Lessons I wish I had learned before I started teaching differential equations [pdf] (1997)
The differential equations course was one of the most baffling experiences I ever had. The professor on the first day told us the course would be rote, as opposed to proofs, and every day he copied the methods to the chalkboard. He specifically instructed us to copy them verbatim into our notebooks. In this way there was not a lot to discuss, and from time to time the professor would gently steer us back to merely copying and memorizing the methods, even though no one had questioned him out loud. The methods were entirely disconnected. I had no indication of how they were derived or what the original motivation might be. What a differential equation was or why I wanted to "solve" one---this generated a second equation---was a mystery. None of the problem in the physics sequence looked like this. The engineering students claimed to have them, but reminded me that "this was all done by computers now." In the textbook there were no word problems, only formulas, and so I was never able to infer what this might all be about. The problems gave no opportunity for insight beyond recognizing the form. On the homework I manipulated one formula into another. On the test I did the same thing. Through memorization, I got an A in the course. I never encountered a differential equation before or since.