It’s fascinating to observe how my children mathematize the world! Where do they see mathematics? How do they interact with it?
Doing Puzzles in a Ballgown
This morning, my 5yo (S) and I worked on a puzzle: 200 pieces depicting the main characters from Frozen. She started on it while I made breakfast for the family. By the time I was ready to join her, she had completed several of the faces from the tableau. “You can sort some of the pieces by color,” S offered me.
Soon, she had completed a long strip with Elsa’s face, stretching from the top edge of the puzzle down to the bottom.
“This looks like a column!” Her eyes twinkled. Her teacher had mentioned rows and columns in school a few months ago, and ever since then she’s spotted them everywhere: windows, a tray of baked goods, playground structures, etc. I think she finds the organization soothing. It’s so beautifully predictable, so satisfyingly even.
“1, 2, 3, 4…” she counted aloud while pointing to each puzzle piece in the column. “There are 10 pieces in the column! I bet there are ten pieces in every column,” she mused. YES. We still had dozens of pieces to go before we’d be able to prove this hypothesis correct, so instead I pushed her to think about the total.
“Ooh, I wonder how many columns we will have if that’s right!” We had a long way to go. Given my vast experience with arrays, I could start to visualize a 10 x 20 outline in my head. I wondered what S was thinking about.
“Ten… and another ten… twenty!” She announced. It felt like we were engaging in. parallel conversation, both focused on the array but on different quantities.
“Yes! Wait, was that more than one column that you just counted?” I asked, while channeling my teacher self. As her mom, I kind of wanted to tell her that she was crushing it.
“That would be two columns,” she clarified, smoothing out the seams of a few haphazardly joined pieces. “Two columns, twenty pieces!”
I picked up the cover of the box, and indicated the total number of pieces. “There are 200 pieces. How many times would we count by 10s to get to 200?” It was a leading question, but I didn’t want enthusiasm to wane.
“10, 20, 30, 40, 50…” she started to count. I helped her keep track of the 10s with my fingers.
She triumphantly announced 100, thrusting her arms into the air. “Is that 100 columns so far?” She looked down at my fingers, without us so much as exchanging looks. “Oh, no, it’s 10 columns. So ten columns!”
“Yes, ten columns would be 100 puzzle pieces!” I tried to keep my face neutral, but I’m sure I betrayed that this wasn’t quite right. So I continued: “this puzzle has 200 pieces.”
“So… 11 columns? That was fast.’
I see this a lot with students who are starting to grapple with place value. 97, 98, 99… 100! 200! 300! However, S fluently counts 100, 101, 102, etc. I imagine she’d never tried skip counting past 100, though, so I decided to show her a little. “100, 110, 120…”
She continued the pattern with me: “130, 140, 150, 160, 170, 180, 190, 200!”
“That was another 10 fingers! Ten fingers and ten fingers is 20 fingers!”
I’m not even certain she remembered we were talking about columns still, but she was making some solid connections.
I continued to help her with the edge piece, and pretty soon we had the full top edge. S counted each piece, one by one, all the way to 20.
“Mama! I think there are 20 columns and 10 in each column.”
The Non-Multiplicative Benefit of Arrays
S loves the structure of an array, but the way that she interacts with them is different from the way that many of my (older) students interact with them. She doesn’t typically skip count to find the total. She firmly counts by ones. (I mean, she’s 5 years old.)
My theory is that she finds arrays a helpful way to chunk quantities. For example, instead of counting a long line of 24, she has points where she naturally pauses and can confirm, for herself, that she’s on the right track.
Early Thoughts about Groups
The other day, a friend mentioned that she was looking for work samples from 2nd or 3rd graders for a grouping problem. My children are 5 and 3 years old, so not exactly a good fit for the task, but, given S’s fascination with arrays, I decided to see what she might do.
Arthur has 4 boxes of chocolates. Each box has 6 chocolates inside. How many chocolates does Arthur have altogether?
I read the first two sentences to S aloud, without reading the question. I wanted to see what she was thinking about. For me, this moment was mostly about learning about her thinking, rather than S learning a specific mathematical idea.
She started out drawing a row of 6 dots, to represent the six chocolates. Then she extended the array with more rows of six dots, trying to line them up as best she could. She counted each number aloud as she drew the circle.
“22, 23… 24!”
“Ooh, what does the 24 tell us?”
She blinked. Her big, brown eyes stared at me expectingly. “What?”
“Does the 24 tell us how many boxes of chocolates there are?”
“No, there are 4 boxes.”
“Does it tell us how many are in each box?”
“No?” S was losing her confidence. Units are tricky! She’d already mathematized the situation; the metacognitive ability to track her thinking and what everything means is developed over time.
“How many chocolates are there in all?”
“ohhhh! Is it 24?” She asked. Then she labeled them all again.
“Twenty four! Twenty four! Twenty four! Twenty four!” She chanted, radiating pride.
Representations and Organization
I don’t expect S to use arrays to support multiplication for a while. We don’t do “school-ish” math at home — unless she’s trying out something for one of my friends! — and I appreciate and respect that boundary.
There’s no rush.
However, I am endlessly fascinated by how arrays have the power to organize and chunk thinking. She’s sees how groups can be a powerful organizing principle, and this will help her later as she builds more and more efficient ways to count. (This also makes me wish I hadn’t left my CGI book at school! I’d love to reread parts about this progression.)
Kids innately see patterns and math within our world. Sometimes, we can nudge them to make sense of it and figure out more connected ideas, like my interactions with S here.
But sometimes, we should just sit back and let the kids mathematize. They have so many brilliant thoughts. When S notices an array at the playground, I don’t push her to count anything. We just marvel at how the world is full of beauty.
Embrace the Challenge 
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