each TBI task. I found students mobilized non-mathematical practices: what they activated was conditioned by and delimited to knowledge normally expected of students in the course, and their mobilization contrasted in various ways with mathematics intrinsic to the tasks they were offered. I also propose an operationalization of the institutional notion of positioning previously proposed and examined as a mechanism regulating students’ activity in didactic institutions." -- provided by author