By Bobby Neal Winters
These days I spent a lot of time thinking about teaching, about learning, about setting up systems wherein the first will facilitate the second. We want to teach our students certain skills and certain content, but there are other things, things more mysterious that we want to happen too. One of these things is called knowledge integration.
When I was in college in the early 1980s, there was a group of us who were being educated in the sciences. This was on the down-side of the wave that was caused by Sputnik and the Cold War. I still think of that era as the good old days. Sure we were worried about nuclear annihilation, but we were working.
There was a group of use who were all taking the same classes. We would go from computer programming to calculus and from calculus to physics. Occasionally there were those in the group who were more experienced and worldly who would give the rest of us the low-down on how the world worked. It was much like learning about sex. There was the official story that the grown-ups gave us, about doing things responsibly and preparing for the future, that didn’t sound all that exciting at all, but there is the unofficial story from our near-peers which is grittier and somehow more attractive as it is full of all sorts of shortcuts and inside information.
For one thing, if you were in engineering you wanted to be in civil engineering if the democrats were in, aeronautical engineering if the republicans were, and electrical engineering if you weren’t sure, but that electrical engineering was hard. They would also talk about the hard classes, the ones that you should put off as soon as possible: COBOL, Organic Chemistry, and Differential Equations.
These classes were so hard that you begin to hear about them from your college-age friends while you were still in high school. They were the Unholy Trinity of the Sciences. Of these, I took Differential Equations and I took it the first semester of my Sophomore year. I’d had every intention of putting it off, honest, but Ken Brady, who was the Acting Chair of the Department of Mathematics when I started college wouldn’t hear of that. I needed to get it in as early as possible so I could go on to take more challenging courses.
I will grant my near-peers one thing. Differential Equations was one of the most challenging courses I’d had in my life up to that point in spite of having been very well prepared for it. Its one and only thing in common with sex is that you can never understand the experience until you’ve been through it. Indeed, this is even more so in the case of Differential Equations as nature has prepared us for sex in a way that it hasn’t Differential Equations. This having been said, let me try to explain it in non-technical terms.
You start mathematics with algebra. Then you take trigonometry which uses algebra. Then you go into Calculus which in those days was divided into Calculus I and Calculus II. In calculus, you do use some algebra and some trigonometry. There will be sections here and there where you as a student are required to recall some algebraic trivia or some arcane formulas from trigonometry, but those instances are fairly well quarantined from each other. There is breathing room around them. There is time to sit back and say, “Yep, that was kind of hard, but I lived through it.”
Differential Equations is different. To begin with, there is the tacit assumption on part of the teacher that you remember with perfect precision every mathematical activity you’ve ever participated in in your entire life. You know how to solve every polynomial equations; you know how to evaluate every obscure integral; you are comfortable, nay, accomplished with the arithmetic of complex numbers.
I’ve since had the opportunity to teach this course, and I stand amazed at the amount of work that my teacher, Mr. Phillip Briggs, was able to get out of us. The man didn’t have a doctorate, but that didn’t matter. He had the knack of getting us to work. You may remember the character Fezziwig from Charles Dickens’ A Christmas Carol. Scrooge tells the Ghost of Christmas Past, “He has the power to make us happy or unhappy.” Well, Mr. Briggs had the power to make us work our backsides off.
In Differential Equations, you are learning some new concepts, but those new concepts require that you remember some old ones. In algebra, we learn about solving polynomial equations and obtaining their roots. We also learn about exponential functions. In Differential Equations there is a technique where in you use both of those things, plus keep track of some completely new and arcane rules at the same time. Then they throw complex numbers into the mix just for good measure.
Then there is the amount of work involved. In algebra, most problems can be solved in a few lines. In calculus, most can be completed in half a page. In Differential Equations, especially when you start using infinite series, the solution of one problem can literally go on for several pages, and on any line of those several pages, your solution might easily go awry. The text we used had the answer to every problem, so that when you were done with your several pages of work you could check to see in you were right. If you weren’t--which did happen with astonishing frequency--you had to start all over. Which I did, even though--and this was part of Mr. Briggs’ genius--the teacher never took up homework!
I knew there was something special happening at the time, but I didn’t know the name for it and didn’t learn for many years later. I was integrating my knowledge. We all were. We were taking things that we had learned in separate, isolated settings and bringing them together in a new setting. In applying our algebra in a new setting, we were making it a part of a larger world.
To be fair, this was happening in lesser degrees in other courses like physics where we applied math to physical problems, but that didn’t use such a broad variety of mathematics and didn’t use it so intensely.
Differential Equations served as a crucible for the Knowledge Integration, but the work that Mr. Briggs got out of use was the sine qua non. Like Jewel said, “There ain’t nothing for free.”