Friday, January 14, 2011

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 – 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.


Debra Lilley said...

This is on your blog, which is read by IT people around the world. If it was read by a few teachers you would be inspiring their generation in a time when there is very little appreciation of what they do.

Cary Millsap said...

That would be nice. I'd like to learn how to reach them.

Debra Lilley said...

Ask Alex's teacher, they must belong to a group of some kind, have their own forums, associations. Or just try Google

Simon said...

Excellent. Thanks for this Cary - and of course Mr Harkey...

Rodger said...

Teachers are so important, but we pay teachers so little. It's good to see credit where it is due.

Do you think IT workers and managers could also do their homework, rather than using shortcuts and trial and error methods so common now?

Cary Millsap said...

Rodger, if you want to contribute a lot to humanity and make a lot of money, that's often two distinctly different jobs. And of course I believe the answer to your question is yes.

Massimo said...

Your story and comments about Mr. Harkey resounded Taylor Mali's inspirational poem about teachers:

Clayton said...

Very interesting blog. Finally, a section on your site that isn't too technical for me!

Hope all is well... as a person working in the education industry now I appreciate this post quite a bit!

Allen said...

Cary - I am going to show my very bright, but sometimes stubborn (read - won't write down all of the steps) 13 year old son this post. Thanks for the insight; brillant, as usual!

Cary Millsap said...

Massimo, thank you; I enjoyed that.
Clayton, thank you.
Allen, good luck; I hope it helps!

Born2Conquer said...

Good one Cary, with your permission I have saved the Properties A pdf file and will be sharing it with my son.

Cary Millsap said...

Born2Conquer: Regarding permission, absolutely yes. Use and enjoy.

Joel Garry said...

I did the same as Allen with my 14-year-old. He's in his high school's Honors Math Analysis, the only freshman among juniors and seniors. He's at the top of the class, even impressing his hyper-enthusiastic teacher with a new proof of something in the textbook. I've long had a fear of him being too smart to have the necessary discipline - but that's more my problem than his, I hope.

The response was predictable - that it applies more to those who are learning rather than those who get it. I tried to point out that this was about a disciplined approach to problem solving, without success.

He has several times demonstrated to the skeptical he is able to retain the skills over time, without continual practice, and is second in his class of 667, taking all honors classes (PE excepted, of course). So sometimes you just have to sit back and let them succeed! He sees tattooed 14 year old girls pushing baby strollers around campus, so there is some scary negative motivation there, too.

I could go on about how bad my high school math teachers were, confusing rigidity with discipline, but that would be pointless. I can't tell if Mr. Harkey would have influenced me for good or rubbed me the wrong way. But I can say my kids teachers have been way better than mine were, even in the face of severe (and still increasing) budget cutbacks. I still blame Howard Jarvis for that.

word: droven

Cary Millsap said...

Joel, we've had lots of kids in little-boy baseball who were so big and strong compared to their little-boy peers that they never had to focus on fundamentals. They had so much capacity that they dominated their peers, even when they did things wrong. They had no interest in learning how to do things better, because in their little pond, they were already the best.

But it has proven to be an unsustainable strategy.

Sizes and speeds tend to level out as little boys grow up, and it's the kids with the proper fundamentals who start winning. When it starts, it's an avalanche. It's a lifetime of winning after that. The high-capacity guys with no fundamentals get to have some fun for a few early years, but it's the other guys who get to have the fun for the decades to follow.

The best results, of course, await those who have both extraordinarily high capacity, passion, and the drive to truly master the subjects of that passion (or those passions).

Mahesh said...

Nice article. I printed the pdf so I can use it for my kids in the future as well. Slightly not related but relevant is the Dunning-Kruger experiment.
A rejection of hard work and striving to learn the fundamentals is not only just laziness but ignorance as well and it is difficult to break the vicious circle unless you have a great teacher like you had.

Chiefy said...

Every time I try to download the .pdf file so I can use it with my daughter who is having a hard time with algebra, a message appears saying that there was a "drawing error." Is there somewhere else I can get access to it?

Cary Millsap said...

Chiefly: Not good! If you'll send your email id to me using the Contact Us form at, my colleagues and I will help.

mmcneely said...

Cary, this is funny - I didn't read this post until after I posted on my blog about how your Dallas Oracle Users Group presentations remind me that it's important to honor and remember the basic "knowledge" building blocks of tuning. Each of your presentations make me remember it.

oscrub said...

Your comment re: little-boy baseball analogy really resonated with me. It's true that fundamentals are at the core of everything, yet somehow we always gloss over them to get at the good stuff. I'll be sure to save this pdf for when I have kids of my own. Thanks :)

Steven Hall said...

This is an excellent post, and makes me acutely aware of my failings.

Uday said...

Marvelous post!. You are so lucky Cary, to learn mathematics, with Mr. Harkey. Subsequently sharing the same with the rest of the world, is a sign of a good student in you,as well. I read your post, along with my son and we were excited and moved. Many thanks, Uday Kanhere, Bangalore

Cary Millsap said...

Uday, for you to be moved, ...moves me.

Thank you for your kind words.


pgus said...

Cary, This is great, I am currently in a teaching program after years of being in construction. I am looking forward to adapting this approach to middle and high school social studies. If anybody has any suggestions, I would love to hear. I am also currently coaching wrestling and the little boy baseball analogy will definately be used today at practice.

Les Cargill said...

Cary: Les Cargill here - we were in the same graduating class.

I did not see where anyone mentioned possibly using Kahn Academy to propagate this information. Might be problematic since Kahn is not about teaching teachers, but rather teaching students, but still - teachers *use* Kahn Academy.

Cary Millsap said...

Hi Les! I agree with you wholeheartedly. For me, Kahn Academy has been a force for good in the world and in my home.

jeff6times7 said...

I just reminded Carlie that 4 of the rules are required to solve 30x = 0.8. Fascinating.

Nicholas Stanek said...

I was wondering if you could provide any more information on how this method is applied to actual problems. More of an explanation would be greatly appreciated.

zsoltkormany said...

Cary, could you please make the "Properties A", and/or your modified version of it downloadable again? I'd appreciate it.

Cary Millsap said...

@zsoltkormany, I have just replaced the files on the new web page.

Thank you for your interest.

Cary Millsap said...

@Nicholas, the example I gave, where the problem is “7/x – 6 = 20; find the value of x” is pretty typical of the algebra problems we saw, and, as you can see in the post, it took 14 steps to complete it.

Even a simpler problem like “x + 5 = 10; solve for x” would have produced several steps:

1. x + 5 = 10 // given
2. (x + 5) + (-5) = 10 + (-5) // 18 addition property of equality
3. (x + 5) - 5 = 10 - 5 // 6A subtraction (def.)
4. x + (5 - 5) = 5 // 2A associativity
5. x + 0 = 5 // 3A inverse property
6. x = 5 // 4A identity element

Now, the really cool thing about this axiomatic approach is that there's no single right way to apply the rules to solve a problem. (There might be an optimal way, but there's not always a need to find that.) I could have, for example, applied these rules instead:

1. x + 5 = 10 // given
2. (x + 5) * 2 = 10 * 2 // 20 multiplication property of equality
3. 2(x + 5) = 20 // 1M commutativity
4. 2x + 10 = 20 // 10 distributive property
5. (2x + 10) + (-10) = 20 + (-10) // 18 addition property of equality
6. (2x + 10) - 10 = 20 - 10 // 6A subtraction (def.)
7. 2x + (10 - 10) = 10 // 2A associativity
8. 2x + 0 = 10 // 3A inverse property
9. 2x = 10 // 4A identity element
10. 2 * x * 1/2 = 10 * (1/2) // 20 multiplication property of equality
11. 2 * 1/2 * x = 5 // 1M commutativity
12. 1 * x = 5 // 3M inverse property
13. x = 5 // 4M identity element

Phew. There at step 2, you probably thought “D’oh! What’s he doing?!” But there’s nothing illegal about multiplying both sides by 2. It’s not the fastest way to solve the problem, but it’s perfectly fine to do it that way. You get the right answer, as long as you follow the rules.

This demonstrates a dyad of mathematics: there are strict rules to be followed, AND within those rules we can still play with our creativity. I could have used a 42-step proof that x = 5 if I’d wanted. But I would have preferred playing outside over writing down 36 extra steps on every homework problem.

So I learned through practice (homework) that I could recognize certain patterns, to which I could apply certain patterns of rules that got me finished more quickly. For example, when I see x plus some stuff on one side of an equation, I know I can isolate the x and get closer to the finish line by subtracting that stuff from both sides of the equation (applying “6A subtraction (def.)” and “18 addition property of equality”).

With practice, the patterns in my mind accumulated, and I got faster. Which is exactly what has happened in my career as a software performance optimizer. It’s rules, and patterns.