Phase plotting is a useful way of illustrating the behavior of functions on the complex plane. Exploiting natural connections between the complex plane and hyperbolic geometry, we introduce a similar method of plotting for differential geometry by assigning unique colors to the preimages of geodesics. Interestingly, for some representations the portraits are also phase portraits in the classical sense. This work is a collaboration with Paul Vrbik.

Abstract algebra is an essential part of undergraduate mathematical learning and yet this subject is also known for its high level of difficulty at the collegiate level. A comprehensive understanding of molecular symmetry and group theory is also an important part of undergraduate chemistry curriculum. In this presentation, I will describe a preliminary local instructional theory which aims to engage students in learning abstract algebra, specifically group theory, within the context of undergraduate chemistry by supporting students in their guided reinvention of the classification of chemically important point groups. Using the emergent models heuristic from the instructional design theory of realistic mathematics education and data from a recent pilot study with a pair of students, I will illustrate how the students engaged with the context of molecular shapes to move through various levels of mathematical activity; from a very situated activity, considering specific molecules, to a more general level of activity, considering any molecule.