Geometries and Education:
In this course, we will explore a variety of topics from geometry, including axiomatic and analytic treatment of incident geometry, plane geometry, spherical geometry, hyperbolic geometry, and the Poincaré disk model from Riemannian geometry. My primary goal for this course is to make the problem-solving approach accessible and easy to apply in various situations, and to develop the fundamental lifelong skills of solving problems and interpreting solutions in the real world. I anticipate that the course will help you develop and expand your capacity in Spatial Reasoning and broaden your knowledge from Euclidean/spherical/hyperbolical surfaces to several axiomatic geometries. More importantly, this course will help you to change the way you see the world around you, as it is the key to many of how we work in mathematics and beyond.
Prerequisite: SC/MATH 1021 3.00 or SC/MATH 1025 3.00 or GL/MATH 1660 3.00 and a minimum of 21 credits in SC/MATH courses without second digit "5".
Course credit exclusion: SC/MATH 3052 6.00 and SC/MATH 3050 6.00.
Prerequisites / Co-requisites / Exclusions: Please consult https://mathstats.info.yorku.ca/supplemental-calendar/ to ensure you have the required prerequisites for the course, and that getting a credit course does not exclude you from another credit.
Official Course Calendar Description: Axiomatic and analytic treatment of various geometries, including incident geometry, plane geometry, spherical geometry, hyperbolic geometry, and Poincaré disk model from Riemannian geometry. Students will also reflect on the teaching and learning of geometry and spatial reasoning.
Expanded Course Description:
Geometry is one of the oldest fields of mathematics that goes back to the times of Euclid, Pythagoras, and other famous ancient Greek mathematicians. Initially concerned with fundamental concepts of point, line, plane, distance, angle, surface, and curve, the scope of geometry has been expanded during the last two centuries, leading to the creation of several subfields that include Riemannian geometry, algebraic geometry, etc.
Geometry is a good subject to explore the role of visual and spatial reasoning in the practice of mathematics: generating insights, problem-solving, communication, remembering, proofs, etc.
In this course, students will explore a variety of topics in the exciting field of geometry, including axiomatic and analytic treatment of incident geometry, plane geometry, spherical geometry, hyperbolic geometry, and Poincaré disk model from Riemannian geometry. Important themes for each of the topics include axioms & common notions, geodesics & straightness, congruency & similarity, area & holonomy, isometries, projections, parallel postulates in different geometries, isometries with matrices & eigenvectors, etc.
Coursework will explore geometric questions through investigation, hands-on materials and dynamic geometry software such as Geometer's Sketchpad, GeoGebra 3D, GeoGebra Geometry, Desmos, Maple, and also emphasize the many applications of geometry in the areas of computer-aided design, computer graphics, robotics, architecture, virtual reality, video game programming, and engineering.
Geometry also has important modern applications, to such areas as Computer Aided Geometric Design, computer graphics, computational geometry, robotics, modern physics, biology and engineering. How we practice Euclidean geometry (and teach geometry) is being changed by computer programs, both symbolic (such as Maple) and visual (such as Geometer's Sketchpad) and 3D printing. We will incorporate technology and look at geometric questions raised by how we might do geometry with computers.
This course is designed to further reflection on the teaching and learning of geometry. Connections to the Ontario curriculum will be made, from time to time, and students are encouraged to bring their own connections into classroom discussions. You are expected to raise your own questions and explore ways to answer them. Since a project is a critical part of the course, your own questions should strongly support the development of possible project topics. Value your questions – and ask about connections. Developing your questions and these connections is an important objective of the course.
A related goal of the course is to help students develop and expand their capacity in Spatial Reasoning – a form of reasoning which recent research has highlighted as well connected to many subject areas, as well as general performance in mathematics, and general creativity and problem solving. An underlying goal of the course is to 'change the way we see'. How we see is key to many of how we work in geometry (and in other parts of mathematics and beyond).
