Ep 60: Number Bonds and Tape Diagrams

Pam Harris  00:01

Hey fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam. 

Kim Montague  00:09

And I'm Kim. 

Pam Harris  00:10

And we make the case that mathematizing is not about mimicking steps or rote memorizing facts, but it's about thinking and reasoning; about creating and using mental relationships. That math class can be less like it has been for so many of us and more like mathematicians working together. We take the strong stance that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keeps students from being the mathematicians they can be. We answer the question, if not algorithms and step by step procedures, then what?

Kim Montague  00:51

So in the last two episodes, we began a conversation around models and modeling, right, and how it's important to consider the progression of modeling First, the model the situation, then the model of student thinking, and then using that model as a tool.

Pam Harris  01:06

And in this progression of modeling, notice that it's not about any model. It's about models that can become tools, important tools. So in this episode, we're gonna get specific about a couple of recently popular models that we think should be de-emphasized and what we can do instead. So y'all, I often get asked about what I think about number bonds and tape diagrams, or sometimes called strip diagrams, and how they play out in some current textbooks. People wonder about that. People are really curious about how they fit. I think people might be noticing that we don't use those two models... at all.

Kim Montague  01:55

At all. So Pam, let's start with number bonds. What is a number bond? And will you describe a number bond for the listeners?

Pam Harris  02:03

Okay, so I think a number bond is a well intentioned idea of helping kids focus on the relationships at hand, focus on what's sort of happening in the problem. And so it looks like this: Often, it looks like circles and there's two circles that are sort of the parts. And those circles are kind of connected to this larger, the whole. And so there's like a big circle, that's the whole, and two circles that are the parts. And so - like explaining this over a podcast is tricky. So Kim, pick a number.

Kim Montague  02:38

65.

Pam Harris  02:39

Okay, so a number bond for 65, could be I put 65 in sort of the total place. And then I kind of draw two lines that come off of that circle, and then I draw two smaller circles. And I could say 60 in one and five in the other. Or I could do a different number bond where 65 in the total. And I can have 50 in one and 15 in the other. Or I can even have one where I have 65 in the total and I could have 70 in one and five in the other because 70 minus five is 65. And then I might have to play around with which one's bigger and all the things. Okay, so it's like sort of like, how are these numbers related to each other? Alright, so let's also talk about the other one. So we talked about number bonds and what they look like. Kim, what's a tape diagram? Or strip diagram?

Kim Montague  03:23

So a tape diagram is a rectangular box, maybe, with a couple of empty spaces where students would take numbers from a problem they're given. And they are supposed to put the numbers in the correct spot. So like, if it was a missing add-in problem, they might fill in one part and the total from the numbers in the problem. And then they need to find the other part. It's a way to organize information from the problem.

Pam Harris  03:51

Yeah, exactly. So I might have the total kind of outside the rectangle. I've seen where I might have one rectangle, and like you just said, the parts are inside and the total is kind of outside the rectangle. I've also seen it where the total is its own rectangle and then the parts are kind of a rectangle below that cut in pieces, and you sort of put the parts in that second rectangle that is the same size as the total. Okay, so let's talk about how those two models are, in a way, a model of the situation. So we've talked about our modeling context that we like, where kind of the beginning step is a model of the situation. And so in a big way, we feel like number bonds and strip diagrams are models of the situation. They are an attempt to get the relationships out there to make them visible. So which one's the part? Which one's the whole? Oh, do you have two parts and you need to find the whole? And so both of those models can sort of help kind of get that understood. If you think about a problem solving step-by-step procedure -which we don't advocate - but often that first thing is understand the problem, right? 

Kim Montague  05:06

Yep.

Pam Harris  05:06

It's like important. And why is that the first thing? Well, because it's important to understand the problem, or you're probably not going to solve it correctly. So of course, we need to understand the problem. And so this is a well, my number bonds and strip diagrams, are well intentioned attempts to say to kids, "Understand the problem". Where it can go awry is when we demand that kids draw it every time. If I'm a kid who already understand what's happening in the problem, I don't think it's very helpful to demand that I draw the number bond or the strip diagram. It's just like an extra step that doesn't help me; it's not improving my understanding. If I'm the kid who doesn't understand what's going on in the problem, it might help to draw the number diagram or the strip diagram, it might help to do that. But often, it also becomes like sort of this extra thing to do. It's like, how big do I draw the rectangle? And does it need to be tall?Kids like to pay attention to all this weird stuff that may not have to do anything with the problem itself. So it kind of becomes like this extraneous thing that may or may not be helpful. So do we want kids to understand what's happening? Absolutely. So also another thing that can kind of get in the way. So think CGI, have we ever done a podcast episode on CGI? 

Kim Montague  06:18

Surely we've mentioned it. 

Pam Harris  06:19

If we haven't, we'll do one soon. But if you're not familiar with Cognitively Guided Instruction, and we really appreciate Cognitively Guided Instruction - which is referred to as CGI - because they did a lot of really groundbreaking research that then has allowed us to build on that research. But sometimes we might have the opportunity to get a little bit mixed up here. In CGI, there's a strategy called direct modeling. Direct modeling is all about acting out what's actually happening in the problem, a good thing, right? Another well meaning like, let's model what's going on in the problem. That can be a very necessary thing for kids to do when they are beginning problem solvers. They need to like actually act out what's happening. And then they use that acting out to help them solve the problem. But that can be different than a model of the situation. Because it can be... how do I say this? Direct modeling is where you're actually like acting out the problem, what's actually happening in the problem. Strip diagrams and number bonds are less of acting out the problem and more of a model of the relationships at hand, or a model of the relationships that are happening in the problem. But what's interesting is, all of those are a model of the situation. In our framework, whether I'm acting out the problem, to understand it, and to solve it, maybe just say to understand it, whether acting it out of direct modeling, or I'm drawing a number bond, a strip diagram as getting the relationships out of what's happening in the problem, we would consider all of that the model of the situation. 

Kim Montague  08:01

Right.

Pam Harris  08:01

That's sort of the beginning of our kind of modeling scenario here, or the way that we think about modeling. So it might be true that a student needs to act out the problem, directly model the problem with actions or drawing what's actually happening. And that's fine. And we would call that a model of the situation. But we can also be more abstract in general, like the number bond and strip or tape diagram, where it's more of a graphic of the relationships involved, it's less the action that's happening. Either way, these models of the situation set up the relationships. What they don't do is represent what you're actually doing to solve the problem. How you're actually using the relationships to compute.

Kim Montague  08:44

Yeah.

Pam Harris  08:45

So let me get a little bit more specific. I know I'm doing a lot of like, not very exampled discussion here. So let's get examples. Let's get to some examples. So what's an example of a number bond and how it can be helpful, and maybe less helpful than we might have been led to believe? So here's a problem. Let's say, "Hey, we're reading. We like to read. Kim and I both read, reading's good. We're on page 195 of the book. And there are 303 pages in the book. Hey, how much do we have left to read?" That could be a great problem, right? There's the problem. We need kids to first understand the problem. We don't want kids flipping a coin, and just grabbing the numbers 195, 303, and like adding them or multiplying or dividing. We don't want that to happen. 

Kim Montague  09:33

Sure.

Pam Harris  09:33

We want them to actually understand what's happening. So kids could draw a number bond or a tape diagram with, let's do a number bond first. They could put 195 in one little circle and 303 in the total circle, and then a blank in the other sort of part-circle, the other smaller circle. Like, that's how they can sort of say, "Okay, I know what's happening. I've got 303 in the total, that 303 is the total number of pages and I'm at 195. So that's part and then I have this other part that's the pages I have left to read." I could also do that on a tape diagram. I could have the whole rectangle be 303. And then I could have that second rectangle where I've cut it and I've put 195 in part of it and I need to find the other part of that second rectangle, the empty part to find. But now what? 

Kim Montague  10:21

Yeah.

Pam Harris  10:21

We've represented the situation, the scenario, we've got it, we understand what's happening. But now how do we find that missing part? Okay, so Kim, how might the kids solve that problem?

Kim Montague  10:31

Right, and here's the important part, right? So if I'm thinking about 303 minus 195, I might say 303 minus 200, and get 103. And then I've subtracted too much. So I can add back that five, to get 108. 

Pam Harris  10:49

Bam, and you use a nice over strategy. 

Kim Montague  10:52

Right.

Pam Harris  10:52

You thought about that problem as subtraction, and you subtracted a bit too much. And then you add it back, nice, super. That's the strategy. We're focusing on how students are solving that problem. And that was a very nice, sophisticated strategy. Super. So now as a teacher in our sort of way that we think about modeling, now it's time for our second thing, where I would take what Kim just did and I would model that. I would make it visible. I would draw an open number line. And I would start at 303. And I would draw this big jump back of 200. And I would say, "Where did you land again?" Land on 103. And then I would say, "Kim, what did you do again?" Depending on how much I'm working with the student. And she says, "I subtracted too much." And so I'm like, "How much too much?" She said, "Five too much." So then I would draw back. The jumps shouldn't have been as long so I would back up five? And then okay, so what is that? 103 and then up that five. So now we're at 108. And I would draw that 108 and I would go, "Hey, does this represent what you just did?" Look at it, check it out the relationships you just used, now they are visible. We can use this open number line to make that visible. Cool. Then over time, I can ask that kid, "Hey, you've seen me represent your thinking, you've seen me do that a few times. Now the next time you do that over strategy, I want you to represent your thinking, now you make your thinking visible, just like we've done, cool." Then over time, and with lots of experience, that model becomes a tool. And kids might actually use that model to solve that problem. That's our hope. That's our modeling sequence. Okay, so Kim, what if a kid used a different strategy? Same problem. 

Kim Montague  12:33

Sure. So a couple of other strategies, right? So here's one. What if I found the difference between 195 and 303. So that might look like the teacher drawing a number line. So let me describe what I would do first. So if I had 195, then I could add five to get to 200. And then I could just add 103, to get to 303. So add five to 200, and then add 103 to 303. And then that would look like a teacher drawing the number line with the 195 on one end, and the 303 on the other end, and making those hops that I just described, kind of on top of the number line to show the plus five and the plus 103, which is the total of 108.

Pam Harris  13:21

Excellent. And you would write all that out. The teacher would model that student's thinking, represent that student's thinking. Sometimes people call it annotating student's thinking, I don't really like that so much. Maybe we'll bring that up in a minute. I like representing the student's thinking using that number line, because we know that's an important powerful tool. But you said a couple strategies. All right, give me another one.

Kim Montague  13:44

So how about you give me one? Maybe two.

Pam Harris  13:47

Okay. So if I'm thinking about 195 and 303, and the distance between them, I might choose to find that somewhere slightly different. I might put them both on the number line, and nudge them up both five. 

Kim Montague  14:04

Oh, okay.

Pam Harris  14:05

So now I'm finding the distance or difference between 200 and 308, because I've moved both of them up five, because 308 minus 200, bam is just 108. And I'm just there. And I'm sort of done. And then we call that the Constant Difference strategy. So as we just talked about this kind of sequence of modeling, I have gotten to the point now where when I solve subtraction problems, I think on an open number line. It has become a tool for reasoning for me. But funny, are you ready? When I just did that, if you guys could have seen me, I literally had my hands in the air. And I put one hand where I could see 195 and I put another hand where I could see 303 and I move them both to the right up five. So that I could see that 200 and see that 308 and then I could solve the problem. Now, it took me way longer to discuss what I did, than it did for me to just, like my hands were there. And it took me way shorter to solve that problem, way more efficient solve that problem than it would have - could you imagine the traditional algorithm for 303 minus 195? Wow, all those opportunities for error.

Kim Montague  15:16

Lots of crossy-outies.

Pam Harris  15:17

Yeah, lots of crossy-outies. Again, over time, and with lots of experience, that model becomes a tool and then kids, and me, actually use it to solve problems. Notice that the model that we're advocating is an open number line, that is a tool worth building.

Kim Montague  15:37

It is. And I'm going to go back for a second because I think you mentioned something that we kind of glossed over. And the really, really important thing that I think you just said, was that over time you ask a kid, "Hey, when your brain does that, it can look like this. Does this match what your brain did?" Because we want to represent their thinking. So we just talked about a model that we love for subtraction and really addition too, and that's the open number line. But for multiplication, division and proportional reasoning problems, Pam, what's your preferred model?

Pam Harris  16:09

Yeah, that's a really good question, because we started talking about strip diagrams and tape diagrams. And that's often where we see tape and strip diagrams,  I'm saying them like, they're two different things. They're just called those so maybe I should choose one, strip diagrams. We see those often in proportional reasoning problems, where we sort of set up what's happening in the proportion, and then we can kind of think about the relationship and those ratios, and we can kind of solve them. But how do we solve them, that's the rub, right? Like, it's just the setup, it's just - the strip diagram only helps me get a feel for what's happening. It doesn't then help me use the numbers in the structure to solve the problem. It definitely doesn't model what I do to solve the problem, it just sort of sets up the scenario, the situation. So in Episode 58, we actually walked through kind of a way that we use ratio tables, we use our modeling paradigm to think about setting up a ratio table. And I use sticks of gum. So if I have sticks of gum in a pack, then I can think about how I can model that situation by I start putting down in a table. Okay, I've got one pack to 17 sticks, I've got two packs to how many sticks. As kids say those numbers, I put them in the table, and it's a model of the situation. And then as they solve problems, I model their thinking. I represent their thinking using scaling, or using lines to sort of, I'm adding packs together, so I add the six together. Or I'm scaling as I double the packs. Or as I multiply the packs times 10, then I multiply the sticks times 10. And I represent all that thinking on the ratio table. That's the model of thinking. And the more that we do that and through time and experience, then that tool becomes a model for thinking. Well, I can do the same kind of a thing with a non unit rate scenario. So if I have something like four slices of pizza for $5. I represent that scenario. First it's a model of the scenario, of the context, where I put four slices of pizza for $5. And then as students think about different numbers of slices, and the cost or different amounts of money, and the number of slices of pizza I get - as they solve those problems I represent their thinking using that ratio table. And our goal is then to transition to where that ratio table becomes a tool for thinking, in order to think. And as students when they see a proportion, they think to themselves "Oh, well, you know, how how does that fit in a ratio table? Oh, I can use that ratio table as a tool for thinking."

Kim Montague  18:39

So in a nutshell, these other models that we've been talking about are not bad or wrong. They're just really limited. They can help students slow down and evaluate what's happening. It's just a format, though. And in any case, it doesn't help you decide what to do with the numbers, like you said. But teachers think it does. It's like manipulatives where we think the math is embodied in the manipulative but that's only because we've already schematized it, not because we can now just show what we worked out and have it magically transfer to our students. 

Pam Harris  19:12

Oh that's so well said.

Kim Montague  19:13

Yeah, teachers have probably seen this a ton, right? You might have seen the student who may or may not be able to take the numbers from a word problem and figure out where to place them in a tape diagram. Maybe they can do that. But even if they're able to, many of them get stuck in knowing how they want to mess with the numbers. 

Pam Harris  19:32

Yeah, even if they get to that point. Now they're there. What do we do next? And what we haven't done is help teachers know what to do next. Well, woolah! We are helping you hopefully know what to do next. Ask lots of good problems, pull out that thinking from kids, you represent their thinking using a very important model that then can become a tool for them to solve problems with. So it's helpful to figure out what's in a problem. Okay, you could do it in other ways. But tape diagrams, number bonds is a fine way, that's fine. But don't get caught in demanding that from students because number bonds and tape diagrams are not particularly helpful to know how to mess with the numbers after that. And they're also not the models that we believe will become the tools we want students to become really familiar with, so that they can use them as tools for thinking and reasoning and mathematizing. Alright, so if you want to learn more mathematics and refine your math teaching so that you and students are mathematizing more and more, then join the Math is Figure-Out-Able movement and help us spread the word that Math is Figure-Out-Able!