I had a small math interaction with my kindergartener yesterday while we were doing some much-needed toy decluttering. We have a collection of toy cars that she plays with occasionally, but not often enough that it makes sense to keep all of them, so I told her she could keep ten. She pulled out a handful of favorites and counted: “One, two, three.” Then she added a couple more to the pile and counted them all again: “One, two, three, four, five.” Then two more: “One, two, three, four, five, six, seven.”

“Wait,” I said, seeing an opportunity. “You have seven already, and you get to keep ten. How many more can you choose?”

“I wish you hadn’t of asked me that!” she said, with a big sigh of 5-year-old exasperation. But she took up the problem. She began holding fingers up one at a time while quietly counting them, but stopped after a moment and just held up all the fingers on one hand and two fingers on the other. Then she looked at her hands and said, “Three!”

“Great!” I said. “Go ahead and choose three more cars that you want to keep.” 

“Do you know what I did?” she asked, before she started picking up the cars. “I looked at my fingers and I saw that I had three fingers down and I knew that’s what I needed.”

This was a brief interaction…but there was so much in that interaction for me to see! 

If you spend any time in the math education world these days, you’ll probably come across the questions What do you notice? and What do you wonder? The questions, when used well, can turn a mathematical activity from a teacher-driven activity to a student-driven activity, by stirring up and amplifying students’ inherent curiosity.

But these terms, notice and wonder, are also helpful in thinking about how we as parents or caregivers interact with children around their mathematical thinking. Instead of trying too hard to direct their mathematical learning, we often do more good by developing our own curiosity about their mathematical thinking, and our ability to simply notice what is happening in a given interaction.

Here are some of the things I noticed as I watched my daughter solve this problem: 

I noticed that she was smiling when she said, “I wish you hadn’t of asked me that!” Her big performative sigh told me that she knew I was asking her to do something she considered harder than simply counting the cars over and over until she reached ten. But this is a child who will not do something that she doesn’t want or have to do, and the smile told me she was up for the challenge.

I noticed that she didn’t count out all seven fingers. She started to, but quickly realized that she could just hold up five and two. This is a new development. She’s been counting on her fingers enough now that she is beginning to know what seven looks and feels like. Seven is more than just a count that ends at seven. It is also a relationship: a five and a two. 

I noticed that she also didn’t count the three or the ten. Once she had seven fingers up, she could see everything she needed to solve the problem: seven fingers up, three down, and ten in all. This is another relationship: seven fingers up and three down forms a ten. But also, she didn’t have to model all the actions of the story. She could look at a static image of seven and three and ten and relate it to a story with a beginning (cars she’s already chosen), middle (cars she still needs to choose), and end (all the cars she gets to keep).

I noticed that she explained her thinking unprompted. I almost always ask my kids, “How did you think about that?” and my husband does this, too. It’s just something we do in our household. So it wasn’t the explaining that was noteworthy here, but rather the fact that when I didn’t ask her (because my adult brain was already moving back the goal of decluttering the play space), she launched into an explanation anyway. It’s no longer just a parent-driven part of a math interaction–she now expects and wants to share her thinking.

One of the best things we can do to lay the groundwork for a love of mathematics is to notice what our children are thinking and seeing and doing (not just what we want them to think or see or do), to wonder with them, and to truly experience the wonder of watching them. When we approach our mathematical interactions with children with genuine curiosity, we see more, and children see that we see them.