by NCTM President Henry (Hank) Kepner
NCTM Summing Up, February 2010
For many of us, our focus on mathematics learning is tied to the grade levels of the students whom we teach. For a moment, however, if early childhood education is not our area of expertise, let’s expand our vision of mathematics learning to encompass the earliest learning. Prior to kindergarten, many children have the interest and capacity to learn meaningful math and acquire considerable mathematical knowledge. Regrettably, too many preschool children are not exposed to the kind of sequenced experiences that can support and guide their development of important mathematical concepts over time.
The recently released National Research Council (NRC) report Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity identifies the need to focus on children’s learning paths in number, geometry, and measurement. In our roles as parents, family members, caregivers, or teachers, we can expose young children to math opportunities that abound in everyday experiences. Early interactions can focus on learning to count, naming quantities, and pointing out shapes in the environment.
From the beginning, we should help children go beyond the vocabulary and rote elements to focus on mathematical relationships. The age-old question, “Who has more?” is foundational to all of mathematics. I contend that our human fixation on comparison was a fundamental impetus for our invention of numbers. Initially, determining who was taller was settled by standing back-to-back, who was heavier was established by using balance pans, and who was faster was decided by running a race. As comparisons became more complex, numbers and units became more useful in comparing, In fact, we are not able to compare today’s marathon winner with the first Greek Olympic champion without numbers and common units of measure.
Developing proficiency in counting has several parallel components. The most recognized is learning the number names in sequence. Having a child count to 5, to 10, or to 20 is a rote and sequenced routine. But building on this achievement requires that the child bring mathematical concepts and reasoning into play. Adults can provide direction and opportunities in this learning process by engaging the child in making one-to-one correspondences in increasingly challenging contexts. (Note that mathematicians use one-to-one correspondence as the pivotal concept and tool in dealing with infinity in advanced mathematics!)
To me, a child’s development of the concept of cardinality is one of the strongest and most visible indicators of his or her growth in mathematical knowledge. I find evidence of a child’s progress in acquiring this understanding by observing his or her response to my question, “How many?”
Consider a simple formative assessment task: Have the student count a number of objects. Observe the child using one-to-one correspondence, pointing to each object in turn and saying the appropriate number name. Then ask, “How many?” A child who has not grasped the concept of cardinality typically simply repeats the counting technique.
By contrast, a child who has developed the idea of cardinality often simply smiles and repeats the number name paired with the last object! The clear implication is that this child knows that the last number name also represents a name for the quantity of the entire collection. This task provides a formative assessment of the fact that a child has formulated the idea of cardinality—one of the most important mathematical concepts. For me, this is one of the high points in observing a learner putting big ideas together!
The NRC study reports that many early childhood programs do not extend children’s mathematical knowledge. Instead, they have these young students repeat the same tasks in varied settings without posing challenges that would push them to the next level. Analyses of numerous early childhood “complete” curricula indicate that a 360-minute weekly program typically includes only 58 seconds of mathematics. Although these programs frequently claim math connections, they are not pushing children’s knowledge and reasoning but merely repeating in varied settings concepts that the children already know.
Against the backdrop of my struggles on behalf of equity during my presidency, I find the evidence in the NRC report most discouraging. Young children from low socioeconomic backgrounds are already vulnerable and at risk, demonstrating lower mathematical knowledge than their peers from higher socioeconomic backgrounds. Regardless of the ages of our students, we are challenged to make up for the inequities that our students have experienced.
Beyond our grade-level responsibilities and expertise, I encourage each of us to push for solid preschool mathematics education. This is not a push for hundreds of minutes of child time in formal mathematics learning, but rather an appeal to all of us to direct young children’s attention to mathematics in more thoughtful ways as we help our youngest students to look at, and learn about, the world around them. By expanding their mathematical view of the world, we also expand their worldview!