5 – (-2): Taking Away -2 from 5 , or Comparing -2 and 5?

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.

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!


How I Learned Something About Negative Integers

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.

A common problem geometry teachers face is that people have overly narrow perceptions about what counts as math. This makes sense. 90% of what kids learn in math class is arithmetic or arithmetic’s older sister algebra. These associations often don’t fit with what goes on in geometry, to the detriment of geometry. For some kids, these perceptions of unmathiness undercut their motivation to try to understand the central skills of geometry. For others it can lead to a very, very slow start to the year, where they feel as if perpetually stuck in an introduction to a subject they never get to. (When are we going to do the real math?)

One approach to this problem is to be explicit and unwavering. This is math, and we will spend class working on drawing, defining, sketching, debating, justifying and explaining. This is what math is about, just as much as it’s about practicing skills or developing procedures. Don’t like it? Tough cookies. This is geometry, and you need to learn what geometry is all about.

I don’t like this approach. It doesn’t fit with what I know about learning and teaching. I don’t teach kids to solve equations by being explicit and unwavering: This is how you solve an equation. What is this guess and check stuff? Algebra isn’t arithmetic, time to get with the program kids! Instead, we take what kids know about arithmetic and we build on it. This is one way to solve an equation, but it’s inefficient. Here’s a new tool, it’s called algebra.

I’d argue that just as we ought to take a developmental approach to teaching any mathematical topic, we ought to take a developmental approach to developing students’ perceptions of math. All we can expect by being staunch and strident is for lots of kids to not actually understand what we’re talking about.

If we want to take a development approach to solving equations then we’re tasked with articulating students’ existing thinking, eventual thinking, and proposing thinking that could bridge those two poles. If there’s an analogy here, then those are the questions we need to answer about how geometry students are likely to perceive math.

What will my geometry students likely perceive as legitimately mathematical coming in to my class? What is my goal for their mathematical perceptions?

I don’t know. Evidence — student work or research — would be helpful here. And maybe I should find a way to smoke out their beliefs about math early on in the school year. In the meantime, here’s a guess as to what students’ think counts as mathematical:

  • Numbers are involved
  • After learning some math you can solve a new type of problem
  • Math questions are either right or wrong
  • Math is essentially communicated in symbols or technical language

Now, all of these beliefs are wrong in general for mathematics. That doesn’t mean these beliefs are entirely wrong. These beliefs are true for most of the math students are likely to have seen in their schooling. A shame, yes, but that it’s a shame has nothing to do with kids’ thinking, which is as legitimate as it is wrong.

Prior beliefs aren’t valuable so that we can lambaste kids for having them. Imagine starting a unit with a formative assessment that reveals that kids have all these wrong beliefs about exponents. Next class, everyone sits down and then I say, “Well so many interesting beliefs about exponents. But they’re all wrong! Here, let me show you.” I know that some teachers do this, but it’s a silly activity. You aren’t really helping kids develop those ideas and see their limitations, you’re telling them not to think, that their opinions aren’t worthwhile. You’re not giving them a chance to be smart. And yet this is conventional wisdom about how to deal with students’ beliefs about the nature of math and geometry.

What should we do once we know what students beliefs about exponents are? We should creative opportunities for kids to understand the limitations of their beliefs, give them time to formulate new theories, make explicit the truth about exponents once they have the experiences that can help them make sense of that. And shouldn’t we do that with students’ beliefs about math also?

I have a favorite style of activity. It’s an activity that kids can do with their current knowledge, but can also be done with more sophisticated knowledge, and it can be done better with the more sophisticated knowledge. In this case, that would mean that I’m looking for math that is both recognizably geometric but also makes sense as math in the limited theories that my students are likely to come into the year with.

What would that mean? My favorite geometry textbook spends its first major of unit focused on constructions. But the above puts some constraints on what sorts of constructions it would be a good idea to start with:

  • Constructions that include numbers
  • Constructions that can help students solve a new type of problem
  • Constructions that are either right or wrong
  • Constructions that involve symbolic language

Do such constructions exist? There are certainly constructions that include numbers — my favorite textbook includes a bunch of them, things like “make a triangle whose sides are exactly 3 in, 5 in and 7 in long.”  And there are certainly types of construction problems that are awfully similar: “construct a triangle with these sides”; “draw the angle bisector of this angle”; etc. And these are either right or wrong, in a sense. I could also introduce precise, symbolic and pictorial language to describe these constructions. This would feel like math.

But that’s not going to be enough. I also want to nudge my students beyond these conceptions. Math involves explanations and proofs. It often is done in informal, everyday language. It’s often more than just right or wrong. How do we get at this?

I think my plan is to build this on top of the other constructions. We’ll ask questions about the numerical constructions — why does this work? why doesn’t this work? why can these sides make a triangle? why can’t these? — and in dong so we’ll make conjectures, create informal proofs of theorems. It will be important for me to make explicit that we will spend much of the year focusing on these sorts of arguments.

Experience before formality. That’s a great principle, and one that we should use not just for mathematical content but also to help students develop their perceptions of what it means to be mathematical.