In the late 90’s, at a different college, I taught Linear Algebra a few times. I wasn’t satisfied with my teaching. I could see that the students were struggling – they really couldn’t do proofs – and I had no idea how to help them.
When I decided to teach Linear this semester, I spent a lot of time beforehand studying the material. After a 12-year gap, I knew I needed to refresh my understanding. Although I still wasn’t sure exactly how I would help my students learn to prove theorems, I trusted that I’d be able to figure it out.
Just like my former students, my current students have struggled with proof. One thing that has helped is the true-false questions Lay provides in every section. I often have my students vote: “Just guess, it’s ok to guess wrong. Then we’ll discuss it.” We don’t have clickers, but they’re pretty willing to do it, and I keep finding out how much harder the material is for them than I had expected.
I use the true-false questions to make quizzes too. All my students aced the first quiz (on consistent systems and general solutions, no proofs), so I knew I had a great group. I made the second quiz a bit harder (1 question: does the span of two given vectors include a third vector given?). They still did pretty well, so on their third quiz I asked them to ‘prove or disprove’ one statement. Most of the class failed that quiz, so I gave them an alternate version the next day. They still mostly failed. I made a third version a few days later, and they finally improved.
On their first test, the ‘prove or disprove’ question had the lowest success rate of all the questions, but many of them did get it. Most of my students are used to acing their math classes, so I find myself reassuring them that this is a journey, and that I trust that they’ll get good at this before the course is over.
I think there’s one other big difference, though it’s hard to pinpoint it. The work I’ve done over the past 4 years, working on math that challenges me, and writing about mathematics, has made me more of a mathematician, and I'm sure that's helping me teach this course better. I have Bob and Ellen Kaplan, Amanda Serenevy, and a few other great teachers at the Summer Math Circle Institute to thank, along with Josh Zucker and Paul Zeitz who work with the Bay Area math circles. I also have my blog readers to thank for motivating me to keep writing here. Thank you all!
This class is the most exciting class I’ve taught in a long time. For the first time in my life, we’re ahead of the schedule I’ve set myself. That’s how good my students are. And I’m getting to talk with students about what it means to do mathematics. I think they’re learning something new, and I’m grateful to be a part of that.