Bringing Addition to a Multiplication Party

I just realized that two things I had thought to be quite different might, actually, be really similar.

First, a series of mistakes my 4th Graders make when they use addition thinking for multiplication problems:

Second, my 4th Graders’ thought that multiplication by a negative would make a number positive, but smaller:

Yesterday a couple of 4th Graders asked, “Wait can you multiply by a negative?”

Any guesses as to what prompted this question?

Kids had been working on a multiplication puzzle and (accidentally) gotten themselves into a position where they needed to solve ___ x 20 = 10. If positive numbers make multiplication bigger, then shouldn’t negative multiplication make things smaller?

What is this mistake? Why should multiplication by a negative make a number smaller, but positive?

Here’s what I’m realizing: it comes from the thought that positive/negatives have opposite effects in multiplication/division. Which isn’t true, but it is true that positives/negatives have opposite effects in addition/subtraction.

The relevant opposites when it comes to multiplying aren’t positives/negatives, but instead numbers greater/less than 1. To draw the contrast really clearly, when it comes to adding the relevant opposites are numbers greater/less than 0.

This is not some out-there and abstract idea, though. When kids work with negative numbers they regularly reveal an understanding that positives and negatives should have opposite effects, as with 3 – (-5):

We talk a lot about opposite operations, but do we talk enough about opposite numbers? We talk a lot about negatives as opposites to positives, but do we talk enough about numbers less than 1 as opposites to numbers greater than 1? How much of learning is trying to figure out the limits of thinking like addition?

 

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6 + -2 as “Adding a Minusing 2”

For the last month or so since I read this article, I’ve been unable to shake the idea of “mathematical worlds”:

Rather than describing the challenges of integer learning in terms of a transition from positive to negative numbers, we have arrived at a different perspective: We view students as inhabiting distinct mathematical worlds consisting of particular types of numbers…Proficient students and adults may also inhabit multiple mathematical worlds from moment to moment.

This metaphor completely fits my experience with children. There is often doubt about what “world” we’re operating in. Are fractions allowed? Are we using i right now, or does x^2 = -9 have no solutions?

(The other relevant metaphor is of a game, as in “I didn’t realize fractions were allowed.” Is there a connection between the metaphors or world and of game?)

One thing this metaphor helped me realize is that my students might bring up negatives in my class more often if I simply gave them permission to. How do you give permission? I decided to bring them up right away with my 4th Graders. (I culled the questions from here.)

I handed out a piece of paper to each student. I asked them to work completely on their own. The first question was 9 – 10 = __. Here were the responses:

• 1 (7 kids)
• -1 (4 kid)
• ? (1 kid)

Next I asked, 3 – 5 = ___, and received similar responses. So far, kids were pretty confident.

The next question led to protests, though: 6 + __ = 4. Some protests, some talk of -2. (Wait, are we allowed to use negatives?) I collected their papers and, since kids seemed to have what to say about this, I asked volunteers to share their thinking.

Isn’t it interesting how the first two questions have positive answers but the third has a negative response? I think this student swapped worlds between #2 and #3.

First went E, who explained that since 6 – 2 = 4, this had to be -2. I wasn’t sure exactly what she meant by this, but then H objected to this whole thing in an incredibly helpful way:

H: “Wait, I don’t think that makes sense. How can we say that we’re doing 6…plus minus 2?”

Me: I have no idea what you mean so I’m going to offer a belabored explanation of how we’re not adding “minus 2,” rather we’re adding “negative 2.”

H: “I mean like I don’t see how you can add minus like that.”

Me: Wait, I think I understand what you’re trying to say. You’re saying it’s like if we said 3 times divide 2, like, what would that mean?

H: Yeah, that doesn’t make any sense.

But then a bunch of other kids jumped in. They were totally able to make sense of “6 plus minus 2.” They said you were adding a minusing of 2. This is subtly different than anything I’d ever before understood kids to say. They weren’t treating “-2” as a point on the number line or even as a single number at all. They were making sense of -2, in this context, as an action. The action of subtracting 2.

All the while this was making me nervous. Middle and high school students have trouble distinguishing between the three slightly different ways in which we use the minus sign in mathematics. (See: “High School Students’ Conceptions of the Minus Sign”)

Should I have said something to keep kids from thinking of “6 + -2” as “6 plus a minusing of 2”?

The more I thought about it, though, the more confident I became that this was an incredibly productive way to start thinking about adding a negative. Kids who think about 6 + -2 as “adding a minus 2” are already along the road towards thinking of the process of subtracting as an object (a subtracting). There is more work for such a thinker to do — they have to come to think of this subtracting of 2 as the sort of thing that you can operate on. (What does it mean to add 3 to a minusing of 2, as in -2 + 3?) But there is a clear conceptual path for these students to take.

(I’m thinking here of Anna Sfard’s idea of interiorization, condensation, and reification.)

This all is a brand new idea to me, a way of making sense of negative arithmetic that I’d never understood before. I continue to be blown away by the variety of ways that kids make sense of arithmetic with negatives — each problem situation seems to allow for slightly different conceptions. Really making sense of negative arithmetic might involve negotiating many different ways of making sense of many different types of problems. It’s only once kids have a few different mental models that they can start uniting their thoughts into one big idea of negative numbers.

We’re going to keep this up in 4th Grade. Actually, this all happened yesterday, and today kids ended up digging into the number line, its mirror-like symmetry and the “realm of the negatives” (which J connected to Minecraft and Portal, god save the children.)

Our notes from the discussion today.

 

Kids seem to be having fun exploring this realm, and they’re doing good math. Why not give kids permission to enter this world?

But They’ve Never Heard of Positive Numbers

Ancient Greek mathematicians had no notion of negative — or positive — numbers. Similarly, for most young children, numbers are not signed. Children learn to count with natural numbers. At some point, they learn about zero. Later, they encounter (regular) fractions, decimals, and percentages. Typically, a child’s formal introduction to the notion of sign comes after all this experience. Interestingly, we have found that many students in the elementary grades have some familiarity with negative numbers but have never heard of positive numbers. These children inhabit intermediate worlds that consist of regular numbers and negative numbers before they begin to (intermittently) inhabit worlds of positively and negatively signed numbers.

Regular Numbers and Mathematical Worlds

Chatting with @KentHaines @AranGlancy about Contexts for Subtracting a Negative

Test run of my integer unit is going well! We are climbing the ladder of abstraction. @aranglancy @mpershan pic.twitter.com/kXYpKGZmmS

— Kent Haines (@KentHaines) September 1, 2016

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.

 

 

Three Ways Kids Might Think About 3 – (-5)

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.

—

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:

I can’t believe I didn’t realize you could think of subtracting negatives like this

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.

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:

Just saw my new favorite explanation for subtracting a negative: "5 – 3 = 2, so -5 – -3 = -2."

— Michael Pershan (@mpershan) September 22, 2015

(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:

Kids consistently find 2 + (-4) much easier than 2 – (-4).

Why?

— Michael Pershan (@mpershan) September 27, 2015

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.

Jackpot! http://t.co/UsJhodAjn6 https://t.co/QDENxKh84U

— Michael Pershan (@mpershan) September 27, 2015

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:

When I help 3rd/4th Graders learn to multiply, we work on juggling lots of tricks, strategies and moves. Ought to be same for negatives?

— Michael Pershan (@mpershan) September 27, 2015

There's no one thing we use for multiplying. We use contexts, doubling, doubling/halving, skip-counting, break apart by 10s…

— Michael Pershan (@mpershan) September 27, 2015

Likewise, with negatives we'll need contexts, analogies to whole numbers, making zeroes, counting on, adding a negative as subtracting…

— Michael Pershan (@mpershan) September 27, 2015

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.