Geometry education is an important aspect of STEM education and career development, but it is often overlooked in K-12 education in the United States. Although there is some research on teaching geometry to students with learning difficulties at the elementary level, there is a lack of research on teaching advanced geometry skills at high school…

In this thesis, we apply model theory to Lie theory and geometric group theory. These applications of model theory come via nonstandard analysis. In Lie theory, we use nonstandard methods to prove two results. First, we give a positive solution to the local form of Hilbert's Fifth Problem, which asks whether every locally euclidean local…

How do people learn novel mathematical information that contradicts prior knowledge? The focus of this thesis is the role of structure in the acquisition of knowledge about hyperbolic geometry, a non-Euclidean geometry. In a series of three experiments, I contrast a more holistic structure--training based on closed figures--with a mathematically…