It wasn’t long after I asked students to share their thinking about the problem above that one of the students started commenting on her classmate’s solutions.
“That is wrong. You are wrong.”
I was recording possible solutions on the board while she was holding court behind me.
I wrote :
“Those are wrong. The right answer is 10 1/2. Those other answers don’t even make sense.”
This voice had been the most prominent voice since I walked into the 6th grade classroom. It was loud and disruptive. It overpowered the other voices and it demanded attention. It was accompanied by evasive eyes, sneering, and whispers to a friend. I had been trying not to pay attention to it. I was waiting for her to contribute something authentic so I could recognize her as a contributing member of our community, but it wasn’t happening.
I asked the class, “does anyone want to defend any of these answers?”
One girl came up and drew a number line on the board. She said she thought the answer was 10 and one half. She circled each of the ten groups of two-thirds. She circled the one-third cup at the end of the number line and told us she thought it represented one half of a meal.
One boy, Max, raised his hand. He said, “I don’t think it matters if you say 10 1/2 or 10 1/3. You can call it one half or you can call it one-third. Maybe they are the same thing?”
“What?!” She interrupted, “That doesn’t even make sense. That is wrong.” Her voice was getting louder and less respectful.
At this point, I had to say something. This one voice was starting to supersede all of the other voices in the classroom. It was eroding the fabric of trust faster than I could establish it.
I raised my voice. I said, “We are sharing our math thinking. You are being disrespectful. I need you to listen to your classmate’s ideas without putting them down.”
I made eye contact.
She looked away.
I felt bad. I didn’t want to call negative attention to her. I think she felt embarrassed because I called her out in front of her peers. Did I lose her? I wanted to build her trust, but the time and space that I was offering to her was being used to damage the relationships I was trying to build with the other students. I don’t think it was intentional. I think it was coming from a place of mistrust. What did she mistrust?
The math problem I proposed.
The model her classmate was drawing.
This girl does not have an easy life. I don’t need to go into the details. We all know this girl. She has every reason not to trust me, her classmates, or the math we are doing.
What does she trust?
I think, in this particular instance, the one thing she might trust is the algorithm. That is all that was on her paper. In the beginning of class, I asked the students to draw a picture of the situation described above. She didn’t. Maybe she couldn’t.
Before I arrived in this math class, these students were taught the algorithm for dividing fractions. Their math teacher asked me to help him teach these students how to model fractional division. He said they all know the algorithm, but they are struggling with the modeling part. My goal was to teach them how to model a fractional division situation so they could explain and interpret the units they were working with.
As I sit here, I wonder if my goal should have been to get them to trust themselves as mathematicians.
At this point, we had three different answers on the board. It was up to the students to decide which answer made the most sense.
I asked Max if he would come up to the board and show us his thinking. He did. He drew seven circles. He partitioned each circle into three parts. He colored in groups of two-thirds. He counted 10 groups of two thirds and then said,
“See. There are ten. And, then, there is one-third of a cup left so I think the answer might be 10 and one-third, not 10 and one half.”
“I got 10 and one half,” she said. “I don’t get it. This math class is making me feel retarded.” Her voice was softer than it was before. She was talking to her teacher now- my colleague.
I felt uncomfortable again. I don’t like it when people use the “r” word. I don’t think she meant to be disrespectful when she said it. I think she meant to be self deprecating. I don’t like self deprecation either. I was trying to think of what I should say to her – should I call her out on using the “r” word? Would that make things worse? Before I could say anything, my colleague spoke:
“Don’t use that word. Can you think of another word?”
“Fine. This math class is making me feel stupid.”
Ouch. That is just about the worst thing I could hear in a math class. She sat down next to her teacher.
I wondered, what should I do?
I looked right at her. “Did I hear you say that you are not sure why your classmate is getting 10 and one third for an answer because you got 10 and one half?”
“THAT is a really interesting question. Let’s talk more about that. Can anyone answer that question?”
A different boy volunteered to come up to the board and share his thinking. His writing is in the dark blue at the bottom of the picture. He drew rectangles.
As he presented, we discussed our thinking. I asked him, “What does the one half represent in the story and the picture?”
“One half of a meal.”
“Some people are saying one half is the same as one-third because 1/3 is half of 2/3. What do you think about that?”
Other students chimed in, “Yes. But the question asked about meals and you are going to have one half of a meal, not one-third of a meal.”
“Oh.” I said, “So one-third is one half of two-thirds but the answer to this question is 10 and one half because that is how many meals we will be able to feed the dog.”
Some nods of agreement.
The girl was quiet for the rest of class. When I left class that day, I worried that I should have done something different. I decided that, no matter what, when I went back to class the next day, I was going to try to help her believe in herself as a mathematician.
The next day, we were using virtual Pattern Blocks to model how many sixths were in one-third. I introduced the Illuminations website to the students and encouraged them to explore. Then, I asked them to use the Pattern Blocks to show how many sixths were in one third.
I immediately went over to my friend. This is what was on her screen:
My first instinct was to say, “You can’t use the squares.” or “Don’t you want to use the triangles?” My second instinct – my growing intuition – reminded me to listen.
I asked, “What are you thinking?”
She said, “There are six of them.”
“Do you think you could use the squares to show one sixth.”
She repeated, “There are six.”
I was thinking, “don’t shut down. Please don’t shut down.” What can I say to get her to trust herself? I wondered, “So… what is one sixth?”
She pointed to one of the squares.
“Okay. So there are 6 squares and you are saying one of them would be one sixth. What was the other part of the question that I asked?”
She looked at the board. “How many sixths are in one-third?”
“So where is one-third in your picture?”
“I don’t know.” She gestured towards three of the blocks. “Forget it. I don’t know how to do it.”
“Yes you do. Can I try something? Do you mind if I move your blocks around a little bit?”
“Show me one sixth again.”
She pointed to one of the squares.
I asked, “How do you know that is one sixth?”
She said, “Because it is one out of 6 parts.”
“What if we wanted to show one out of three equal parts of the same rectangle.”
She thought about it for a while. Then, she slowly traced the line that marked off one of the thirds of the rectangle.
I wanted to jump up and down and make a really big deal out of her success, but something inside of me told me not too.
Then, I asked her, “Can you see how many sixths are in one-third?”
“One? No. Two! There are two.”
“Are you sure?”
“I am going to go check in with some other kids. Why don’t you share your thinking with Jamie. How do you feel about sharing what you found with the class?”
“I understand. I only ask because you taught me something. I hadn’t thought to use the squares to answer this question. I thought you could only use the triangles and the rhombus. I am wondering if other people thought the same thing. They might learn something from your strategy.”
“I don’t think I want to share.”
“Okay. If you decide to change your mind, let me know. You can bring a friend if you want.”
The next pair of students that I checked in with had this on their screen:
Wow. I was starting to wonder if anyone was going to use the ol’ triangle and rhombus combination.
It was time to share our thinking. I decided to check in one more time with my new math friend.
“Do you want to share your thinking?”
“Okay. But can I bring Jamie?”
She and Jamie projected her computer screen up on the white board. She started to explain her thinking. Two students kept whispering. She tried to talk over them. Another two students started a side conversation. She looked down at the ground and said to her shoes, “no one is listening to me.”
“Hey!” I raised my voice again. “She didn’t want to come up here. I asked her to share her thinking and she originally said, ‘no’. Then, she changed her mind. She is taking a huge risk right now and you are being disrespectful. Stop talking and listen.”
“Well. I thought I needed six squares so that I could show one sixth.”
“And how many sixths are in one-third?”
“One. No.. two.”
“One or two?”
“Two. It is two. See them?” She pointed. “One. Two. Two sixths.”
“Does that make sense to you?”
“Yes.” Was that a smile I saw? I can’t remember for sure, but what I do remember is that she went to her seat in the back of the room, picked up her belongings, and moved to an open seat in the second row.
She moved closer to me.