The main goal of this course is to examine the fundamental concepts of geometry under different perspectives, and to learn how to solve problems in different ways. In particular, we use techniques from elementary Euclidean geometry, coordinate geometry and linear algebra, and discuss how each of these techniques are more or less effective or efficient, depending on the specific context. At the end of the course, we discuss situations where none of the above geometric frameworks provides an accurate description of reality and step into the fascinating world of non-Euclidean geometries.
Previous experience with mathematical proofs is not required for the course. In fact, learning how to write proofs of simple geometric statements is one of the main objectives of this course. This happens through a series of step-by-step problems that gradually guide students through the proof-writing process.
Prerequisites: MATH 031 (Applied Linear Algebra)
Textbook: "The Four Pillars of Geometry" by John Stillwell
While maintaining a traditional lecture format, the course has an inquiry-based approach. A typical lesson includes several polls or open-ended questions aiming to stimulate the students' intuition on the topics, followed by guided step-by-step solutions or guided proofs that aim to make the intuition formally precise. The homework problems are designed to both foster further reflection on the topics discussed in previous lessons and build new intuition for topics to be treated later in the course.
I structured the course around group work. During every class meeting I briefly introduced some new topics, then let the students work in small groups on problems about the same topics, and finally have a class discussion about the problems, with students from all groups contributing to the class solution.