Math Journal: Playing with My Own Ignorance

Mary Everest Boole, wife of English mathematician George Boole, once described algebra as “thinking logically about the fact of our own ignorance.”

This definition made me chuckle. Like any human being, I am ignorant on many things, but I usually avoid thinking about that.

So I wondered what would happen if I took Mrs. Boole’s advice and tried thinking logically about my ignorance.

How far could I go?

Perhaps you’d like to try this experiment with your children. All you need is a pen and paper or a whiteboard and markers and a bit of curiosity.

And if you enjoy this exploration, check out my Math Journaling Adventures project to discover how playful writing activities can help your students learn mathematics. Preorder your books today!

A Mathematical Experiment

There is a number.

The number might be positive or negative, monstrously big or unimaginably small. I don’t know what that number is, only that it exists.

But while I am ignorant about this mystery number, I do know some things about math and logic. Using the things I know about how numbers work, can I say anything true about my mystery number?

Whatever the number might be, I know that adding one would make a bigger number.

Am I just guessing, or can I know for sure?

Even though I don’t know this mystery number, I do know how adding works. So I can deal logically with my ignorance and say that adding one HAS to make a greater amount.

The Language of Math

I can even write this fact in the language of math, if I give my unknown number a name, like “George” or “X” or “blob”:

blob + 1 > blob

Is there anything else can say for sure about our mystery number? Well, there are a lot of other numbers I could add:

blob + 27 > blob

blob + 1,000,000,000 > blob

blob + 0.001 > blob

If I keep going, I can create a whole family of true statements about my mystery number. In fact, I could add any positive number — anything bigger than zero, even just the tiniest bit bigger — and the total would have to be larger than what I started with.

And I can write this fact in math language, too, if I give the number I’m adding a mane:

blob + pony > blob (if pony > 0)

Brainstorm with Your Students

• How many other things can you say that have to be true about our unknown, mystery number?

• Are you just guessing, or do you know for sure your statement is true?

• How do you know?

So far, in my own mathematical experiment, I’ve managed to fill a whole page with true statements about blob and pony.

I haven’t gotten any closer to solving the mystery. I still don’t know whether blob is large or small, positive or negative. But I’ve had fun anyway, just playing around, trying to think logically about the fact of my own ignorance.

How far can you go?

From Arithmetic to Algebra

by Mary Everest Boole

Arithmetic means dealing logically with facts which we know (about questions of number).

“Logically”; that is to say, in accordance with the “Logos” or hidden wisdom, i.e. the laws of normal action of the human mind.

For instance, you are asked what will have to be paid for six pounds of sugar at 3 pence a pound. You multiply the six by the three. That is not because of any property of sugar, or of the copper of which the pennies are made.

You would have done the same if the thing bought had been starch or apples. You would have done just the same if the material had been tea at 3 shillings a pound. Moreover, you would have done just the same kind of action if you had been asked the price of seven pounds of tea at 2 shillings a pound.

You do what you do under direction of the Logos or hidden wisdom.

And this law of the Logos is made not by any King or Parliament, but by whoever or whatever created the human mind.

Arithmetic, then, means dealing logically with certain facts that we know, about number, with a view to arriving at knowledge which as yet we do not possess.

When people had only arithmetic and not algebra, they found out a surprising amount of things about numbers and quantities. But there remained problems which they very much needed to solve and could not. They had to guess the answer; and, of course, they usually guessed wrong.

And I am inclined to think they disagreed. Each person, of course, thought his own guess was nearest to the truth. Probably they quarreled, and got nervous and overstrained and miserable, and said things which hurt the feelings of their friends, and which they saw afterwards they had better not have said — things which threw no light on the problem, and only upset everybody’s mind more than ever.

I was not there, so I cannot tell you exactly what happened; but quarrelling and disagreeing and nerve-strain always do go on in such cases.

At last (at least I should suppose this is what happened) some man, or perhaps some woman, suddenly said: “How stupid we’ve all been! We have been dealing logically with all the facts we knew about this problem, except the most important fact of all, the fact of our own ignorance. Let us include that among the facts we have to be logical about, and see where we get to then.

“In this problem, besides the numbers which we do know, there is one which we do not know, and which we want to know.

“Instead of guessing whether we are to call it nine, or seven, or a hundred and twenty, or a thousand and fifty, let us agree to call it x, and let us always remember that x stands for the Unknown. Let us write x in among all our other numbers, and deal logically with it according to exactly the same laws as we deal with six, or nine, or a hundred, or a thousand.”

As soon as this method was adopted, many difficulties which had been puzzling everybody fell to pieces like a Rupert’s drop when you nip its tail, or disappeared like bats when the sun rises. Nobody knew where they had gone to, and I should think that nobody cared. The main fact was that they were no longer there to puzzle people.

This method of solving problems by honest confession of one’s ignorance is called Algebra.

Addendum

I’d never heard of a Rupert’s drop. Have you? It’s a bit of glass that, while it was still molten, was dropped into a bucket of cold water.

If you hit the drop on its bulbous head, you can’t smash it, even with a hammer. But if you snip the drop’s thin tail, it shatters into lots of tiny bits.

And that which makes their Fame ring louder,
With much adoe they shew’d the King
To make glasse Buttons turn to powder,
If off them their tayles you doe but wring.
How this was donne by soe small Force
Did cost the Colledg a Month’s discourse.

—Anonymous, The Ballad of Gresham College (1663)