Several teachers I’ve worked with want to set a culture in their classroom that increases not just the quantity but also the quality of mathematical talk. It made me wonder… how do I recognize good mathematical talk and how would I help students see they’re doing it right?

Good mathematical arguers:

- use definitions to justify their use of mathematical objects & processes (e.g. when deciding if something is a rectangle or not, they cite the definition)
- clarify the assumptions they’re making or the criteria they’re using (e.g. I decided to do this because…)
- are accountable not just to why they think they are right, but also why other solutions are wrong, or why their answer may not be unique
- consider multiple cases
- use simpler examples to justify themselves
- think about whether their answer is reasonable and fits the original context of the problem
- is accountable to everything they noticed in the problem
- uses units and names of quantities, not just values (numbers) to talk about their thinking (e.g. I divided the total number of apples by the number of bags, not I divided 12 by 4 and got 3).

What else?