Unit 3: Reasoning and Explaining (MP2 and MP3)

3.3.2 Language

In previous work analyzing mathematical language (see section 3.2), patterns of language become apparent. As you learn to listen for and recognize these patterns, they can be modeled by "revoicing" (Chapin, et al., 2012) student language. You can provide "just in time" language support to students as they are explaining their thinking. This enables teachers to "support English learners as they engage in complex mathematical language" (Moschkovich, 2012).

The language of explanations uses conjunctions (because, but, so, and, etc.).

 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.

The language of justification uses an “If…. then… because…” structure. In many cases, “then” is implied.

 Max: I think what it's trying to say is that if it's 3, 3, and 3, (then) those are all equal numbers, but 3 is odd.

Justifications often include modals (would, should, could, might have, must). This language may need some additional support.

 Chris: If you count by twos, (then) you would just get an even number, and you'll never land on an odd. But like Tristin said, "…it goes in a pattern: odd, even, odd, even. And, if one number was both, (then) it would break the pattern. I don't think a number can be both even and odd."

Teachers may respond to these types of student language patterns by:

• Scaffolding student responses. A teacher may listen to students present their explanations and add “because...”, thus prompting students to add to their response.
• Using sentence frames such as “If ... then... because...”. This frame communicates the parts of the response that are needed and the language structure, but does not limit student thinking.
• Moving up and down the questioning hierarchy, helping students to fill in gaps in knowledge and moving the class to the levels of discourse where students develop meaning.

A persistent argument among students was that a number could be both odd and even. In the following video clip, Brandon, Christopher, and Nathan explain the flaw in the others' reasoning.