Bill McCallum was leading a large-ish session on building a culture of collaboration through jointly investigating student work.
The task for the teachers in the room involved making sense of middle school students' written-out interpretations of different possible takes on how to simplify the expression
7 – 2 ( 3 – 8x)
Being experienced teachers of middle school math students, Fawn and I were both immediately captivated.
"Look at how this student identified right away that the value being distributed is a negative 2 — not just a 2," she said. "They noticed that part of it right away."
I noticed the student's language, which indicated a little mid-process magical thinking about the how to distribute multiplication over subtraction: "...because you use order of operations"; "you always do the problem inside the parentheses first"; "...but then "it's a problem that you've got – 2 on the outside and – 8x on the inside."
"The student is using these phrases as magical incantations," I said. "The rules are still spells to him or her." Fawn agreed.
We both recognized these pieces of productive struggle from our own students's journeys. We dissolved into flow as we started talking about different ways to provoke authentic insight and discovery in our students. This is what is fun about getting together with kindred teacher spirits. It gives us the chance to share a deep kind of noticing that happens automatically during the school year, when we are trying to avoid drowning in the sheer overwhelming volume of student work.
While we'd been lost in analyzing, noticing, and wondering — and unnoticed by us — Bill had stepped closer to eavesdrop on our conversation and to join in the fun. At a certain point, he stepped right into the flow of conversation, offering his own noticings and wonderings about the students' wordings and insights. Several times we all burst out laughing — not at the student's work but at our own pure delight in it. Even after all this time, we can all still be captivated by adolescent mathematical thinking.
Well into his late 80s, Michelangelo was often heard to repeat the motto, "Ancora imparo" — "I am still learning." That is a concise summary of the delight that all teachers feel when we get the chance to sit together as a part of a learning community and think about teaching and learning together. This is the best teaching and learning reminder I know, and I always feel blessed when I have one of these flashes of self-remembering during one of these moments. So I wanted to capture this one.