Student understanding of unit vectors and coordinate systems beyond cartesian coordinates in upper division physics courses

Abstract

In upper division physics courses students are required to work with various coordinate systems. This skill becomes particularly important when learning Electricity and Magnetism, where the most appropriate coordinate system will often depend on the geometry and symmetry of a problem. This study aims to identify and describe "resources" used by students when answering physics questions regarding unit vectors in non-Cartesian coordinate systems, specifically polar coordinates [1]. Data were collected in the form of written responses and interviews in upper division physics courses at two universities. After deeper analysis we identified several resources that students use in ways that can be productive and unproductive.

Mapping activation of resources among upper division physics students in non-Cartesian coordinate systems: A case study

“…This pattern matching resulted in the inclusion of unnecessary unit vectors, which frequently appeared as students writing spherical position vectors in the incorrect (but Cartesian-imitating) form ⃗ r ¼ rr þ θθ þ ϕφ. Our initial work on students' thinking on unit vectors in non-Cartesian coordinate systems categorized student thinking into "clusters" of similar but not identical ways that students thought about unit vectors [13]. Specifically, we found that students conflated unit vectors at a point with position vectors to a point.…”

“…Additionally, there was a tendency for students to conflate ideas about the direction of a unit vector having to do with either the location of an object at a point or with the direction of motion of an object at a point. Vega et al [13] further describe a "Unit Vector Cluster" that has essential resources to fit a canonical definition of a unit vector: a unit vector is a vector; has a length of one unit; points in the direction of increasing coordinate; and is dimensionless. The naming of these resources is done for completeness and from an expert perspective, and not all of these resources were identified by the students in the interview sample.…”

“…Using the second protocol, three subjects were interviewed; one subject a junior undergraduate physics major, one a second-year physics graduate student, and the third a physics faculty member. Across the two protocols, a total of three of the seven subjects responded to unit vector questions in a manner consistent with the resource clusters described in our initial work [13] that were not productive for these particular questions. The remaining four subjects answered the initial unit vector questions in our protocol successfully.…”

“…Some students would report correct answers but with unclear or incomplete reasoning. Some incorrect answers appeared superficially similar but presented slight differences that made it unclear what the underlying difficulty might be [13]. Very often, it simply wasn't clear what the student was trying to do, and a considerable amount of interpretation and inference on the part of the researcher would have been required to categorize a response into any category.…”

Mathematization and the ‘Boas course’

“…Sayre and Wittmann used a resource framework and described 'plasticity' of resources in the process of choice of coordinate system. [8] Vega described student resources in unit vectors in Math Methods [9].…”