Counting To Infinity
By Tanya E. Friedman
First and second graders gathered in a tight circle on the ratty quilt of carpet remnants that covered the floor in the math area. Knees overlapped and elbows kissed, but everyone squeezed in tight to discuss fourteen, the number of the day.
“How many ways can we make fourteen?” I inflected my voice to make the question into a compelling dare.
“It’s the same as ten plus four,” Lila said. Her voice carried a trace of boredom. We’d done this the day before with thirteen, the day before that with twelve. She knew she could find all the ways by herself. But for some faces in the circle, recognizing that ten plus four is fourteen didn’t automatically lead them to eleven plus three or twelve plus two. The numbers still full of mystery and magic. That morning I wrote their answers on a large piece of newsprint with a thick purple marker, carefully arranging their ideas, hoping to make the patterns more evident so that by the end of the lesson we could come up with a rule. I wanted to lose the mystery and keep the magic.
Jason’s arm pulled dramatically toward the sky. “Seven plus seven!” As if the numbers might choke him if he had to hold them in his throat one more moment. I wrote seven plus seven at the bottom of the left side of the chart.
Alejandro gave us, “Four plus ten.” He flipped around Lila’s ten plus four. Above his wide smile his eyes flickered because he’d caught onto this part of the puzzle and thought he was being sneaky, finding solutions without doing the work. I had secretly hoped they might accidentally discover the commutative property of addition and I loved that we both thought we were being sneaky.
“Fourteen plus zero,” Darya said. A few days earlier as the class moved from the math area to line up for recess, she’d lingered and then confessed her love for zero.
“One plus thirteen,” Charlotte offered, and I remembered she’d given us one plus twelve the day before.
I reminded them we had found fourteen ways to make thirteen. Alan thought we’d definitely have more ways today because “Fourteen is more than thirteen.”
Victoria noticed the way I arranged the equations on the newsprint. I left space below one plus thirteen and above four plus ten. She raised her hand tentatively. “Two,” she said, and then counted up on her fingers, “Two plus twelve?” She offered her words carefully, as if we were building a house of cards and her idea might topple it over. A small smile of relief shaded her face when I wrote her solution just exactly where she thought it would go.
When everyone saw two plus twelve slide in right underneath one plus thirteen and a whole space above four plus ten, the pattern became visible. A charge of comprehension traveled around the circle like electricity, lighting up faces. Eyes focused. Rocket hands shot into the air.
When we were done, I asked if that was all. They answered, “Yeee-eess” in robotic unison, responding, I thought, to the intonation in my question and not necessarily consulting the evidence. We counted. “Fifteen!” Davon yelled out.
I asked again, “Is that all the ways?” When they nodded, I pushed, “How can we be sure?” I waited for them to say something about using every number from zero to fourteen and there being no other numbers to use. I hoped we could generate a rule.
I noticed Elijah’s furrowed brow and called on him. At six, he was an impressive mathematician.
I often let other students try out their ideas before they heard his and accepted them as truth. “Well,” he said in his most serious voice, “it’s not exactly what you asked.” He looked up to confirm that it was okay to lead us off topic. “But, I think there are other ways…if we’re allowed to use subtraction.”
I paused. If we used subtraction, we wouldn’t get to the rule. I asked myself what mattered in this lesson and as soon as I did, I knew that their construction of ideas, their ownership of our class discussions, of what numbers mean and how they work had to outweigh my plan. We could get to the rule tomorrow with fifteen.
“What do you mean?” I asked.
“Well, fifteen take away one is fourteen, and so is twenty take away six,” Elijah explained.
I wrote his equations on the paper. I wondered which kids would follow this change of course.
A few hands went up. “Sixteen minus two,” from Charlotte.
“Seventeen minus three,” from Danesha.
“Twenty-four take away ten,” Arlo added.
Then their arms all lifted, pulling toward the sun, pulsing with the power of knowing something. Quickly our chart filled. I squeezed their ideas in tiny numbers between the others. They fed each other with numbers, tossed the equations back and forth, playful and proud.
“One hundred fourteen minus one hundred,” Alejandro said.
“Two hundred fourteen minus two hundred,” Lila added.
“Three hundred fourteen minus three hundred,” from Victoria. Then all three started to laugh. More and more poured forth.
When the parade of hands slowed and the edge of attention began to fade, I asked, “Now how many ways do you think there are?”
“A hundred,” Libby said, but the other kids shook their heads. There had to be more than that.
Our big piece of paper overflowed with ways and there were still so many more.
“A thousand,” Diamond tried.
“A million,” Darya said.
“How could we find out for sure?” I asked.
“We have to name all the ways and then count them,” Jason said.
Victoria reached forward and began to count the equations on the paper, silently mouthing numbers as she touched each one.
“How long do you think that would take?” I asked.
Elijah’s hand rose slowly. His brown eyes looked to the ceiling briefly. He started carefully. “If we count all the ways, we would have to count,” he stopped to check his own thinking, to ensure he said it just right, “we would have to count…forever.”
He spoke slowly, uncovering a mystery, placing each word meaningfully into the open space in our circle. “There would always be another way to count because there would always be a higher number.”
And then, as an afterthought, almost to himself, Elijah added, “We could never stop counting. We could never finish.”
We were all still. Listening. The room cracked open. The classroom walls melted. We floated in infinite space. In the cartoon in my mind, the students, now brightly colored cartoons of themselves, loose in the universe, gently hovered between planets, between streams of flying numbers.
I was out of questions. “I think you’re right,” I said. “I think that’s the definition of infinity.” And I realized that I had never really understood infinity, never felt it in its ever expanding, never ending, stretching everywhere foreverness.
We sat quietly, all of us, in awe. Of fourteen. Of Elijah. Of infinity.
Tanya E. Friedman is a teacher, writer, and equity coach. She facilitates groups of teachers and parents who want to incorporate anti-racist practices into their teaching and parenting. Her memoir about learning to teach, to be an adult, and to work for racial justice as a white person is almost done. Tanya lives in Brooklyn with her partner and daughter. Find her on Twitter @tanyaef.