In my previous life as a teacher, I had little to no doubt how to present the material in class. I recognized the fact that students have different learning styles, and knew that it was up to me to give a couple of different explanations. But I stopped there.
After the brain surgeries, my ability to think sequentially was compromised, I came to appreciate the variety of thinking styles among my students. I delighted in the challenge of adjusting my explanations to address this diversity. Having struggled to relearn and improve my own sequential thinking skills, I recognized many of theissues that students faced as they learned. I enjoyed the process of tracking down the source of the problem, and finding an approach that worked.
Working with Rebecca was always fun--in order to understand the material I teach, she has to fully understand the fundamentals behind. A week ago we were talking about infinitesimal numbers. I said, "If you accept the notion of infinity..." she nodded. "then you can view and infinitesimal as its reciprocal."
She frowned. "But infinity is not considered a real number. Why is that?"
I tried another tack. “Whatever large number you can think of, say 1,000,000,000,000, there's always a larger number, 1,000,000,000,000,000). Similarly, infinitesimal numbers—however small a positive number you can think of, like 0.00000000001, there's always a smaller positive number, 0.00000000000000001, without actually reaching zero."
I saw her “eureka” moment in her eyes. “Oh, so infinity and infinitesimal are like processes.”
From now on in my calculus classes, when I introduce infinitesimal numbers, I'll be using the term “process.”
Most of my students are humanities majors. Not as “pattern fluent” as the engineering students, they think outside the box, partly because they've never really been in the box of linear processing, or if they have, their box has porous walls. It is not unusual for students' questions to lead me to reiterate and expand explanations and approaches to various topics.
Ariel was one of my students several years ago. He had a lot of trouble with the material on limits (where you get as close as you want to a number without actually reaching it). We both struggled to understand what particular obstacle lay in his way.
When we moved on to the next topic in lecture, continuity, he asked, “Why do you teach limits before continuity?”
Ariel's question caught me by surprise, not because I thought he was challenging me personally, but because I really hadn't thought about the issue. That's the way it's been taught for a long time, it was certainly the way I learnt it. I hadn't thought to question it. I had a pretty good guess why, but I wanted to confirm it through some research. And once I had the answer, I also understood Ariel's broader issue with the mathematical notion of limits and was able to resolve it. Ariel's question completely changed the way I explained calculus. In particular, I now use infinitesimal numbers when I teach limits (i.e. you can get to within an infinitesimal number of the value).
My humanities students' input often leads me to fresh ways of looking at mathematics. Robert taught me to think more visually than I used to about graphing. David showed me a better way of chunking down the material on integrals, and from Rebecca I learnt a completely new way of viewing some of the fundamental concepts, such as trigonometric functions, the number “e,” and infinity.
As I work to hone their analytical skills, they work to improve my teaching skills. They are teaching as I teach, and I am learning as they learn.