The Method of Partial Fractions

 The Method of Partial Fractions

By Bobby Neal Winters

I am teaching Calculus II again after 22 years.  None of my students had been born yet the last time I taught the course.  They are all young, fresh, and full of energy. Their brains are so sharp. They keep me honest on my “arithmetic.”

I put the scare quotes around “arithmetic” because most people would call it algebra, but to a mathematician algebra is something different.  If you are not a mathematician, my telling you what we mean by algebra wouldn’t help; if you are, you already know.

I try to make it easy to tell me when I’ve made a mistake and how to do that kindly because that is a skill that will be useful to them even if they don’t use a single thing they’ve learned in Calculus II.

That does bring me to a question that does come up from time to time: When will I ever use this?

It is a legitimate question.  They--and the Great State of Kansas, bless it from border to border--are paying good money for them to learn this material.  When will they use it?

An easy answer is that I don’t know.  There are a lot of students, and there is a lot of material.  They have different backgrounds, different talents, different ambitions, and different plans.  I don’t know what those plans are, what all of them entail, nor how they might change.

What I do have is a collection of mathematical material that was invented by Isaac Newton and Gottfried Leibniz about 300 years ago, has been found useful by scientists and engineers, and has been refined over the course of a couple of centuries.

It is like a huge toolbox that is full of expensive, well-used tools. I am teaching them that the tools exist and the best way that I know to use them.

To continue with this metaphor, I’ve been having to clean up and sharpen the tools because I’ve had the opportunity to use them for the last two decades.  

Well, that is not entirely true.  While I was in administration, I didn’t continue to teach calculus, but I did continue to teach a course called Introduction to Analysis.  This is a course in which the theoretical foundations of Calculus are taught. To really explain what this means in a way that would satisfy a fellow mathematician would require a lengthy article that not many would read.  Not “many might” be a great exaggeration of the number. For the current readership, let me just say that Newton and Leibniz were scary smart and did things that folks like you and me have difficulty understanding.  The theoretical framework that has been set up makes it accessible to a few more.

Recently, I caught myself with a topic I had not seen at all in the last two decades: the Method of Partial Fractions.  Again, this is one of those things I am not going into detail for the current readership, but folks who’ve had a course in high school algebra (fairly recently) would be able to understand.

Metaphorically, this is a chisel in the toolbox that is Calculus II, and I needed to sharpen mine.  Because of this, I came into the office before church on Sunday morning and spent a couple of hours with the equivalent of a whetstone and a leather strop. It was a yellow pad and a pencil, but a whetstone and a leather strop sounds much cooler.

It brought back a bunch of memories, not all of them comfortable.  Believe it or not, I was something of a know-it-all in my college days. (There will be a pause here to let those who knew me during those days spew whatever they are sipping as they read this through their noses.) I remember when my Calculus II teacher was teaching me this method.  I thought I knew a better way to do it than he did.

To his credit, he said nothing and let me do it my way.  I got the right answers; the math was correct; but it was a lot more work. 

It took me 44 years to figure that out.

I understand why he let me do it that way; I believe it was the right thing to do; I would do the same thing myself in the same circumstances.

But it does make me smile a little right now.

But that is neither here nor there.

The alert reader will notice a couple of things.  The first of these is that I’ve not used this particular technique in more than 20 years and before that I only used it whenever I was teaching it.

These alert readers will also notice that we learn other things while wrestling with hard material:  We learn to wrestle with hard material.

Things will be easy for these students that other people think are hard.

I believe that it has helped me, at least.

So, if you’ve had algebra within the last couple of years, you can bring your yellow pad with you up to me at the coffee shop, and I will tell you more about the Method of Partial Fractions than you ever wanted to know.

Maybe I already have.

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.