The Historical Approach

By Bobby Neal Winters

Introduction: The Historical Approach

As with so many things, it came into my hands through the recommendation by a colleague of a book she had not actually read.  The book was A Radical Approach to Real Analysis by David Bressoud. The thing contained therein was the historical approach to teaching mathematics.  

Mathematicians are by their natures optimizers.  Perhaps the greatest mathematician of all time, Karl Friedrich Gauss (1777-1855), said, “A cathedral isn’t a cathedral until the last piece of scaffolding is removed.”  His personal motto was “Pauca et Matura,” few but ripe.  This is his testimony to the fact that after mathematicians have scaled to the top of the mountain with ropes and spikes and what not, they try to build trails, roads, or even rails roads up for the rest of us.

This is a great service for the rest of us, but in looking at the road, there is a tendency to forget that this is simply the endpoint of a process which as included numerous researchers, teachers, and students.

Claude Shannon (1916-2001) is known as the father of information theory.  In his theory, he imagined information going from one location to another via symbols.  The teacher/learner relationship is a system of communication, but not necessarily in the way one might naively imagine it following on the heels of that sentence. A teacher is a facilitator of the communication process. It is here that I want to thread carefully, because I dwell among those who kill upon hearing the phrase “Not a sage on the stage but a guide on the side.”

In communicating from a transmitter to a receiver, each has his (or her) on set of symbols.  Practical communication theory tells us there will probably be information lost even if both sets of symbols are the same.   In the teaching/learning process, the symbol sets are probably not the same.  The transmitter may be a German scholar who was doing his best work when Napoleon was marching across Europe and the learner might be a wannabe scholar growing up in the Oklahoma oil fields when Jimmy Carter was in the White House.

My point is there is a living system in place that transmits between those two points and more.  The Gaussian attitude of removing the scaffolding is an important part of presenting that beautiful cathedral, but we as teachers must keep in mind that the scaffolding is still out back in the shed.  Knowing that the beautification of mathematical results (or indeed the corresponding acts in any discipline) is an important part of pedagogy, should empower us as teacher-scholars.

Origins in Applications

One piece of scaffolding that often remains hidden is the Primum Movens in mathematics is physics.  I use the word physics here to be broad enough to include engineering.  This Prime Mover has interfered with mathematics multiple times in history.

The Greeks did pursue geometry for its own sake as an intellectual game, but Euclid (circa 300 BC) was known as Euclid of Alexandria.  Alexandria was the city of Alexander the Great who, while not a Greek’s Greek, did spread Greek culture to lands the old fashioned way: by conquering them.  It is not difficult to imagine Euclid and his like codifying the discoveries of those who’d been involved the many construction activities of the Egyptians. In effect, they were creating an orderly way for students to learn the geometry without to actually build the pyramid first.

Fourier Series were developed by Joseph Fourier (1768-1830) for problems connected with the theory of heat. Fourier, while getting great results, didn’t have a theoretical foundation for his mathematics.  Providing those foundations gave birth to new fields and new directions to old.

We like to teach the pursuit of knowledge for its own sake, but knowledge has an end, a purpose.  Mathematics need not teach mental masturbation. It should, in fact, teach the opposite: the delay of gratification.  The delay of gratification, I am convinced, is one of the cornerstones of civilization.

While in the process of learning to delay gratification, it is helpful to the student to know there is some gratification to be had at the end of the process.  The engineering student will endure Heat and Thermodynamics because he knows he must have it to attain his engineering credential.  He will endure Engineering Mathematics because he knows he must have it to pass Heat and Thermo.  He will endure Fourier series because he has to understand them to pass Engineering math.

As a mathematics teacher, I can help him to build that narrative. That process will be aided if I know the story myself in broad terms.  Whereas the details of the story are not a necessary part of my intellectual equipment as a mathematician, per se, they are an important part of my tool box of teacher as scholar.

Bridges

Another value to the use of the historical approach is in helping the student cross the bridge from where he is now to where he needs to be.  One commonality shared by a modern student of mathematics before taking up Fourier series and a historical figure like Fourier he began his work is an ignorance of Fourier series.  We can begin at the same place Fourier did and go from there.

Fourier didn’t spend much time on the theory.  The results he got worked very well for him when they worked. When they didn’t work, well, that could be taken care of later.  

And it was.

The first part of the mathematical program consists of calculus and other courses that are of use to engineers, physicists, chemists, biologists, and economists who are interested in mathematics as a tool rather than as an end in itself.  Mathematicians go through these ourselves because we do value the utility of the subject and want to equip ourselves in its more practical aspects; this is true, but there is more.  In addition, the number mathematics majors is typically too small to pay for a separate track for our majors.

This having been said, our majors get a somewhat skewed view of the subject.  Many absorb the view of a subject that consists of methods that have already been worked-out, methods that they need only memorize and master.  Mathematics is a living subject and our students need a bridge from the civilized, cultural center of the subject across the river of uncertainty to the frontier.

The historical approach supplies such a bridge.

As someone who has lived most of his life in the center of the country, first Oklahoma and then Kansas, I will claim some familiarity in the frontier.  Just like in the movies, there are times on the frontier where the rules are temporarily...um...ignored.  We can see Newton, Euler, and Cauchy doing things we would rap our student’s knuckles for.

Isaac Newton (1642-1727) developed series presentations for sine and cosine.  His work is absolutely brilliant, but it’s also stupid.  This is an exercise today for a freshman using Taylor’s Series.  Seeing the insight behind Taylor’s Series takes half a second, but Newton, as brilliant as he was, didn’t have it.  This didn’t stop Newton, however, as he experimented, found patterns, and verified the patterns to his satisfaction.

Leonhard Euler (1707-1783) played--and I will stick by that verb--with infinite series and obtained tremendous results.  Toward the end of his career, some of his contemporaries thought he was going off the deep end because some of his series didn’t converge.  It was more than one hundred years after his death before the foundations were laid that justified these wild calculations.

Augustin Cauchy (1789-1857) began to put analysis such as was done by Newton and Euler on firmer foundations, but in doing so he was himself faced with the dangers of virgin territory.  In Calculus I, we teach the Mean Value Theorem whose formulation is due to Cauchy.  When we prove it in the introductory analysis course, its prove can be contained in a single paragraph because the modern theory is so well developed.  Yet Cauchy’s proof is much longer and contains some jumps within it that would cause us to paint a student’s paper red: but it’s brilliant!

It is good for students to see that our subject is an adventure.  While as undergraduates, they might not have received the preparation to go into the subjects of current research interests, they have had enough to appreciate the research of the eighteenth century.  Seeing the spirit of that frontier might inspire them to explore current frontiers.

The Lasting Effect of History

As scholars who teach, we should be aware of the effects of history upon us.  We are part of a larger world that has been growing and changing for many centuries.  In many cases we bemoan the publish or perish culture that has grown up around us in modern academe, but, in doing so, we forget that publishing results is our gift to civilization.  Great minds like Archimedes (c. 287 BC – c. 212 BC) sent letters to others whom they thought might understand them.  Many times they played tricks upon their rivals to confuse them.

Others like Niccolò Fontana Tartaglia (1499/1500–1557) solved important problems like the general cubic equation, but kept the results to themselves so that they could pose challenges to others and build up their own reputations.

When Isaac Newton wrote Principia Mathematica he invented the language to describe his new system of the world, and while there might be a jot or a tittle moved here or there, we still use basically the same language today.  I am not sure he was thinking about this three hundred years ago, but the effect of his work, his words still stand.  Without learning history, however, we might not understand that he had to be pushed, bullied, and cajoled into writing his book and he made Edmund Halley pay to have it published for him.

When groups such as the Royal Society and the various journals arose as means of disseminating research results and establishing priority, progress hastened. It is difficult to over-emphasize the value of these steps toward openness and publication.  Yet, history should also show us that we are continually changing.

I believe we need to broaden our ideas of scholarship to include activies which are, in fact, necessary to teaching, learning, and research continuing in our universities.  Ernest Boyer explored this generally in his book Scholarship Reconsidered: Priorities of the Professoriate.  I believe that an exploration of the history of my subject, mathematics, vindicates this idea.  

Research and teaching are inextricably linked. Research discovers the mysteries of creation, but teaching, in a continuum of forms, transmits those discoveries to the ages.  I can have conversations with Archimedes, Newton, Euler, and Gauss because teachers who were themselves scholars have worked to make that happen.  They have learned the language of the countries of the past and have shaped the language of a boy from the Oklahoma oil field so that he may understand, if only just a little.

That is what teachers can do and that is little short of magic.

Elementary Statistics

By Bobby Neal Winters

The Learning Phase

Mathematics and Statistics are separate disciplines. This is something that both mathematicians and statisticians are insistent upon. Statisticians use mathematical tools much in the same way physicists and chemists do, and many mathematicians take statistic for the same reasons they take physics: as an application of their mathematics and to make themselves more marketable when it comes time to look for a job.

My undergraduate degree included a course in Probability and Statistics, but it was very theoretical in nature without much indication of how one might actually apply this in the real world. This is common among undergraduate math degrees. Also commonly offered is a class in elementary statistics that is rather short on theory but long on practical applications.

Fifteen or twenty years ago, as a part of my responsibilities for teaching service courses in the math department, I was assigned elementary statistics. It was a course I’d never had. The first time I taught the course, being a team player at heart, I used the text that had been adopted by the department and, as I taught, staying a few sections ahead of the students, following the syllabus that had been determined by the department.

It was in this way I learned statistics. As I learned it from the book the department was using, that book must have done something correctly, but--and you saw this coming--I hated the book.

Why?

You may be familiar with the French post-Impressionist painter Georges Pierre Seurat. If you are, then you probably know more about him than I do. What I know is that he painted pictures using dots. If you stand with your eye just an inch or two away all you see are dots; if you stand back there is a park or dogs playing poker or something else.

This text was written like that. When I was done teaching the course, I took a step back and saw the whole picture at once, but I was not convinced that any of the students would do that. For them, it was all dots. Or, if I my phrase it another way, it was just one damn thing after another without any connection between them. Nevertheless, being a team player, I continued in this way.

The Synthesizing Phase

In thinking about the course, it occurred to me there were four parts: descriptive statistics, basic probability theory, applications of the Central Limit Theorem, and advanced applications.

The descriptive statistics consisted of the topics that one, for the most part, could cover in a high school or even middle school class: Make a frequency distribution; list numbers from largest to smallest; find the average; find the standard deviation; draw a bar graph; draw a pie chart. Students, even the ones who were ultimately going to fail the class, rocked on this. Among those who showed up for the exam, it was rare to see anything less that a B. This made the second part of the class even more tragic.

Students find what I refer to as basic probability theory to be difficult. Topics in this section can be anything from if you have a can that has 300 red beads and 700 blue beads, then what is the probability of drawing a single blue bead (Yes, it’s 0.7.) to the hypergeometric distribution. This section cleaned the students clocks, which is to say they uniformly found it to be quite difficult.

The portion of the course I describe as “applications of the Central Limit Theorem” is actually the meat of the course. This is the part of the course that our clients in the university, i.e. those who use statistics in their classes, want their students to know. This includes estimation and hypothesis testing.

The final portion of the course consisted of the sample proportion, which is used to estimate the percentage of the vote a politician will get, and the chi-square test, where the question of whether particular models fit is discussed.

Teaching as Literature

After I’d taught the course a number of times, it was clear that it all did fit together like one of Seurat’s paintings, but that the students might be better served if it were taught like a classic work of literature. By this I mean, I thought I should take advantage of certain opportunities to foreshadow concepts.

For example, there is something called Chebyshev’s Rule. It is the most difficult topic covered in the first section. To use layman’s terms, it says that only a small proportion of the data can be very far away from the average. This opens up the opportunity to mention proportion, which has already come up with relative frequency, and foreshadows the central means of hypothesis testing. In terms of literature, it helps to tie the course together.

Spending more time on Chebyshev’s Rule also serves to make that first section a bit more challenging. If it is de-emphasized, as I tried a few times, there is little to give the students that they are now in a college course because so much of the material in that first part of the course is at a middle school level. Chebyshelv’s Rule, by way of contrast, can be challenging in a number of ways.

There are other opportunities to create ties they run the length of the course. I use the jar of colored beads as a recurring example. It’s used when I introduce probability, when I talk about Bernoulli Trials, and when I talk about the sample mean. By this means, and others, the student has a better opportunity to see that the course is unified and that it’s not just one damn thing after another.

Embedding in Time

Shortly after I’d separated the Elementary Statistics course into four parts, I took a year of sabbatical a Brigham Young University. They had a testing center that was very popular with the students. Teachers left copies of the exam with the testing center for an period of time--I think as much as a week--and during that interval of time, students could come and take the exam when they felt as if they were ready. Staff at the center proctored the exam, so pressure to be a policeman was removed from the professor. As I said, the students loved it and I grew to love it, but there was a catch. In order to use it, you had to schedule your exams ahead of time.

Up until that point in my career, I had not. I had exams when I felt we’d covered enough material, and the idea that this could be a predictable thing was foreign to me. In order to use the testing center, I scheduled my exams and discovered that this was not such a difficult thing to do.

When I can back from sabbatical, I scheduled my exams for Elementary Statistics and discovered a number of things. One of them was that the students didn’t complain. Indeed, while my students have never said so directly to me, I’ve read studies that support the idea that students like structure.

More important to me, however, was the discovery of how easy it made everything. Instead of having to weigh the decision of whether to have a test on a particular day, there it was on my calendar! I could make out the damn test and not procrastinate. Once those dates are set, they serve a similar function for the course as the arbor does for a grape vine: the course just grows around them.

Quite frankly, in some very real sense, when the dates are set for the exams and other assignments there is nothing much else for me to do.

Assessing and Modifying

As you’ve seen, course design is a dynamic process. I’ve modified my course as I’ve learned new things. I have changed as I’ve observed that worked and what didn’t. I was doing course design before I new the name for it. I was doing assessment before I heard the word.

That having been said, those words and phrases have power. When we teach a course, there are certain things that we want students to learn or we wouldn’t be teaching the course. Those items are called student learning outcomes. When we’ve taught those things we want to determine whether the student has learned them. This is called assessment.

When we assess, we are assessing a communication channel. The channel has two ends, the transmitter and the receiver. We give a lot of attention to the receiving end of the channel in that we assign a grade of A, B, C, D, or F to the student, and this is right in that it is the student who is paying money to learn and those who hire that student will want some measure of how well the learning has taken place.

But.

But there are two ends of that channel. For my own sake as a teacher, I need to know how well I am doing and I need to change what I do in order to make it better. When I learn what I need to change, I should carry it out.

Technology

Over the last twenty-odd years there have been numerous changes in technology. When I began teaching, I had to walk two blocks across campus to check my e-mail. I did it once a week, and frequently I didn’t have any. Now, God help me if I skip a day of clearing out the junk mail.

Regardless of the downside, technology has provided more tools to reach out to the student. I would like to mention three of these: PowerPoint, lecture-capture, and the learning management system.

Much has been made of PowerPoint and the effect it has had on bullet-pointing the learning process. There is a danger than the medium will affect the message. That is a legitimate concern. PowerPoint is not a panacea. I still use the chalkboard in many classes as I believe that the students need to see the practitioner at his (or her!) craft so that they may be empowered themselves.

If I’ve gotten my hands dirty, then they know they might have to get their hands dirty.

Yet there is so much material that simply needs to be put in front of the student and talked about. This can be done badly with PowerPoint, yes, but it can also be done badly with acetate slides and with chalk on the board. What I put on PowerPoint, I used to put on sheets from a yellow-pad and just transfer it to the board. There is no loss of empowerment and there is a great gain in efficiency.

As far as the bullet-point-ization goes, this is taken care by talking and adding meaning while you lecture over the slides. The audio can be recorded and synchronized with the slides on your computer by what is called--straightforwardly enough--lecture capture. It can be done in an almost effortless fashion and uploaded to the Internet.

On the Internet, we has access to our university’s learning management system (LMS). We can upload the PowerPoints ahead of time for the students to print-off and take notes on. We can put the captured lectures there by them for the students to listen to.

I have my whole course online, organized in four parts. In each part there is a schedule of what I plan to do everyday. All of the deadlines are there. I’ve interspersed quizzes for the students to take online, where they are automatically graded and recorded.

For the students who want to learn, there is unparalleled opportunity. The students who don’t want to learn are forced to be more creative in their excuses. It’s win-win in other words.

The Human Touch

Yet with all of this, we cannot forget the human touch. If we don’t care, the students won’t care. If we aren’t excited, the students won’t be. If we don’t think the material is important, then the students won’t.

In other words, you still have to teach.

A Certain Open Manifold Whose Group is Unity

By Bobby Neal Winters

One thing I’ve learned over the course of my university career is that every discipline is different.  We’ve different interests, different methods, different strengths, and different personalities.  This became clearer to me when I visited a meaning of the Student Advisory Board for the College of Arts and Sciences.  Each of the departments has a student representative and each of those students reminded me of faculty I knew in the department.  I think particular personalities might be drawn to particular disciplines, but I also believe that the activities demanded by a discipline has an effect upon the practitioners.

Pure mathematics is a lonely discipline.  You spent a lot of time at your desk staring at a yellow pad.  You spend a lot of time staring out into space.  You spend a lot of time staring at a black board.  Staring seems to be a key ingredient.

I am under the impression that in some disciplines you spend a lot of time in libraries reading books, taking notes, and copying quotes.  You are interested in a particular topic and you look up things, read them, try to understand them, and create arguments or form a synthesis or something. Regardless, the library is an indispensable element to your research.

I do not believe this is as true for mathematics as it is other disciplines.  I did read some books.  Indeed, there were about three books that I lived with while working on my doctorate.  They were my own personal Torah, Nevi’im, and Ketuvim.  There were also a few papers.  Together these formed not so much the material for my doctoral thesis but the foundation upon which I was to build my thesis.

My thesis topic was decided upon in a bar.  Napkins were taken out as they traditionally are in such places and my thesis advisor outlined two or three problems.  These were problems that my advisor didn’t know the solution to but suspected could be solved.  I then began working on them.

The way I approached this process was to take field trips in my head to look over the mathematical territory.  This is about as good an explanation as I can make at this point.  It is a very visual process.  Where the pictures come from, I don’t know but I can surmise they are products of the mathematical experiences of listening to lectures, reading articles, and imagining.  I cannot over emphasize the imagining part.  What differentiates this from writing fiction--which I have done as well--is that the things you imagine must be translated into statements in written language which can be verified by others independently.

We mathematicians are a bit arrogant.

I thought that last sentence was so true it deserved a whole paragraph by itself.  Within the Academy, scientists are considered arrogant, but I believe mathematicians are arrogant even among scientists, putting aside for the moment whether mathematicians are actually scientists or simply a breed of our own.  One way we express this is by saying that mathematicians can’t get a publication out of a failed experiment.

When mathematicians set out of work a problem, we either solve it or don’t solve it.  If we don’t solve it, we don’t necessarily have anything left over that we can use.  We are making the assumption that scientists can at least publish that a particular technique doesn’t work, but I think that is more a function of our ignorance of other subjects.

In any case, my thesis involved Whitehead manifolds.  This is a class of topological spaces defined by J.H.C. Whitehead in 1935 in a paper entitled “A Certain Open Manifold Whose Group Is Unity.”  There may be some of you whose curiosity is piqued by that title.  It is mysterious.  Every word in the title is an English word which could be used in ordinary conversation, but certain ones have been hijacked by mathematical highwaymen: open, manifold, group, and unity.  

The standard mathematical response to curious enquiry here is “This is technical. You wouldn’t understand.”

I wonder how we got the reputation of arrogance.

The truth is that mathematical research is a two-fold process. The first part is taking an excursion into the imagination I told you about.  The second part is describing that excursion.  You are entering a world that exists within your mind--I will avoid a discussion of Platonic philosophy by not saying “only in your mind”--and you are then having to describe it.  When the explorers came to the new world and found plants and animals that didn’t exist in the old world, they had the advantage of using the Indian name for them.  

There are no natives living in our heads--modulo a schizophrenic here and there--so we have to load new meanings onto existing words.  As we are going to have to use these words in proofs that will have to be verified by others, we tend toward precision.  The art of backing off that precision to explain what is not necessarily a difficult idea to a layman is more difficult that is generally realized. Few mathematicians take the time to learn it.

All of this to say, I am confident enquiring minds such as yours could understand, but I will save that for a later date as it would take us too far afield.

Mathematics is full of people who have spent a lot of time in their own heads and like to play with language. We tend to like making puns though, that having been said, we are not that much fonder of hearing them than anyone else.

The time we spend in our own heads tends to exact a cost in people skills.  Though I have known some mathematicians who are extroverted in the classical sense, there is something to be said for the definition of an extroverted mathematician being someone who stares at the top of your shoes.

A positive trait we have is the desire to take the complicated and simplify it.  Good mathematics is mathematics made simple.  Well-written mathematics is mathematics wherein the mathematician, having made his own journey in perhaps a quite arduous fashion, retraces the territory to blaze a trail so that others can make the trip in safety with conveniently located rest-stops along the way.

The communication aspect of mathematics is, in fact, the most difficult, but it is the most important part. Historically, there were those who did their work in isolation, for their own pleasure, and did not share.  When this practice was common, mathematical progress was slow, but as the sharing of information became more common, progress hastened.  

Mathematicians can transfer their skills to other areas of university life.  They can use their explanatory and organizational skills in lower level mathematics in their teaching.  They can bring the ethic of hard work and ability to deconstruct complex tasks into simply ones to committee work.  They can, but, as with anything else, the challenge is in the doing.

The weakness comes with the tunnel vision that is a part of the focus necessary to do research mathematics.  There is a pronounced tendency to denigrate anything that isn’t math.

Maturity as a scholar within the university community comes with the recognition that the rest of the people there have something of value they are bringing to the table as well and that you can learn from them. This maturity is a goal that is worthy of our aim.

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.

Shall We Gather at the River

By Bobby Neal Winters

Cowboy looked up from the seat of his skiff at the clear, blue sky and felt the chill wind at it whipped past his face.  It was cold, but no colder than would be expected on a mid-December day.  They’d been no snow yet, but the early morning’s frost, now erased by the sun’s rays, had hinted at things yet to be.

Whenever people asked Cowboy what he was doing out on days like this, he said he was out looking for arrowheads along the creek.  It was a believable story as it was the sort of thing that people did in this part of the country.  And today was a good day for arrowhead hunting in any case. There’d been a long, dry summer which had been relieved recently by some heavy fall rains.  With the grass killed out and the erosion of the rains, there might be some new arrowheads exposed.  

And who knows, he thought, I might even find a few arrowheads.  It was the communing with nature, he liked though.  He loved being in the out of doors with only the sky for a roof.

He pulled his skiff up to the creek bank and carefully stepped out.  It was a beautiful day to be out in the open. The only cloud he saw wasn’t actually a cloud; it was smoke rising from a fire up on the hill a short way ahead.  In other places and times he might’ve thought it was hunters, but here and now Cowboy knew it belonged to Wendell Warthind, who’d been a childhood friend of his father’s.  He had a pretty good idea what Wendell was doing too and he made up his mind to ignore it.  Not ignoring it would make life complicated.  He had every intention of following through with this purposeful ignorance, but events weren’t going to allow it.

Cowboy planted his oar in the mud of the bank and tied his boat to it.  

He’d taken one step, maybe two up the bank when he heard the blast.

The sound could’ve been an explosion.  Cowboy knew what Wendell was doing up there and an explosion was one possible outcome of that.  That wouldn’t happen if Wendel was at his best, but he’d been slipping lately.

But Cowboy’s practiced ear knew differently. While it was technically an explosion, it was one that was more precisely characterized as a shotgun blast.  He pulled is 44 magnum from its holster and began running toward the sound when he heard profanity issuing in Wendell’s voice and several sharper explosions which Cowboy recognized as rifle shots.

He came in sight of an opening in the trees where in he saw Wendall hiding behind a large stump over which he was pointing a thirty-ought-six hunting rifle.  In the middle of the clearing was the campfire whose smoke Cowboy had seen. It was beneath a device that Cowboy recognized as a still.

Wendell hadn’t seen Cowboy yet, so Cowboy slipped behind a tree and looked in the direction that Wendell’s rifle was pointed.  The first thing he saw was a jeep with two flat tires and a couple of bullet holes in the side.  Then he saw a pair of feet beneath the jeep.  He followed those feet up when he saw the top of a blond head.    

When he repositioned himself a little, he saw the whole face and his jaw dropped.  The face belonged to the new preacher over at the Pentecostal Church, the Reverend Mosley.

Mary Beth Mosley.

Cowboy turned his attention back to Wendell  It looked to Cowboy that Wendell had a bead on Mary Beth and that if something weren’t done pretty quick there was going to be a-killing.  It was then that Cowboy pulled out his badge and started talking loud.

“Okay, there, Wendell,” he said with his 44 pointing straight at his father’s old friend.  “This has gone far enough.  Put the gun down.”

Wendell was caught off guard.

“What the hell?” he said as he turned.

“Put that rifle down!”

It came with enough force that Wendell put it down.

Cowboy now turned his attention to the Reverend Mosley.

“Mary Beth,” he said. “Throw that shotgun down.”

Mary Beth stood up from behind the jeep revealing a figure that caused Cowboy to curse it being wasted on a lady preacher.  

She had a double barrel shotgun in her hand.

“Oh, Sheriff,” she said brightly, “I am so glad to see you.”

Cowboy didn’t respond to her brightness.

“Put that shotgun down,” he said.  There wasn’t a hint of a smile.

Her brightness dimmed considerably as she put her shotgun down.

“Okay,” Cowboy said. “You come over here.”

As she got closer to him and farther from the gun, he moved toward Wendell, grabbed his rifle, and tossed it into the woods.  As he did this, he noticed that behind Wendell was a glass three-gallon jug of a clear liquid which he appeared to be protecting like the apple of his eye.

Just at the word was forming in Cowboy’s brain, he heard it coming out of Mary Beth’s mouth.

“Moonshine,” she said. “He’s been making moonshine up here.  I figured that I’d bring my shotgun up here and blow some holes in his still, but he was here.  I was trying to shoot that big jug, but I got him instead.  He had it coming, though, because that’s Satan’s brew.”

It was then that Cowboy noticed that Wendall had been wounded in the leg.

“You all right, Wendell?” he asked.

“I’ve been better,” he said.

Cowboy then went to the jug and smelled it.  

Heaven.

He picked up a tin cup from the ground, tipped a little of the jug’s contents into it, and took a swig.

Rapture.

He looked at the jug, he looked at Wendell’s leg, and he looked at the lady preacher’s jeep.  He finished his cup of whisky with a cough.

“Okay, I need to get you all back to town,” he said.  He reached down and picked up the jug.  With it in one hand and his 44 in the other, he said, “Follow me.”

It only took them a short time to get to the river, but the amount and the viciousness of the fighting made it clear that he wasn’t going to be able to leave them alone together.  Once they arrived, another problem became apparent.

“That is surely Satan’s brew,” the Reverend Mosley said. “We ought to just pour it into the river.”

This caused Cowboy to lick his lips as it was the finest tasting brew he’d had in a long time.  There was no way he could let that happen.  He then looked at the creek and his small boat and a new problem occurred to him.  How was he going to get these two people and the jug of whiskey across the river?  The skiff was only big enough for him and one other thing.

If he took Wendell across first, Mary Beth would dump the whiskey into the creek, and Lord Almighty, that would be a crime.  If he took the whiskey across first and left Mary Beth and Wendell alone together, one might kill the other.  There was only one thing he could do.

“Okay, Mary Beth,” he said, “you come with me.”

He put the jug down, sat in the back of the boat, and invited Mary Beth to sit between his knees which--after looking a little suspicious--she did.

It was a cold day, and it warmed Cowboy up--in more ways than one--to have Mary Beth snuggled between his knees as he paddled across.  Once there, he put her out and made the return trip.  He’d been doing a little thinking on his way across and he’d gotten an idea.  When he got to the other side he had a plan.

“Wendell,” he said, “hand me the jug.”

“Ain’t you afraid she’s going to dump it out?”

“Hand me the jug,” Cowboy repeated.

Wendell gave him the jug, and he went back over.  Once there, he got out, put the jug out of the way on the far bank, and turned to Mary Beth.

“Okay, come back with me,” he said.

“Don’t you trust me with the whiskey?” she asked.  The way she was smiling, she new the answer.

“No,” he said.

She climbed back into the boat and he slipped his knees around her again.  This time there was no protest.  Cowboy rowed back over, and when they arrived at the other side, Cowboy, directed his attention to Wendell again.

“Okay,” he said, “you trade places with her.”

This was done with no more problem than an exchange of hateful stares.

When he rowed Wendell to the other side, it wasn’t nearly as pleasant as it had been with the lady preacher.

“Ain’t you afraid I’m going to run off with the whiskey?” Wendell asked when he got out.  

“Not the way that leg is looking,” Cowboy replied.

He then headed back, fetched Mary Beth, and brought her back, all the while regretting that this would be the last time.  When they arrived at the other side, she didn’t seem to eager to get out either.

Wendell broke the spell, however.

“Where do we go now?” he asked.

“My truck’s up there,” Wendell answered. “I think I’ll take you to the emergency room first and once that’s squared away, I think I’ll take Mary Beth...I mean Reverend Mosley home.”

Cowboy thought he saw Wendell’s eyebrow move ever so slightly.

“Is he going to jail after that?” Mary Beth seemed eager to know.

“You never can tell what’s going to happen next,” he said.  “You never can tell.”