Unit 3: Reasoning and Explaining (MP2 and MP3)

# 3.3.1 Discourse

In the following transcript, three students share definitions and their reasoning behind the concepts of odd and even. A challenging class discussion follows. In this exercise, attend to the culture of the classroom and to the reasoning of the students.

 Student Statements Amity: Definition of "even": Even is a group of numbers, but every group has the same amount in each group. Alma: Definition of "odd": Odd is a group of numbers that doesn't have the same amount of numbers in each group. Amity: For our work, we've been having a hard time agreeing with each other, so we've been using cubes [shows cubes]. So if we were to do 14... if you put them into groups of three, it won't work because... Lorena: ...you have two extras. If you make it into.... Amity: ...groups of two... Lorena: ... groups of two. It would be even, because they all have one partner. Alma: And then if you, if you put it into a four, it's... they have two.... Amity: That's two left... Alma: There are two left over. So... Lorena: So.... Alma: It would be odd. It would be odd or even.

 Class  discussion: Jessica: You said, "If you group it into twos, 9 is odd, but if you group it into threes, it is even." But, I'm getting confused, because you say that it is even, and then you agree with the odd. Chris: I don't agree that the number 3 decides if it's even or odd. I think the number 2 decides, because 2 is always even. If you count by twos, you would just get an even number, and you'll never land on an odd. Gabby: The number 3 + 3 + 3. It's not even, but it's equal. Teacher You just said something really important. What did you say about it's not even but equal? Tristin: When you count by threes it goes odd (3), even (6), odd (9), even (12), until you stop. Daniela: But you're saying that 3 is odd, and 6 is even, and 9 is odd. So 3, even though the next one, 6 is even, you start with odd (3). Tristin: 'Cause usually when you count by fours, it's even: 4, 8, 12, 16, 20 ... Those are all even. Then 3, 6, 9, 12, 15 is odd, even, odd, even. Because usually like 2, 4, 6, 8 is all even. It just keeps going in a pattern. Daniela: Yep, well every time you count by twos, it's even. But, the rest of the numbers are switching: odd, even, odd, even. Max: I think what it's trying to say is that if it's 3, 3, and 3, those are all equal numbers but 3 is odd. Daniela: But what about what Amity said (that) if you go to one group and it's only 3 then it's odd? Max: It doesn't matter. Wouldn't you agree that 3, 3, and 3, it'd be even? Daniela: The groups have the same amount. Max: Exactly! 9 is both even and odd. Chris: But like Tristin said, it goes in a pattern: odd, even, odd, even…and, if one number was both, it would break the pattern. I don't think a number can be both even and odd. Teacher: Why not? Chris: Because an odd number like 3, you can't split it in half equally. 'Cause like one team would have two, and the other would have one. Daniela: It's kind of like how Mr. Asturias began with 50 and then the argument, "Is it odd or even?" begins.

(Teacher's work is to) establish a classroom culture permeated with serious interest in and respect for others' mathematical ideas. Deliberate attention is required for students to learn to attend and respond to as well as use, others' solutions or proposals as a means of strengthening their own understanding and the subsequent contributions they can make to the class' work."

Ball 2002

In your Metacognitive Journal, reflect on the following :

Community of Reasoners:

Do you see a respect for others’ mathematical ideas? What is the evidence?

How often does the teacher intervene and for what purpose(s)?

Where do you see evidence of students referring to, building on, and/or challenging other’s thinking?