Anton’s creative way to derive multiplication facts
“I have no idea what he’s saying,” I thought to myself as I scanned the faces of the other nineteen 4th graders, hoping one of them had a glimmer of understanding in their eye I could latch on to.
The young mathematician persisted.
“You can subtract to multiply” he said as he continued on about a messy, confusing sounding idea.
“Interesting,” I said, a catch all word I used when I wanted to keep a poker face, which at this moment concealed that I had no sense of how this student was reasoning.
I desperately hoped to do two things. First make sense of what this young mathematician was saying. Second, to orchestrate a productive discussion that would help the entire class learn. But this was early in my first year teaching 4th grade, my pedagogical toolbox was sparse.
The faces, I scanned them again, nothing to grasp at. We were still gelling as a class and the trust wasn’t there to put another student on the spot hoping they might have an idea what he meant.
I didn’t know it at the time but Anton’s thinking was original. While I didn’t know much about teaching I did have Steven Reinhart’s Never Say Anything a Kid Can Say, Van de Walle and co., and Lainie Schuster in my head. At the time, ‘student-centered’ was my mindset with a mantra of ‘first do no harm’, which I hear framed today as foster a productive disposition.
Anton’s voice would be heard.
“Come on up to the chart paper and show us how you’re thinking” I said. Was this the right move? The class had disengaged. Even if I could make sense of Anton’s thinking was it worth it to lose the nineteen other students? Should I have refocused our exploration on more conventional ways of deriving multiplication facts?
This was the fall of 2013, my first year really teaching, and what I didn’t know, in regards to pedagogy, content, and math practices, outweighed what I did. However, I knew exactly what I didn’t want to do: lecture, make kids hate math, destroy their sense of worth in math class, build a house of cards based on procedural proficiency without understanding the concepts—in short, run a traditional classroom that serves no one well. I was following the playbook of Charlie Munger and applying the powerful tool of inversion. I knew how to create an awful math class. Start by not doing those things. Instead play in the wide open world beyond the teaching I had experienced.
Experiment. Reflect. Improve. Repeat.
It takes luck…
At the chart paper Anton began scratching out what would become a wonder of creative thought.
Shortly, another young mathematician was on his feet at the chart paper talking through Anton’s procedure with him. I used what pedagogical know-how I had to get them to repeat, clarify, and converse with the class. The beginnings of a procedure were codified.
A trickle of students had retrieved their math journals. Then others jumped up to get their own. Suddenly, we were hitting on all cylinders as the class was engaged in making sense of and applying Anton’s procedure. The tenor of the room had taken a 180 from earlier.
Before long the class had tested all 9’s, then 8’s, then 10’s using the procedure they codified before we ran out of time. I didn’t know it at the time but thanks to Anton’s creativity, we had stumbled into an early version of conjecture and test—finding the boundaries and conditions for when something is true. Just as important, “The Mathematics Laboratory” was born, a foundation for what would emerge as our classroom culture for the year.
In the moment, I was surprised as I watched this unfold. Reflecting on this lesson over the years, I have become even more impressed. Anton’s work was truly original. I have never seen anything like it. It was beyond our scope to do so at the time but wouldn’t it be wonderful if he goes back to his conjecture some day to try and derive why this procedure works?
Luck played a part in this day’s seminal events. I didn’t ask Anton or any other student to “be creative” or “think innovatively” while deriving facts. However, to use the garden analogy, the soil was fertile. Beginning on the first day of school the soil had been tilled, seeds planted, and watered. The luck was that upon seeing a sprout, even when it was ugly and unexpected, it wasn’t confused for a weed and ripped from the dirt. It was allowed to grow. I’m confident if not this day, a similar catalytic event would have happened another day, given I didn’t mess it up.
I think back on this moment often. It ties back to many ideas, big and small that I think about—not just in regards to teaching and education but about listening, developing people, and culture. On a micro-level this early success, for the students and for me, laid a positive foundation for our year.
It showed me what *was* possible in seeing/hearing, understanding, and using student thinking to drive learning for an entire classroom. Seeing what was possible gave me a new set of questions to ask and led me to the research and practices that laid the foundation for the exponential leap in pedagogy I would make in my second year. Reflecting on this day at a macro-level causes me to rethink the way education is structured.
I’m curious—what creativity/originality/innovation have you witnessed from young mathematicians and under what circumstances did it happen?