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.
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).
Denise Gaskins' Let's Play Math 
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.
Hi Denise,
You did say in your introduction to your blog that you have taught all levels of math. You also said to write to you with things about math that stump you.
It’s true that mathematics has many facets. When a mathematician specializes in a field of Mathematical sciences, that will be what he/she is good at. There will be other fields that this same person will not be able to explain. It does not make them good or bad at that field of math, just inexperienced or out of practice.
I’m not sure what you are calling nonsense in your comment to Sue,
“You are right that no one is ‘good at math” in the way it’s commonly understood. It’s sort of a nonsense category.”
What do you mean by” in the way it’s commonly understood? ”
How do you determine how math is commonly understood?
Could you elaborate on that?
As far as I can see, the common understanding of being “good at math” is someone who can generate answers (especially on standardized tests) quickly and without effort. I wrote a blog series on the topic, Understanding Math: A Cultural Problem.
I say it’s a “nonsense category” because people apply it based on how they feel as children when learning math in school. But in order to learn, you have to confront new ideas — and math always has topics to offer that are beyond our present understanding. If anyone expects to work quickly and without effort on new topics, that person is expecting something impossible.
The only way to be “good at math” in the common meaning is to work well below one’s appropriate learning level, like a sixth-grader doing first-grade work.
I think of myself as good at math, but I wasn’t “good enough” to get a PhD. And I’m not “good enough” to solve the problem you posed above.
A decision tree is very helpful on this problem.
Sue, I think the key to being truly good at math is your earlier statement: “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.”
No matter how good we think we are at something( especially in academics) there’s always going to be someone who’s better.
That’s actually a good thing! I got better at tennis by playing against a better player.
Knowing the big picture of a math topic, like probability, helps it keep from seeming like nonsense. And Reimann’s conjecture is not something you just jump into the middle of- so of course it might seem like nonsense or make us intimidated.
From now on, I don’t care if anyone, including myself is “good at math”.
You can calculate in your head well, or you have good computational skills is not only better English, but also more meaningful.
Another reason it doesn’t matter if you are “good at math”, is if we refer to a student saying “I am good at fractions. They are easy for me. I understand how they work.” Then, they even show us they know how to properly use them by solving problems. Is that really enough? Say that it is the appropriate level of learning for that student. I just used fractions, but feel free to use whatever topic of math that would be challenging for your student.
If your student says, I understand the math, but I don’t like it, it won’t matter how much skill they have.
We have to make math more intriguing, less boring.
We need to promote”math appreciation”.
I do like how you say you learn when you teach.
I just think the word nonsense is getting over used.
Here’s a good article about being bad at math:
https://www.businessinsider.com/being-good-at-math-is-not-about-natural-ability-2013-11
You’re saying it’s non-sense to have computational skills and decide that you are “good at math”.
You said it’s common to label someone is “good at math” because they do well on standardized tests. But if you take away the testing and let the students learn math in a way that is individualized, they will work at math.
So, like this article explains much better than I can, to do well in math (ie building math computational skills AND understanding) one must do work. And what keeps us going is not bragging that we are good at 1st grade math in 5th grade (someone please tell me why homeschool parents imitate the poor school model for learning and why we call it 1st grade math??)
I’ve only heard that someone who wanted to tutor high school math said they were good at math and when they started tutoring high school math, they couldn’t do it!
Public schools should try to learn from homeschoolers.
What keeps us gaining math skills and understanding math is learning in the way that the individual learns and not comparing and a whole lot of patience, perseverance and hard work.
My best advise is to get a mathematician, with a PhD, someone who teaches math at the University level to learn math from and to consult with about curriculum. The math Ed phds need to consult with them, too.
Does any of this make sense?
I think the term “good at math” is without meaning, which is why I call it nonsense. To have meaning, we must elaborate: “good” in what way? and what does the speaker mean by “math”? Too often, it’s used to justify giving up — I’m not “good at math,” so why even try?
Teaching or tutoring involves a whole new level of things to learn. A teacher or tutor needs to know the math, yes, but also to understand how students learn, what are their common misunderstandings, how to draw out the students’ thoughts and help them put their ideas into words.
There are a lot of positive things happening in math education these days. If you follow math teachers and math professors on Twitter, you will see all sorts of interesting ideas. Creative, enriching approaches to math are spreading. Exciting times!
If you look it up, Keith Devlin is the one who said mathematicians need to be more involved in the curriculum writing for math. I completely agree with him and like the way he explains it. If I find it, I will try to share with you what he said. It was on his blog.
Hope you can get something good out of my comments.
Your comments make me think and push me to clarify my thoughts. Thanks!
Keith Devlin has a lot of insights on math education. Here is his blog, for anyone who wants to check it out.