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.