Using Geometric Habits of Mind to Connect Geometry from a Transformation Perspective to Graph Transformations and Abstract Algebra

The Common Core State Standards for Mathematics advocate for geometry from a transformation approach, and we illustrate how this movement opens two important mathematical opportunities. First, geometry from a transformation approach connects to algebra at the secondary level, especially through transformations of graphs; and second, geometry from a transformation approach connects to abstract algebra at the university level, especially through linear algebra and group theory. To make these connections visible, we propose an extension of Driscoll et al.’s (2007) Geometric Habits of Mind (GHOM) framework that can apply to graph transformations and ideas of abstract algebra. We then describe three specific contexts where the extended GHOMs link problem solving and theory building activities (Gowers, The two cultures of mathematics. Retrieved from https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf, 2000) across geometry and algebra at the secondary level and abstract algebra at the university level. These contexts are: (1) reconciling three definitions of congruence, (2) using transformations to understand conventions for defining families of functions and shapes, and (3) analyzing plane transformations of the plane in terms of linear algebra. We conclude with some implications of our work for the preparation of secondary teachers.