What does it look like when it's really working?

A shoe-less Henry(we are very casual at Long-View!) stands at the whiteboard in this video, taking the lead for the moment as Teal Band works through a part of a Quantitative Comparisons Thought Exercise. We pulled out this approximately 8 minute segment because we felt it was a pretty good example of what we are aiming for when we attempt to foster high-level discourse within our classrooms.

A discourse-rich environment is key to teaching children to understand mathematics as a discipline that is creative and dynamic and collaborative, and it is also key to aiming young mathematicians toward deep understanding. Learning mathematics and learning to think mathematically are entwined with learning to speak mathematically and that's why we emphasize the importance of discourse. It's important to get children to talk often, whether that is in Studio Time with a peer or whether that is in a whole group discussion. And we want that talk to be productive, high-level, and inclusive. We want it to be mathematically sound. We want it to be precise, serious, and attend to deep understanding of concepts. We want to attend to social dynamics and academic identity. And we want to leverage these discussions to push everyone together toward a more sophisticated way of thinking mathematically. This is communal work, and in this video episode Teal Band, a class of 4th-6th graders who are only a few months into the start of the year, are providing an illustrative example for us to analyze together.

As you watch Teal Band at work, you'll see that mathematics is being co-created as they tackle the "problem" (in this case they are working to simplify a series so that they can ultimately compare whether it is greater, equal, or less than the solution for the function they solved in the column to the right). Throughout this rich episode we see them grapple with complexity, make their thinking visible, challenge misconceptions or fragile understandings, reason, offer justification and proof for their reasoning, and ultimately come to consensus as a group.

As you watch, notice:

• Henry leads off and invites his band into the way he is thinking about tackling this problem by referencing “about a month ago we did a similar problem” and proceeds to explain to his peers what he recalls from that problem-process that might be helpful this time.

• Billings comes into the conversation and notes that he is on the same page as Henry because he remembers that particular problem and understands Henry’s thinking, but adds that Astrid probably doesn’t (because she joined Teal Band very recently, several months after the year started).

• As more students come into the discussion by offering ideas or strategies or debating, Finley asks “How are we exactly going to compare the answers to this problem to the other one?” at which time the teacher, Eric, interjects to move Finley (and anyone else) to more clarity (i.e. they will end up with a sum that will then be compared).

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