Integer Arithmetic - Contexts Aren't Enough, So Which One Should We Use?

Michael, the Project Z research you found on the ways students view integers has shaken my preconceptions as much as it has shaken yours. They have led me to take a big step back and try to look at this unit more holistically.

As an algebra teacher, I think that integer addition and subtraction boils down to three big, interconnected ideas:

  1. Addition and subtraction are opposite operations, or inverse operations. That means that they have the opposite effect when operating upon two numbers
  2. Positive and negative numbers are opposites, or additive inverses. That means that they have the opposite effect when added to a number
  3. Subtracting a number has the same effect as adding its opposite

Everything in my teaching of integers is aimed at these three big ideas. And this research has really helped me think through the ways that students come to understand these ideas.

In your last post you did yeoman’s work trying to connect each of CGI’s addition and subtraction problem types to word problems involving integers.  The issue, as you discovered, is that not every problem type lends itself to easy interpretation using a real-world context. This is also discussed in one of the Project Z presentations, in which the authors state:

"These problems involve context, and when we set out to think about integers, we looked at contexts. But interestingly, we found that when we gave students contexts, such as owing money or increasing or decreasing elevation, they generally avoided using negative numbers. I can talk about a debt as negative dollars or a loss of yards in football as negative yards, but when was the last time you watched a football game and someone said, “Wow, that guy just gained negative 3 yards?.”

Well, damn. So at best, these contexts can get our students part of the way toward a comprehensive understanding of integers. The rest of the battle, which is a subject for a later post, probably has something to do with number lines and open questions like 5 + ___ = 2.

Still, there is at least some value to finding a context that builds a basic understanding of how negative and positive numbers interrelate. And I think I have found one. Or at least, I have found a context that seems very promising.

But first, a game.

You are in a hot air balloon. Sort of. This hot air balloon is different from normal hot air balloons. It is a lawn chair is held in the air by a series of small balloons, each of which can raise your lawn chair by 1 foot. It is also held down by several sandbags, each of which lower the height of the lawn chair by 1 foot.  For the sake of consistency, let’s start the balloon at a starting height that we will call 0, and let’s say that your lawn chair currently has 5 balloons and 5 sandbags attached.

Your opponent also has a lawn chair held up by 5 balloons and held down by 5 sandbags, also starting at a height of 0. Your goal is to raise your lawn chair up to a height of 10 feet above the starting position. To do this, you and your opponent take turns drawing cards. You do what it says on the card and change the height of the lawn chair accordingly. The cards look like this:

But there are also some wild cards that are a bit more complicated:

The first person to raise her lawn chair to 10 feet above starting height wins!

You have a player token and a vertical number line to keep track of your progress. You can use 2-color tokens to represent balloons and sandbags, or any other manipulative you wish.

I have no idea if the mechanics of this game make it fun, or whether ten feet is too easy/too hard to achieve. That part I can tweak later. This game has the main thing I want, which is students grappling with the effect of adding and removing (adding and subtracting) balloons and sandbags (positive and negative numbers). The formal symbols, as always, can wait. I am trying to build a conceptual framework first.  

So after students play this game a time or two, I would give them the following set of questions:

1)  Go through your deck of cards and pick out the three cards you think are most helpful to your chances of winning the game

List those cards below.

Why did you pick these three cards? Explain

2) Now go through your deck of cards and pick out the three cards that are most harmful to your chances of winning the game.

List those cards below.

Why did you pick these three cards? Explain.

3)  Now go through your cards and pick out two cards that had no effect on your lawn chair’s height.

List those cards below.

Why did you pick these two cards? Explain.

N.B. I asked students to list 3 cards in questions 1 and 2 even though there are 4 cards that are the most helpful: Add 4 balloons, Remove 4 sandbags, Add 3 balloons, Remove 3 sandbags. I do this intentionally with the hope that students will get different answers and end up debating whether removing 3 sandbags is “the same” as adding 3 balloons. Again, no formal symbols yet. Just laying groundwork.

The third question gets to the big concept number 2: Positive and negative numbers are additive inverses. I want students to realize that sandbags and balloons cancel each other out.

From here, I have lots of ideas. Here are a few:

  • Play a second version of the game, but every card has two instructions. For example “Add 3 balloons and add 4 sandbags,” or “Add 1 balloon and remove 3 sandbags.” After the game, have students rank the cards from most helpful to least helpful
  • Ask a series of questions such as “Mr. Haines wants to add more sandbags to his lawn chair, but he doesn’t want his lawn chair to go down in height. How can he accomplish this?
  • Ask a series of questions such as “My lawn chair is at a height of 4. I want to get it down to a height of -2. What are some ways I could do that? List as many as possible”

Then, at some point, we transition to problems such as 4 + (-7) = ____ and 5 + ____ = 2. But maybe, just maybe, the students will come to understand those ideas more quickly because of the significant time they spent analyzing balloons and sandbags. How, you say? I have no idea. I haven’t tried it yet. But I think it’s at least worth investigating.

In order to make this transition, you need a lot more than one day of connection. It’s not as simple as Monday: Sandbags and Tuesday: Negative Numbers. Students need to make their way up and down the ladder of abstraction until they know where the rungs are without looking down. Give students a problem such as “add 3 balloons and 4 sandbags” and ask them how you could represent it with math symbols. Give students a problem like “-2 + 5 = ___ ” and ask them to write a word problem about hot air balloons to match it.

Not everyone is going to reach 100% fluency with this model. As I mentioned above, students can and will avoid negative numbers whenever possible. But this might be a good starting point for a unit on integers.

Or maybe not! But I'll play the role of context optimist for the time being. It's more fun to argue.