0.999…. and the Debate that Repeats Forever

In Calculus class, it was often hard to resist distractions. I’d taught those kids for four years, and there was too much fondness built up between us. We swapped movie suggestions, debated Beatles albums. I’m surprised we got anything done.

But one day’s digression stayed surprisingly on-topic. (Well, at least it had to do with math.) The students got curious about repeating decimals.

Adrian led a pack of disbelievers in the claim that 0.999… = 1. (By 0.999…, I mean the decimal number in which the 9’s go on forever and ever.) “You’re crazy,” Adrian said. “You’ve lost your grip on reality, Orlin. Snap out of it.”

With my sanity under challenge (for neither the first nor the last time), I pushed back by offering the standard proof of the fact: Continue reading


Three Sentiments (or, Ode to the School Year)

5 - gradingFast year, right? Summer is here. The seniors can be found draped across their desks, exploring stages of hibernation so deep that they are yet uncharted by the medical community. It’s all very festive.

And into this start-of-summer breeze, I’d like to offer three sentiments.

First, guys, let’s be honest. We didn’t always getting along. I assigned too much homework, gave too many quizzes, wrote tests that stumped you. When I relented even slightly, you celebrated like a labor union that had scored a victory against the cigar-smoking management. I pushed you too hard, expected too much, demanded the unfair or even the impossible. This brings me to my first sentiment: Continue reading

A Ray of Light

There are moments of teaching I like to remember – episodes of cleverness, compassion, success. And then there are the other moments, the ones that my thoughts tend to flee, the ones I prefer not to think about. This is a story about both.

One Friday after school, a student came to me with questions. As a 12th-grade transfer, she found herself struggling to catch up with the students who had already spent years in the crucible of our intense charter school. To graduate that year, she needed to take my Statistics class concurrently with Algebra 2. She was failing them both. Continue reading

Stupid Graphs!

Dear Students Who Think Graphing is Stupid,

Right on! Graphs are stupid. Cosmically stupid. Deliberately stupid. In fact – and I hate to pull this rhetorical trick on you, but you leave me with no choice – that’s kind of the point.

A graph is not an end product. It’s more like a map – a simplified picture of something big and complex, a schematic diagram that shows important features and omits distracting details.

Take the function f(x) = x2. This relationship includes a dizzying number of input-output pairs, more than you could ever hope to list. Even with every atom in the cosmos at your disposal, you could never create a “complete” graph of it: to fill in the details, you’d need particles smaller than quarks, and to reach the extremities, you’d have to extend far past the most distant galaxies.

So when you graph f(x) = x2, you’re not depicting the whole function. You’re drawing a simplified version, a pocket-sized map. “Function for Dummies,” you might call it, or “Idiot’s Guide to This Function.” Continue reading

Fistfuls of Sand (or, Why It Pays to Be a Stubborn Teacher)

Teaching is full of compromises. This is a story about one small compromise that I refused to make, a stubborn act that paid off, though I didn’t expect it to. The setting is a Calculus classroom, but I hope the story will resonate with anyone who spies something dubious in the rigid and widespread assumption that learning can be endlessly itemized, carefully quantized, and instantaneously measured. This story has a moral, which I’ll tell you up front: Some lessons don’t sink in right away.


By my third year of teaching, I expected my classes to go all right. Not great, mind you: I might stumble over a definition, or botch the phrasing of a question, or optimistically allocate 5 minutes for an example that takes 15. Many days, I still made a minor idiot of myself. But I had put the fiascos of my first year behind me: no more droning 20-minute lectures, no more kids nodding off in the front row, no more pleading for their attention or castigating them for losing focus, as if my sloppy lessons were their fault.

Best of all – perhaps my only real strength as a teacher – I knew the terrain of their minds, how much mathematical territory we could cover in a day together.

So I was perfectly confident when I allotted one day for the Intermediate Value Theorem. The IVT captures a perfectly obvious idea: If at one time you’re 4 feet tall, and later on you’re 6 feet tall, then at some point in between you must be 5 feet tall. In other words: if you reach two different values (e.g., 4 and 6), you must also reach any “intermediate value” between them (e.g., 5, or 4.2, or 5.97).

Of course, the theorem frames this in the rather technical language of the mathematician: Continue reading