Is Zero a Pile?
The need for imagination, a sense of truth, and a feeling of responsibility—these three forces are the very nerve of education.
It’s mid-main lesson, and students are passing out red and blue velvet bags to each other (the bags made lovingly by their eighth grade buddies!), singing a song about grasshoppers that I taught them earlier in the year. It’s one we sing regularly when we hand out materials: “Grasshoppers three a fiddlin’ went, hey! Ho! Never be still….” The lilting and delightful first graders sing along joyfully, some belting it out with vigor, some humming the tune while they pass to their friends. When all bags are handed out and silence settles across the classroom, I instruct them to take out their math mats and open the bags.
A hum of work ripples amongst the children as they discover what is in the bag. “I have beautiful stones!” one child discovers. “Shells, sea shells,” another child exclaims. “I think mine are almost gems!” a third child ponders. And so begins a math lesson in grade 1. Yes, we are using manipulatives, as so many schools do, but these manipulatives are from the natural world and were actually collected by the children and their families over the summer. They turn them over in their hands, admire them, and lay them out lovingly. And so begins the child’s engagement and love of the materials and the work they are doing.
I begin very simply, asking questions that draw the children in. What do you have? How many are there? Their enthusiasm is palpable. Some children recognize the treasures. And the classroom is quite abuzz with answers and chattering to each other. Some have 10, some 13, some 15, some 9. Some are counting correctly, some not quite, but it’s all fine at this stage of the lesson. And this variety of quantity in materials is all done with conscious effort on the teacher’s part.
Because then, my next request: Children, please place 10 treasures on your mat. Now, this may seem a simple request, but in early first grade it is anything but. The children get to work, counting out their stones or shells or gems. A hand shoots up: “But I only have 8!” The child looks puzzled. Her desk partner’s brow wrinkles and some concern is shown on other faces. “Oh my,” I say, with genuine concern. “How many more does she need?” I ask the class. For a moment, confusion rules. This is the first time we’ve encountered this problem. I wait, knowing that the class can solve this problem together. Learning through questions, “inquiry-based learning,” is one that research shows drives student curiosity, knowledge retention, and, frankly, from what I’ve seen as a Waldorf teacher, genuine desire to solve a problem.
After a short silence and some quick thinking, a student hand goes up.
“She needs two more!”
“Oh good, yes, two more,” I reply. “Can anyone share?”
Her desk partner looks at her own pile and says, “But I only have 10, so I can’t give any away.”
Another child next to her says, “I have a lot!”
“Oh? How many?” I ask, genuinely curious. The child counts ever so carefully, placing a small finger on each stone, and the rest of the class is silent, anticipating the answer, listening to the careful counting.
“15!” he says.
“Oh, that’s quite a lot,” I reply. A child in front of her says, “But if you give her two of yours, she’ll have 10 and you will still have enough too.”
The child with 15 sits for a moment, and depending on personality, it might be easy to give some away, or it might be hard. I don’t pressure, but wait for a moment, and then praise the kind act. The children’s eyes twinkle. The problem has been solved. And now, the class “gets it.” They begin sharing and trading as needed, helping each other count, and I move about the room, helping correct the counting if needed, until we all have 10.
Now, at the front of the room, I say, “Children, you have one pile of 10 on your math mats. Now I’d like you to make two piles out of that one.” At first, again, a little confusion (but not too much), which is good for inquiry and encourages persistent thinking. “What do you mean?” They might ask. Then one child catches on, and explains it, and then they all catch on.
And what do you have? I ask. How many are in your piles? Nearly all the hands in the class go up.
“2 and 8”
“5 and 5”
“3 and 7”
As the children share their responses I write them on the board, quickly. I am as excited and enthusiastic as they are. “Oh my, another!” I might respond. And, “There are so many ways to make 10!” I might muse.
And then, when we think we have all the possibilities, a boy raises his hand.
“Zero and 10,” he says.
“Oh my,” I say. And I genuinely bask in the moment of uncertainty. I pause, chalk poised, ready to write something, but I’m not really sure what to say. I love these moments as a teacher.
For a moment, there is a pause, and silence. Then, children trickle in comments, without hands raised.
“But zero isn’t a pile!” One student says.
“Isn’t it?” I ask. “I’m really not sure!”
“What?” another child says, “You’re not sure?” Because the teacher is always supposed to be certain, they must be thinking.
“Children,” I say, “This is a mystery of math.”
“Zero isn’t a pile!” another student says, “There’s nothing there!”
“Maybe not!” I say, noncommittally.
“But maybe there is!” another girl says. “I can imagine the pile of zero!” I had chills as she said this, knowing that, there, is one of the true powers of the intellect, in a 7 year old child, and one that I want to foster and develop, and that Waldorf education holds dear.
The classroom nearly erupts into discussion and agreement and disagreement, and I have to laugh with delight at the scene. As a colleague said recently, “Math isn’t quiet!”
I continue to think on this lesson for weeks afterward, the question of “is zero a pile.” And I can’t help but recognize that these first graders, when given the freedom to discover math through inquiry, landed upon a question that top mathematicians are currently researching, and that there truly isn’t an answer to.
–Ashley Umlauf, Grade 1 Class Teacher