Linear Algebra is Scary, Even for Future Mathematicians

The Calculus of Friendship: What a Teacher and a Student Learned about Live While Corresponding about Math (2009) is a delightful book by Steven Strogatz. Strogatz is Jacob Gould Schurman Professor of Applied Mathematics at Cornell University, and he writes beautifully. The book is about his decades-long correspondence with one of his high-school mathematics teachers, Don Joffray, who was inspirational to Strogatz. It’s a great book about two people who take pleasure in looking at the world mathematically and can become totally engrossed by considering mathematics problems. I highly recommend it to anyone who likes mathematics and knows a little bit about calculus. The book is ultimately about love, both the love that these two men have for each other, and the love that they have for solving mathematics problems.

One passage that struck me starts on page 15, and describes Strogatz’s first mathematics course at university:

… my first math course in college at utterly deflated me and changed the way I viewed myself. It was a proof-oriented course on linear algebra, aimed at freshmen who had aspirations of being math majors. It was intended to be a taste of rigorous, abstract math — the sort of thing you’d have to be good at if you wanted to be a pure mathematician. The professor, a renowned topologist, was so shy that he slithered along the wall when he entered the lecture room on the first day, as if hoping to become invisible. He spent the rest of the semester looking down at his Wallabees and tugging at his red beard. The few times I dared to ask him a question, he looked startled and gave a monosyllabic response. I read the book, did the homework, paid careful attention in lecture, and had no idea what was going on. It was terrifying. No matter what I did, I couldn’t get it. The textbook was dry, hyperprecise, and devoid of illustrations. The homework was baffling. And the tests — just thinking about an upcoming test would send me running to the bathroom.

This is the common experience of just about everyone who takes linear algebra in first-year university. Such courses are typically taught way too abstractly, with textbooks that are way too abstract, before just about every student in the course is ready to cope with this level of abstraction. The remedy is clear: Start with lots of concrete examples and gradually build from this core of examples, so that by the time the abstract ideas are presented they make sense in the context of the large body of examples that students have already played with, already understood, already internalized. This is the approach I have taken in teaching linear algebra in the past few years to high-school enrichment groups, and it’s the approach I’ll take in this course too (and in the follow-up course). It works!

I’ll support this course with a concrete, examples-oriented, tutorial-style textbook, which I’ll post here.