I’ve been trying to teach myself some abstract algebra the last few weeks from this great free online text. I’m really enjoying it! It’s fun to learn new math, and I like seeing new ideas as a learner. I never took any abstract algebra in college, and before starting this adventure I knew there were things called groups, rings, and fields but had no idea what they were. Now I know more! Along the way, I’ve been thinking about what learning math can teach me about teaching. Two lessons stick out.
I became interested in learning about groups after reading Patrick Honner’s October article, The (Imaginary) Numbers at the Edge of Reality in Quanta Magazine. It’s a great read, and it positions group theory as part of a larger story, framing different number systems in terms of their connections to physical problems and sharing the stories of the mathematicians who first worked with them. I became fascinated by quaternions, and I’m lucky that the text I’m learning from uses quaternions as an example in different contexts and keeps me connected to a narrative beyond the math itself.
How often does this happen in math classes? Not very often in mine. Now I’m thinking about how I can find ways to position the math that we’re learning as part of a larger story. I don’t think this needs to be a radical change; it can be a quick addendum of historical context, a narrative about a relevant mathematician or mathematicians, an interesting application of a topic, or just taking a moment to share how different concepts are related, framing where students have been and where they’re going. But humans learn from stories, and are motivated to learn from stories, and I think this is something that is underused in my classes.
An example is worth a thousand definitions. You can define “ideal” as carefully as you like and I’m still going to be confused the first time I learn about it. Share a handful of well-chosen examples and non-examples and all of a sudden it makes sense. In math we love definitions. I find many definitions elegant and beautiful. I spend lots of time thinking about how to explain concepts in ways that will make sense to students. These things are important, but it’s possible to overestimate their importance. Examples work with explanations to create students’ mental models of concepts. Examples give something concrete to latch onto, and they can illuminate boundary cases and subtleties that might not make sense in an explanation or be clear from a definition. And as I see more examples, I start to create new generalizations and come up with explanations that make sense to me. When I first read about ideals, they were nonsense. Now I think about them like a magnet — they’re this set of objects that pull other objects in, and once you’re in, you can’t get out. For instance, if I’m working with integers, once something becomes a multiple of 3, no matter what you multiply it by, it stays a multiple of 3. That might not make sense to you, but it makes sense to me. And the more I see new examples and incorporate them into my mental models, the better I can apply that knowledge. Examples give me a chance to test my understanding and see whether my ideas make sense in a new context.
I don’t think I do this very well with students. Student understanding often happens within the paradigm of my explanations and my ways of looking at mathematical ideas. There’s a place for that, especially to minimize confusion and misconceptions. But there’s also an place to give students lots of examples to work with, to ask them to come up with explanations that make sense to them, and to embrace their ideas and perspectives. I can explain the end behavior of rational functions until I’m blue in the face talking about top-heavy and bottom-heavy functions, and students are often still confused. Offering a set of well-chosen examples and letting students come up with language and analogies that make sense with their experience could be a much less painful way to do it.
There’s nothing groundbreaking about either of these ideas, but as someone who knows a lot of math and isn’t often in the position of learning new math, they’re easy to forget. A constant challenge of teaching is the curse of my own knowledge, and learning something new, even when it’s hard, is a great way for me to see learning from a new perspective and push myself to teach in ways that are accessible and engaging for all students.