Everyday Mathematics

Scene: Harriet’s car. Harriet and AJ are en route to the local ice rink, as they are at this time every Friday morning.

AJ: You know what I think?

Harriet: What?

AJ: I think that if there’s no end to how big numbers can get, then there’s no end to how small they get either.

Harriet: Interesting theory. Why do you think that?

AJ: Well, because if a number can always have a bigger number, then there should always be a smaller number too. But I can’t figure out zero.

Harriet: (parking in front of the ice rink). There’s another way to think of it. What if you were trying to go somewhere but you could only go halfway at a time?

AJ: Then it would take you two times. You’d go halfway and then you’d go the other half.

Harriet: But what if you could only go half of the distance to the door. Here, I’ll show you what I mean. (They grab their skates and gear and lock the car). We’re going to walk halfway to the door of the skating rink and then stop.

AJ: O.K. (He eyeballs the distance and walks to the halfway point). I’m halfway.

Harriet: Good. Now we’re going to walk halfway between here and the door.

AJ: Oh, I see. And then we’ll go halfway again.

Harriet: That’s right. What’s going to happen?

AJ: We’re never getting to the door. The door is zero!

Harriet: What happens to the distance we walk?

AJ: It gets smaller and smaller to infinity!

[AJ runs into the ice rink and spots his friend D., who is waiting for him inside the glass doors.]

AJ: Hey, D.! I just ran infinity!

* * * * *

Blazingstar recently left me a link to a contentious debate about the teaching of math in elementary school that is being waged on YouTube. One of the subjects of the debate is Everyday Mathematics, the curriculum that has been adopted by AJ’s school. The first of the videos can be seen here. There is also a series of responses, most of which I’ve had a chance to watch yet. The basic problems the first video has with Everyday Mathematics is that 1) it doesn’t teach the traditional algorithms for mathmatics, 2) it teaches math in a way the parents don’t always understand 3) it forces the students to come up with more than one way to do a problem (which apparently prevents “mastery” that is better acquired through rote memorization) and 4) it relies on calculator use.

Everyday Mathematics’ take on the teaching of curriculum as well as the rebuttals to the video, believe that the problem with teaching rote memorization is that we’re not teaching comprehension but the following of instructions. Everyday Math seeks to teach children how to problem solve. They spend a lot of time on the practical uses of math, having children figure out how to find math problems in everyday life.

I find the debate interesting, because I was one of those kids who balked at rote memorization. I’m still like that. If I didn’t know why I was supposed to memorize something, then I just wouldn’t remember it. AJ is the same way. AJ is, however, quite good at traditional algorithmic-based math. He likes practicing counting by 3s or 5s or 10s. He likes doing pages of problems. And he also craves the sense of mastery that the completion of a task, like the memorization of times tables or finding quick and accurate solutions to a page of problems provides. The big question I have is why does asking “why” and showing “how” necessarily mean that no memorization is happening? Can't they be used in tandem?

I feel like AJ’s current way of learning math is ideal for him. It is “Everyday mathemetics” at its fundamental sense. He thinks about something he notices in the world and we try to figure it out together through numbers. Numbers, to AJ, are yet another way of exploring meaning, the same a language or art or music. He plays in the real world with numbers, but he also likes to play with the numbers themselves to see what they will do. It's like the way he plays with Legos, sometimes building buildings -- a house, the Sears Tower-- sometimes just experimenting to see how he can put them together

After the conversation about infinite smallness, I found myself noticing AJ’s mathematical mind at work on the ice rink. Or rather, what I noticed was that other kids run into each other all the time but AJ almost never does, unless someone gets him from behind, out of his line of vision. Why? Because AJ triangulates with amazing accuracy. I’m sure this is more an issue of his attention than his ability – he doesn’t like to get run into. He’s never been the kind of kid who hurls his body around without fear of the consequences. But it struck me what a complicated skill it is to figure out how you, a moving object, might intersect another moving object, where that intersection is likely to take place, and then adjust your own speed and direction to avoid an intersection. That could easily be another math problem from the everyday, although I’m not sure I’m qualified to do the calculations.

But it is still a mathematical skill, and a much more useful application of math than pretty much anything I remember ever doing in an elementary math class. You do the math, you stay on your feet. You fail the problem, you hit the ice. It’s that simple.

[Crossposted at Spynotes]