In Kent’s post (clicky), he detailed four major contexts that are commonly used to help students get a handle on negative arithmetic:
- Piles and Holes
These are contexts, though, not problem types. A single context might contain lots of different problem types, some of which might have nothing to do with negative numbers at all.
How can we turn Kent’s contexts into problem types?
I often come back to Children’s Mathematics and CGI (clicky) as a model for this sort of work. Their work focused on the addition/subtraction word problems that students solve in elementary school. So, no negative numbers in sight. Still, maybe their categories could be helpful to us here. Here they are:
The differences between some of the CGI problem types are somewhat subtle. For example, consider the difference (offered in Children’s Mathematics) between Join, Separate, Part-Part-Whole and Compare problems:
“Join problems involve a direct or implied action in which a set is increased by a particular amount…[Example:] 4 birds were sitting in a tree, 8 more birds flew onto the tree. How many birds were in the tree then.”
“Separate problems are similar to Join problems in many respects. There is an action that takes place over time, but in this case the action in the problem is one in which the initial quantity is decreased rather than increased…[Example:] Colleen had 13 pencils. She gave 4 pencils to Roger. How many pencils does Colleen have left?””
“Part-Part-Whole (PPW) problems involve static relationships among a particular set and its two disjoint subsets. Unlike the Join and Separate problems, there is no direct or implied action, and there is no change over time…[Example:] 8 boys and 7 girls were playing soccer. How many children were playing soccer?”
“Compare problems, like Part-Part-Whole problems, involve relationships between quantitites rather than a joining or separating action, but Compare problems involve the comparision of two distinct, disjoint sets rather that the relationship between a set and its subsets…[Example:] Mark has 8 mice, Joy has 12 mice. Joy has how many more mice than Mark?”
The key difference between Join and PPW problems is mirrored by the difference between Separate and Compare problems. That key difference is an action that kids can represent, and the absence of any such action. In a Join problem, Tom gets more apples, while in PPW instead we’d just want to know how many apples Tom and Jane together have. In Separate Tom loses some apples, while in Compare we just want to know how many more apples Jane has than Tom.
I tried to come up with some integer word problems that map the Join/PPW and Separate/Compare distinctions. Here’s what I came up with:
If you think the above table makes sense, read it again. I’ve read through these word problems over and over, and I’ve lost a lot of confidence that the distinction between Join/PPW and Separate/Compare makes a ton of sense with integers. Why not? A few reasons:
- The whole point of the Join/PPW distinction is that kids can come up with strategies for Join that take advantage of representing the action. For example, kids might draw a picture of Tom having 5 apples, then they’d draw a few more apples to Tom’s picture-pile. Though it’s formally tenable, that representation is harder to come up with in PPW problems because the action isn’t staring you in the face. But is the action really staring you in the face when we’re talking about “getting more debt” or “adding particles”? Do kids obviously know how to represent these actions?
- Also, do older kids have the same hang-ups over different sorts of addition and subtraction story problems? I don’t know.
- It’s incredibly difficult to pin these down as negative number word problems, because kids will solve many of them by using whole numbers to model the scenario. It’s possible to think of the net worth of someone with $2 in debt and $5 in savings as 5 – 2, not 5 + (-2). Which gets at an important point: integer arithmetic is used to represent scenarios. The word problems should not be seen as representations of the arithmetic. That’s getting things badly backwards.
- There is a ton of difference between these word problems depending on the magnitude and particular values involved. Maybe it’s more important to track the differences between word problems with different values than it is to track the stories presented in the problems?
Despite all these worries, I think there might be something salvageable in the distinction between Separate/Compare and Join/PPW. It seems to me much easier to come up with integer word problems that are clearly Compare and PPW than Separate and Join problems. And Join is easier than Separate. Really, I find it tricky to find problems that Separate problems that I think really are well represented by negative arithmetic.
Even that’s not quite right, because it’s fairly easy to find word problems that involve taking a positive quantity away from some starting quantity. What’s tricky to find are non-ridiculous story problems that involve taking away a negative quantity.
Of course this was going to be the hard part. We can come up with Compare problems all day that can be represented as a- (-b) (differences in elevation, differences in net worth, differences in temperature, etc.). The hard thing is finding Separate problems that don’t sound ridiculous and contrived that can be modeled as a – (-b).
Here’s my concluding take: there’s a basic tension in the way teachers and curriculum writers approach integer work. Everyone knows that the trickiest thing to make sense of is subtracting a negative quantity, as in a – (-b). Educators either take a Separate or a Compare approach to this sense-making, and there are trade-offs and advantages to each approach.
Go with Compare: There are lots and lots of Compare problems that can be represented as a – (-b). The problem is that kids primarily think of subtraction as taking away, not comparing. Teachers who go with Compare as their instructional strategy have to spend a lot of time helping students understand subtraction as comparison in non-negative contexts. The other disadvantage is that there’s an essentially arbitrary thing to remember about the Compare interpretation of subtraction, and that’s the sign. Is 5 – (-2) going to be 7 or -7? Are we starting from (-2) or 5?
If you go with “Compare,” you spend a lot of your time building the Compare conception of subtraction.
Go with Separate: When kids make sense of subtraction, they usually interpret subtraction as Separate. Huzzah! The problem is that it’s very hard to formulate Separate word problems that (a) involve subtracting a negative quantity and (b) make any sense at all and (c) preserve subtraction. Possible, but it’s a narrow path to take. Yes, cutting off 5 sandbags can be modeled as subtracting -5, but students won’t necessarily see this.
The difficulty of finding contexts that support the finicky notion of taking away a negative quantity means you often are stuck working in relatively few contexts, which is sub-optimal for building understanding. (Imagine learning to add whole numbers but only working in apple scenarios! You would probably have trouble with non-apple problems.) If you want to use students’ Separate understandings of subtraction, you’re spending a lot of time driving home the connection between your context and integer subtraction, since you can’t depend on students’ drawing connections across many different contexts or their naturally seeing their word problem work as connected to integer arithmetic.
If you go with “Separate,” you spend a lot of your time driving home the connection between your context and subtraction.
There are a few ways to end this line of thought. One is, “and that’s why Separate problems are so important!” Another is the same, swap “Compare” with “Separate.” A third is a sort of pessimism of the value of word problems for supporting integer arithmetic. I’m leaning towards pessimism at the moment, but we’ll see if that sticks!