Effective inquiry-oriented instruction demands sensitivity toward students' common ways of thinking about content, as does the design of tasks to support that instruction. In the research literature, little is known about the variety of ways in which students think about matrix multiplication, which is a fundamental part of any introductory undergraduate linear algebra course. The data for my study comes from the first in a series of classroom teaching experiments (CTEs) in undergraduate linear algebra. In my talk, I will discuss the ways in which the analysis of data from a set of semi-structured clinical interviews informed the revision of the initial instructional sequence in subsequent CTEs. Specifically, the revised instructional sequence is designed to support students' learning of span, linear dependence and independence - central ideas to the course that are typically introduced later in the semester.
Finally, I will discuss how this work fits in with my broader research program, in which I aim to develop a framework for designing and scaling up implementation of research-based, inquiry-oriented mathematics curricula in the post secondary setting. Specifically, implementing such curricula requires a great deal of learning on the part of the instructor, and considerable changes from traditional lecture-based instructional practices are entailed. As such, I am interested in better understanding ways of supporting teacher learning of both domain-specific pedagogical content knowledge and specific instructional practices that support the learning of all students.