Thinking About Children’s Thinking

Over the last month, for the start of the school year, my AIMS Center colleagues and I have had the privilege of working with teachers and observing the interactions of 3- and 4-year-old children, something our team has done for the last two years.

As we observe these little ones, we have learned how to see their mathematics through their eyes. We temporarily push back our adult thinking, almost as if we are novices seeing things for the first time. This is by no means an easy task, but in doing so we are noticing things that we may have previously overlooked or construed as either a child’s lack of understanding or lack of mathematical knowledge.

We are also finding that children really see the world differently than we do, and that they readily see the world through a mathematical lens.

While observing one student in an outside environment over the course of two days, I noticed that he would randomly call my attention to the shapes in his environment without prompting. He identified and counted out 17 rhombi on the fence, as well as the shapes he saw in the garden – ovals for petals and leaves, and circles for seeds. He counted the number of laps he took around the play structure. He noticed the leaves on the tree were made up of different sizes. He was able to quantify cars in the parking lot and kids on the play structure.

This same child, observed in an inside environment working with magnetic tiles, created a structure resembling a house with walls and a roof. When he ran out of square tiles, he began picking up other shapes: rectangles, large and small triangles, and putting them together in different configurations, trying to construct a square so he could continue building. After about 2 minutes, he formed a square with 2 triangles and said in an inquisitive voice, “triangles make squares?”

Later, I observed this little guy and his teacher engaged in a subitizing activity. I have highlighted a few of his responses and encourage readers to try putting aside their adult mathematical knowledge and consider what these interactions might suggest about his thinking.

Child: That’s 4.

Teacher: How did you know that?

Child: I just know!

[Using one of his fingers he tapped out 1, 2, 3, 4]  




Child: That’s 5.

Teacher: How did you see that?

Child: I saw 2 dots on top, 1 dot in the middle, and 2 dots on the bottom.



Child: That’s 6.

Teacher: Tell me how did you see this one.

Child: I saw 2 and 4, and I see 3 and 3.


As you read the three above situations, what did you think about the child’s thinking? How might you expect other children to respond? Would it be the same or different?

Every day I see children engage in math, it’s just naturally a part of their world. If students are thinking mathematically during their early years, what gets in the way of this thinking as they get older? Is it school mathematics? Is it learning algorithms before they have a conceptual understanding? Something else? I would love to hear your thoughts!

Next week, my colleague Aileen Rizo will share what she has observed children doing when engaged with an activity we developed.

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