Conceptualizing Multiple Coordinate Systems in Linear Algebra

Linear algebra is gaining increased attention due to its applications in many modern high-tech industries, such as artificial intelligence, machine learning, and robotics. Coordinate systems underlie many concepts in linear algebra and promote students’ spatial thinking of key concepts by imposing a visual aspect to courses that may otherwise be abstract and computational. Despite the importance of coordinate systems, there is surprisingly little research on them within the context of linear algebra. This dissertation reports the results of one-on-one clinical interviews involving a series of ten tasks with five undergraduate STEM students who had taken a linear algebra course. The goal of the clinical interviews was to explore student thinking of a location and its coordinate pair when space is described by a single coordinate system as well as multiple coordinate systems. I identified that students engage with three entities: Magnitude/location, Coordinates, and Unit lengths when they name a location (Naming and Re-Naming) and they locate a name in space (Locating and Re-Locating). When interacting with multiple coordinate systems, I found that students reasoned either proportionally or reciprocally to determine what operations to perform numerically in a situation. This dissertation also presents the results of teaching experiment interviews involving a series of instructional tasks and a GeoGebra applet. These interviews were conducted with three selected students out of five who completed the clinical interviews. The goal of the teaching experiments was to support students in extending their interpretations of matrix entries and in reasoning with matrix multiplication under non-parallel linear coordinate systems. I describe both their initial and progressive interpretations in three main ways. First, students extended their ways of interpreting entries of a matrix from a string of numbers facilitating a coordinate conversion process to a set of basis vectors that constructs a coordinate system. Second, when interacting with multiple coordinate systems, students extended their proportional and reciprocal reasoning to involve matrix multiplication. Third, students developed two ways of representing coordinate pairs using their extended interpretations of matrix entries: coordinate as a coefficient in linear combination, or coordinate as a component in an ordered list.