In this month’s Something to Talk About blog Amanda Jansen and Molly Brnich share how to build a classroom discourse community in the math classroom through the practice of making and evaluating claims.
This fall, as students and teachers have been returning to school, I have been catching up with some of the teachers with whom I collaborate. I reflected recently with high school math teacher, Molly Brnich. We talked about the ways she supports students as they jump into math content but also as they develop as a math community.
Molly works to build a classroom in which students explore mathematical situations and generate claims. A claim is an idea of what might be true. Students’ claims are publicly displayed on a chart in her classroom, written as they emerge from the students. Then the class works on exploring whether the claims are true by investigating evidence. They work together to refute or justify their claims, which can also lead to refining and revising their claims. I wrote about Molly and her classroom’s revision practice of Create, Justify, or Refute a Claim in Chapter Four of Rough Draft Math.
Molly and I talked about how descriptions
of teaching in books can sound so idealized,
and the first weeks of the school year feel so
different from those descriptions.
Molly and I talked about how descriptions of teaching in books can sound so idealized, and the first weeks of the school year feel so different from those descriptions. How do teachers establish norms and practices with their students? If teachers want their students to create claims and explore them, how do they get started?
At the beginning of the school year, it takes time and effort to support students as they learn to participate in this mathematical practice. Earlier in my own teaching career, I underestimated how much students benefitted from opportunities to learn how to engage in the discourse practices I hoped to foster. When I tried to enact practices that didn’t go as well as I wanted, I would question the teaching practices themselves. I have learned so much from Molly about how students benefit from experiences to help them (1) understand what a claim could be, (2) become comfortable with sharing their thinking with the class, (3) recognize that they can question the validity of a statement, and (4) revise claims as they refine their understandings.
Understanding What a Claim Could Be
To support students with making claims about what could be true in a mathematical situation, Molly starts by inviting her students to share what they notice and wonder. Molly helps students see that their noticings and wonderings are implicit statements about what they think could be true in their explorations.
At the beginning of this school year, Molly wanted to support her students’ understandings of integer operations. She had her students explore a task about a chef who had to frequently change the temperature of a soup with special cubes. The chef could change the temperature by adding hot or cold cubes to the pot or by removing hot or cold cubes that were already in the pot. The cold cubes were like ice cubes, but they did not melt. The hot cubes were like charcoal briquettes, except that they did not lose their heat. Students considered how the temperature of the soup would change in various situations, such as: (a) ten cold cubes were added and six cold cubes were removed or (b) five hot cubes were added and four cold cubes were removed. This exploration moves students toward understanding the meanings of integer operations.
Let’s take a look into Molly’s classroom as students worked on the chef task.
Molly: So, how are we thinking about this problem?
Nora: Wait, if you have a hot cube and a cold cube, it’s zero?
Molly: So, you’re wondering that?
Molly wanted to introduce the idea that students’ wonderings should be documented and considered by the class. She wrote Nora’s question on the board: I wonder if adding one hot cube AND one cold cube would be the same as not changing the temperature or the same as adding zero.
Another student, Madison, had written and drawn about her wondering, but was hesitant to share it with the whole class. Molly asked, “Can I share it for you?” Madison had created a drawing with zero pairs.
Molly told me later that she worked with her students to connect the wondering to the situation by talking about Madison’s drawing. She said that they could turn Nora’s wondering into claim, and the evidence to support the claim could be Madison’s drawing with the zero pairs that showed that the temperatures weren’t changing. She wanted to show students how they could go from a wonder or a noticing to an exploration to a claim to evidence.
Later in the class, Molly asked her students to write a reflection in response to this question:
How did you contribute to the class’s learning today?
Nora, who posed the original question, wrote, “I contributed to the learning by getting out a question that eventually helped solve a problem.”
Becoming Comfortable with Sharing Thinking with the Class
Cesar wrote, “I guess I did contribute
to the learning, because when Vivian
didn’t want to read what she wrote,
I was okay with reading it to the class.”
When students do not generate claims at first, Molly provides the students with claims to evaluate. She then asks them to generate evidence to decide whether or not the claim is true. On this day, Molly wrote some claims on the board that she had in her back pocket in case no student came up with claims or in case no students were comfortable sharing their claims with the class. She asked students to pick a claim and tell her if they thought it was true or not.
On this day, Molly asked a student named Vivian to share her thinking about one of the claims, and Vivian was hesitant. Molly asked Vivian if she would be comfortable with someone else in the class sharing for her. Vivian agreed. Molly asked her group members if anyone would help Vivian out and share her thinking. Molly asked another student in that group, Cesar, to share, and Cesar hadn’t really contributed at their table at first, because Cesar didn’t seem to understand what they were doing. But Cesar was able to read Vivian’s ideas to the class.
Later, in response to the reflection question about contributing to the class’s learning, Cesar wrote, “I guess I did contribute to the learning, because when Vivian didn’t want to read what she wrote, I was okay with reading it to the class.”
Molly helped both Vivian and Cesar contribute to the discussion. Vivian’s ideas were amplified, and Cesar could recognize that he can be a contributor to the learning process, which Molly hoped could lead to more contributions from both of them in the future.
Recognizing That They Can Question the Validity of a Statement
When we reflected together later, Molly said, “What I’ve noticed is that the more that I push students to at least share their thinking or highlight it even if it’s not yet a claim, just saying, ‘Hey, so and so noticed this. Can you take one minute and decide if you think it’s true or false?’ Just deciding if facts are true or false is helping them start to make claims. Because I told them, ‘I could be telling you straight up lies, and you’re believing it, you’re not questioning anything.’”
During the chef task, Molly deliberately shared a false claim with her students and stated it as a true fact, as an answer. “I was like, all right, since I’m a nice person, I’m going to tell you this… and they wrote that down. And I didn’t say a word. I noticed one girl kind of wondered about it. She was confused, like, ‘huh?’ And then she didn’t say anything, but she didn’t write down what I said. So I went over to her group, and I was like, ‘you didn’t write that down. Weren’t you paying attention? I just gave the answer.’ She was like, ‘I don’t think that’s the answer.’ And another student was like, ‘yes, she gave it to us!’ And I was like, ‘You guys! I just lied and you guys all bought it!’”
From that point forward, Molly started to hear her students use the language of “I’m not convinced! Convince me. I need to be convinced.”
Molly went back to the list of claims that the class had been thinking about, such as two hot and cold cubes not changing the temperature. She reminded her class that they created evidence and then decided that a wondering was a true claim. Molly told her class, “That is the point of thinking about claims, to try to be convinced about whether these things are true. Claims allow us to do that.”
Revising Claims as Students Refine their Understanding
Molly’s students considered this situation: Five hot cubes were added, and four cold cubes were removed.
Molly: Let’s talk about what you’re wondering.
Ray: Well, I don’t know how you can subtract. You can’t take out cold cubes. There’s none of them in the pot.
Molly: How can we model taking out colds? What is really happening? If I am taking out colds, what is that doing to the temperature? It’s going up. Is there another way to say that?
Ray: So, taking out cold is really the same as adding in hot.
Molly wrote what Ray said on the board.
Molly: What happens when I go into life, and I don’t have hot and cold cubes? How can I make this more general?
Ray: Is it something like subtracting a negative maybe?
Molly: I don’t know.
Molly returned to his table, and Ray had written, “Subtracting a negative is the same as adding a positive.” Ray’s words became a claim that the class went back to discuss the next day. They looked at his first claim, then he edited it with the class’s help so that they could see how the operation was related to the metaphor of hot and cold cubes and zero pairs.
Working with claims puts
students’ thinking in the center of the
learning process, which shows students
that they have powerful ideas.
Molly used these practices of (1) supporting students with understanding what a claim could be, (2) becoming comfortable with sharing their thinking with the class, (3) recognizing that they can question the validity of a statement, and (4) revising claims as they refine their understandings to help her students learn more about integer operations, but she also helped them think about their thinking. Her students experienced the power of learning through revising their thinking as they considered what might be true and why. Working with claims puts students’ thinking in the center of the learning process, which shows students that they have powerful ideas.
About the authors
Amanda Jansen is a professor in the mathematics education program at the University of Delaware where she has worked since 2004. Prior to her current position, she was a middle school mathematics teacher in Arizona. She is the author of Rough Draft Math, published by Stenhouse in 2020. You can connect with Mandy on Twitter @MandyMathEd.
Molly Brnich is a high school mathematics teacher at Las Américas ASPIRA Academy in Newark, Delaware. This is her sixth year teaching. Prior to her current position, she taught middle school mathematics in Delaware and South Carolina. You can find Molly on Twitter @MsBr2250.