One bag of cheese puffs. Four hungry children.
My two children — ages 4 and 2 — had been playing with friends — ages 4 and 8 — when my four-year-old slipped away from the group to ask about food options.
“Mama… did you bring any… snacks?” She asked with a coy smile, eyeing my purse.
That’s when I handed her the single, individually-sized bag of Pirate’s Booty (cheese puffs).
At this point, the other kids started to show some interest. “Do you have another bag?” the 8-year-old inquired, without missing a beat.
I explained that not only did I not have another — okay, I did, but that would have altered my mathematical experiment — but that my 4-year-old daughter would be put in charge of splitting the snacks fairly.
She looked at me with wide eyes. Blink. Blink.
She returned to the band of hungry children with the open bag. “We have to share,” she announced.
“Hmm. How will you share this fairly? Does anyone have a plan?” I asked the lot of them, noticeably too interested in the outcome.
The other 4-year-old rose his hand, as if to share an idea, and then smiled sweetly and silently. My daughter ran her finger tips along the seam of the bag of cheese puffs.
Just like I would in the classroom, I offered some tools to help: some small plastic bowls. The 8-year-old immediately took four of them, one for each kid.
“Does that help?” I asked.
“Now everyone needs some,” my daughter started.
“Oh! I know! Let’s all take a handful,” 8-year-old suggested.
“What does everyone think about this plan?” I prodded.
My daughter smiled. “Everyone take a handful!” she summarily called out. The other 4-year-old went first, dipping his fist into the bag. My daughter’s fist followed. Then came the 8-year-old, who illuminated the fatal flaw in this plan. He managed to extract almost twice as many puffs than the four-year-olds had. Meanwhile, the hand of my poor 2-year-old son emerged with only two stray puffs, like he had played a rigged claw game in a dusty arcade.
“Do you think that was fair?” I asked.
My daughter considered this, gently biting her lip. “Not really,” she said. Then she turned away to snack on the puffs. She didn’t seem as bothered about the glaring inequities as I would have assumed a four-year-old would be. Meanwhile, her two little brother sadly plucked at his two little puffs.
The following day, my daughter had a playdate with one of her classmates. They worked up an appetite building tall magnatile structures and spirited dramatic play about a fictitious ice cream shop. For lunch, we decorated a pizza with a mess of sauce and mozzarella cheese. (I made a salad for the friend’s mother and I, so that we could pretend to be ‘adults’ who do rational things, like include vegetables at every meal.) The pizza cooked quickly, but not quickly enough for the kids.
“We’re huuuuuuuuungry!” my daughter lamented.
“Yeah! We’re staaaaaaaaarving,” her friend added.
“Do you want some salad?”
Ha. Why did I even ask.
“How about strawberries?”
I saw both kids pause to consider this option, so I pulled the pint of strawberries out of the fridge before they could decline.
“Do you want any, Elena?” I asked the friend’s mother.
“Oh, yes, strawberries sound delicious.” I appreciated how she used her response to market them to the kids.
I washed 10 strawberries, and dried them with a paper towel. “Hmm. I wonder how we can split these up,” I mused aloud, at an entirely unnatural volume.
My daughter lit up with memories from the previous day. “Get out some bowls,” she demanded. She pointed to herself, then to Leo, and Leo’s mom. “One, two, three. Three bowls.” I thought she might place a fistful of strawberries in the bowl, but instead I saw her start to partition them into groups.
“Two for me… two for Leo… two for Leo’s mom…”
She dropped two strawberries into each bowl.
“Okay, now one more for me, one more for Leo, and one more for Leo’s mom,” my daughter continued.
“How come you gave only gave one more for everyone, instead of two more?”
“There were only four strawberries left. Not everyone can get two. Only two of us could get two,” she replied.
“Well. What do you want to do with the leftover one?” I asked my daughter.
“Umm… nothing,” she replied. “Let’s leave it there.”
In this way, she’s like so many upper elementary students I work with. Oh, is there a remainder? Let’s pretend that never happened.
My daughter carried the bowls with the finesse of an experienced waitress over the table. “You know, mama, maybe whoever eats their strawberries first can eat the leftover strawberry,” my daughter suggested. Clearly, she had plans for that strawberry.
I can’t know for certain what was going on in my daughter’s head when she said that “not everyone can get two. Only two of us could get two.” Did she imagine rings around each group of two strawberries? Did she think of a line partitioning these four strawberries in half, revealing two groups of two? Did she even consider giving everyone two again, or was she reverse engineering her response to me?
When people explain their math thinking, it often leaves me hungry to know even more. I am totally hooked on kids’ mathematical thinking.
Why did my daughter use two different strategies to share the food? The Fistfuls of Pirate Booty Strategy, from our 8-year-old family friend, felt like a quotative (grouping) way of thinking about division. It focuses on the formation of equal groups — or groups that were intended to be equal — and thinking about how many of those equal size groups fit into the total amount (dividend). This strategy would have been more successful if the groups had been truly equal in size, of course.
Meanwhile, the Sharing Out Strawberry Strategy felt like a partitive (sharing) way of thinking about division. She distributed parts of the whole into each bowl, a piecemeal way to create an equal share.
Did she learn from the first strategy to determine the second? I don’t know.
I think that the quantity and shape of the items also determined her strategy. The Pirate Booty cheese puffs are small, and they come in an opaque bag. There must have been at least 60 of them in the bag — probably more. The puffs leave behind a dusty white residue on your hand, so there’s not a lot of incentive to touch more of the them than necessary.
The strawberries also varied in size, but were generally larger than the puffs. There were only 10 of them. I had laid them out on table so that my daughter had the opportunity to get her bearings: determine the total quantity, and even see whether giving two strawberries to each person would solve the problem.
This reminds me of the Pennies Paper.
Nicholas C. Johnson, Angela C. Turrou, Brandon G. McMillan, Mary C. Raygoza & Megan L. Franke (2019): “Can you help me count these pennies?”: Surfacing preschoolers’ understandings of counting,” Mathematical Thinking and Learning. https://doi.org/10.1080/10986065.2019.1588206
In this paper, published in the journal Mathematical Thinking and Learning, the impressive team of authors detail a study of preschooler’s counting. It’s a beautiful look at the complexity of early mathematical skills. The 3 – 5 year old participants were given three tasks: count out loud, count 8 toy bears, all lined up in a row, and count a pile of 31 pennies.
One would assume that children that did not accurately count eight bears in a line would not be able to be accurate with a loose pile of 31 pennies, but, somehow, 40.5% of children counted higher on the pennies task. (64.4% of preschoolers counted either as well or higher on the 31 pennies task than they did on the oral counting task, in fact.) The authors were careful to offer the disclaimer that “there was a broad range of performances across the two tasks, with no single outcome accounting for even half of the sample.” They offer some interesting analysis about different students.
One major takeaway that I have is that contexts matter, and that we should dial back our own confidence about ‘knowing’ what a student is thinking. We may be able to predict what strategy they might employ, but mathematical thinking is fluid and flexible.
I pulled out one last bag of pirate booty, and asked my daughter to share it with me and her brother.
This time, I poured all of the pirate booty onto a plate so that she could see each individual puff.
She scooped handfuls of puffs into each bowl. “These look about the same.”
I asked her why she chose to count the strawberries, but not to count the pirate booty. She offered the following explanation: “I didn’t want to count. I just wanted to keep getting the pirate booty for everyone. It would be helpful.”
Maybe she thought that counting would be too laborious. (Agreed!) Honestly, her handfuls strategy makes more sense for puffs, since the total volume matters more than the number of individual puffs.
Another parent watched this scene unfurl. “That’s so interesting! I should take notes. You’re teaching her problem solving skills!” I shrugged off the compliment. I try to give my kids space to wrestle with their own ideas — and my motivations are a mixture of educational philosophy and selfish research. I want to learn more about what they think, too!