“Can I light the candle tonight?” she asked. Every night we do this, my nine-year-old and I. Weekdays I’m at work and she’s with her grandmother, and on weekends we do chores around our homestead. Every night, though, we light a candle, sit in meditation for a moment, and go over another some idea that will be useful to her later.
A few nights ago, we began talking about exponential growth.
I began with a story. A wizard invented a game for the emperor of Persia to play, I said, and called it chess. The emperor loved it, and was so grateful that he offered the wizard anything he wanted in the kingdom. The cheeky wizard told the emperor that all he wanted was some grains of wheat on a chessboard – one grain on the first square, two on the second, four on the third and so on.
The emperor was puzzled – he was prepared to give the wizard vast regions for serving him. He had sworn to give the wizard what he wanted, though, so he started filling up the grains of the chessboard.
So my question to you, I asked The Girl, is this: if there are 64 squares on a chessboard, how many grains of wheat were there in all?
The Girl dutifully began working it out with pencil and paper. “One ... two ... four ... eight ... sixteen ... thirty-two ..,” and paused.
“Hang on,” she said, “I’m not even to the end of the first row. This is going to be a lot.”
At 1,024 I told her she could start rounding off – one thousand, two thousand, four thousand...
“How high does it get?” she asked.
The final square, I said, carries more grains of wheat than exist in the world, I said. The wizard bankrupted the empire.
“What did the king say?” she asked.
I don’t know, I said, but I hope the king had a sense of humour.
On the second night we drew graphs. I had asked her to draw a graph of her height at age two, age three and so on – we have them all drawn on a ruler on the wall.
“It makes a straight line,” she said. More or less, I said – that’s called arithmetic growth. Now what would happen if you took some number, no matter how small, and doubled it, and doubled it again, and again?
She plotted it out. “It curves,” she said. “It starts out at the bottom and goes straight up.”
I pointed to the part where the curve shoots up. See this part? I said. It will look like that’s where everything goes wrong. But this is the same curve as back here, I said, pointing to where the amount was small. This is just where you can’t ignore it any more.
A few nights later we talked about percentages. She knew that one per cent was one in a hundred, and ten per cent was ten in a hundred, or one in ten.
“So what happens if something increases by seven per cent?” I asked. If you start with a euro – 100 cents – and you increase it by seven per cent, what do you have?
“A hundred and seven per cent?” she said. Right, I said. What happens if you do it again?
She thought a moment. “A hundred and fourteen per cent?”
That’s close, I said – that’s where it gets interesting. See, you’re not just adding seven each time – you’re multiplying a bit more than one by a bit more than one. What you get, in the end, is a bit more than a bit more than one, so it ends up being a hundred and fourteen .... and a half.
The Girl and I did it eight more times until we reached two – in ten moves, I told her, it’s gone from one to two. And in ten more moves to ...
“Three?” The Girl said, and I smiled. It doubles, I said.
“No – wait! Four!” The Girl said. “Then eight! Than sixteen! It’s exponential growth again!”
My daughter flopped down face-first on the bed. “No matter what I do, I can’t get away from exponential growth!”
Yes, I said – remember that in a few years, when someone offers you a student loan.
A few nights later we talked about life we couldn’t see. You know when I make wine? I asked. I boil flowers, throw in lemons and sugar, wait for it to cool to blood temperature, and throw in yeast. The yeast eat the sugar and wee alcohol, and they divide: one yeast cell becomes two, becomes four, becomes eight ...
“Let me guess - exponential growth?” she said.
What happened if it kept going forever? I asked, and she made an upward curve with her finger.
Has that ever happened? I asked, and she shook her head.
How do you know it hasn’t? I asked.
“Um ... “ she said. “The Anthropic Principle?”
Right! I said – I’m really proud of you. We know that never happened because we’re here, and the world isn’t covered in yeast, so we know it never happened.
“Could it ever happen?” she asked.
No, I said – you don’t need to worry about that. Whenever someone tells you something’s going to keep going forever – in this world – they’re wrong. Tomorrow we’ll talk about negative feedback.
These aren’t normal things that most nine-year-olds know, but it’s what I wish every child knew, what I wish someone would have taught me. I don’t just prepare these lessons for no reason, or to show her off – I discourage her, in fact, from appearing different.
All these lessons, though, are jigsaw pieces, and before she endures the storms of adolescence I want her to see what picture it creates. Sooner or later, you see, she will look at a graph of the last century or two – population, waste, pollution, temperature, chemicals, extinctions, or any number of other factors – and see instantly what that upward curve means.
And I want her to be able to look around at a homestead, with the skills we teach her, and know why we chose to have a child anyway, and why the jigsaw also offers a picture of hope.