In 1967, when I was first-year student in engineering, I took physics. To derive a mathematical equation to describe how a pendulum sways back and forth, the professor began with Newton's Laws. Then he write line of line of algebra, crawling down the blackboard, each line representing a step in the reasoning. The climax at the bottom of the board: a surprisingly short equation which, the professor told us, was called the Equation of Simple Harmonic Motion. T is the time it takes the pendulum to swing, L is the length of the pendulum, and g represents the strength of gravity. One implication: the time it takes a pendulum to swing doesn't depend on how hard you push it, just on how long the pendulum is.
The professor then pointed out that this equation doesn't just describe the swinging of a pendulum. The same equation, he said, also describes the vibration of a weight suspended between two springs. And it also describes the flow of electricity in a simple electrical circuit that consists of a battery and three elementary components (a resistor, a capacitor, and an inductance, as I recall) linked together with wire. Then the professor leaned forward over the lectern and asked passionately, "Isn't that beautiful?" I was in the second row and I wrote it all down, concluding my notes with the word 'beautiful,' just in case "beautiful" turned up on the next quiz.
Several months later I learned the same equation again, this time in a calculus course, and the lecturer said the same thing. And "beautiful!" I wrote once again in my notes. And I heard "beautiful" down a third time, two years later, in an introduction to electrical engineering.
Two years later, I had become a doctoral student in management at that same university. I shared an office with Lew Erwin, a doctoral student in mechanical engineering. We had been undergraduates together and were good friends.
"So why did you leave engineering?" Lew asked me one day. I was already enough of a manager to answer a question with a question, so I retorted, "Why did you stay in engineering?"
He thought for a moment and then responded, "Well, take something like the Equation of Simple Harmonic Motion. It describes the motion of a pendulum, and a mass vibrating between two springs, and electricity flowing through a simple RLC circuit, and I think that's beautiful!" What happened to me at that moment has happened again every time I've told the story. And it's happening as I write these words. My eyes teared up, my jaw dropped, and I said in awe, "My god, it is beautiful."
Almost twenty years later, I was listening to a couple of physicists at the University of Maryland, Joe Redish and Jack Wilson, talk with one another. By this time I was a program officer with responsibilities to find and then support innovative work in higher education. I'm afraid my attention wandered after awhile as they talked with one another. One of them said something like, "Yahda yahda yahda equation of simple harmonic motion." And suddenly all of those things that happened in college came up back to me and for the first time I wondered, "Why was it that three excellent instructors worked so hard to teach me something, and failed completely, when a couple years later a simple remark did the trick?"
So I wrote a little paper for myself, comparing two quite different notions about why this might be so.
- Maybe I'd matured, in the way that William Perry once described college students maturing. Perry's research suggested that younger students tend to see the world in black and white terms, with the professors the sources of truth and knowledge. After a couple of years of development, they can conceive that there might be more than one truth but, at this point, they have only themselves as a point of reference. "Everyone has a right to his own opinion," is a comment that's a hall of this stage. Only later, often not until after graduation, can students use evidence and their own values to choose among several 'truths,' using evidence reason to take a stand and to act.
- Or perhaps I'd seen the beauty so easily because, by this time, I'd done research myself and had learned how hard it is to describing something complicated and real in a simple, useful way, and how much harder it is to come up with such a simple, useful conceptual description that would 'work' for three different situations that, on the surface, looked completely different. (By that argument, the way to help freshmen see the beauty is to make sure they do this kind of research when in high school).
I sent a copy to Lew Erwin, by now a professor himself at our old university. Next time I saw him, I asked "So which of these two theories is right?"
"You're wrong," he laughed. "Both your theories are wrong. You learned about the beauty of the equation from me because it was me who told you. I'm your friend. That's certainly how I learned it. At nights sometimes, I would sit on the roof of our fraternity with my friend Phil Abbot, looking at the stars and talking about things like this."
Lew went on, "I've been teaching for a while now and I've figured out that a teacher may be able to teach what to think. But only a friend can teach you how to feel about it."
I think there's some truth in all three explanations. College does help some students develop through a very complex process of reorganizing the ways they understand the world. And research -- research that encourages students to develop explanations and then test those explanations -- can help. But Lew was right, too. Our relations and conversations with friends, often outside classrooms, can change how we feel about things
Lew died young. It was the most horrible of ironies: late one night, as he lay sleeping with his wife, his wonderful heart just stopped beating. I told this story about him, me, and the equation of simple harmonic motion at his memorial service. Ever since, I've told it to others, to let them know what Lew taught me. Please pass it on.
PS. Want to learn about the Equation of Simple Harmonic Motion? Here's one of many sources on the Web.
PPS. What does all this have to do with educational uses of technology? We know that technology, whether that technology is a computer or a piece of chalk, doesn't cause learning. However, technology can serve the cause of learning by enabling people to learn in ways that might otherwise be difficult or impossible.
- So, if you believe the Perry explanation, technology can give students more choices in how to learn, choices that stretch but don't go beyond the student's stage of understanding. That's a strategy that Perry scholars have recommended.
- If you think that research experience was the key, computers and the Internet have vastly widened the scope of meaningful research open to undergraduates and students in K-12 schools.
- And if you think that being with friends is key, consider how to use modern technology to increase the ways in which people can bump into one another, and commune.