Square Peg Thinking, Round Hole Problems: An Investigation of Student Thinking about and Mathematical Preparation for Vector Concepts in Cartesian and Non-Cartesian Coordinates Used in Upper-Division Physics

Abstract

As part of the broader effort to explore the math-physics interface in the upper-division our research team seeks to develop research-based curriculum to aid students in translating middle-division mathematics course content to middle- and upper-division physics courses. Toward that goal, an investigation into student thinking regarding vector concepts, such as basis unit vectors, position vectors, and velocity vectors in both Cartesian and non-Cartesian coordinates was performed. Through analysis of students’ written responses to in-class assessments, we identified several themes of student thinking that were emergent. Think-aloud interview protocols targeting those themes were developed. Seven subjects were interviewed and their responses were analyzed through a Resources and Framing theoretical framework. Analyzing interview responses allowed us to name specific resources that were activated and categorize those resources into thematic clusters or groups. A case-study provided the opportunity to map how the coordination of a single student’s resources can activate together and how the non-activation of a key resource can cause a dramatic shift in a student’s thinking. The interview data also revealed a propensity for “pattern-matching”: writing the algebraic expressions of position vectors in spherical coordinates in a form morphologically similar to that of Cartesian coordinates. The data showed that many resources that are productive in Cartesian coordinates are inappropriately applied to non-Cartesian coordinates, although rarely in the same way across students. The consistency with which such resource activations across coordinate systems occurred led to an investigation of what Calculus I-III students are taught and what they learn about these vector concepts in various coordinate systems by the end of their multivariable calculus courses. Content that used Cartesian coordinates dominated textbook material, both quantitatively by proportion, and qualitatively through the nature of the presentation. Students were surveyed at the end of Calculus III courses using questions with notation consistent with both physics expectations and that used in those calculus texts, revealing an emerging understanding of vector concepts both in Cartesian and non-Cartesian coordinate systems. These findings will be further elaborated upon in this dissertation. Commentary on how these findings can inform instructional material development will also be presented.