Number Smoothies

or, How to Avoid Thinking in Math Class #2
(See Also Parts 1, 3, and 4)

This September, I gave my 7th-graders an elegant little problem about a 12-step staircase. You’re climbing from the bottom to the top, using combinations of single and double steps. The question is, how many ways can you do this?


I was stunned when some of my students offered answers almost immediately. “145!” one screamed, as if he had just gotten bingo. “Am I right?”

“Whoa, that was fast!” I said. “Why 145?”

“12 times 12, plus 1!” he announced. “Am I right?”

“But…” I hesitated. “But why 12 times 12? Why plus 1? Are we just doing random computations that sound like fun?”

He listened to my questioning with the same patience you’d give a friend’s mediocre guitar solo. Then he launched right back into his chorus: “So,” he said, “am I right?”

To him, at that moment, “doing math” meant “making a number smoothie.” You take the numbers in front of you, throw them all into the blender, and mash the “pulse” button until you get something.

The funny thing is, in our classes, this often works. You see a thick block of text; you pick out the numbers; you run them through the formula; and voila, you’ve got a solution, no thinking required!
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How to Avoid Thinking in Math Class

the first post in a series
(see also parts 2, 3, and 4)


“I’m planning a series of posts,” I told my dad the other day as he drove me home from the airport. “The title is How to Avoid Thinking in Math Class.”

Before I could get any further, he rubber-stamped the idea. “That sounds great! I always tell people, the point of school is to help you not to think.”

It’s a good thing he was the one behind the wheel, because if it were me, I’d have slammed the brakes and spat my latte all over the windshield.


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And Equality for All

a brief biography of the equals sign

Like Roald Dahl and Catherine-Zeta Jones, the equals sign was born in Wales.

map of UK

It was 1557—not that long ago, in the scheme of things. Just a few years before the birth of Shakespeare. In fact, the Danish prince and the Scottish king captivated the public long before their humble Welsh neighbor reached wide renown.

The early equals sign was a lovely but ungainly thing, a long pair of parallels that its inventor called Gemowe Lines:


Over the centuries, this stilt-legged creature shortened into the compact and tidy symbol we know today.

And before that? Well, mathematicians simply spelled out equalities with the phrase “is equal to.”

10 is equal to 7 + 3.

8 x 9 is equal to 72.

And of course, a2 + b2 is equal to c2.

The equals sign offered a way to avoid the tedious repetition of these words. Or, as Robert Recorde, the father of the symbol, put it: to auoide the tedioufe repetition of thefe woords.


An equals sign, then, is a verb. It’s the mathematical equivalent of “to be”—just as common, just as concise, and just as powerful.

But that’s not what kids see, is it?

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The Religions of Academia

Mathematics: God laid down axioms, and all else followed trivially.


Law: In the beginning, God gave His creatures free will, wisely limiting His own liability for any damage they might cause.

Computer Science: God threw something together under a 7-day deadline. He’s still debugging.


History: God wrote the Bible, which claims that the heavens and earth were created by God. This is exactly why you can’t always trust primary sources. Continue reading

The Sixth Sense for Multiplication

 Or, Seeing Arrays (Less Cinematic Than Seeing Dead People, But More Useful)

This year, I’m teaching younger students than I’ve ever taught before. These guys are 11 and 12. They’re newer than iPods. They watched YouTube before they learned to read.

And so, instead of derivatives and arctangents, I find myself pondering more elemental ideas. Stuff I haven’t thought about in ages. Decimals. Perimeters. Rounding.

And most of all: Multiplication.

It’s dawning on me what a rich, complex idea multiplication is. It’s basic, but it isn’t easy. So many of the troubles that rattle and unsettle older students (factorization, square roots, compound fractions, etc.) can be traced back to a shaky foundation in this humble operation.

What’s so subtle about multiplication? Well, rather than just tell you, I’ll try to show you, by using a simple visualization of what it means to multiply.

Multiplication is making an array.


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