I’m teaching topology for the first time this semester, so I’ve been poking around the blogosphere for ideas of different ways to explain some of the ideas in this class to my students.
Luckily, right before I started the semester, I ran across this post on College Math Teaching: Bad notation drove me nuts….(and still does). Quick, what’s the area of a circle? πr2. What’s the fundamental group of the circle? ℤ. Those aren’t the same circles. The circle with fundamental group ℤ has area 0. It’s the boundary of the circle with area πr2. Unlike the author of College Math Teaching, I wasn’t confused about the circle when I first saw it in topology, and it never occurred to me that that particular example of sloppy nomenclature could be a stumbling block for my students. I think that reading that post has helped me be more thoughtful and careful about notation and terminology when I’m teaching this class.
Another post that has shaped my thinking about teaching upper-level undergraduate classes is a guest post on the AMS math education blog: Mathematics professors and mathematics majors’ expectations of lectures in advanced mathematics. The short version: those expectations are different. For example, I (and apparently many other math teachers) think that leaving some details of proofs to students can be a good way for them to solidify their understanding of a proof in class. After reading that post, I’ve decided to include more of those details in classes if I think they’re important, and if I want students to fill in details on their own, I am having them do it in class or on homework.
I’m teaching algebraic topology, and the author of College Math Teaching is teaching point-set topology, so there aren’t a lot of posts directly about the material I’m teaching, but I’m enjoying a feeling of camaraderie when I read their posts about their class this semester. My students are fairly experienced proof writers at this point, but the author’s students are not. I can’t imagine trying to teach some of these concepts at the same time as teaching proofs writing! On a companion blog, there are course notes for the author’s class. I could probably use some brushing up on the separation axioms. I’ve been interested in counterexamples in topology recently because I wrote about the π-Base, an online version of the classic Steen and Seebach book. College Math Teaching has nice posts about two interesting counterexamples, the Alexandroff Square and the Topologists’ Sine Curve.
I’ve also found Math ∩ Programming, Jeremy Kun’s blog, to be valuable for my teaching. I read over his post about the fundamental group at the beginning of the semester when I was starting to teach it. I’m doing a lot of my teaching from Munkres, which can be a bit too formal at the expense of intuition, so seeing an explanation that is a little more idea-focused is helpful. Kun is a very good math writer, and I’m sure his primers would be useful for students and teachers of other subjects as well. There’s plenty more to enjoy in his blog, and I can’t help but point you to his guest post on Baking and Math. After all, torus knot baklava is much tastier than programming. (Sorry, programmers. I’m never going to be as interested in programming as I am in pastry.)
I recently ran across The Geometric Viewpoint, a blog about geometry and topology aimed at undergraduates and written by Colby College faculty and students. I thought about having some writing assignments in my topology class and ultimately decided against it, but maybe reading the student posts will make me think about it more for next time.
There are a few defunct or rarely updated blogs I’ve found useful. Sketches of Topology has some interesting topology constructions with great illustrations. Dan Ma’s Topology Blog has accessible posts about point-set topology. I haven’t seen much about undergraduate algebraic topology in the blogosphere, though.
Any other recommendations for topology blogs that focus on undergraduate topology?