I probably learned about variables from playing around with a Commodore 64 when I was about the age you are now. But I didn't see them used in mathematics for many years, until they were finally introduced, in about 7th grade, as a tool for solving problems. Take a problem, write down the equation, putting variables for the unknown quantities, and then you have something you can solve.
A little while ago I realized that this isn't the only way, or even the most important way, that variables are used in math. Variables are used to write laws.
The other day you were getting a shower, and I told you that letters could stand for numbers, that you can use letters to write rules about numbers. The letter could stand for any number, and the rule would always be true. I wrote in the condensation on the glass:
Then I stopped and said, well, A plus zero equals what? You said zero. I said I didn't think that was right, because what if A was seven? Seven plus zero equals zero? So then you said A. And I couldn't be sure but I thought you really got it. That's right, I said. I'm sure you could tell I was very pleased. Actually I was surprised and excited.
Later I wrote some math pages for you to solve. The first one said,
Here is a rule:
Do all the numbers follow this rule?
If there is a number that doesn't, that's called a counterexample. It means the rule is false.
In math, a true rule is always true, for all numbers.
The other one had some mathematical statements on it and asked you which ones were true.
You surprised me.
You got them all right. I asked you about the rule on the first page, A < 0, and you had to look up what the < symbol meant, but then you told me right away that it was false, because zero isn't less than zero.
I was amazed. I asked your mother, “Did you see those pages J. did today? What does this mean?” She wasn't surprised. “It means first-graders can learn pre-algebra,” I said, insistent.
“Some can,” she said.
She is half right: you are special; you are bright; and you are interested. But I know there are millions of special, bright, curious kids like you in this country, and I think by and large their schools are selling them short. You sure are lucky you've got me, kid. But not as lucky as I am to have you.