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?

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US vs. UK: Who Grades Students?

When I began teaching in the UK, I discovered that the word “grade” belongs on that List of Words That Change Meaning Dramatically When You Cross the Pond, along with “vest,” “pants,” and “rubber.”

What’s the difference? Well, in the UK, a “rubber” is an eraser, and in the US…

Oh! You mean for “grades.” Well, in the US, “grades” are given by teachers. They aim to assess the quality of your work throughout an entire term. In the UK, your “grades”—the ones that matter to universities—are your scores on a handful of high-stakes exams.

In the US, the scores you get from your teachers form the bulk of your permanent academic record; in the UK, those scores don’t even appear.

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In my experience, both American and British educators react with utter horror to the opposite system. And, weirdly, many objections are mirror images of one another.

Objection #1: Your System Drowns Students in Testing

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A Technique is Just a Trick That Went Viral

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In February, I was throwing together a geometry test for my 12-year-olds. I wanted a standard angle-chasing problem, but – and here’s the trick – I’m lazy. So I grabbed a Google image result, checked that I could do it in my head, and pasted it into the document.

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But when I started writing up an answer key, I ran into a wall. Wait… how did I solve this last time? I trotted out all the standard techniques. They weren’t enough. A rung of the logical ladder seemed to have vanished overnight, and now I was stuck, grasping at air.

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One-Word Math Classes

You know what’s often missing from math class? Yes, candy bars, but even more important than that: coherence.

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Math class shouldn’t be a mishmash pile of facts, thrown together haphazardly, like an academic version of The White Album. It should be a perfectly interlocking tower of truths, climbing upwards with singular purpose—an academic Sgt. Pepper or Abbey Road.

A good class isn’t a greatest hits record. It’s a concept album.

In that spirit, I’ve been taking each topic in the secondary math curriculum—algebra, geometry, calculus, etc.—and trying to boil it down to its one-word essence. Here are the rules of the game:

  1. You must choose a single word to complete the sentence, “[Branch of math] is the mathematics of _____.”

For example, you might say, “Topology is the mathematics of dinosaurs,” or “Category theory is the mathematics of abstraction,” or “Combinatorics is the mathematics of sadness.” (To be clear, only one of those is remotely accurate; you have my sympathy, combinatorists.)

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US vs. UK: Mathematical Terminology

I’m an American, born and bred. I revere the 14th Amendment, root for the New England Patriots [dodges rotten fruit], and can rattle off all 44 presidents.

Yet here I find myself, in Birmingham.

Not Alabama. England.

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In some ways, it’s not so different. As my friend John advised before I moved: “The British speak English, care about money, and yell about politics. You’ll barely notice you’ve left.” But it’s not quite like home: the spellings, the roundabouts, the big red buses—and, most relevant for a teacher like me, the sometimes startling differences in the ways our two countries educate kids.

In less than a year of teaching, I’ve encountered some surprising disparities, each of which prompts the obvious question: Which way is better, the British or the American?

I have nothing to gain by these comparisons. If I favor America, my judgments will be dismissed as jingoism (just like 97% of the other things I say). And if I favor Britain, I will have my surrender lorded over me for months to come by English teenagers. Thus, I find myself in a predicament similar to many facing America today: a lose-lose situation. And, in the true American spirit, I shall plunge forward anyway.

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