Rethinking Early Years Math Assessment in the First Days of School
Assessment: What research says
Children are intuitive mathematical thinkers
The importance of early mathematics is increasingly recognized with a growing body of evidence linking early math skills to later academic achievement. In children’s first years of schooling, teachers are therefore challenged to consider how best to capitalize upon and further develop their students’ knowledge in math areas. Recent work has demonstrated the impressive amount of mathematical experience and knowledge children acquire simply by virtue of maneuvering in a physical, social, and cultural world. As Herbert Ginsburg (2008) puts it, “Before the onset of formal schooling, young children… are splendid little mathematicians. They deal spontaneously and sometimes joyfully with mathematical ideas. This is what real mathematicians do.”
Clements and Sarama (e.g., 2009) describe a process of “mathematization” by which the intuitive, informal understandings of everyday life become generalized, represented and abstracted, transforming concrete, context-specific pieces of pragmatic know-how into flexible, fluid concepts that can be reflected upon, manipulated and
applied across a range of problems and situations.
In classroom settings, as children develop a common mathematical language for communicating and thinking about concepts, they gain a common pool of knowledge that integrates and further develops the more idiosyncratic experience-based ideas gained in the course of living. At the same time, while Ginsburg and others have observed that all children come to school with considerable intuitive math understanding, the demands for more formalized school mathematics can quickly expose a widening divide based upon class, race, and other entrenched hierarchies of power and privilege.
As a matter of equity, it is essential to help each child build a bridge between their out-of-school knowledge and what they’re expected to learn at school. To effectively build upon children’s intuitive understanding, a math educator must find out more about that knowledge and consider its connections and discontinuities with established curriculum.
Assessing math in the early years
Educators have varied reasons for, and approaches to, assessing early math knowledge. One aim is to identify what students are ready for – what will be interesting enough to challenge them without overwhelming them with confusion? Another purpose may be to create groups of students based on homogenous or heterogenous math skills. We may seek to establish a baseline against which new learning can be measured. While there are solid arguments for assessment as and for learning to take priority, there may also be times when we need to succinctly communicate a child’s capabilities to others such as next year’s teacher, a consultant, or families. In the kindergarten years, where much of the learning is play-based and child-driven, assessment may lean towards the informal, observational side of things,
during both play and instruction (Ontario Ministry of Education, 2016b). Knowledge of the math and careful questioning play a role here, always recognizing that a child engrossed in play may have more pressing non-mathematical purposes at that moment. Sometimes a piece of information obtained through observation or overhearing is better filed away – literally or mentally – to be revisited at a more receptive time.
Educators can also learn a great deal from more constrained measures that ask students to all perform a similar task, especially if that task is open enough to accommodate a range of responses and skill levels.
Examples on video
Longstanding math educator Marilyn Burns demonstrates assessment
A. A student identifies the tenths
B. A student explains the meaning of the digits
C. A student incorrectly identifies the tenths
D. A student incorrectly identifies the tenths
It is crucial that a teacher retain a sense of curiosity about the meaning of a child’s response. One mother of a number-obsessed four-year-old was disconcerted when informed in a parent-teacher interview that her child was nicely on track and could now count to ten, when the child had been frequently observed in spontaneous, joyful counting well into the 100s. As this mother commented, “Either she [the child] wasn’t acting like herself at school or the teacher wasn’t noticing – either way, I don’t love that my kid is learning to give back the bare minimum asked for and nothing else.”
Another parent heard that his six-year-old’s apparent number facility was illusory because the child had “zero understanding of place value”. When questioned later by his father, the child explained that when asked how many “ones” there were in 375, he had replied there could be many answers: 375 ones, or 37 tens and five ones, or three hundreds and 75 ones, and so on. He gleefully added, “There are even 75 fives!”
Such examples serve as a caution to keep an open mind about failures to perform as expected, especially with young children who haven’t always figured out what we’re asking or why we’re asking it. As Clements and Sarama (2009) remind us, educators “must be particularly careful not to assume that children see situations, problems, or solutions as adults do. Instead, good teachers interpret what the child is doing and thinking and attempt to see the situation from the child’s point of view.” (p. 4)
Following Piaget and his heirs in developmental math research, some teachers tout the benefits of focused “clinical interviews” to probe areas of understanding and misunderstanding beyond the correctness of an answer. Often these pose a simple question or task accompanied by (non-instructive) conversation that reveals a child’s unique approaches, problem-solving strategies, and ways of thinking about math.
Examples from Stanford
Videos highlighting assessing students using the measurement assessment protocol
Videos examining the processes of interviewing and interpreting children’s number knowledge
An extensive counting module for early years teachers to reference ahead of working with their students.
We live and breathe in a society saturated with comparisons. From the day they are born, children develop within this context. We see this early on in the family unit – “He has his father’s brains”; “She’s the difficult one”; “He’s creative like me” – then within expanding levels of social complexity that include schools, career fields, and social media networks. For many parents, the start of schooling can offer an abrupt awakening into a social world founded upon measurement, in which the unique brilliance of their child as they intimately know them all-too-quickly dissipates into a series of leveling contrasts: average here, not quite up to par there, possibly with “special needs” of one sort or another. Sadly, the wondrous miracles of culturally-informed individual development can start to pale when measured against a never-ending series of evaluative norms.
As Indigenous educator Dr. Evelyn Steinhauer writes, “We are judged from the day we are born… there’s an energy associated with that judgment that’s internalized… As educators, we need to design assessment that nurtures and assigns value to a broad range of intelligences and processes of analysis” (Steinhauer et al., 2020, p. 78).
Welcoming young students for the first time into the world of school, early years teachers may wonder how they can both get to know children as they really are, on their own terms, in their
own way, at their own pace, while still helping them develop the common school-oriented skills, knowledge and forms of communication that will enable them to fully contribute as members of their classroom community.
This dilemma speaks to a fundamental tension surrounding assessment. In the early days of kindergarten, with at least half the class new to school, we can’t underestimate the sheer amount of work children are putting into figuring out their strange new environment – mulling over why, for example, you can only eat at certain times, hungry or not, or sit in a circle with legs crossed talking about the obvious (“it’s sunny today”) when you have a Lego castle on the go, or are asked endless questions by teachers who already know the answers. With all that’s going on in a child’s life, assessment is a complex interpretive endeavor. Kindergarten children are just learning to use the language of western schooling, and what they mean by what they say doesn’t always mesh with adult understandings. Correct answers may mask confusion, while incorrect responses hint at emerging insights. Especially in these early years, development is circuitous and inconsistent; skills that appear in one situation, at one slice of time, may appear quite differently in another. Things change so quickly that it takes ongoing observation, interaction, and conversation to properly get to know a young child over time. And still they will surprise us!
Teacher Insight: nursery to SK
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.
Clements, D., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York: Routledge.
Ginsburg, H. P., Lee, J. S., & Boyd, J. S. (2008). Mathematics education for young children: What it is and how to promote it. Social Policy Report: Giving child and youth development knowledge away, 22(1), 1–23.
Ontario Ministry of Education (2016b). The kindergarten program. Toronto, ON: Queen’s Printer for Ontario.
Purpura, D.J. & Lonigan, C.J. (2015). Early numeracy assessment: The development of the preschool numeracy scales. Early Education Development, 26(2), 286-313.
Steinhauer, E., Cardinal, T., Higgins, M, et al. (2020). Thinking with Kihkipiw: Exploring an Indigenous theory of assessment and evaluation for teacher education. In Cote-Meek, S. and Moeke-Pickering, T. (Eds.) Decolonizing and Indigenizing education in Canada, Toronto, ON: Canadian Scholars.