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:

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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.

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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.

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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.

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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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Once, There Weren’t Numbers

a fable about the origins of those helpful counting thingies

Once there weren’t numbers,
and life was cold and sad.
You might say “I’ve got lots of stuff!”
but not how much you had.

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You could gather flowers,
but you couldn’t count them up.
You could ask for chocolate milk,
but not a “second” cup.
And though their eyes could see just fine,
the people still were blind.
They held things in their arms and hands,
but never in their minds.

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