On Friday, I asked my Algebra 1 students to make sense of 3 – (-5) and problems like it. This is the holy grail of integer instruction. It’s the hardest type of problem to make sense of, harder than (-5) – (-3), harder even than (-3) – (-5).

My students had two productive ways to make sense of 3 – (-5), and it seems to me there’s at least another way they didn’t use. That makes three ways to think about 3 – (-5). Maybe there are more ways out there, but we’re constrained by the ways in which it makes sense to think about subtraction, so the possibilities aren’t endless.

What are these three ways to make sense of subtraction? Subtraction can be seen as a “Take Away,” “Compare,” or “Additive Inverse” operation. Here’s what I mean by each, as applied to 10 – 2:

1. 10 – 2 might mean “you start with 10 things, you take away 2 of them.” It’s dynamic, meaning, there’s change, action. This is how most of our young friends think about subtraction.
2. You might also think of 10 – 2 as a comparison between two quantities, “how much more 10 things is than 2 things.” Older students are sometimes taught to interpret this as the distance from 2 to 10. Either way, there is no action, rather there is a measurement in place of change.
3. Finally, you might think of subtraction as doing the opposite of addition. 10 – 2 means, then, do the opposite of what 10 + 2 does. 10 + 2 means (whatever it means but possibly) you add 2 more to 10, so 10 – 2 means take 2 away from 10.

My students adapted the “take away” and “additive inverse” to make sense of 3 – (-5). They might have used comparison, but they didn’t. Here’s what those first two models looked like:

Take Away:

Here’s one way to use the take away interpretation for 3 – (-5).

You start with 3 positives. Really, though, we’re going to want to think of this as starting not with 3 positives but with a net charge of positive 3.

And this is still a net charge of positive 3…

Now, take away 5 negatives. Or, take away a charge of -5? Anyway…

…now you have 8. Tada!

Kids in my class weren’t explicit about that middle step, but they referred to it obliquely. They said things like, “well you have to imagine a lot of negatives and positives hanging around there.” I’ve seen curricula where you talk about a soup of charges. That works!

Of course, this needn’t be a model limited to charges. Swap charges with money/debt, or with helium/sandbags, floats/anchors. While kids might have different familiarity or comfort with different contexts, they all seem to help support the same kind of thinking.

Opposite of Addition

My heart’s with this one. I think it’s very promising.

It depends on knowing what 3 + (-5) is, because the idea is that 3 – (-5) should do the opposite of it.

So, when presented with 3 – (-5), you think, OK, what does 3 + (-5) do. Oh, that’s the same thing as moving to the left/taking away from 3:

Now grounded, you reason that 3 – (-5) does the opposite. So it makes 3 more positive, moves to the right, etc.

(N started talking about this in class in a mumbled way that I didn’t completely understand. Then, later, I offered the number line representation on the board, presenting it as a way that no one had mentioned yet. But N said, “That’s exactly what I was saying!” I found that heartening.)

Comparision

This is where the word problems can come, in full force. How much does the temperature change if it starts at -4 and moves to 10? What’s the difference in elevation between -10 and 20?

I’ve claimed before that kids don’t tend to come into their work with this interpretation of subtraction in their back pockets. That’s fine, it still works, potentially.

You say, 3 – (-5) means “how do you get from (-5) to 3?” The answer is, you add 8 to (-5).

Why does 3 – (-5) mean getting from (-5) to 3? Not much we can say here, ‘cept that it’s consistent with 5 – 2.

Comparison seems like the least promising interpretation for me, but, hey, some people seem to swear by it.

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Like I said, my heart is with the additive inverse approach at the moment. My representation of it was triggered my by kids’ work with a problem from Transition to Algebra (which I recommend highly). This was the problem, and I think it helped my students develop the ideas needed for that additive inverse interpretation:

A week ago, I thought I knew all the ways that kids could and would think about subtracting negatives. Then this happened and my world is a brighter place.

Today in class I asked kids to solve these in their heads, and to share how they thought about them:

The big question in my mind was, do kids make sense of (-5) – (-1) as taking one negative away from five negatives? In short, the answer was “yes.”

As I often do when I want to talk about strategies, I asked kids to shout out the numerical answer they got to (-5) – (-1) right away so it would be out of the way. Then, I asked for strategies.

The first strategy shared was the “two negatives make a positive” and variations on that. Then I called on J:

If 5 – 1 = 4, then you just flip the signs to get (-5) – (-1) = -4.

OK, great! Why? J struggled to articulate the reasoning. N stepped up to the plate, and put it plainly:

It’s like taking 1 negative away from 5 negatives.

L asked for clarification. N offered it. L, who hadn’t had a way to make sense of this, had a moment. I asked her to revise her quiz later in class, and she had no issue applying this reasoning to the relevant subtraction problems on the quiz.

Great!

Every new strategy delimits a problem type, because every strategy works for some problems and not others. The type of arithmetic problems that the above strategy (let’s call it “Taking Like from Like” or “Whole Number Analogy”) helps with is fairly specific:

• It works for (-5) – (-1) = (-4), but it doesn’t quite work well for (-1) – (-5) = (-4). You can’t really think of that as taking 5 negatives away from 1 negative without running into some issues.
• Of course, the model can be stretched to encompass (-1) – (-5), just as all models can. You just say you get -4 negatives. I wonder if my kids will find that useful? I don’t expect that it will, but I would love to be surprised!
• So the problem type this “Taking Like From Like” works for is subtracting a negative from a negative where the first term is larger than the second. It’s not really a useful interpretation for 5 – (-1) or (-1) – 5.

If we’re wondering what contexts could help develop this “Taking Like from Like” reasoning, they would have to be contexts in which this strategy is easier to come up with. Then the plan would be to draw on experiences with these contextualized problems in making sense of the formal arithmetic.

So, what sorts of contexts could support this? Elevation seems useless here, since the elevations aren’t really objects in the sense needed to “take away a negative from a negative.” For the same reason, temperature seems useless here. Money might be helpful here, since if you’re adding debt with debt (and savings with savings) you’re essentially adding/subtracting like from like. Maybe a problem such as “You owe 5 dollars and then someone takes $2 of IOUs away. How much do you owe?” would do the trick? Particles/charge problems, I think, are the most promising here. Both because the context unambiguously involves negatives and also because it’s easy to represent taking those charges away.

In sum: “Taking Like From Like” is a powerful strategy; it helps for (-a) – (-b) where |a| > |b|; particle/charge problems (and maybe debt/savings problems) can help support the development of this strategy by furnishing students with contexts they can later turn into metaphors and interpretations.

In Kent’s post (clicky), he detailed four major contexts that are commonly used to help students get a handle on negative arithmetic:

1.       Elevation
2.       Temperature
3.       Money
4.       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.

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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!

Apologies, Kent, for not posting my response to your piece yet. It’s in the works. Part of the reason why I haven’t responded is (sick baby/jewish holidays/new school year/baby gets sick again) your normal life stuff. But I also feel like my understanding of integers is changing every few days right now. I need to capture a bit of learning I’ve had over the past week or so, to remember what this sort of learning about teaching math can be like.

This is a story about how I learned something new about how kids can think about subtracting negatives.

It started two weeks ago. I asked my Algebra 1 kids to answer some questions about integer addition/subtraction (taken from the Shell Center).

I was talking about this in the math office, and two other Algebra I teachers decided to give the assessment to their classes and (knowing the nerd that I am) share the results with me. There was one thing that I was truly surprised to find. I liked it so much, I tweeted about it:

(I think it’s really important to this story that I was genuinely surprised by this.)

I explained this to myself as a fairly sophisticated understanding of negative numbers. When I imagined what it would look like in class, I imagined relating the idea that (-8) – (-3) = (-5) to the idea that you can multiply both sides of an equation by -1. I thought to myself, hmm, that would be tricky to bring up in class, but it’s a very cool, sophisticated way to think about it.

I moved on, though the idea was still in the back of my head.

(I think it’s really important to this story that I am actively engaged in an attempt to better understand teaching/learning negative arithmetic with Kent. If I wasn’t engaged in this project, I might never have gone any further.)

The next step was murky. I was teaching my kids negative numbers (a little bit here and there, never our main curricular focus) and I was also reading articles and swapping emails with Kent.

Last night I threw out a question on twitter:

Kevin Moore, a researcher, replied with some names of researchers who work on understanding how kids think about negative arithmetic. This pointed me to Project Z, a big CGI-style research project about how kids think about negative arithmetic before and after formal instruction.

After a few hours of poking around (I got really into it!) I landed on this video of a kid (“Jacob”) thinking about -7 – x = -5.

Here’s the key part of his thinking, the thing that excited and surprised me:

• He takes 5 cubes, counts them. “Pretend this is negative 5.”
• Puts 2 more cubes on top. “Plus 2.
• Then he says a bunch of interesting, incomplete sentence stuff.
• “7 – 2 = 5.”
• After some thinking, he writes, cautiously, “(-7) – (-2) = (-5).”

That made sense with something I had seen the Project Z researchers describe in a handout. They’d call this “analogical reasoning,” where the analogy is between negative numbers and whole numbers.

Now, I’m not saying this is some genius clicking on my part. But I was genuinely and honestly excited and surprised by all this. And it made me realize that the thing I had seen the kid do on the Shell Center assessment probably wasn’t the sophisticated equation manipulating that I saw. Probably, instead, they were describing some analogical thinking.

All this is making me realize that I’d never encouraged or helped my students think about negative numbers in this way. Would it work for every case? No, of course not, but thinking about (-8) – (-3) is hard enough for kids that having a way to think about it would be immensely helpful.

Then I connected this with a feeling I’ve had for a while. When looking at contexts for negative numbers with Kent I’ve felt that there really can’t be just one approach, that at best we’d need some sort of case-by-case instruction. Different models, analogies, strategies are helpful for different types of integer problems.

I had thought this through for contexts, but I hadn’t connected it with contextless thinking about negative numbers. With this, I was able to make a connection to what I know about teaching whole number arithmetic in early grades. I tweeted about it:

This has implications for my teaching of integers. I have a new strategy I can throw into the mix, new strategy I can keep my eyes out for in student work. I have a new focus for integer subtraction.

There’s always the meta-question: what allowed this learning to happen? Here’s what I’m seeing as significant in this story:

• Having a project helped.
• That the project was about student thinking about a particular topic was important for a lot of reasons. First, it’s the sort of thing different teachers on twitter can reasonably talk about. Second, student thinking about particular topics has obvious implications for teaching. Third, it’s something that researchers also work on, so I was able to be connected to Project Z by Kevin.
• It was important, I think, to be surprised. To be surprised you have to have expectations, though. I think my expectations were formed by looking at the large collection of student responses to the Shell Center task. That made this kid’s work stand out.
• I didn’t really understand the research until I watched the video of Jacob working on -7 – x = -5. I was surprised by that because I had tried to explain the surprising thing before.
• Being on twitter helped for two reasons in this process. First, twitter helped me to mark a that initial observation of (-8) – (-3) = (-11) as surprising. I used twitter as a sort of bookmark for an interesting idea. Second, without twitter Kevin wouldn’t have shared Project Z with me.

This story is about changing the way I see student learning, but it’s supporting the way I had been thinking about my own personal learning. This story increases my confidence in the potential of focusing on understanding student thinking for improving my teaching.