In Topology, students are exploring a strange, beautiful world in which we consider the shape of space without the notions of distances and angles.
Imagine you and a collection of friends have a large rubber band. By grabbing points on the rubber band and stretching away from each other, you can form a polygon. In fact, given enough friends and a stretchy enough rubber band, you could create any polygon you'd like. If you have infinitely many infinitely small friends, you could even imagine that you could create shapes like circles and ellipses.
There are a lot of interesting questions we could ask about the differences between these shapes—these questions are the jumping-off point for the field of geometry. But there is also one thing that all of these shapes have in common: they can all be formed by stretching out a rubber band. You could say that topology is the study of the rubber band.
Topology is a field of mathematics that explores the shape of space without the notions of distances and angles. This seems like it should make things simpler, but in fact it opens the door to a strange, beautiful world where things don't always behave the way we think they should. Our goal this block is to build a tool kit for exploring this world.
Students are put in the driver's seat as much as possible in this class. At the beginning of each class a student presents a solution to a problem that they found particularly interesting or challenging. These presentations often become student-led class discussions, with other students asking questions about the argument, comparing it to their own, and working together toward a common understanding of the problem and its solution. After the student presentation we have a brief discussion of the definitions for the day, and then we transition to problem-solving. In our midterm preparation, students created their own study-guide, and collaboratively solved and presented potential exam problems.
Moving forward, we will maintain our focus on learning through discovery and collaboration. Students will continue to present solutions orally to each other, but we will also add a writing component. Students will be working on a short paper about a topological counterexample. I have been extremely impressed with the way these students have risen to the challenge of this field throughout the first half of the block, and I'm looking forward to seeing them continue to grow in the coming weeks.
-- Susan Durst