I want to tell you a story. Everyone likes a story, right? But at the heart of my story lies a confession that I am afraid will shock many readers. People assume that because I teach math, blog about math, give advice about math on internet forums, and present workshops about teaching math — because I do all this, I must be good at math.
Apply logic to that statement. The conclusion simply isn’t valid.
Horrifying Gaps in My Knowledge
My mathematical understanding is stuck in the early-to-mid 17th century.
After reading several intriguing quotes about the Riemann Hypothesis, I was overcome by curiosity. I looked it up. The Riemann Hypothesis is a string of nonsense syllables surrounding one magic phrase: non-trivial zeros. Those words create surreal images in my brain.
I cannot reliably remember pi past three digits. Four if you count the decimal point.
In my world, groups are friends who hang out together. People who are good at math talk about groups, and I will sometimes almost believe that I am close to understanding at least part of what they mean. Then it all slips away again.
To me, combinatorics sounds like something done by a less-than-respectable woman in studded-leather underwear and spiked heels. The story I want to tell involves combinatorics, but only the G-rated kind.
I have forgotten most of the mathematics I ever learned. Some of it I never understood, so it passed away painlessly, without regrets. Other math I did enjoy at one time, but it perished from extended lack of use. Most of calculus is that way. I mourn its loss. Even in the math that I normally teach — and therefore that I should be good at — I occasionally stumble into chasms of appalling ignorance. My story begins with one of these.
If, in reading my blog, you discover more evidence of mathematical ineptitude, please deal gently with me. I know I am not good at math. I am just a dabbler, but I’m eager to learn.
Then Why Am I Here?
You may be wondering, if I am not good at math, then how dare I teach it, or blog about it, or offer advice to others?
I love mathematics. I can’t stay away from it. Like Isaac Newton’s boy at the beach, I want to grab every ocean-splashed pebble I can reach. My reach does not extend very far, and my stones are not as good as his, but they are my treasures nonetheless. I understand them.
And there is one thing I am good at. When I understand something, I can see how to explain it to others. For me, this is the definition of understanding: to be able to see connections and illustrations, elaborations and parables. This is what makes me a teacher.
Which brings me (at last!) to my story.
Once Upon a Time…
One of the parents from my MathCounts class brought in a combinatorics problem, and it stumped me. I was forced to invoke the Teacher’s Emergency Response: “I don’t know. Let me do some research, and I will get back to you.”
Here is the problem, for those who are curious (from the 2006-2007 MathCounts Handbook, Workout 9):
Four people sit around a circular table, and each person will roll a standard six-sided die. What is the probability that no two people sitting next to each other will roll the same number after they each roll the die once? Express your answer as a common fraction.
At home, I worked through the problem and got an answer that I recognized as patently ridiculous. I worked it another way and got the same answer. I left the problem on my desk and went to bed. I am not Maria Agnesi. No one solved the problem while I was asleep.
When I tried again the next morning, my wrong answer came back like a summer fly determined to sit on my forehead and rest its wings.
Online, I checked the MathCounts website. They host a forum for coaches, which may contain a discussion of this problem. But I am not an official coach, and the forum is closed to the general public. I belong to another forum, however, where I often give math advice to struggling homeschool parents. On that forum, someone who is better at math than I am was running a diagnostic workshop. You bring the problem, and he would teach you how to solve it.
Well, I had a problem. Was I brave enough to share it? These people thought I was good at math. This was going to be embarrassing.
I humbled myself and submitted the problem. The “professor” suggested an approach I hadn’t tried. I misinterpreted his suggestion and set off on a wild goose chase, only to find my familiar answer waiting at the end of the trail. The professor asked specific, pointed questions. I saw that his questions went straight to the heart of my problem. I couldn’t answer them. I explained my reasoning step by step, showing the most logical way to derive my wrong answer. There is was — my ignorance on display, naked and quivering, ready for dissection.
The professor had pity on me, pointed out the step where I had gone wrong, and gave me the correct step. I could see that his method worked, but it sat like a fig leaf over my still-shivering ignorance. Why would his step work when mine would not? How could I know what to do the next time a combinatorics problem came up?
I was too tired to deal with it any more. A nasty germ had dropped in and made itself at home. I thanked the professor for his help and went back to bed.
And the Miracle Happened
Sometime during the night, as I tossed around unable to sleep, I saw it all. I understood both the how and the why of the professor’s solution. I knew the prerequisites, the things a student would have to master before even attempting the problem. I saw how to explain the key insight that solved the problem. I sketched all the diagrams and calculations on my mental chalkboard. I could teach this problem. Victory tasted sweet.
As soon as I felt well enough, I asked the professor to find me another, similar problem. I want to make sure I can generalize my insight and apply it in a new context. But I have no doubt about that, really. I have found a new beach pebble for my collection, and I will not let it get away.
This is what learning math feels like.
Next weekend, we will hear plenty of talk about “the Agony and the Ecstasy” of the Big Game. I say, football is nothing compared to mathematics.
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