Algebra Redux

Algebra Redux

By Gene Wilburn

“On Algebra — We’re a month into it, and I’m planning to start a real protest movement, one to have X and Y removed from the alphabet. Z is also suspect as far as I’m concerned…Damn it! They put a man on the moon; can’t they find some way to end the scourge of Algebra?” ~ Huston Piner, My Life as a Myth

“I can explain to you why algebra is useful. But that is not what algebra is really for.” He moved his fingers gently on my temples. “It’s to keep what is in here healthy. PE [exercise] for the head. And the great thing is you can do it sitting down” ~ Mal Peet, Tamar

It pains me that math gets so little cred as a form of mental satisfaction. Oh, everyone respects it, in the way you respect quantum physics — good stuff, yup, the stuff of the universe, yup — just don’t get it near me! Crikey, there’s a horse that can count better than I can. I mean, are you serious? Mental satisfaction?

I don’t mean the satisfaction of the accountant whose books balanced, though there is likely a smattering of it there as well. I mean the pure pleasure of climbing the mountain of numeric relationships and reaching an understanding, and a point of view, you never thought you could. It doesn’t have to be advanced calculus. Mere algebra will do.

But first, a disclaimer. I really like algebra, though I’m aware that many of my friends would consider this a personal failing on my part. Yet when I was in grade school I deeply disliked math because I found arithmetic, such as doing long division on paper, indescribably boring. And frustrating, because I’ve never been very good at arithmetic. I’d nickel and dime myself on tests, making little arithmetic errors here and there even though I knew how to solve the problems. And I’ve never been very solid on the times tables either. Some part of my brain just doesn’t take to arithmetic, and those were the days before electronic calculators were invented. All arithmetic was done by hand.

So, what happened to make me like algebra? Two things: the math itself, and a fantastic high-school math teacher. When it’s introduced to you by an articulate, witty and cool teacher, algebra becomes almost electric.

I just got it, right off. It was ‘a’, ‘b’, ‘c’ and ‘x’, ‘y’, and ‘z’ used symbolically to store values. Just like numeric variables in a computer language (though I didn’t know about them at the time). That plus the sheer power of the equal sign (=). If you can determine equality in some mathematical relationship, you can then solve for its components.

And you can do anything to the equation and it’s still true as long as both sides get the same treatment. Factoring is just a way of simplifying the equation. And sometimes when two equations are related, you can work out the variables in common based on some tricky, but nifty logic. And then there are inequalities like “<”, “>”, “=<”, or “=>”, not to mention getting involved with “nots” and “ands”. Talk about sharpening up your logic circuits.

Of course algebra was considered an essential skill for anyone going into science or engineering. I thought, like many others in the late 50s, that I was Cape Canaveral bound as an engineer or scientist. There were two kinds of teenage boys when I was one: those who wanted to be James Dean, and those who wanted to be a younger version of Werner Von Braun. Alas, I was one of the latter. I was not one of the cool kids.

But, it was not to be. As Mae West said, “I used to be Snow White, but I drifted.” I spent a year in engineering, studying calculus among other challenging studies, but the tidal currents of my mind pulled me to the Humanities, especially literature, philosophy, and art, and I never saw advanced math again. At least not for many years.

Then I retired. Casting about for things that could challenge my brain to keep it healthy during retirement, I began doing sudoku and crossword puzzles, and although I liked them, I didn’t feel I was getting the right kind of challenge. A small tendril of memory teased my brain into thinking about math again and I began to wonder if I could get back into it. So I tried it.

My favourite book series in math and engineering is the Schaum’s Outline series. They’re terse but full of problems to be solved. They guide you through the first steps in solving equations after every new concept, then leave you on your own, only providing the correct answers. You learn a lot trying to reason out why your answer isn’t right, and feel good when you’re on target. It’s totally hands-on learning. So I started with Schaum’s Outline of College Algebra, regaining familiarity with math and undoubtedly growing a set of new neurons.

What I noticed, parenthetically, was that the more I studied algebra, the sharper my mind felt. It’s as if my brain highly welcomed a return to this side of its operations. But then, after a couple of years, for reasons I cannot remember, I drifted again, and quit studying math.

Recently I’ve begun to feel mentally sluggish, beyond just forgetting things and wondering why I’m staring at the pantry shelves. I’ve started to feel not as sharp. Then another new tendril of thought got through to me — I think tendrils may be how the subconscious communicates with the conscious mind — and I remembered how good studying math made me feel.

And so, the arithmetically clumsy me is going to return to studying algebra. Completing the square, here I come.

Slide Rule (I)


Autumn 1959.

I’d never been truly inspired by a teacher until I entered high school (Lyndon, Illinois) and got Mr. Buikema for general science and algebra.

He was a newly minted teacher from Northern Illinois University at Dekalb. With his blond crewcut, dark horn-rim glasses, and enthusiasm for science and math, he epitomized the collegiate, scientific look of the late 50’s. Not to mention he was a fan of the Kingston Trio, my favourite music group, and he had built his own Heathkit amp which he connected to an AR-3 speaker (this was before stereo). I idolized him.

I liked science and enjoyed his lab, but what really excited me was his algebra class. I’d disliked arithmetic because it was so boring, but algebra — it was a new language, the beginnings of real mathematics. I was hooked. The symbolic notation and equations appealed to me from the start. Factoring seemed intuitive and beautiful. I loved story problems of the type “if Train A is going west at X miles per hour, and Train B is going east at Y miles per hour, etc., how far will each have traveled when they pass each other.”

One day while we were working on our algebra exercises, Mr. Buikema pulled a slide rule from his leather briefcase to do some grade calculations. I was smitten. I’d heard of slide rules but I’d never seen one up close, much less met anyone who knew how to use one. The slide rule, or “slipstick” as it was affectionately called, was the icon of science and engineering. Wernher van Braun, the German, then American, rocket scientist was photographed using one.

After class I asked Mr. Buikema if they were hard to learn, and he said if I picked up an inexpensive one, he’d teach me how to use it. The next week my Mom bought me a plastic 10-inch slide rule for $1.99 at Walgreens. It had the most-often used scales and while it lacked the precision of an engineer’s metal or laminated bamboo slide rule, it was more than adequate as a learning tool.

Mr. Buikema taught me how to do multiplication, division, squares and cubes. It was my first calculating machine, and I loved its elegance and what would be called, a half century later, its geekiness. To my delight I found I could use the slide rule effectively in solving physics problems, calculating forces on an inclined plane, and velocities of objects in motion.

I had joined the Space Age.

Taking Stock: Facing 2012

iPhone Selfie

I hope you all had a good Christmas and New Year season!

Traditionally New Years Day is a time for resolutions that will largely be unkept in the months that follow, so I’ll refrain from making any. Besides some of them are ongoing no matter what time of year: lose weight, exercise more, write more.

Looking back to 2011, I’ve had a Macbook Air (11″) for a year now and it’s so slick and useful it still feels new. As such it’s an incentive to get down to the task of writing just so I can use it. I enjoy my technologies, but it’s been a long time since one has stayed so fresh. Kudos to Apple for another brilliant design and execution.

There are rumours of a new iPad in the works some time 2012. If it turns out to be true I might be ready to pick one up. I gave my previous one to Marion after getting the more writer-friendly Macbook Air, but I confess I miss the iPad experience. I get a miniature version of it with my iPhone 4 but it’s not the same without the large viewing screen.

As I’ve mentioned in the past, I’m a fan of podcasts and I’d like to pay tribute to my two favourites: I Should Be Writing, by Mur Lafferty, and Brain Science Podcast, by Dr. Ginger Campbell. You ladies have allowed me to listen in on hours of intelligent conversation. Thank you.

I have a couple of directions I may take my writing in 2012. One idea I’ve been kicking around is putting together a series of personal essays into a Kindle book. The other is to write on a couple of subjects that interest me, but as extended feature articles that could be published as Kindle Shorts.

I don’t have any special photo projects in mind for the year. I’m content to carry a camera around with me and take shots of this and that as I see things. I plan to post a new photo on my Flickr photostream every day, if possible. The camera in my iPhone 4 increases my odds of meeting this goal.

One of the things I may do more of in 2012 is post short reviews of books I’ve read. My current reading is Intellectual Curiosity and the Scientific Revolution, by Toby E. Huff. I’m about 25% into it and already it’s shaping up as the best science book I’ve read in the past year.

Currently listening to The Harrow & The Harvest, by Gillian Welch. Indispensable if you like a traditional folk sound.

My other two goals for the New Year are to study more philosophy and mathematics. I’m nearly ready to tackle my Algebra II course and I have a good Teaching Company Great Lectures course Modern Intellectual Tradition: From Descartes to Derrida that I’ve started. Staying intellectually active is less a goal than a deep-seated need. I suspect it’s the same for you.

I look forward to seeing and hearing from friends in 2012. May your 2012 be a wonderful year.

Algebra Redux

Basic Algebra Review (by StarbuckGuy)

When I entered the 9th grade, age 14, I left behind the boring world of arithmetic and entered the new world of algebra. I was transfixed. It switched on something in my brain that validated all my thoughts about studying math and science. I felt I had arrived.

The experience was heightened by lucking into a great teacher at just the right time. My algebra teacher, Roger Buikema, was fresh out of university on his first teaching assignment. He was late-50’s, early-60’s cool, sporting a crew cut, black-rimmed glasses, and a fondness for the Kingston Trio. His enthusiasm for math and science was infectious.

Algebra was the most elegant thing I had yet encountered in school, and even now, fifty years later, the study of algebra, then advanced algebra, remains as one of the highlights of my education. I was never a math wizard or math precocious, but I liked math and did moderately well at it.

I later studied plane and solid geometry, and trig, which I also loved. I dipped my toe into calculus, but by the time I reached that point in my study of math, my primary interests had shifted to folk music, girls, and literature. I always regretted I hadn’t taken math a little further.

A few weeks ago it occurred to me (I’m a slow thinker) that there was nothing preventing me from restudying math. I was already doing puzzles — crosswords and sudokus — and math exercises are the same: puzzles to be solved. I thought I’d start with trig, but when I looked at the subject anew, I realized my math infrastructure had rusted out with age and disuse. I needed to rebuild the scaffolding in my brain by backing up and starting with algebra.

In my brief one-year stint as an engineering student, I had discovered the Schaum’s Outlines series of publications on math and engineering topics. They were excellent as supplementary material to the main textbooks, and served as inexpensive tutors. I liked them because they were clear, but terse. They didn’t muck about with long explanations of things. They gave the essential information then presented a bunch of exercises and review exercises. They also provided the answers so you could check your work as you went.

Just the thing, I thought. I checked and the Schaum’s Outlines are still going strong, so I sent for Elementary Algebra. The covers are much slicker than they were in the mid-60’s, but the content is much the same.

As I started my review, I’ll admit outright that my brain hurt from working on the exercises. I could no longer “think math” the way I once did. I kept mucking up signed number operations and even simple arithmetic. But I’ve kept at it, relying on brain plasticity to work up some new neurons or perhaps reassign some old ones to get me back into math think. (Note to brain: I really don’t need the complete lyrics of all those 50’s rock songs in my head. Perhaps you could reassign them?)

Slowly, it’s beginning to work. Each lesson becomes a little less strained. I picked up a nice Texas Instruments scientific calculator for $20 to use to check my work, as well as to learn its higher functions. Back in the day I used a Post Versalog slide rule. Calculators hadn’t yet been invented.

As of today I’m working on the review exercises for chapter five, “First Degree Equations.” Chapter six, “Formulas,” appears to be a review of plane and perhaps solid geometry.

I won’t say it’s been easy, but it’s been worthwhile. Some of the excitement of a long-ago fourteen year old is returning. Each exercise is another puzzle to solve. They feel good when I get them right with no errors. They make me think hard when I get them wrong and have to backtrack through my work to see where I went off the rails. That’s good exercise too.

My brain’s capacity for math is improving daily. Not that it’ll help me tie my shoelaces in the morning, but I’ve recovered a lost love. I have no idea where this might lead, if anywhere, but the journey is stimulating.

Basic Algebra Review (by StarbuckGuy)

Pi Day


Until I saw this beautifully crafted Google logo in my browser, I didn’t know it was Pi Day. In fact, I didn’t know there was a Pi Day. An official holiday in the US.

Evidently it is celebrated on March 14th because the first three numbers of Pi are 3 1 and 4. March 14th also happens to be Einstein’s birthday, so the two are sometimes celebrated together.

Pi, known for centuries as Archimedes Constant, is a critical, irrational constant used in mathematical calculations in geometry, trigonometry, physics, chaos theory, and probably many other subjects.

This was a timely bit of serendipity for me. I’ve lately been enjoying a new Teaching Company course called Chaos, and I’ve commenced a review of Elementary Algebra, both of which I may blog about in the near future.

So let us all celebrate these two coinciding things: pi and Einstein’s birthday!

200px-Pi-symbol (by StarbuckGuy)