An Axiomatic Approach to Algebra and Other Aspects of Life

Not many days pass that I don’t think a time or two about James R. Harkey. Mr. Harkey was my high school mathematics teacher. He taught me algebra, geometry, analytic geometry, trigonometry, and calculus. What I learned from Mr. Harkey influences—to this day—how I write, how I teach, how I plead a case, how I troubleshoot, .... These are the skills I’ve used to earn everything I own.

Prior to Mr. Harkey’s algebra class, algebra for me just was a morass of tricks to memorize: “Take the constant to the other side...”; “Cancel the common factors...”; “Flip the fraction and multiply...” I could practice for a while and then solve problems just like the ones I had been practicing, by applying memorized transformations to superficial patterns that I recognized, but I didn’t understand what I had been taught to do. Without continual practice, the rules I had memorized would evaporate, and then once more I’d be able to solve only those problems for which I could intuit the answer: “7x + 6 = 20” would have been easy, but “7/x – 6 = 20” would have stumped me. This made, for example, studying for final exams quite difficult.

On the first day of Mr. Harkey’s class, he gave us his rules. First, his strict rules of conduct in the classroom lived up to his quite sinister reputation, which was important. Our studies began with a single 8.5" × 14" sheet of paper that apparently he asked us to label “Properties A” (because that’s what I wrote in the upper right-hand corner; and yes, I still have it). He told us that we could consult this sheet of paper on every homework assignment and every exam he’d give. And here’s how we were to use it: every problem would be executed one step at a time; every step would be written down; and beside every step we would write the name of the rule from Properties A that we invoked to perform that step.

You can still hear us now: Holy cow, that’s going to be a lot of extra work.

Well, that’s how it was going to be. Here’s what each homework and test problem had to look like:

The first few days of class, we spent time reviewing every single item on Properties A. Mr. Harkey made sure we all agreed that each axiom and property was true before we moved on to the real work. He was filling our toolbox.

And then we worked problem after problem after problem.

Throughout the year, we did get to shift gears a few times. Not every ax + b = c problem required fourteen steps all year long. After some sequence of accomplishments (I don’t remember what it was—maybe some set number of ‘A’ grades on homework?), I remember being allowed to write the number of the rule instead of the whole name. (When did you first learn about foreign keys? ☺) Some accomplishments after that, we’d be allowed to combine steps like 3, 4 and 5 into one. But we had to demonstrate a pattern of consistent mastery to earn a privilege like that.

Mr. Harkey taught algebra as most teachers teach geometry or predicate logic. Every problem was a proof, documented one logical step at a time. In Mr. Harkey’s algebra class, your “answer” to a homework problem or test question wasn’t the number that x equals, it was the whole proof of how you arrived at the value of x in your answer. Mr. Harkey wasn’t interested in grading your answers. He was going to grade how you got your answers.

The result? After a whole semester of this, I understood algebra, and I mean thoroughly. You couldn’t make a good grade in Mr. Harkey’s algebra class without creating an intimate comprehension of why algebra works the way it does. Learning that way supplies you for a whole lifetime: I still understand it. I can make dimensioned drawings of the things I’m going to build in my shop. I can calculate the tax implications of my business decisions. I can predict the response time behavior of computer software. I can even help my children with their algebra. Nothing about algebra scares me, because I still understand all the rules.

When I help my boys with their homework, I make them use Mr. Harkey’s axiomatic approach with my own Properties A that I made for them. (I rearranged Mr. Harkey’s rules to better illuminate the symmetries among them. If Mr. Harkey had been handy with the laptop computer, which didn’t exist when I was in school, I imagine he’d have done the same thing.)

Invariably, when my one of boys misses a math problem, it’s for the same stupid reason that I make mistakes in my shop or at work. It’s because he’s tried to do steps in his head instead of writing them all down, and of course he’s accidentally integrated an assumption into his work that’s not true. When you don’t have a neat and orderly audit trail to debug, the only way you can fix your work is to start over, which takes more time (which itself increases frustration levels and degrades learning) and which bypasses perhaps the most important technical skill in all of Life today: the ability to troubleshoot.

Theory: Redoing an n-step math problem instead of learning how to propagate a correction to an error made in step n – k through step n is how we get to a society in which our support analysts know only two solutions to any problem: (a) reboot, and (b) reinstall.

It’s difficult to teach people the value of mastering the basics. It’s difficult enough with children, and it’s even worse with adults, but great teachers and great coaches understand how important it is. I’m grateful to have met my share, and I love meeting new ones. Actually, I believe my 11-year old son has a baseball practice with one tomorrow. We’ll have to check his blog in about 30 years.