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.

First thing: As always, credit to Aran Glancy and Christy Pettis for the game.

I think I agree with the end of your post (the narrow view of contexts), which allows me to agree with the rest of it while still typing a long reply! Here goes.

The area model of multiplication is a great context for teaching multiplication strategies because it provides a visual model for something that likely already makes sense to kids. What I mean is, kids can look at the problem 12*34 and explain that we are finding 12 groups of 34 (or 34 groups of 12) even if they don’t know the answer. So the visual model of area helps them by making the task of counting those groups more accessible, and by helping them find strategies such as decomposing and using the distributive property.

Conversely, if you give a middle school kid the problem -7 – (-24) and ask them what they are finding, they will have a difficult time explaining what that problem means, even if they can solve it. You are taking away 24 negatives from 7 negatives? It’s confusing.

So then giving students the strategy “This is the same as adding a positive” is doubly confusing because now they have the problem -7 + 24, which feels more comprehensible but doesn’t feel connected to the original problem, other than by the teacher’s insistence that it is equivalent. (I am oversimplifying and condensing lots and lots of conversations with students here, to be sure)

So what does the context of balloons and sandbags, or anchors and floats, or debts and credits, do for these students? First, I think it provides a mental architecture for how negative numbers relate to each other. We know that young kids can learn how to count, but that comparing numbers takes more time. My son could count to 10 months before he could decide whether 6 or 8 was bigger. Middle schoolers, I think, have the same struggle with negative integers. They can count them and even label them on a number line, but they still have trouble answering questions like “Which is greater, -6 or -8?” correctly. The answer is counterintuitive because it goes against their longstanding experience in comparing numbers.

So by playing the game, kids learn how negative numbers relate to each other. They count up and down repeatedly and start to get a feel for the number line. They can more easily answer questions like “Which is greater, -6 or -8?” because they can mentally change the problem to “Which is higher, -6 or -8?”

I don’t know if that counts as a strategy, but it forms the basis of a lot of other strategies. By playing on the number line, kids can start to notice patterns within the integers. For example, the numbers -5 and 3 are 8 away from each other, and that is deeply related to the fact that 5+3=8. It took me years as a teacher to realize that kids don’t always notice that pattern at first. But over time they can, especially if they’ve actively played around with the number line a lot. Then you can give them problems like 17 + (-25) and ask them to model that on an open number line. They don’t want to draw 17 dashes, so instead they can just draw a big hop from 17 to 0, and then determine that they have 8 remaining to go, so their final value is -8.

I think that process of using zero as a benchmark is a strategy in the way you are describing. So is the strategy you wrote about, where kids treat subtracting a negative as adding a positive. And I think kids can develop that strategy within the context of the vertical number line, while also being able to describe why it works.

I have asked students. “I am at 7 and I want to go to 10. What are the two easiest ways to get there?” The students mentioned adding 3 balloons and removing 3 sandbags as the most direct options. So then I ask them to write these scenarios as expressions. They write 7 + 3 and 7 – (-3), and I think they can explain why these are equivalent.

I really don’t want kids to think of the additive inverses as solely being a way to solve subtraction problems. I want them to think of the additive inverses as the core concept they are learning from this unit. The whole unit is built to engrain the idea of equivalence here. Yes, it’s a strategy for the problem -7 – (-24), sure, but hopefully it goes deeper than that.

To get back to area, ideally we want students to be able to explain the distributive property using the area model, so that they have a solid foundation for using the distributive property in Algebra and beyond. I want my students to be able to explain the equivalence of additive inverses using the number line so they have a foundation for using the inverses in Algebra and beyond.

Whew. Maybe I should post this on my blog too.

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I also think building up the notion of subtraction meaning finding the difference/distance between two numbers on the numberline is helpful here. In -7 – (-24), I can think about that as the distance between -7 & -24 on the number line. I haven’t quite figured out how to keep track of whether the distance should be positive or negative here, except to build off the pattern of positives: if you subtract the smaller number from the bigger number, it’s positive, whereas if you subtract a larger number from a smaller number, it’s negative (always true, I think, regardless of the original signs?).

This is going to be my first year teaching negatives to students for their first exposure (previously, I taught 8th grade and I just remediated negatives). In the past, I’ve used two color chips/algebra tiles, and I’ve seen student success with the idea of “removing zero pairs”. But I am beginning to recognize the power of the numberline. One thing I especially like about this context is the inclusion of the numberline.

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Fantastic comment. I certainly agree that “finding the 0” is a strategy that can be developed in context and applied out of the context.

What you’re saying, I think, is that kids think about taking away a weight and realize that this is the same as adding a lift, and that this is the same strategy that kids need to use for subtracting a negative.

What I’ve been saying, I think, is that “taking away a weight adding a lift” is not how kids will approach the problem in context. They’ll use different mental moves to make sense of taking away a weight, and these mental moves don’t map on to those needed to make sense of subtracting a negative. (For example: “taking away a weight makes things go up, so it’ll go up by the amount you take away.” Negatives don’t appear in that line of thought, I don’t think.)

Now, though, I’m less sure. I’m back to seeing it your way.

Why do I keep going back-and-forth on this? I don’t know. I think it’s very difficult to represent negative quantities in a context, and so it’s very hard to represent taking away negative quantities. Like, when kids are working on area problems, they’re using the same numbers that they’d use in multiplication. So those strategies have some face similarity to work with. But since negative quantities aren’t explicitly represented in the hot-air balloon context, I don’t know if kids are thinking about subtracting a negative. And so I have a hard time feeling sure that kids are thinking about negatives, as opposed to positive quantities that have some opposing effects.

But does this make it different than the area situation? If so, is this difference important? If it isn’t an important difference, I see it your way. If it is an important difference, I see this context as providing an analogy for subtracting a negative.

This line of thinking is making me curious about a context where the amounts are explicitly labeled as negatives. Is there any? Would that help?

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This was a fruitful conversation to watch unfold. Kent, I’m in awe of how easily you put your ideas out there, gets lots of feedback, and improve them so rapidly. It’s this wonderful mix of courage, humility, and confidence. Michael, you’re making a good point about contexts and I appreciate how you find ways to broach challenging subjects on Twitter (and supplement with your blog).

Kent, something that strikes me in your reply is “play”. It pops up repeatedly in your reply as a sense making tool for your students. I’m not surprised. Much like Michael is asking about different ways we think about context in a math classroom, I’m curious, what different ways we think about play.

I’m not tilling any new soil here, but the idea of play as a way to learn, and develop new ideas keeps showing up in a variety of different fields, like a big flashing sign that I can’t ignore. Most recently when I read The Idea Factory about Bell Labs and it chronicled Claude Shannon, the creator of Information Theory, and how he developed new ideas. The guy literally played around and not just in one domain. Thanks to his play we have Twitter, this website, and everything else that is encoded as 1s and 0s. This confused me, because the idea of ‘play’ seems to hold a juvenile connotation in schools, and is even used as a pejorative in anything but kindergarten. I’m glad to see you and your students harnessing play to learn about integers and arithmetic with them.

Keep up the great collaboration.

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Michael,

I think you make an interesting point about sandbags and negatives. I agree with you that kids wouldn’t naturally make the connection, and that it has to be explicitly noted by the teacher in order to be effective. You’re making me wonder whether I should get out a Sharpie and put 1s on all the yellow tokens and -1s on all the red tokens just so kids make that connection more easily between sandbags and negative numbers. I’ll try it out!

Mark,

You are totally right that play is a big part of my rationale for the game, a part that I haven’t written about yet. I should flesh this out further on my blog, but basically I think that games provide a great motivation for understanding the underlying math.

I wrote about my son’s love of Uncle Wiggily before (http://www.kenthaines.com/blog/2016/6/7/uncle-wiggily) and I think it provides an excellent example of how games can promote learning. Joel used to have trouble determining which was more, 6 or 8, for example. But by playing Uncle Wiggily over and over, he learned which cards were good and which were bad. So 12 is better than 3 because you get to move farther. The game gave him a reason to attend to the relative size of the various numbers he could draw, which meant that he started celebrating or pouting the second he looked at his newest card.

My hope is that the balloon game gives middle school kids that same sensation – after you’ve played a time or two, “Remove 3 sandbags” gives the same sense of elation that “Add 3 balloons” does. Conversely, adding sandbags and removing balloons are bad cards. I want kids to make a connection to these four moves on an emotional level, based on how they affect the gameplay.

So yes, play is important because it keeps kids engaged and happy in school. But it can also directly improve learning outcomes, as long as the underlying math is made explicit at some point.

Like I said, I need to flesh this idea out further.

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Do you have resources to share so I can try out the same game with my students?

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I really wish I had been more involved in this conversation as it happened, but I’ll leave my better-late-than-never thoughts here for anyone still following along. Thanks Kent for both running with some of the ideas we’ve discussed and for openly sharing how it went in your classes so we can have this discussion. And thanks Michael for not letting us take anything for granted.

I want to start with something Michael said in the original post:

“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.”

This is a key point and is precisely why I originally set out to create a game and context for integer operations. Integer arithmetic is one of the first places where common sense and intuition start to fail students. Whole number arithmetic and even fractions match up nicely with our experiences counting, grouping, or even distributing ‘things’ like apples, rocks, chips, etc. Even when negative numbers are introduced, they can be thought of as counting things on the other side of zero, i.e. floors below ground, apples owed, etc. But 2 floors below zero is really not the best way to think about integers. For starters, -2 + 2 should equal zero, but what is 2 floors below zero + 2 floors above zero? And another problem, as Kent so insightfully said, is that once we start subtracting these negatives we are asking students to completely change a bedrock principle of whole number arithmetic, namely subtraction makes smaller. This is no small task, and in my opinion, requires that we provide kids with stories that help them make sense of this non-intuitive fact. So Michael, I think you’ve hit on something really important (although I also think you’ve understated the importance) when you talked about how these stories help make meaning of non-intuitive ideas. At the same time, you also pointed out that there are very few stories that do hold together well with integer arithmetic. This is definitely true, and after spending a lot of time thinking about this, I really think that stories like floats & anchors, or Kent’s balloons & sandbags are a better places to start than things like temperature (or just number line in general) or even money.

So let me talk about this story for a second and how it maps to integer arithmetic. Michael you described your thinking for “My elevation is 10 and I throw out 5 sandbags.” How I would hope that a student approaches this (and how I’ve seen many, many students who’ve played Floats & Anchors approach it) would be to say, “throwing out 5 sandbags will make the balloon go up five. Up five from 10 is 15.” Now how does that translate to integer arithmetic? The corresponding problem is 10 – (-5), and I would hope that a student reasons that subtracting -5 will “raise” the starting value (10) by 5 resulting in 15. Essentially, this is saying that 10-(-5) = 10+5. But think about how different this is in the mind of the student than if they had had the rules of integer arithmetic explained to them. In one case the student says, 10-(-5) = 10+5 because (based on this story) that’s the only thing that makes sense. How could it be anything else? On the other hand, we have that 10-(-5) = 10+5 because my teacher told me that mathematicians all got together and decided that subtracting a negative is the same as adding the additive inverse. Which is more satisfying? Which has the potential to leave students with an uneasy feeling about math? I’ve watched teachers say “subtracting a negative means you actually go in the positive direction” (using the number line) and heard students respond with, “but shouldn’t subtracting a negative be even more negative?” On the other hand, the very first time I introduced floats and anchors to a class of students, I asked them what happens when we try to remove anchors when we don’t have enough (i.e. 3 -(-4)). Almost immediately a student responded with “just add floats, ’cause it does the same thing.” In this context (and Kent’s), the equivalence between subtracting one object and adding the other is obvious, intuitive, and really couldn’t behave in any other way. And the context is completely isomorphic to integer addition and subtraction.

All that being said, Michael and Kent bring up a good point about connecting the story (game, context, whatever) to the actual arithmetic problems. This is not at all a trivial task. Students can understand and make sense of the context while at the same time not being able to apply it to integer arithmetic at all. When I work with teachers to implement our game, we spend a minimal amount of time actually playing the game. Really, we’ve seen that 15 or 20 minutes of game play is usually enough for kids to get the idea and understand the context well enough to move forward. The real work is getting them to see the mapping between the game (whether it be floats & anchors or balloons & sandbags) and the arithmetic. They can’t use any intuition they have or have developed about the game in solving arithmetic unless they understand that mapping. That’s what Christy & I have been working on over the last year or so. Things that we’ve had success with include making the mapping explicit (anchors/sandbags = negative numbers, floats/balloons = positive, height above or below sea level/elevation = net value, subtraction = removing, adding = well, adding), but beyond that we work with students to 1) translate symbolic problems into the context of the story and 2) work with open number lines (like Kent) with the game pieces to model problems. The first is to allow them to use the intuition they have from the game in solving the problems, while the second is actually to help they build understanding of a more powerful model–directed line segments on the number line.

Yes, despite my evangelism for Floats & Anchors and the like, I think a number line model is ultimately a better one. The problem is that, in my experience, kids think of numbers (positive and/or negative) as discrete, countable things, but to really make sense of the number line model you need to understand numbers as measures. This is (again in my experience) a much harder thing than people often think. So Kit, although I agree that the number line is powerful, I do not agree that it is the place to start with integer arithmetic. I think kids are better off getting a meaningful, intuitive understanding of integer arithmetic based on the “comfortable” ideas of numbers as counts of things and then learning how those ideas are 100% consistent with a number line representation.

Two additional notes: Although it has come up throughout the conversation, I think we have really glossed over the importance of sea level in my model and elevation in Kent’s. This is what sets these two approaches apart from the traditional chip model or from straight forward number line models like temperature. Here is why: a red and yellow chip may cancel, but you are still left with two chips sitting on the table. Yet that is meant to be zero? Why? Because the teacher said, those are the rules? In the game, a balloon and a sandbag (or a float and an anchor) cancel, but the zero has meaning–it’s the elevation (or position above or below sea level). 3 balloons and 3 sandbags don’t make zero in the sense that you have “nothing” but they do put the balloon at an elevation of zero. Now 0 has new meaning–not just “nothing” but also balance, or equilibrium, or the position between positive (up) and negative (down). Also, sentences like 3 – (-5) are hard to make sense of thinking only of balloons and sandbags, but easy if you include elevation (I start at an elevation of 3 then remove 5 sandbags).

And my second note is in regard to Kent’s idea of adding -1s to all the sandbag chips. I’m not saying you shouldn’t try it, but I don’t think it will make things better. Connecting the game to integer arithmetic is non-trivial and needs to be made explicit, but I’m a firm believer that this should be done after students have made sense of the game on its own. I’ve played my floats & anchors game with very young children who have little experience with negative numbers, yet they can make sense of the game and the context with little difficulty. The idea is to build the arithmetic on the context that makes sense. My concern is that by explicitly labeling the negatives within the game you may detract from their ability to understand the concept. In my game we don’t even put negatives on the numbers marking feet below sea level. Sure, some students almost immediately start referring to those positions as negative feet, but some do not. These students just talk about feet above and feet below. In my opinion this is totally fine and does not interfere in any way with their ability to understand how floats, anchors, and position relative to sea level relate.

And Mr. Kit, I’m sure Kent will share his stuff, but you can find all the materials for my floats & anchors game here https://sites.google.com/a/umn.edu/integer-games/

(And sorry for such a long comment. I really need to get a blog going to I have a space to vent some of these ideas. It’s on my to do list.)

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I’m sorry if I made it sound as if I understated the importance of these sorts of stories. Really, really, really I don’t mean to. I think they are very important. It just took me a while to understand that this was what the game was doing.

As this conversation has gone on, I feel as if I’ve gotten more focused about my worry. And my worry really is about connecting the game to negative arithmetic.

I understand this. But — and I think we’re all on the same page here — there is absolutely nothing in the “throwing out 5 sandbags makes the balloon go up five” that involves negatives. And that strikes me as a big difference in the way this context functions, compared to how contexts function in learning non-negative arithmetic. That’s my whole point.

I don’t think this is quite right. What I’ve seen is that teachers typically give an explanation that isn’t so different from the explanation you end up giving when you start with the game:

So what’s different about your approach? First, that you are really making sure that everyone has experience with the context that’s going to be part of this explanation. Second, you’re going to spend a lot of time using that context to show how, in the terms of this context and your interpretation of it, subtracting a negative is adding a positive.

But there’s still a bit of math teacher magic here:

you ask kids to believe that the hot air balloon context faithfully represents negative numbers, even though they do not appear in the hot air balloon context.This is just my point above, stated in a different way. Because negatives don’t appear naturally in this context, it functions differently than contexts in other areas of math.I think this is a good piece of math teacher magic. But it’s still there, and it’s there in a way that it isn’t when I use contexts to teach multiplication, or use specific proportion problems to develop ideas of linearity. That’s my whole point, restated.

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Ok, so now I’m starting to understand where you are coming from, and you raise a really interesting point. I’m going to disagree; however, probably not in the way you would think. First, I’ll grant you that we ask kids to believe that sandbags faithfully represent negative numbers, but I don’t think I would go so far as to say that negative numbers don’t naturally appear in the context. When we start working on making the connections between the context and the arithmetic, we don’t *tell* them that anchors represent negatives, we *ask* them what numbers would make sense to use to represent anchors. Invariably, they immediately say negatives. (Sorry for switching metaphors, but now I’m talking about my experiences, not Kent’s.) So sure negative numbers don’t appear explicitly in the context, but neither do rational numbers. The leap from context to abstract is not a hard one for the students to make, at least in my experience.

But that’s just a specific point about this model. Let me step back for a second to a bigger point regarding the difference between this model and the area model. Here is where I think I really disagree with you. Like I said, we ask kids to believe that integers behave the way floats & anchors do, but I think we do exactly the same thing with the area model. Experience composing and decomposing areas does not help with multiplication until you see the connection to multiplication. Even a simple array model of 3×4 isn’t going to be helpful unless students see the squares in the array as groups of rows or columns. Sure this connection is really straightforward, but it is not a given. Additionally, the area model eventually breaks down just like all of our models. How do you apply an area model to multiplying irrational numbers? Or even negative numbers? How does an area model help explain the scaling interpretation of multiplication? So yes, looking at anchors and seeing negatives is a leap we ask students to make, but so is looking at an m x n array and seeing m groups of n. Once students learn to see it that way, they can use that model to help them think about problems, but seeing the connection is always the first step.

So although I agree that there is a little bit of “math teacher magic” in the floats & anchors/balloons & bags approach, I think there is just as much teacher magic in most (if not all?) the other models and metaphors we use to help students think. But I don’t think that’s a bad thing, and I don’t even think it’s “hand waving.” I actually think it is the work of mathematics. Some of the most exciting results in mathematics, at least that I’ve heard of recently, have come from making connections between different things that behave isomorphically. These connections help mathematicians gain new perspective on each of the original concepts. When we ask students to connect things like arithmetic to other models, whether counting sets, area models, number lines, or integer stories, we are asking them to do the same thing.

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Really thought provoking comment! It’s pushing me to develop my view a bit more. Here’s an updated take.

I completely agree with this, Aran. Every situation or context is different from any other. Every context or situation also contains similarities to others. Math is about drawing out important connections, but it’s always going to involve some assertion (or convincing) that the differences aren’t crucial.

But, clearly, this can go wrong in teaching. I remember in college (that unending source of bad math instruction) I experienced these sort of “intuition building” metaphors that often left me clueless. Here’s an example: two light switches. There are four positions they can be in. One can be on while the other off, both off, etc.

See? Now you understand the Klein four-group. Intuitively!

I think the point is that similarity/difference of two contexts is not an objective matter — it depends on the learner’s point of view. For someone who knows a bit about group theory (like me, now) it’s possible to see two light switches as having the structure of the group. This allows me to look at the context and use it to think through aspects of group structure.

I feel confident that, for most of my 4th Graders, it’s relatively easy to see multiplication in an area context. I have sometimes taught children who didn’t see this, and I think for these kids, there is a bit of (problematic) magic with using area contexts to learn about multiplication.

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In the context of subtracting negatives, what does this mean?

My worry was that kids don’t see negatives in these games — they just see various positive quantities. But how you see these contexts is dependent on learning and instruction, so let me take my comment and turn it into a hypothesis.

My hypothesis is that learning about negative subtraction mostly happens when kids are explicitly taught to see negatives in anchors/sandbags. Earlier, Aran, you said that it’s important to hold off from negatives when introducing the game. That’s fine, but it seems to me that it would be important to play the game after we encourage kids to see negatives in these contexts. Otherwise, we might lose the benefit of these contexts for negatives.

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That Klein four-group example is just perfect. I think you are absolutely correct. A context can only be helpful if you have the right prerequisite knowledge. That knowledge may be of some prerequisite math, or knowledge of the context, or (more likely) both. I’ve never really thought of it this way, but I think this has a lot to do with how we’ve ultimately structured our instruction for integers operations. We do start by letting kids play the game without forcing integer arithmetic on them, but then, as you suggest, to truly capitalize on the context, we need some explicit instruction. We’ve done this by turning the game pieces (floats, anchors, a vertical number line, and a boat) into manipulatives that we have students use to directly model integer arithmetic problems. I still think that that initial play phase is essential for letting students explore and understand the context, but you are right, without some direct connection to the integer arithmetic it doesn’t work. We actually have data from our initial attempts which bears that out!

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