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Nursery: Math is already on the mind

Norah L’Espérance, who teaches the three-year-olds at the Dr. Eric Jackman Institute of Child Study lab school, spells out the challenge: “One-off assessments on their own aren’t going to be very meaningful until I know a child well enough to have context for what she does or says, until I know where she’s come from, how she’s responding, and why. You need to first build trust and a shared experience of the school routine. When a child counts at a low level, making sense of that is complicated. Is it the testing situation, are they just not caring at the moment, or asserting their autonomy? Is accuracy just not their priority? Or have they not yet mastered basic counting?”

Norah recalls a student who was an accomplished storyteller: “At some point, late in the year, I called her over for a one-on-one counting assessment where I tell a story with plastic bears and ask them to count in various ways – seeing how high they can go, whether they count them in a bunch or in a line, whether the counting is organized or random. Have they mastered cardinality, conservation or one-to-one correspondence? I sometimes prompt them to count again or touch each bear as they count. I was surprised this very capable child seemed only able to count four objects accurately. It was interesting and a little weird, but she so liked the story aspect that she was eager for another turn.  This time, she had trouble even with just four bears! The third time, instead of my story, I followed her lead and used a story she created about the bears. This time, she counted beautifully, amazingly well. I learned a lot about assessment that day.”

While such puzzling assessments can be illuminating in their own way, Norah generally avoids anything that could be construed as a test until her students are all comfortably settled.

“My priority in the first months is establishing my relationship with the kids and their sense that school is a safe place and not a performance. In the beginning, you’re focused on socio-emotional things, saying goodbye to parents, playing with others, that kind of thing. I worry more about those foundational strengths: Can you communicate? Can you solve problems? Are you curious and interested? When I think of assessment in Nursery it’s those developmental quirks we want to catch – the speech issues, the social issues, the motor issues – the academic piece, the math, we’re going to see its beginnings in everything they do.”

Junior Kindergarten: Different ways of counting emerge

When Norah’s students enter JK a year later, their teacher Marcia Bumbury similarly stresses relationship building in the first days and weeks of school. She’s also looking in a targeted way at math, especially numeracy, across a range of both teacher-led and more informal play settings.

“There’s so much math happening in everything they do. I like to keep it casual. Let’s say we’re looking at names, I might remark, ‘Oh, I wonder how many letters are in your name’ or when we meet in a circle, ‘Hmm I wonder how many kids we have in our class’. Then as the kids count together, I look around to see who’s eager to count, who’s engaged, who’s not.”

In this way, Marcia learns a great deal about children’s math before ever having formally tested a child.

“How kids count tells you so much. Once the year gets going, I might call them over and ask them to count something. How do they organize materials? Do they count from left to right or right to left or are they all over the place? Do they line objects up? Sort them before counting? How do they keep track? How high can they recite numbers? Do they understand that the last number counted represents the number of objects? You learn so much by just watching. And as they play, you’re listening to what they’re saying to their friends. Things just happen naturally in their play. Sometimes it’s hard for teachers to see that. Sometimes you also have to show this to families, especially if they grew up in a different kind of school system, so having concrete examples is very helpful.”

All this data informs Marcia’s planning.

“For example, if they can see that their name has four letters without counting, if they can subitize, I take note of it and think about what’s next for that child,” Marcia says. “Okay, they can count this high, let’s give them other things to count that they might have to organize differently. I might work individually with the child or I might put a couple of similarly skilled kids together, though anyone else who’s interested is always welcome to join.”

Senior Kindergarten: Making sense of strategies

By the time this group of children reaches Carol Stephenson’s Senior Kindergarten class, they have two years of school routines under their belt and a common stock of school-based knowledge. They have also come to know Carol well over two years on a shared playground and are more than ready to engage in teacher-led instruction along with independent play. In the first days of September, Carol engages each child individually in math and literacy assessments that help to provide a pedagogical starting place and guide her formation of skill-based half-class groupings for regularly scheduled math instruction.

Like Marcia, Carol initially focuses on counting, offering an open-ended task with rich possibilities. Dumping a pile of 63 pennies on the table, she simply asks the student to count them, noting how they move the pennies, who’s making rows or grids to structure their counting, who’s grouping by tens or fives, who can keep track with or without spatial scaffolding, and so on. Watching children in this simple counting exercise turns out to be incredibly interesting. Even among children who all perform the task successfully, the different strategies that come to light reveal an intriguingly wide range of conceptualizations and approaches to math.

In a second task, Carol carefully preselects seven numbers on a 100s chart (choosing, for example, 6, 8, 12, 15, 20, 56, and 77). She asks the child to identify each number in turn; if they correctly identify all of them, she gives them larger numerals on paper (125, 206, etc.) The problem is then reversed: “I’ll say the number, you point to it.” She gleans rich information about a child’s ability to recognize number patterns, implicit knowledge of place value, number knowledge and strategies for figuring out unknown numbers.

Two weeks later, Carol presents children with an empty (unmarked) number line and asks them to place the numbers from 0 to 10 along this line, doing their best to space them out evenly. How they resolve or fail to resolve this spatial challenge gives her valuable insight into how children organize space and number and bring them together, a skill that will be key to much of their math learning. She is, however, quick to point out that assessment is less about method and more about a deeper ongoing conversation. It both depends upon a solid, comfortable relationship between child and assessor and further helps to build that relationship. It can be a revelation for children to encounter an adult who attends so closely and without judgment to what they do and how they think.

What’s striking as Carol, Marcia and Norah describe their varied approaches is how thoughtfully each teacher navigates the tension between getting to know children as human beings and getting to know them as mathematical thinkers. How a child thinks mathematically is not all that matters, but it is one important facet of who children are. A teacher’s care for and ongoing curiosity about each child in their full individuality provides an essential foundation for pedagogic assessment in the early years.