Axiomatizing Geometry from a Transformation Approach
Summary: Historically, educators have held up the study of Euclidean geometry as a model of learning deductive reasoning rooted in axioms. However, axioms based solely on the Elements may not get a high school student all the way to clean, rigorous proofs of congruence and similarity from a transformation approach. While Euclid did allude to superposition by isometries, he did not axiomatize this idea. In this talk, I will discuss an axiomatic system for College Geometry that satisfies the following properties: (1) The system supports proofs of congruence and similarity of all polygons as well as curves defined as a locus of points such as conic sections; (2) it has the potential, with minimal modification, be suitable for use with high school students; (3) it treats angles in a way that generalize easily to angle usage in trigonometry and precalculus. This system combines and adapts Wu’s (2013) and Picciotto and Douglas’s (2017) systems.
Duration: 60 minutes
Format: Online seminar via Zoom web meeting software with questions and discussion. Detailed instructions for joining the seminar will be emailed to registered participants.
Yvonne Lai has been fascinated by geometry since she was young, and wrote her doctoral dissertation in the area of hyperbolic geometry and geometric group theory — in other words, an area where geometric transformations are key. Following her PhD, she studied under Deborah Ball as a post-doctoral fellow to learn the research area of Mathematical Knowledge for Teaching. This project combines her love of geometry and her passion for teaching prospective teachers.