I love Kent’s work with integers. But I can’t stop thinking that this lesson uses contexts in a slightly different way than contexts are usually used to support the learning of arithmetic.
In my 4th Grade classes I teach kids how to multiply larger numbers. For this learning, I often use an area context to help. How does it help? Essentially, it’s a scaffold. For example, it’s often easier for kids to find the area of a 12 x 34 field than it is for them to calculate 12 x 34. This is, I think, because of two things many kids know about fields. First, that if you split up the field into a bunch of parts, the field is just made up of those parts. Second, when you split up the field on a page it’s easier to keep track of how some dimensions change while others stay the same. These are two benefits that come from treating 12 x 34 as an area problem.
But it does us no good to just ask kids to solve an easier problem for the sake of ease. We use this easier problem to help kids solve a more difficult problem, 12 x 34, without the context. What makes the area problem such a helpful context is that area is a perfect model for multiplication. So the area context is very close to the arithmetic problem.
What is it, then, that my kids can bring with them from the area problem to the multiplication problem? Strategies, or put another way, actions. There are things they know how to do with area problems, and extraordinarily similar actions will apply to the multiplication context. The strategies are the same in the two problems. You develop a strategy in an easier context, and then bring it over to the tougher context. Roughly, I think this is how it works.
How about subtracting a negative? The context that Kent has us using involves balloons and sandbags. We start with 10 balloons and 5 sandbags on-board our hot air balloon. What’s our height? Then we take away 3 sandbags. What’s our new height?
Here are my questions:
- How similar is this context to the integer context?
- What is the strategy/actions that kids are learning how to do in this context?
And I don’t know the answer to these questions. That’s why I’m stuck in trying to understand how the integer game works.
While thinking about this, I’ve been tempted to say that the strategy is “treating subtraction of a negative as adding a positive.” The issue is that it’s unclear to me whether this is a strategy that kids are really using in the hot-air balloon context. It’s certainly now how I think about all of these problems. Take this one: “You start at a height of 10, and you throw away 5 sandbags. What’s your new height?”
Here’s how I think about this problem:
- Throwing away 5 sandbags? That means that my equilibrium is off by 5 things.
- 5 things that make me go up.
- So I’ll go up 5.
- That puts me at 15.
It’s unclear to me what this habit of mental calculations looks like in the realm of negative integers. I’m wondering if this way of thinking doesn’t carry over to negative arithmetic particularly well.
Part of the problem is that there aren’t a lot of “strategies” available for us as we think about 10 – (-5). There only seems to be one way: treat it like adding a positive. How do you create a context that makes this easier?
(Though I’m a fan of the “subtraction is the opposite of addition” approach. This seems like a strategy to me.)
But I do love Kent’s game. I bet it does help kids, and I would use it to introduce subtracting a negative.
I think this game functions differently than the area context, though I struggle to define exactly how. (Your thoughts?) I’ll take a shot at expressing one half-formed idea I had about how the game might work.
It could be that the game is particularly helpful as an introduction because it gives kids a story to tell about how come this all makes sense. I can think of areas of math where I’ve found stories like this helpful for remembering things that otherwise I can’t make meaning of. So maybe this game functions more like an analogy or heuristic explanation than a context. Putting that another way, maybe I’ve just been thinking too narrowly about how contexts can help learning.