A praxeological investigation of divergence- Exploring challenges of teaching and learning math-in-physics.

Abstract

This thesis undertakes a study of challenges with teaching and learning the vector calculus concept of divergence in the context of undergraduate physics education. Based in the Anthropological Theory of the Didactic, a case study of two university physics courses is conducted. Commencing with an examination of undergraduate physics textbooks, an outlook on divergence as portrayed in the context of undergraduate physics education is obtained. Subsequently, lecture and exercise session observations as well as student interviews are conducted and analysed. This leads to the identification of a number of challenges for teachers and students with making sense of the concept. A central challenge disclosed is to align two conceptually different perspectives on divergence: As a particular sum of partial derivatives and as the volume density of flux out of a volume at a point. Whereas the latter prompts both teachers and students towards describing the divergence of a vector field as simply the net flux out of a volume, the former gives rise to viewing it as an unspecified "total change" of the field. It is further indicated that portraying divergence in the physical con- text of fluid mechanics presents the challenge of clarifying the distinction between the divergence of vector fields on R2 and R3. Also, making sense of the divergence of vector fields containing singularities, for example at regions of charge for the electrostatic field, introduces a significant didactic challenge. Whereas the divergence of vector fields at such problematic points may be easily derived, it necessitates an extension of its definition from a sum of partial derivatives to what the "Divergence Theorem intuitively implies". A discussion of why these challenges are pertinent to attend to, and how they can be addressed by alternative teaching strategies, is conducted ultimately.

Master’s student: Anders Wolfsberg (MATH)

External examiner (censor): Claus Michelsen (SDU)

Main supervisor: Ricardo Karam (IND)

Co-supervisor: Carl Winsløw (IND)