Task Design: Towards Promoting a Geometric Conceptualization of Linear Transformation and Change of Basis

Reform-oriented mathematics education calls for instruction that assists students in developing from their current ways of reasoning into more complex and formal mathematical reasoning (e.g., Gravemeijer, 2004). Such instruction centers on a host of theoretical and pragmatic concerns, and our research explores many of these concerns in a first course in undergraduate linear algebra. Our work draws on the theory of Realistic Mathematics Education (Freudenthal, 1991) to design instructional sequences that build on student concepts and reasoning as the starting point from which more complex and formal reasoning develops. This emphasis on creating a classroom environment that fosters students ’ ability to reinvent advanced mathematical concepts served as inspiration for the development of an instructional sequence with the particular goal of fostering a geometric conceptualization of linear transformation and change of basis in R2. This paper will focus on a particular task from the aforementioned instructional sequence, the Change of Basis Task, that was developed with the intent of leading to a student-driven reinvention of the well-known equation A = PDP-1, for the case where A, P, P-1and D are 2x2 matrices. In this paper, I will detail some of the student responses to and interactions with the Change of Basis Task, as well as elaborate upon what affordances th