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
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!