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 beautiful as his, but they are my treasures nonetheless. I understand them.

And there is one thing I am relatively good at. When I understand something, I can see how to explain it to others. Usually several ways, in multiple representations. 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 was not an official coach, and the forum is closed to the general public. I did belong to another [no longer active] forum, however, where I often gave 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 it 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 think. A nasty germ had dropped into my life 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 broke through confusion. 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 wanted to make sure I could generalize my insight and apply it in a new context. But I had no doubt of my success.

I had found a new beach pebble for my collection, and I would not let it get away.

This is what learning math feels like.

Next weekend, we will probably hear plenty of talk about “the Agony and the Ecstasy” of the Big Game. I say, football is nothing compared to mathematics.

This post is my too-late entry for Week Four of the #MTBoS #MtbosBlogsplosion blogging challenge. It’s an expanded reblog of an article that originally appeared in 2007.

CREDITS: Feature photo (top) by One Laptop Per Child. Spiral Fractal by Kent Schimke. Child on Beach photo by Dennis Wong. Dice photo by Ella’s Dad. Embarrassed Lion photo by Charles Barilleaux. Stones on Beach photo by Moyan Brenn. All via Flickr (CC BY 2.0). “Pieces of Math” poster from Loopspace (CC-BY-NC-ND).

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I get you. Oh, how I get you. I was a classroom math teacher for most of my career. Do you know why? So I could actually LEARN it that time around. It worked. I knew where, why, and when kids messed up. Teaching and learning go together. Knowing means it’s time to move on to a bigger challenge.

Love what you do.

Teaching and learning go together. Knowing means it’s time to move on to a bigger challenge.So true! Reminds me of a quote from Karl Friedrich Gauss:

That was very brave and honest to share. I’d push back on the way you framed it though. “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?” You and everyone else who stops short of some internal standard of perfection are not bad at math. That framework is so hard to really break. The struggle you describe is part of thinking mathematically and a state I want all the kids I work with to reach and not be afraid of.

I’m not sure I know I know how to move beyond the false “good at math”/”bad at math” dichotomy but its always on my mind. As always I really enjoy reading your postings.

Ben

Thanks, Ben. I debated using the “good at math” phrase, but decided to stick with it because that is how people think. We can push a positive, growth mindset till the sun turns cold, but I don’t suppose we’ll change that. And my goal was to show that one doesn’t have to be “good” at math to enjoy it. Even when we struggle, we can ALL enjoy learning new things.

But the problem with that framing is that no one is “good at math”. We all get stuck somewhere. The struggle is part of what math is about.

You are confessing insecurity, but imo you have not really confessed to being bad at math. Did you know that probability is especially tricky? Have you read about all the math professors who told Marilyn VosSavant that she was wrong in her reasoning on the Three-Door problem (or Monty Hall)? http://marilynvossavant.com/game-show-problem/ (They were maybe being sexist. She is/was too full of herself imo, calling herself the smartest woman in the world.)

I get excited about problems that stump me. This is a good one. I want the answer to be (5/6)^4. But I suspect it’s not, since I’m not taking account of the fact that the people are in a circle. (And the probability of all being different than your neighbor in a 5 person circle tossing a coin is 0. That leads me to think I need to do more… But what?)

I thought I wasn’t so good at math after my undergrad program at the University of Michigan. I now think it’s important for me to do it at my own pace. I am only good at math when I am intrigued enough to push myself.

You are right that no one is “good at math” in the way it’s commonly understood. It’s sort of a nonsense category. That’s not insecurity as much as recognizing that math is a great, infinite ocean, and my mind is finite. The more I learn, the more I see in the distance. It can be intimidating, but it’s also exciting to grasp a shiny new bit of truth.