Grayson Looks to His Peers As He Justifies His Thinking
This is video of Grayson orally reasoning through the simplification of an expression during a Quantitative Comparisons thought exercise. As Grayson tackles the simplification of 1/9(-8 – -11)27 he justifies his thinking to his peers, looking to them to evaluate whether his reasoning is sound.
This is a key element of a productive culture: children should not feel that the teacher is the "sole arbiter" of mathematical correctness in the classroom. Instead, the norm should be well-established that when justifying one's work, one should look to the mathematical community, and not just to the teacher, as the audience for those justifications. Mathematical justification is not just about understanding why a solution is correct, but about convincing others. It's important that this justification is "public, objective, and open to debate." It's also important that one's argument is based in sound mathematical justification, such as when Grayson explains how he simplified 1/9 * 3 * 27 through one of the key ideas that supports multiplication: association. He uses association very strategically when confronted with a series of multiplication.
Creating a classroom culture that is non-authoritarian, built on the understanding that correctness lies not in just what the teacher says to be correct but in the ability to convince a peer community using mathematical justification can be difficult.
We've worked on this by:
• Ensuring that our content is built on conceptual understanding, not procedural understanding, so that that mathematical proof can be offered.
• Teaching kids to speak directly to their peers, using eye contact.
• Stepping back as a teacher, physically when possible, and by moving the discussion to the community by using phrases such as, "Do you all agree or disagree?" or “Do you agree with Grayson?” or “Does anyone not understand what Grayson said?”
• Scaffolding children as they learn to explain clearly, with sound logic.
• Refraining from asking leading questions.
• Ensuring the focus in the classroom is not on “answer-getting.”
• Facilitating debates when they arise, rather than resolving them for students.
• Being open to student choices, rather than funneling them toward “your way” or what you believe is the “right way.” At the same time, efficiency and elegance is held as goals, along with sense-making.
• Responding to students’ questions by probing their thinking, and asking them to check their own correctness.
• Making sure rich discourse is the primary teaching pedagogy.
• Cultivating an intellectual-ness in the classroom by speaking to the kids at a high level, expecting high quality interactions, and investigating the richness of the discipline.
We suggest you read this article, by Guershon Harel and Jeffrey M. Rabin, if you’d like to understand more about authoritative vs. non-authoritative proof scheme: Teaching Practices Associated with the Authoritative Proof Scheme.