2. Teaching
3. MAT441

Welcome

Welcome to the course web page for the Spring 2019 manifestation of MAT 441: Introduction to Topology at Northern Arizona University.

What is This Course All About?

Loosely speaking, topology is the branch of mathematics concerned with the properties of space that are preserved under continuous deformations, such as stretching, bending, and twisting, but not ripping or gluing. Topology is sometimes referred to as “rubber sheet geometry”, but truth be told, this only vaguely captures one aspect of the subject. The driving force behind topology is a desire to distill familiar mathematical concepts such as convergence, connectedness, continuity, and dimension down to their essence. Formally, topology is the study of properties of topological spaces that are invariant under continuous maps. Of particular interest are the properties that are preserved under homeomorphisms, which are invertible continuous maps with continuous inverses. Homeomorphisms are isomorphisms between topological spaces. Chapter 1 of our textbook “Topology Through Inquiry” contains the following beautiful quote:

Topology is a subject whose power arises from the impulse to abstract essential features from complex situations and then to let our curiosity roam while striving to truly understand what is essential about fundamental ideas.

Here’s another wonderful quote about mathematics in general:

The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.

An Inquiry-Based Approach

This is not a lecture-oriented class or one in which mimicking prefabricated examples will lead you to success. You will be expected to work actively to construct your own understanding of the topics at hand with the readily available help of me and your classmates. Many of the concepts you learn and problems you work on will be new to you and ask you to stretch your thinking. You will experience frustration and failure before you experience understanding. This is part of the normal learning process. If you are doing things well, you should be confused at different points in the semester. The material is too rich for a human being to completely understand it immediately. Your viability as a professional in the modern workforce depends on your ability to embrace this learning process and make it work for you.

In order to promote a more active participation in your learning, we will incorporate ideas from an educational philosophy called inquiry-based learning (IBL). Loosely speaking, IBL is a student-centered method of teaching mathematics that engages students in sense-making activities. Students are given tasks requiring them to solve problems, conjecture, experiment, explore, create, and communicate. Rather than showing facts or a clear, smooth path to a solution, the instructor guides and mentors students via well-crafted problems through an adventure in mathematical discovery. Effective IBL courses encourage deep engagement in rich mathematical activities and provide opportunities to collaborate with peers (either through class presentations or group-oriented work). If you want to learn more about IBL, read my blog post titled What the Heck is IBL?.

Much of the course will be devoted to students presenting their proposed solutions/proofs on the board and a significant portion of your grade will be determined by how much mathematics you produce. I use the word “produce” because I believe that the best way to learn mathematics is by doing mathematics. Someone cannot master a musical instrument or a martial art by simply watching, and in a similar fashion, you cannot master mathematics by simply watching; you must do mathematics!

In any act of creation, there must be room for experimentation, and thus allowance for mistakes, even failure. A key goal of our community is that we support each other—sharpening each other’s thinking but also bolstering each other’s confidence—so that we can make failure a productive experience. Mistakes are inevitable, and they should not be an obstacle to further progress. It’s normal to struggle and be confused as you work through new material. Accepting that means you can keep working even while feeling stuck, until you overcome and reach even greater accomplishments.

Don’t fear failure. Not failure, but low aim, is the crime. In great attempts it is glorious even to fail.

Furthermore, it is important to understand that solving genuine problems is difficult and takes time. You shouldn’t expect to complete each problem in 10 minutes or less. Sometimes, you might have to stare at the problem for an hour before even understanding how to get started. In fact, solving difficult problems can be a lot like the clip from the Big Bang Theory located here.

In this course, everyone will be required to

• read and interact with course notes and textbook on your own;
• write up quality solutions/proofs to assigned problems;
• present solutions/proofs on the board to the rest of the class;
• participate in discussions centered around a student’s presented solution/proof;
• call upon your own prodigious mental faculties to respond in flexible, thoughtful, and creative ways to problems that may seem unfamiliar at first glance.

As the semester progresses, it should become clear to you what the expectations are.