This week in math I did something I have deeply mixed feelings about … I introduced the tape diagram to my 2nd graders. For those of you not familiar, the tape diagram is a model for representing a mathematical situation that is a favorite of the EngageNY/Eureka Math curriculum, which my school started using about four years ago. While EngageNY loves the tape diagram, I can tell you from experience with both 2nd and 4th graders that kids remain unconvinced of its magical qualities. It’s pretty common to hear groans when you say, “Can you make a tape diagram to show what’s happening in that problem?” Many, many 4th graders I taught repeatedly told me, “Tape diagrams don’t help me.”

There is so much confusion around the tape diagram, for kids and teachers alike. Kids seem to find it extremely onerous to represent the action of a story problem in a tape diagram, even when they understand what’s happening in the problem and can write equations that match the steps needed to solve it. They don’t WANT to make a model. They just want to get the answer!

Teachers wonder how much value they should assign to a child’s ability to make a tape diagram that accurately represents the problem. If a kid can solve the problem but can’t make a tape diagram that matches, what does that mean? Is a tape diagram a tool for problem solving, or a way to represent what’s happening in the problem after you’ve solved it, or a way to make sense of what’s happening in the problem? Is there an audience for the tape diagram — like is it something that you use to communicate your understanding of the problem to others, or is it a tool for deconstructing the problem and understanding it yourself? These are some of the questions my colleagues and I have tossed around during our conflicted relationship with the tape diagram.

A big, big question for me as a 2nd grade teacher is whether or not making tape diagrams is developmentally appropriate for my kids. At times I’ve thought that it’s an abstraction of the problem that just doesn’t make sense to my students, because they are not yet able to think abstractly. I worry about what happens when we push kids to do things that they aren’t ready to do.

Enter my classroom on Wednesday morning. EngageNY has offered my students, on their first foray into making tape diagrams, the following problem:

*Shawn and James had a contest to see who could jump farther. Shawn jumped 75 centimeters. James jumped 23 more centimeters than Shawn. Draw a tape diagram to compare the lengths that Shawn and James jumped.*

After introducing the tape diagram to the whole class, I sent students off to work on making diagrams for a set of word problems. I was working with two students on the rug, and slowly but surely, my group grew and grew as students trickled over for help. “I don’t get it,” said Veronica, a feisty seven-year-old with a flair for the dramatic. She rubbed her head and frowned.

“OK, come join us, we’re all reading the problem together,” I tried. I read the problem out loud to the group. I got up and acted it out, jumping Shawn’s jump, having the kids mark where I landed, then working together with them to actually measure out 23 centimeters beyond that to show how far James jumped. I said, “Do you see it? Do you see each of the jumps here?” I gestured at the space on the carpet where I had just jumped. “Can you try to draw a tape diagram that shows Shawn’s jump and James’ jump?” I asked them, my tone revealing both hope and doubt.

Veronica frowned again. “I can’t,” she said. She looked up at me. “My brain can’t do all of these things.” (Big Feeling #1 – frustration.) She looked down at the paper. “It’s too small,” she said, pointing to the space allotted for drawing the dastardly diagram.

“You need more space? Sure, go get another piece of paper,” I suggested. But the whole time, I was thinking, “Yup, she’s right. This is just way too hard.” I hate the feeling when I see the work through my kids’ eyes and I feel how hard it feels and I think, “Is what I am doing totally inappropriate?” and then I slog forward anyway, because that’s what we’re doing that day. I really hate those moments. (Big Feeling #2 – self-doubt.)

Veronica came back with her new piece of paper, but she never made a tape diagram on it. Instead, she poked holes in it with her pencil until it resembled Swiss cheese. Then she put her pencil near her top lip and pushed it up above her teeth to make a funny face, distracting another kid in the group (who was really just attempting to copy the tape diagram I had drawn on the board during the lesson, because she too couldn’t figure out how to do anything else). I started getting that feeling like, “Now I’m going to lose the whole group and no one even understands this anyway!” (Big Feeling #3 – anxiety.) “Veronica, go back to your table,” I said. She sighed and stomped off. (Big Feeling #4 – anger.)

Can you believe it? I sent that poor girl back to her table. I’m ashamed to tell you, world, that I did this, but that is what happens when tape diagrams come out. Big Feelings ensue. And sometimes they make you do things you wish later you hadn’t done.

About fifteen minutes later, it was time for lunch and recess. The students cleaned up their math materials, got jackets, and lined up. Veronica’s face said it all. She pulled her hood up over her head. The tape diagram Big Feelings had not gone away just because math was over. We started walking down the stairs to the lunchroom. I heard some commotion in the middle of the line. When I turned to look, Veronica was coming around the corner, and then I saw her push the boy in front of her. Pretty hard. Hard enough to make him fall down the stairs (which, thankfully, he didn’t). I gasped, audibly. (Big Feeling # 5 – terror.) The rest of the class turned to see what had happened. Veronica froze. Everyone was OK. But wow.

When we got down to the lunchroom (safely), I pulled Veronica aside. She immediately started weeping. And then I apologized to her. (Big Feeling #6 was the same for both of us – remorse.) I told her that I was so sorry that I had sent her away from the group during math. I explained that my Big Feelings had gotten the better of me, just like hers had on the stairs. We hugged. She joined the class for lunch, seeming a little bit lighter.

What I thought about after this experience was how much Big Feelings impacted our ability to teach and learn math that day. I thought about something I had read recently from David Cohen’s *Teaching and Its Predicaments*:

“To turn up evidence that students have not learned is one of the most threatening things teachers can do; a student who fails to comprehend is an actual or potential failure for the teacher. The more vivid the evidence that students did not learn, the more troublesome it can be. This is another predicament of teaching: acquaintance with students’ knowledge is full of promise but loaded with problems.”

When we teachers feel that our students don’t understand something we’ve attempted to teach, we have choices. We can think that we need to try harder, or change the way we taught it. We can think, “This wasn’t developmentally appropriate. They’ll get it when they’re ready.” We can think that we tried our best and if that kid is just going to fool around, it’s on them. Each of these possible interpretations reflects the teacher’s emotional response to the deeply threatening possibility that a student has not learned. I think I went through all of those with Veronica on Wednesday. Teaching is just so hard.

I still don’t know whether tape diagrams are worth the Big Feelings they bring up. But what happened on Wednesday reminded me that all teaching and learning, math or otherwise, is emotional business.

What is a tape diagram ?

I think it may be bits of diagrammatic rectangles showing the lengths of two pieces of diagrammatic ribbon, where the two parts are proportional (as in 75 and 23).

Well, that is 10 times more complex than a number line, and oh so tedious.

I am not in the least bit surprised at the “don’t know how to do it/what’s the point anyway” stuff.

A tape itself would be more to the point.

Consign it to the bin !

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And the “area model” is so confusing as well.

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Hi Kim, Thanks for writing such an emotionally raw piece…math, and tape diagrams, can certainly bring out a range of emotions I know! A few things come to mind that I’d like to share, for whatever it’s worth: 1-Have you looked at how TD are introduced at the K and 1st gr levels in EngageNY? If students and teachers have worked with that beginning level then the problem you described is easy to model. Harder when you jump in right there. 2- I’ve taken PD from Eureka writers in the past about TD and it blew my mind! I really walked away with a great understanding and appreciation for where it helps students work through higher level math. It was another one of those “why didn’t I learn it this way?” moments. 3- Have you ever seen/heard of Zearn.org? I encourage you to check it out, especially since you are working with the modules and the teachers I work w use it as a co-teacher so when a lesson calls for modeling, like the one you described, they’re not alone in teaching. PM me if you’d like to know more about how my teachers are using it! 😉

Thanks again for sharing your experience…no doubt countless other teachers have had similar experiences with modeling in math!

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Thanks, Kim, for this. Not least for anatomising those feelings we and the kids get when it’s not quite right. I think all of us – at least, all of us who are close enough to students to be aware of when it’s going wrong – have been through these things, and somehow having them written down like this makes them more face-able, less totally shameful!

I’ve not really used tape diagrams (I think they’re what are called bar models in the UK), although I do use their 3D wooden cousins the Cuisenaire rods. But not really for problems like this. I think it’s when I’m using someone else’s method or material, that I don’t quite believe in or haven’t quite digested, that we haven’t built up from the very easy stages, that this kind of breakdown happens. It’s worth more blog posts, worth being better understood.

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I have a fair amount of experience with bar model/tape diagrams. I feel as though their magical qualities become apparent with higher level problems (fractions, ratio, percentage, and more advanced problems with the 4 operations. The issue is that the students do have to learn the fundamentals of their construction, and often we do these on problems that are probably too easy for them. The students don’t see the beauty of the model until a bit later. It makes me wonder what is going on when a student can’t draw the model and tells me that it doesn’t help them. Why can’t they draw the model exactly? Is it that they don’t really understand the relationships involved and are relying on a key work or two? And why do we ask students to draw models at all- for fractions and other topics? What if a student can divide fractions using a memorized procedure but can’t draw a number line or area model to show the idea?

I think these models are an insight into the child’s understanding. I think 2nd grade might be a bit young for the students to draw them, however.

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Wow. Thanks so much for sharing this. I felt like you were in my head. In the beginning of the year, I was trying to introduce tape diagrams in a 6th grade class while I was teaching ratio and proportion and I felt exactly the same Big Feelings. It felt like I was teaching the students a model just for the sake of teaching them a model because no one was actually using the model to solve the problem. I asked myself, “what does the model show that the other models don’t?” I am not sure I have the answer to this question yet. I played with the model more – using Cuisinaire rods – and it helped me understand multiplicative comparisons and why we start teaching them in 4th grade. The biggest aha moment that I had was this: “we should probably have some K-12 conversations about which models are most important to consistently use , why do we use them, and how do they help students connect big ideas.” Then, we should probably all dig deep into understanding why and how we use them. Thanks again for sharing this. It is one of those posts I should re-read once a month to remind me of my purpose. Keep writing! I would love to hear more about this journey.

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When I think about the tape diagram, I am reminded of how representational it is. When students seem to be struggling with this tool or when I am considering introducing it in a new context, then I prefer to begin with a concrete tool to build some context for the tool. It sounds like you were almost there when you were jumping on the carpet to model the problem. Perhaps doing it outside where chalk can be used to mark start and end points, which can then be converted to a tape diagram would help students see how the tool works. Cathy Fosnot talks about offering tools for students thinking in the beginning as students are becoming familiar with the tool. As time goes on, I have found that these tools become models of students’ thinking. They are actually seeing the models in their mind and using them to solve! My advice–Persevere through the big feelings and know that research supports your decision to help students “see” the math through the use of tools.

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I love this post. Thank you. I am always drawn to honest posts that connect emotion to mathematics teaching and learning. I wonder what BIG FEELINGS Veronica went through. I think that Engage NY has a very problematic understanding of using models. The models are forced on kids. Having been trained in using models through a realistic mathematics education process (think Cathy Fosnot), I operate using a very specific sequence through which models develop for kids. Through context, then through teacher use of the representation, and then only slowly, over time, for kids. Engage NY took up the idea of models like the number line, and then enforced them.

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Can you have the class work as “open middle” problems instead of using prescribed methods? This means that everyone starts at the same point (closed start), there is a target answer (closed-end), but the kids are allowed/encouraged to choose their own method (open middle). As the teacher, you can pose a problem, show sample methods, and let the kids work through the problems. You can ask them to compare and contrast methods. You can have them explain their approach and then you show your approach (helpful if you have a particular method you are trying to explain).

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Kim, Thanks for your thoughtful, very honest piece! I can really identify with those big feelings and the predicament of student (non-) learning. Doesn’t it feel sometimes that learning isn’t linear but a big messy mix of learning, forgetting, and learning again?

As you can probably guess, I’m a big fan of the tape diagram and how it can help make visual the complex relationships in a math problem. I agree with one of the earlier commenters that it’s possible they didn’t fully make sense of them in 1st grade (or did and then forgot). I wonder if giving them a chance to analyze them in worked examples and to describe the relationships they see might provide a more accessible entry point? I agree that sometimes producing a tape diagram feels like a performance for the teacher, and that question you ask about audience is a great one (I’m going to think about that in my own class). Maybe if we flip it so that students are the audience (either through worked examples or student work samples) we’ll see TDs go from a model of thinking to a tool for thinking? Eager to chat more with you about it!

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