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?

That’s a great list.

I’d add that a good math arguer an engage peers in conversation and debate, and consider and weigh more elegant solutions.

I might add (particularly thinking from an elementary and/or middle school level)

* uses diagrams or drawings to help explain their thinking

* uses manipulatives as they explain why their thinking is reasonable

* is willing to listen

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I would like to post and share these qualities with my students. Successful discussion also takes lots of practice and a classroom community which fosters acceptance and respect for others ideas and strategies. One thought is to share some examples of good math talk (https://www.teachingchannel.org/) with the kids and point out what we notice in regards to good discussion.

Paula, I like your idea of sharing these with students (and adding to the list!). I haven’t used Teaching Channel much and wonder if you have some good example videos or search terms you’d share with us?

Great list. It would be interesting to have our elementary (grades 3,4 and 5) students write their own rubrics for good math discussions (I prefer discussion to argument). I think any discussion needs:

- an open mind

- wait time for the speaker to think

- fair distribution of questions and comments

I notice that you prefer the term discussion to argument, and I definitely do too to describe the math class environment. I was drawing on the language of Common Core: “Construct viable arguments…”

I notice that you’re also focused on creating the environment for a positive, respectful, generative exchange of ideas which is really important!

I wonder what, if anything, you would add in terms of creating “viable arguments” as well as positive discussions?

Good mathematical arguers use rich math language, draw diagrams, pictures, use numbers and if necessary use manipulatives to show their thinking. They are also willing to listen to the complete thoughts of others and adjust/or not their thinking/answer.

They are able to take their answer and apply it to another number/similar problem to prove and justify how their strategy still works.

I noticed this phrase “rich math language” and it resonated with me and reminded me of the Common Core Practice “attend to precision.”

I wondered if it’s the precision of math language & definitions that makes them so important… like the way not all figures with 4 equal sides can be precisely called diamonds, and some might be better examples of diamonds than others. But if you say, “that’s a rhombus” you can be precise and use the definition and no closed figure with 4 congruent sides is “more rhombus-y” than any other.

One way to help students with good mathematical talk is to teach them what good mathematical talk contains, model or show students good mathematical talk , and then process or reflect on the mathematical talk with them. For example with students in grades 3/4, you might ask, “What went well in our math conversation? What do we need to work on?”

I noticed that cycle of looking at models, doing, and reflecting that is such a key part of the problem-solving process coming up again in the math talk process. Cool!

I wonder without a model, as a baseline, what our students would notice about their own math conversations and what they need to work on?

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