Purposeful Ramblings Through My Mind

The grade five curriculum requires kids to develop a basic understanding of a variable for the first time: as an unknown number and as a series of values dependent on each other.

My kids already have the first definition. So yesterday we started on the second.

I put up on the board ___ + ___ = 17 and asked them to find all the ordered pairs that would fit in the blanks. I showed them how to write an ordered pair (0, 17) and then let them have at it in their groups. Meanwhile, on a piece of chart paper, I drew the upper right portion of a co-ordinate grid (because my kids haven't learned integers yet and I didn't have a lot of room, anyway.) When my kids were acting like they were done, I explained that a co-ordinate grid was a way of using math to draw things; it was just a format that everyone would use in much the same way, like a language, so that anyone who saw it would understand how it worked. I rewrote the equation on the board as x + y = 17, and asked one kid to come up and plot a point on the grid. She took (0, 17) and pointed to seventeen on each axis, saying that they were different points that were written the same way. She drew both points. I told the class that this was a problem; how could the same two numbers indicate two different points? How would that help anyone to figure anything out? I let them stew about it for a minute, and discuss how the points were different, until someone finally said, "But in the equation, x came first." I got that person up to the chart to explain what she meant, and we figured out that (0, 17) was a different point from (17, 0).

After that, we plotted each of their ordered pairs on the grid in turn, and I showed them how to label the points correctly. When we had about half the points up there, I got one of the girls who was complaining she didn't get it to come up. I handed her a metre stick and told her to play connect-the-dots.

I had expected the lesson to end there, but it didn't, because when my students notice something, I run with it. Half a dozen of my kids looked at the resulting triangle and said, "That's a right-angled triangle!" One more said, "It's a half-square triangle like in our quilt blocks!" We discussed the other two angles - both 45 degrees - and I asked the kids, "What do you think you'd have to do to get a triangle that wasn't a half-square triangle on this grid? What would the equation look like for that?"

Well, they tried x + y = 12, though my gifted kid told them right off the bat it was going to just make a smaller version of the same triangle. It did. When it seemed they were stumped, I took the metre stick and drew a line from (7,0) to (0,14). Then I asked them to figure out the relationship between seven and fourteen. They eventually got it, with a bit of help to get the form right: 2x=y, where x=7.

Every kid in the class understood the basics of the co-ordinate grid. Every kid as far as I could determine understood the ordered pairs and how they were graphed. A few kids couldn't wait to pull out the graph paper and try some equations on their own. They were tickled by the idea that I did this math in grade nine and it's currently in the curriculum in grade eight.

Today's lesson: Explain why x + y = a whole number is always going to produce a half-square triangle, no matter what the whole number is.

Part of my job involves making sure kids can draw or stack blocks to represent the number 100 000. That's as high as grade fives in Ontario are expected to be able to understand.

Well, here's a much bigger number, represented in stacks of $100 bills.

Probably the most difficult thing about math, either scientific math or advanced economics, is the understanding of quantity that comes with it. One trillion dollars is really hard to visualize. Nobody will ever hold that much cash in their hands, or even in their warehouses. It's 10 to the power of 12, and it's just at the limit of how big something can be and still be relatively simple to represent visually (as this website did.) Once you get past that, it becomes a lot easier to compare it to something else: like, one in ten trillion is one drop of water in an Olympic-size swimming pool, for example. (Note: I don't know if that's true.) I think a big part of the reason most kids have trouble with math in the higher grades is quite simply that nobody takes the time or the energy to draw those comparisons for them in such a way that they understand the concept of "orders of magnitude."

Math gets hard at exactly the rate that it gets abstract. Make it less abstract, and most people can do a lot more of it. The problem with the traditional methods of teaching math has always been that it left math as an abstract concept long, long, long before most people were ready to think, "I can manipulate these numbers in certain ways without thinking about what the numbers actually mean at every point along the way."

Anyhow. An economic visualization and a little plug for concrete methods of teaching mathematics, at the same time.

First, I got all the kids from the grade five class to join my class for half an hour. This lesson is best taught with a large group. (I teach 4/5; I didn't have the grade fours doing something different, because I'm two days back from a maternity leave and didn't have time. Also, they would just have been listening anyway. It won't hurt them to hear this one two years in a row.) I got them sitting on the carpet, and started to teach.

"A long time ago, in the land we now call Iraq, people lived in large family groups under the authority of one patriarch. The word "patriarch" means "rule of the father," and that's exactly what it was. In our patriarchy, the father will be T." (I picked the biggest, baddest of the grade five kids, first to get him out of trouble and second to make him feel important. It worked. He spent most of the class preening.) "Now, in these family groups, bigger was better. The patriarch wanted as many children as he could get, to help him tend his large herds of cattle and sheep. He might have several thousand animals, and they were his wealth. There was no paper money or coins; you were rich if you had a lot of livestock. So the Patriarch had many wives." I paused at this point to let the giggles run their course. "I need seven girls to be wives." I got my seven girls up there, blushing furiously, getting teased a bit by those still sitting. I let that run its course with a few quelling looks, because their turn was coming. "One of the reasons a patriarch would have had many wives was because of the likelihood that some of them would die in childbirth. We are very lucky in Canada, in this century. For the last hundred years or so, it has become more and more rare for women to die in childbirth. Now, most people only have a few children because they're pretty sure all of those children will survive to adulthood. That was not the case for most of history. In a real patriarchy, probably half the women would die in childbirth at some point in their lives. About half the kids would die, too, usually before they were five. If you survived that long, you had a good chance of making it to adulthood."

"The next group of people would be the children of the wives. These were the potential heirs. They were taught how to work from an early age, and they learned how to manage the herds and the servants. The boys might have waited until their late teens or early twenties to marry. The girls were probably married in their early teens.

"I need all the rest of the girls up here now. These girls are servants. Most of them are concubines, who are also going to give birth to children who are the children of the patriarch. These children, though, won't have equal status with the children of the wives. If there is no other male heir, one of them might become the heir. More likely, though, they'll be servants too." I divided up the rest of the kids as children of the servants. Then I got them all sitting down again.

"Now, T is looking for a wife for his eldest son." Snickers in the direction of the eldest son, M, are promptly quelled. "He knows that his youngest wife, A, has a younger sister who would be a perfect wife for his son. But A's family lives a long way away. It takes a lot of land to support herds of the size that T has, so there aren't a lot of other people really nearby. Probably, the nearest other patriarch is a few days' walk away. A's father is further than that. It's going to take a week or two to walk there. The messenger is taking a few head of cattle with him - say, five - as a gift to the bride's family." I pause in my narrative to pick up a piece of modelling clay that I have on my desk, and start working it in my hands.

"The problem is that T doesn't have anyone he really trusts to send this message. There are no phones, no computers, no roads, no cars, no writing, no mail. There's only one way to send a message: have someone go to the other person and repeat it, word for word. None of T's sons are old enough. His servants aren't competent enough. The best person to send is someone T thinks might try to steal from him. So T needs a way to send a message to A's father about how many head of cattle have been sent." I pause a moment to explain that "head of cattle" doesn't, in fact, involve beheading any cows. "He takes some clay, the kind his family makes pots out of, and makes five little pellets out of it. He bakes these so they're hard. Then he makes a hollow ball of clay around them, and bakes that. It's now hard. He gives this to his servant," I have T do all of this, "whose job is to give it to A's father. He in turn will break it open and count the pellets. If the number of pellets matches the number of cattle, then the servant doesn't get killed for stealing from his master." (This catches their slightly-flagging attention again.)

"A's father realizes that there's no need to bake the pellets into a ball. He can just as easily press them into a clay tablet and bake the whole thing. After all, the clay will become just as hard, and T will still know how many are in it. So when he sends his younger daughter back to become M's wife, he also sends six sheep, and a clay tablet with pellets pressed into it," and I demonstrate the technique. T knows then that there are six sheep. No one has cheated him."

Then I come up with another problem: how to tell the difference between cattle and sheep? So you get little clay cows and little clay sheep pressed into tablets. Then someone figures out that all you really have to do is scrape a picture of a sheep or a cow into the tablet, and make marks indicating how many you've got. And so on, and so forth, until you get pictograph writing, ancient Mesopotamian style.

At the end of the lesson, kids get to play work with modelling clay, making a message to send to their father-in-law when they want to trade some of their livestock. They can choose any one of the types of messages. At some point, I point out that this clay tablet serves exactly the same purpose as a cash register receipt in our culture.

And that's my favourite lesson of all time.