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.

The Value of a Picture

 The Value of a Picture 

By Bobby Neal Winters

On a Saturday in early September, in the entrance to Timmons Chapel on the campus of Pittsburg State University, I had a vision.  It was a flash of movie dialog going through my head:

“You come to me on the day of my daughter’s wedding, and you ask of me a favor.’

It was my wife asking me to put together a tripod that was to hold a poster that contained pictures of the bride and groom as babies with baby food smeared across their faces.  

On one hand, it was a worthy thing, well within my skillset, all the parts were there, no tools needed, no glue necessary.

On the other hand, it was less than an hour to the wedding; I was dressed in a coat and tie; and I didn’t have a work bench.  All I had were the “directions.”

(In my wife’s defense, she hadn’t known that the tripod required actual assembly or we would’ve assembled it ahead of time.)

Let me say this: The directions weren’t useless; they were worse than useless. They were written on a tiny piece of thin paper, and consisted of tiny pictures with tiny labels. Connect part G to part H, when you can’t tell what parts G and H are from the instructions.  This sort of thing continued until the last, and perhaps, only useful line of the directions:  Look at the picture on the box and put it together.

So, after 15 minutes of fumbling, I looked at the picture on the box and put it together, with my wife and two of my daughters (not the bride who was otherwise engaged) helping.

A picture is worth a thousand words just as they say.

Description of physical objects is difficult.  When I was doing mathematical research in the topology of 3-manifolds, I sweated blood in describing the pictures I saw in my head.  Was I right in my descriptions?  The first time anyone reads one of my papers, we might know.

But I am not the first person to have had this difficulty, far from it.  This is a problem that goes back literally thousands of years. To know this is true, all you need to do is read Chapter 26 of the Book of Exodus where the construction of the Tabernacle (the tent the Children of Israel used to worship) is described.  If you have a Bible near to hand, I would encourage you to take it in hand, open to that chapter, and read a while.

It reminds me of an exchange from “Paint Your Wagon”:

“You should read the Bible, Mr. Rumson.”

“I have read the Bible, Mrs. Fenty.”

“Didn’t it discourage you about drinking?”

“No, but it sure killed my appetite for readin’!”

My point isn’t to discourage you from reading the Bible but to emphasize how difficult it is to describe reality without pictures.  Even using a lot of words and taking admirable care in description, describing the Tabernacle was hard.  

Can  you make one from reading it?

I am sure you can, maybe several of them that look quite different from each other.  Drawing a picture of the Tabernacle would’ve been helpful with a list of parts and tools needed, but that idea hadn’t come yet apparently.

But things like tents aren’t the only challenge.  A lawyer came to Jesus asking him how he might inherit eternal life.  A discussion ensued in which it was suggested the lawyer should love his neighbor as himself.

This sounds like a simple rule, but the lawyer asked for clarification.

Jesus responded with the story of the Good Samaritan.

A story isn’t literally a picture, but it serves a similar purpose.  It builds a picture in our minds, a picture we can relate the words, the concepts to.

The poster I referred to at the beginning--the one on the tripod--had pictures of the bride and groom with baby food on their faces.  It reminded us that these two responsible, grown-up people were once babies in our arms. To some degree, they are still babies in our hearts, but as parents we need to now let that go.  While that connection can never completely be overcome, we have to know that they must be more important to each other than we, the parents, are to them.

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.

 It’s a title

By Bobby Neal Winters

I’ve been very lucky.

Given that start I could go on to talk about a number of things: My wife; my children; my job.  It could be quite a long list.

But what I am thinking about right now as I write this is my education.

I went to a high school that was small enough that you could know just about anybody, and more than you wanted to in some cases. The universities I attended were not the most prestigious in the country or even the state, not even the state of Oklahoma, but they were places where I could be successful and not get my ego crushed.

I read somewhere that was very important.  I don’t remember where, because I only went to small schools, but that doesn’t bother me.

When I went to graduate school, I was lucky enough that school was my entire life.  I got up in the morning, ate breakfast, and went to teach.  There were mornings when I taught classes at 7:30am. I then went to the classes I was taking the rest of the morning, and I worked on my homework throughout the afternoon and evening.

When you transition from masters’ work to doctoral work, your classes get smaller and smaller until you are the only one in them.  Then, working with your advisor, who is in some sense your only teacher, you choose one homework problem which neither you nor your advisor know the answer to.  You then work on the problem, not knowing whether it even has an answer, until you are done.

This changes you. Not necessarily in good ways.

I do have a doctorate, but I don’t introduce myself that way because a doctorate is a wall. I am already an introvert, so I don’t need another wall around me.

But, like I said, it changes you.  Not the degree, the process. It’s like boot camp in the military, but spread out over a long time.  I don’t mean to say that at any point we had someone like Louis Gossett Jr screaming at us like he did to Richard Gere in “An Officer and a Gentleman,” but some of us have our own little personal Louis Gossett Jr in our heads because of it.  It was laid on in thin, semester-thick slices.

I am in a profession that has titles.  Those titles carry meanings with them: Professor, Doctor, Dean, Chair, Provost. They come down to us from medieval times.  They carry with them expectations accreted over the intervening tim, and when we come into them, we bear the weight of those expectations.

Or maybe, I should say, we try to.  Or some of us try to.

Some try to take advantage.

But there are negative expectations that come along with titles as well.

When people change the way they act around you--one way or the other--because of the title, you have to remember that it’s not about you.  It’s the title.

I’ve seen this in a number of different areas with a number of different titles: Father, Reverend, Doctor, Colonel, Professor.

I listed “Father” first because it is the oldest of those listed. It’s been around for a couple of thousand years and comes with a couple of thousand years of expectations. If a writer introduces a character with the title “Father” there are numerous expectations that are set up which can either be met or subverted.  This is true with the protestant “Reverend” as well. This evokes expectations which are different, sometimes subtly, sometimes not.

The same is true to each of these titles.  The human being steps into them though some sort of rite of passage, but having done so they get the rights, benefits, and responsibilities that have been created by the collective of all who held the title before them.

It’s like a little mask you put on.  The Romans had a name for the little masks that actors used in the theater of their time: Persona.

I can use this as a teacher.  When I am in front of my classes, they react to me according to that persona, that title. I am smarter/weirder, wiser/bigger jerk because of it. It builds a set of expectations that you can either try to inhabit or try to subvert as is best for the teaching process.

Along the way, one of the things that has given me insight on this is attending the ROTC commissioning ceremony. This ceremony is brilliant in its simplicity.  There comes a point where the newly commissioned officer receives his first salute.  He gives a sergeant a coin and receives a salute in return.

The sergeant is paid to do it. It’s his job. He is not saluting you; he’s saluting the rank.  No doubt many an officer goes on to earn the respect through his actions, but the respect he (or she) is given to the thousands before who’ve inhabited the rank.

One of the benefits of growing older is learning that everyone at base is just a human being. Title or no title, you grow-up, you grow old, and you go on.

And if you’ve been as lucky as I’ve been with my wife, my children, and my job, you get to be happy and learn a few things.

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.

The Calculus of Cousins

 The Calculus of Cousins

By Bobby Neal Winters

When I write a test, I type it out on a word-processor (it’s actually a text-editor, but that’s a different rabbit hole for a different day) and I save it on my computer. I put the different classes in different folders.  Within each class’s folder, I make a new fold for each year and I name that using the date.

I am teaching Calculus II this year; it is my practice in Calculus to give the students a test every other Friday, so during the first week of class, I looked up my folder: The last time I’d taught the course was in 2003.

I told this to my students and asked them.  Were any of you yet born in 2003? Raise your hands.

Nary a hand. 

During the span of time since I last taught Calculus II, these students had come to term in their mothers wombs; they had learned how to walk, learned how to talk; gone to pre-school, to grade school, to middle school, to high school; they had learned to drive.

They had done all of this since I last cracked a Calculus textbook with intent to teach.

This should be fun, I thought.

And it has been.  I spend a few odd hours during the day working on problems.

For those of you who teach Calculus, we are using Thomas’s 15 edition Calculus book.  This is special to me because my cousin Gary gave me his 3rd edition of Thomas’s Calculus that he’d used in college in 1961, a year before I was born.

Gary was the first grandchild of Grandpa Sam and Grandma Lora.  Lora was in her mid-thirties when he was born. Don’t do too much math here.  I was Sam and Lora’s last grandchild, so Gary and I were bookends, as it were.

He gave me his copy of Thomas when I started working on the math major with the warning that there was a lot of “blood, sweat, and tears” in it.

In that he was right.  There is a lot of all of that in learning any skill, and mathematics is a skill to be learned.

There was also a lot of coffee in it, for Gary, in particular.  This I know because there are coffee stains throughout the book.  If you drink coffee while studying it is axiomatic that you will spill some of that coffee on your book.

I love that about the book.  I took it to class, and showed it to my students.  It was printed in 1961 and copyrighted in 1951.

Gary would occasionally email me.  He wrote his emails in all caps.  This is because he spoke in all caps.  I don’t mean to say he yelled. His words just carried a weight that required capitalization.

Gary was born in the oil field; served in the military; educated by the GI bill.  He’d worked in aerospace engineering, and transitioned to having his own business of buying, fixing, and selling used airplane parts.

I can still hear him saying, “The people who sell to me are happy, because I give them money; the folks who buy from me are happy because they are getting the part cheaper than they could buy a new one; and I am happy because I am making money.”

Only when he said it, it would’ve come out of his mouth in all caps.

This strikes me as the best business philosophy ever. It makes me proud to have had him as a cousin and proud to be an American.

He helped me.  He reached out to me.  This was at a time when I was not in a position to help him, nor would he have needed my help.

This is a debt that is owed, and the answer to this is known: Pay it forward.

Fortunately, I am in a position where there is ample opportunity to help people, to help young people.

I am blessed because it is my job.

My business model is different. I don’t sell airplane parts; I sell knowledge. 

I get to keep the knowledge that I sell.  In fact, I know it better by virtue of having to teach it.  While money does change hands at some level way above me, the real price they pay is the one my cousin said, “Blood, sweat, and tears.” Somebody might’ve said it before him.

Gary now sleeps with our fathers, as do more and more of Sam and Lora’s grandchildren.

The world is different.  Textbooks are now online.  I don’t like it. It’s unnatural. Try spilling coffee on the Cloud.

But it’s a new semester, and there is Calculus to be learned.

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.

Moving Dirt for Track and Field

 Moving Dirt for Track and Field

By Bobby Neal Winters

Part of my daily routine this summer was to walk from my office in Yates Hall on the PSU campus out past the Crossland Technology Center and back.  This became increasingly difficult over the course of the summer because of all the construction.  

They are fixing the steam tunnels on campus. It’s one of those jobs that will be completely unnoticed by most when they are done, but it needs to be done. By way of contrast, between the Bicknell Family Performing Arts Center and the Crossland Technology Center, they are building a facility for track.

This has been fun to watch.

First they were in with a few dozers; then they put a big fence up around it to keep the “curious” from hurting themselves and suing everybody for their own "curiosity." Then they got busy moving dirt around.

And they did move some dirt, let me tell you, and in a surprising way.  Before the construction, when I looked at this from the north and from the south the field the track is going to occupy appeared to be level.  It turns out they had to move quite a bit before everything was level.

Then they started laying out the track, and there seems to be quite a bit to that as well.

As an Okie, I thought you would just put a couple of barrels on the field a hundred yards apart and let the runners beat the grass down to dirt as they practiced. 

You learn something new every day.

This is going to be an amazing facility when they finish.

I say this will all the authority of someone who has been to fewer than ten university athletic events in the last 36 years.

I want you to know that because I want you to read the rest of this column with the knowledge that I am not a big sports fan and that what I say is not coming from my heart, but from my head.

This track is an important thing.

Pittsburg State has, by all accounts, a great track and field program. Our current coach is said to be quite good, but I don’t know his name. But I’ve soaked up by osmosis over about three decades that Russ Jewett did an amazing job in building the program over the course of time.  (I can remember Russ’s name because he’d been a math major back in the day. We take care of our own.)

We’ve hosted important trackmeets over the years.  They bring people into town who stay at the motels and eat at the restaurants, etc. That’s good for the community at large and good for town-gown relations.

But there is more to it than that.

According to current estimates, next year the number of high school graduates in Kansas and in Missouri begins to decline.  That means the university will be trying to recruit students from a smaller pool.  That is a problem. 

This new track facility will be something that helps Pittsburg State stand apart from the rest.  Those high school students who are looking at PSU as a possibility for their choice in higher education who have an interest in track will find this facility very attractive.

I will probably never go to see a single track meet, unless one happens to be going on while I take my walk, but I am glad that this facility is being built.

I would like to say that this is all part of a plan by the university “brain trust,” but I don’t know that.  I am out of that loop, and the statement assumes the existence of a “brain trust.”  Maybe it is just the work of the Holy Spirit.

That would be okay too.

But dirt is being moved. A track and field facility is being constructed. It’s all looking good.

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.

 What I did on my Summer Vacation

By Bobby Neal Winters

If you want a Bible verse to go along with this column, I suggest Judges 17:6.

Yesterday I finished a side table that I made for my recliner.  I designed it for that very purpose.  I designed a little drawer in it specifically for the purpose of putting the remotes for the TV into.  I know that they will wind up either on the love seat or stuffed between its cushions, and I knew that when I created the design, but I made it as I made it anyway.  After I’m dead, my children and my grandchildren can say that I made that drawer for the remotes as they laugh and stuff the remotes in between the cushions.

But who knows?  Maybe they won’t have remotes then.  Maybe everything will be voice activated.  Maybe the ability to control electronics will be wired into our brains.  

Maybe television will have passed away along with electricity and electronics, and instead of watching TV they will be watching the fire they built to keep the predators away, and my little table will be there to use for fuel.

You never know.

Like I said, I designed this “table” myself.  I put quotes around that because “table” doesn’t quite get it. It’s got drawers in it, so it could pass as a cabinet.  But it’s got a table top on it.

When I say I designed it myself, I am kind of bragging, but if any professional furniture maker looked at it, he would recognize “kind of bragging” translates to “taking the blame.” 

It is a simple design. I started out by making four one-foot by one-foot squares.  I used half-laps to join together wood that was about two inches wide and about one inch thick.  I say “about” because I didn’t measure any of it.  I’d cut the wood from two-by-four studs I bought from Home Despot [sic].

I took two of these squares and made a cube out of them by joining the top square to the bottom square by four 12-inch spindles I’d turned out of the same sort of wood.  This was one of the factors driving the project: I wanted to turn spindles and use them in a piece of furniture.  

I can now take that off my bucket list.

I took the remaining two squares and connected the top to the bottom with four 12-inch stretchers. I cut half-lap dados in the squares and half-lap rabbets in the stretchers, and did the joining that way.  

At this point, there were two cubes with one-foot sides.

I then glued the top one to the bottom (or the other way around it that works better for you).  

Then--and this is very important--in the bottom, I put in my wooden drawer slides.

At this point, every piece of wood I’ve used has been taken from two-by-four studs.  There was possibly ten dollars worth of wood in it.

It then got put aside for a while.

I got involved in making lean-tos for my potting shed.  I learned a good deal more about turning. (Using a wood lathe is addictive.  To me, joinery is to woodturning, as drinking wine is to taking crack cocaine.)

After a number of weeks, I got back to the “table,” and put some side panels on it.  I made my side panels from the six-foot fence pickets you can get for $2 from my favorite DIY store. I milled them up in my planer and on my table saw. They are cheap, but I find them to be quite pretty. 

I glued them into panels and glued them to three sides, leaving the third side open for drawers.

Remember when I said it was very important to note I’d put in my drawer slides?  When I did that, it determined what should be front and what should be back.  

“Should” is such an important word.

When I glued on the panels, I left the wrong side open for the drawers.  I’d set it aside and had forgotten about it.

At this point, everything was glued in place.  There was no going back. On one hand, I had possibly as much as $15 tied up in this project, and I am including glue in that estimate. On the other hand, I saw a way to fix it. 

So I did.

I put in some more drawer slides but in a different way. Not quite optimal, but it pleases my client.

I built the top using some recovered wood given to me by a dear friend.  I don’t remember which dear friend, but I remember the dearness.  Then I made the drawers out of wood I took from a construction grade yellow pine two by twelve. That is to say, except for the bottoms of the drawers which come from plywood that had been part of the shipping package around my new Grizzly bandsaw. (Yes, I am bragging again: I have a Grizzly bandsaw.)

If I have more than $25 worth of wood in this project, I don’t know how that happened.

The tung oil and shellac that I used to finish this project might have cost more than the wood.  They might be worth more than the finished project, but as I said, the client is pleased, and that is what matters.

For those of you who didn’t look up the Bible verse: “In those days there was no king in Israel, but every man did that which was right in his own eyes.”

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.