Take that, Heraclites

By Bobby Neal Winters

Here’s a quote that has been used to start more boring essays than any other: “A man can’t cross the same river twice, because the second time it’s a different river and he’s a different man.”  That’s from the ancient Greek philosopher Heraclites. I am going to disagree. You can. The second time you notice things about the river before that you didn’t the first time because you may have become a little less boneheaded in the interim.

Take that, Heraclites.

As I may have mentioned in this space before, I am teaching Calculus II again for the first time in more than two decades.  The students weren’t yet born the last time I taught this course, but math never changes.

Math never changes, but people do and the way we teach math should too. To a certain extent.

When I say that people change, I mean two things.  One of these is that students are coming from a different environment than I did when I was a student and the earth had only recently cooled.  

But I, personally, am also different. I’ve developed a much broader perspective than I had the last time I taught the course and, certainly, a much broader perspective than when I first took it.  In addition to this, I will say that I am much less bone-headed than when I first took it.  (For those of you whose jaws just dropped to begin a rebuttal, please note that I did not say I am not bone-headed, but “less” bone-headed.)

I thought I knew everything when I started college.  Well, that’s not quite right. I knew I had things to learn, but I thought I knew the best way to go about it.  I thought working through things the hard way was best, but this caused me to miss the point of some of the things I was taught.

My time away from Calculus II has allowed me to let go of some of the misapprehensions and some of the missed points.  Let me now share the best I can to a general audience what I am talking about.

Those of you who’ve gotten over the traumatic effects of algebra may remember something called the General Quadratic Equation.  I capitalized it just to make it more ominous. As scary as it is to many, the sad thing is that that’s just the one variable version, as it usually just has an `x’ in it.  There is also a two variable version with both an ‘x’ and a ‘y’ in it.  Where the `x’ version has coefficients ‘a,’ ‘b,’ and ‘c,’ the ‘xy’ version has `A,’ `B,’ `C,’ `D,’ ‘E,’ and `F.’  

You can make a two-dimensional picture, a graph, from this version and it will be something called a “Conic Section.” These conic sections are pretty, but sometimes, with the general equation, they come out as askew. (Down home, we didn’t use the word “askew.” We would say “whomper-jawed” instead. “Askew” is shorter, but not nearly as colorful.)

There is a way to fix these equations so that we get the same graph, but it’s all less awkward.  We say we are rotating the coordinate system.

Here’s the thing. The initial process to set up the rotation of the coordinate system is nasty.  There is a lot of algebra.  There is trigonometry. You fill page after page after page.

Let me tell you, when I learned this, I embraced that.  I threw myself into it. I figured that I needed to bury myself in the equations.

And there is nothing wrong with not being afraid to work.

But I did this to the extent that I missed the point of it.

At the end of pages and pages of calculations, you get a couple of very simple equations, and these simple equations get you everything that you need.

I discovered this because I came upon a question for one of my other classes, and it sparked a memory.  I went through pages and pages of calculations, to the point my right hand was cramping, to recover the formulas. 

In doing this, I discovered a couple of things.  One being that I’ve learned how to organize my work a lot better in the last few decades. The other was that these calculations were just a means to an end.  That end was that certain combinations of the `A,’ `B,’ `C,’ `D,’ ‘E,’ and `F’ remain invariant under rotation, and that I could get all the graphs in pretty form from that.

I was dumbfounded.

I was dumbfounded, but I know more now than I did before, even after all these years.

Take that, Heraclites.

Bobby Winters, a native of Harden City, Oklahoma, blogs at redneckmath.blogspot.com and okieinexile.blogspot.com. He invites you to “like” the National Association of Lawn Mowers on Facebook. Search for him by name on YouTube.