# The Smartest Dumb Error in the Great State of Colorado

Entering the town of Gold Hill, Colorado, you encounter one of the most extraordinary posted signs in the entire USA:

The founding year: 1859.

The elevation: 8463 feet.

The population: 118 people.

And at the bottom—oh, the glorious bottom—these three numbers have been added together, yielding a total of 10,440. You can check it yourself: 10,440 is exactly correct. The arithmetic is flawless.

It’s perfectly right… and profoundly wrong. It’s a memorable token of a common mathematical mistake: carrying out an operation without investigating its meaning.

I could easily spin out 1000 words bagging on this poor sign-maker. But I’m not going to. (For one thing, there’s a chance the error was a deliberate joke, and even if it wasn’t, there’s enough ridicule out there for bad math.) Instead, I want to argue the opposite.

This error isn’t brainless, stupid, or contemptible. Rather, in several ways, the Gold Hill error is a uniquely sophisticated and modern one.

# Oliver Sacks Knows What It Means to Teach

If you want to see the qualities that make Dr. Oliver Sacks my favorite writer, simply watch what he does when asked to provide grades for the medical students working with him:

I submitted the requisite form, giving all of them A’s. My chairman was indignant. “How can they all be A’s?” he asked. “Is this some kind of joke?”

I said, no, it wasn’t a joke, but that the more I got to know each student, the more he seemed to me distinctive. My A was not some attempt to affirm a spurious equality but rather an acknowledgment of the uniqueness of each student. I felt that a student could not be reduced to a number or a test, any more than a patient could. How could I judge students without seeing them in a variety of situations, how they stood on the ungradable qualities of empathy, concern, responsibility, judgment?

Dr. Sacks is a neurologist. His expertise ranges so far and wide (he has written on autism, Tourette’s, migraines, colorblindness, sign language, musical hallucinations) that the word “specialization” no longer fits.

Now, I’m a teacher, not a doctor. But reading Sacks’ autobiography, I’m struck by how teachers and doctors both feel a crucial tension, confronting the same fundamental choice in how to define our professional selves. Am I a narrow specialist, applying my expertise to address a specific need of the pupil or patient?

Or am I generalist, embracing the full complexity and interconnectedness of the human before me?

# An Open Letter to Benedict Carey

Or, a Dispatch from the Trenches of the “Math Wars”

Dear Benedict Carey,

I very much enjoyed your book How We Learn. It blends the vast and varied harvest of research on learning into something light, flavorful, and nutritious. A psych-berry smoothie, if you will. It’s a lovely summer read for a math teacher like me.

But I’m also a blogger—which is to say, a cave-dwelling troll, forever grumping and griping. And so I’d like to dive into your chapter on practice (“Being Mixed Up: Interleaving as an Aid to Comprehension”). In it, you purport to remain impartial in “the math wars,” but it’s my view that you come down distinctly on one side.

# Where the Laws No Longer Hold

third in a finite series on infinity
(see posts 1 and 2)

Somehow, I suspect I wouldn’t survive long on the frontier.

Drop me in the American West, circa 1850, and I fear my math-blogging and bad-drawing skills might not carry me far. I need indoor plumbing. I need the rule of law. I need chain coffee shops. I’m not cut out for the frontier.

And yet the frontier is exactly where I found myself the other day, when I came across this formula in the wonderful Penguin Book of Curious and Interesting Numbers, by David Wells:

I decided to play around with this product a bit. After all, what are products for, if not playing around?

(Go ahead and play with your Apple products. I’ll play with my infinite ones. We’ll see who has more fun.)

I felt like there should be an easier way to write this expression, exploiting the repetition of factors, so I gave it a shot, and created this:

Then my brain exploded and the universe dissolved around me, because I had just punched logic in the face, and it had punched me back.

The left side of that equation is π/2. It’s roughly 1.57.

The right side of that equation, however, is a product of many numbers—all of them below 1.

What happens when you multiply two numbers smaller than 1? You get another number smaller than 1.

How the heck could that equal 1.57?

# The Strange Music of the Harmonic Series

part 2 in a finite series on infinity

A few weeks ago, the webcomic Saturday Morning Breakfast Cereal posted a cartoon about the harmonic series.

(Obviously it’s a mistake to post an actual cartoonist’s work alongside my own second-grade-quality scrawl, but hey, maybe I’ll benefit from a math humor cheerleader effect.)

Now, what is the harmonic series? It’s this:

The sum never stops. It goes on forever and ever. Lovely, yes, but does it—in any meaningful sense—“equal” anything? Continue reading

# Everything Is Linear (Or, the Ballad of the Symbol Pushers)

What is the biggest problem facing humanity this week?

• A. The threat of Grexit
• B. The bittersweet knowledge that someday, when all of this has passed, we’ll have fewer opportunities to use the amazing word “Grexit”
• C. People thinking functions are linear when they’re SO NOT LINEAR
• D. Other (e.g., cat bites)

If you answered C, then congratulations! You are probably a teacher of math students ages 13 to 20, and we all share in your pain.

For everyone else (including you poor cat-bitten D folk), what are we talking about? We’re talking about errors like these (warning—mathematical profanity ahead):

What’s wrong with these statements? Well… everything. Continue reading

# Infinity Plus One: Please Check Your Intuitions at the Front Desk

the first post in a finite series

If there’s one thing about math that people love—and to make it through the average day, I have to believe there’s at least one—it’s infinity.

Throw the word into a math lesson, and ears perk up. Infinity? Did he say infinity? It’s like a distant celebrity, the subject of endless gossip and rumor. “I heard infinity isn’t even a number!” “Only the universe is really infinite.” “My last teacher said infinity times two is the same as infinity.”  “I can use infinity to prove that 1 = 0!”

Infinity is a sound too high for our ears, a light too bright for our eyes, a taste so sweet that it would tear through our tongues like acid. Basically, it’s mathematical Mountain Dew.

Tellingly, all of our words for infinity define it by what it isn’t. Infinite: not finite. Unlimited: not limited. Boundless: without bounds. It’s hard to articulate what infinity does, so we settle for naming what it doesn’t: end. Infinity is the Anansi of mathematics, a trickster spider weaving baffling webs of paradox and contradiction.

Take this example: which has more numbers, List A or List B?