The Tangent of a Half angle

 See here for an explanation.

Differential Equations

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.”

Trigonometry

By Bobby Neal Winters

Hurt so good

Come on baby, make it hurt so good

Sometimes love don't feel like it should

You make it hurt so good

--John Mellancamp

Before the seventh or eighth grade I would not have said I was good at math.  I was good at science.  I had a great memory. I once destroyed an encyclopedia salesman who came by our rural home when I was ten or twelve years old.  He pulled out his product, opened it to a page that featured a picture of a skeleton, and began is his pitch to my mom.

“This can help your son with is science home work,” he said.  “Every part of the body has a scientific name.  They can’t just call a shoulder blade a shoulder blade.”

“Scapula,” I said.

“What?”

“It’s called a scapula.”

He looked a little thrown off.

“Or a breast bone a breast bone,” he continued.

“Sternum,” I said.

“Or the arm bones,” he proffered.

“Radius, ulna, and humerus.”

He went down and Momma smiled.

But that is just memory.  I loathed mathematics as I knew it.  Arithmetic was my enemy.  Multiplication was hard.  Long division was almost impossible.  By crying, I manipulated Momma into doing it for me.  There was--and is--some block in my head that keeps me from doing it.  Professionals have told me that I have dyslexia, but I’ve never been tested.

The tide in the war between mathematics and me began to turn in the seventh grade when the teachers began to introduce elements of algebra into class. When I took algebra in the ninth grade, I didn’t consider it a chore any more.  In my Sophomore year, I took geometry, and it was as if the scales fell from my eyes.  It was mathematics without arithmetic.  

I thought I was in heaven.

The geometry class had seniors in it, and I cleaned their collective clocks. This made mathematics very important to my self-worth.  I wanted more of it.

My chance came the next year when Algebra II and Trigonometry were both offered. This was a problem because Trigonometry requires the skill set taught in Algebra II, which at that time included the algebra of rational expressions and quadratic equations both of which are needed in Trigonometry.

My teacher, Mr. Sloan, told me that I was doing things out of order but they would let me.  Trigonometry was only offered every other year because I went to a small country high school that simply didn’t have the staff to offer it more frequently.  If I was going to get my trigonometry in before I went to college, I’d just have to do it this way.

For these reasons, Trigonometry became a crucible for my mathematical education.  It was hard because you do need Algebra II in order to do Trigonometry.  There were times when I’d cry while doing my trig homework, but this time Momma couldn’t do it for me because she’d never had it. I had to do it myself.

Two things helped. One of these was that trigonometry has a high content of geometry.  The confidence I’d developed in geometry carried over.  The other was that I was committed to this.  My self-image and ego were on the line.  I did my Algebra II homework first to get it out of the way, and then I did my Trigonometry homework twice.  

This is something that I don’t often share with students but maybe I should.  Doing your homework is good, but doing it twice is better.  It might even be more than twice as good.  You repeat the skill and reinforce it, but you know where it is going and you do it with more confidence.

Writing an assignment the second time is something I’d avoided before then even though it had been suggested to me on multiple occasions. My handwriting is terrible.  I print almost everything and even that is terrible. This is one of the reasons I am suspected of having dyslexia.  So the reason all of my teachers wanted me to recopy was the very reason I wouldn’t.  It was hard.

This time my ego was so tied-up in the subject I finally took the advice.  Aesthetically speaking, the results weren’t good on the second draft, but they were better than the first.

(As an aside, my teachers had always told me to just take my time with my handwriting.  While there is a lot of virtue to that, the subsequent years have proven to me more was needed than just that.  I’ve made new copies of my lecture notes from year-to-year, slowly recopying everything. The results are legible, but barely, and I certainly never have achieved a “good hand.”  There are limits.)

So my course in trigonometry was a struggle for me.  It was an example of what is called productive pain. Okay, what is it and what’s it good for?

Trigonometry is the study of triangles.  Triangles are geometric objects, but in trigonometry we use numbers and algebra to study them.  There are two major aspect to the course: practical and theoretical.  

The practical part consists of learning various techniques, including  the Law of Sines and the Law of Cosines, in order to measure the sides and the angles of a triangle from known information.  There are certain situations where you can get back a whole lot more information than you put in.  Students, especially those who are of a practical turn of mind, seem to appreciate this part of the course as it can be immediately applied.

They are not so sanguine about the theoretical part of the course.  Those who’ve had the course will know that I am referring to the various identities one is force to learn, manipulate, and prove to be true.  Students don’t like trigonometric identities.  Indeed, hate is not too strong a word to use here.

The proofs that we make students perform in these identities are far from intuitive.  They are like mazes in that students can make a wrong turn and have a hard time recovering from their mistakes.  Why, oh, why do we subject students to such pain, other than the native sadism?

Well, in my opinion, our native sadism is reason enough because this sort of pain is good for you, but beyond that, these identities are, in the long-run far more useful than the mensuration formulas we teach.  First, we have to use these identities to prove the mensuration formulas.  There is no royal road to geometry, Mister, if Alexander the Great had to learn it, then you do too.

But more than that, these formulas will be seen again in calculus.  They make certain otherwise impossible problems easy.  In addition,electrical engineers will probably take a course in Theory of Functions of a Complex Variable, and these trigonometric formulas pop up again there.

Indeed, I encountered formulas of trigonometry as deeply in mathematics as algebraic topology, and that is pretty deep indeed.

But the productive pain aspect of it was by far the most important part for me.  School, research, and life itself are places where being able to endure this sort of pain are vital.  In Trigonometry, I learned how to do that and picked up some cool formulas while I was at it.