Counting with Muna
Gibbons and Kendra Lomax, posted December 7, 2015 –
What do children need to know and be able to do
to count quantities? In this post, we look at some of the research in
mathematics education about early number. We describe essential understandings
that children need in order to count. We start by describing the counting of a three-and-a-half-year-old
who is attempting to count a pile of colored chips.
Muna, the daughter of a friend, has been asked
to count for us on multiple occasions. This is what happens when you have mathematics
educators for friends! Whenever Muna comes over, she gets asked such questions as,
“Can you count how many pom-poms there are? What if I give you three more?”
On this particular afternoon, we asked Muna to
count how many chips were in a mound that we had dumped out in front of her.
She happily began by touching her finger to one chip at a time and saying a
number name, “one, two, three, four . . . ” Soon her number sequence diverged
from our traditional number sequence, “eight, nine, ten, eleven, sixteen, nineteen,
fourteen . . . ”
Muna’s counting path and number sequence are
shown below. Notice that she missed some of the chips and also double counted
some of them. When asked how many items she had, Muna identified the last
number, saying, “Sixteen,” as the total amount.
Although it would
be easy to simply say that Muna can’t yet count this many items, we would miss
out on fully uncovering what Muna does
know. To understand what Muna knows, let’s look closely at the important and
intertwined ideas that children must navigate as they learn to count.
Understanding of Number and Learning to Count
Children must coordinate three aspects of
number to be able to count quantity. We list them here separately, but we want
to note that they are interconnected, and children must bring them together to
have a solid understanding of number.
The first aspect of number is quantity. Quantity is related to the number
of objects (Van de Walle, Karp, and Bay-Williams 2012). Children explore
quantity before they can even count. For example, they can identify which bowl
is bigger or which stack has more coins. In our example below, we show quantity
as three circles to represent the number of objects.
The next aspect of number is verbal, or the counting words. Over
time, children learn to produce the standard list of counting words in order (Van
de Walle, Karp, and Bay-Williams 2012), or what is called the forward number word sequence (Wright, Martland,
and Stafford 2006): “One, two, three, four.”
When children initially begin learning the
number word sequence, their performance is quite variable. Young counters may
say number words seemingly arbitrarily as they begin to learn to count, such as,
“One, two, three, nine, fifteen, nineteen . . . ”
With practice and structured supports, students
will come to learn the counting sequence and use it more consistently. The teen
numbers can be particularly tricky because of the number names used in English
(Kilpatrick, Swafford, Findell 2001). Unlike other languages like Spanish or
spoken Chinese, English has unpredictable names for 11 and 12. The English names
for numbers in the teens beyond 12 do have an internal structure, but it is
obscured by phonetic modifications of many of the elements used in the first ten
numbers. Although the teens have some complexities, research on children’s
acquisition of number names suggests that children in the United States learn
to recite the list of English number names through at least the teens as
essentially a rote-learning task. Children can either simultaneously or later
learn to notice structure of the counting words for 13 through 19.
have learned the counting words forward, they also need opportunities to learn
the counting words backward. Counting backward is an important strategy that children
will later use in solving subtraction problems. Counting forward and backward
also help children develop understanding of relative size and how numbers are
related to one another (Van de Walle, Karp, and Bay-WIlliams 2012).
The final aspect is related to the symbolic notation of number, or how to
write or read numbers. Numerals are the written and read symbols for
numbers—for example, “2,” “18,” and “36” (Wright, Martland, and Stafford 2006).
Learning to read and write the ten single digit numerals is similar to learning
to read and write letters of the alphabet (Van de Walle, Karp, and Bay-WIlliams
2012). To gain a full understanding of early number, children learn to
coordinate the quantitative and verbal aspects of number with the numeral
identification. Children will develop these three aspects of number at
different times. For example, some students may have a stable counting sequence
(i.e., they can say the number sequence rote) but struggle when asked to apply
what they know so that they can count a heap of chips. Or students may be able
to successfully count a pile of paperclips, but when asked to represent their
count on paper, they struggle to write the numerals. Although counting beyond
120 is not explicitly named in the Common Core State Standards (CCSSI 2010), we
want to be sure to provide opportunities for young children to explore the
counting sequence with much larger quantities. This supports children in making
connections between what they know about numbers within 120 to the rest of the
base-ten number system.
Another important counting concept is developing
one-to-one correspondence. To accurately
count a set of objects, children must use one-to-one correspondence, assigning each
object with one and only one count (i.e., touching one object and saying one
number name). A challenge that young children encounter when counting items is
how to keep track of which items have been counted already and what is left to
count. Touching items, moving them into piles, and arranging the items to be
counted in a line are examples of strategies that children develop to ensure
they count each item once. Another important idea in counting is that the final
number said when counting represents the cardinality
of the set, or the total number of objects.
As we said above, watching Muna count a pile of
chips allows us to examine what ideas Muna has about counting. We saw that Muna
has a stable number word sequence up to 11 and is familiar with some of the
number words that come after. Through repeated opportunities to hear Muna
count, we hypothesize that her counting sequence tends to follow a pattern. It is
not the standard sequence, but she seems to be starting to understand that,
when counting, numbers should be said in a particular order.
Muna also demonstrates some important understandings
about coordinating the counting sequence with quantity. She attempts to count
each of the chips in her stack and recognizes that the final number will tell
her how many chips she has altogether (cardinality). She tries to keep track of
whether the chips have been counted by touching each one, but because the chips
are in a stack, she does not accurately track each chip. However, when we
prompt her to do careful counting or ask her to count chips that have been
organized in a line, she tends to have more success in keeping track. With time
and support, Muna will learn more of the verbal counting sequence and refine
her strategies for keeping track of what has been counted. For now, we are all
just enjoying counting together.
Stay tuned for the third and fourth blog posts
in this series, where we explore activities that parents and teachers can use
with children to work on these important concepts of counting.
The authors would
like to thank Elham Kazemi for sharing the story of Muna’s counting with us.
The vignette and images from this post are based on video records of Muna
counting over time. Thank you also to our colleague Alison Fox for creating the
figure that shows Muna’s counting path and sequence.
This is the second of a four-part
series that explores counting. You are encouraged to revisit Part 1. We
want to hear from you. Post your comments below or share your thoughts on
Twitter @TCM_at_NCTM using #TCMcounting.
Common Core State Standards
Initiative (CCSSI). 2010. Common Core State Standards for Mathematics (CCSSM).
Washington, DC: National Governors Association Center for Best Practices and
the Council of Chief State School Officers.
Jane Swafford, and Bradford Findell, eds. 2001. Adding It Up. Mathematics
Learning Study Committee, Center for Education, Washington, DC: National Academies
Van de Walle, John
A., Karen S. Karp, and Jennifer M. Bay-Williams. 2012. Elementary and Middle
School Mathematics: Teaching Developmentally. Boston, MA: Pearson.
James Martland, and Ann K. Stafford. 2006. Early Numeracy: Assessment for
Teaching and Intervention. Thousand Oaks, CA: Sage.
Lynsey Gibbons, @lynseymathed, is an assistant
professor in mathematics education at Boston University in Massachusetts.
She is a former elementary school teacher and mathematics coach. Her current
scholarly work seeks to understand how we can reorganize schools to support the
learning of children and adults. Kendra Lomax, @kendralomax, is a math educator
at the University of Washington in Seattle. She designs and facilitates
professional learning opportunities around elementary mathematics through projects
like TEDD.org. Curiosity about
children’s mathematical thinking is at the heart of her work. The authors would
like to note that they are continually learning about children and counting.
They have learned a great deal from their colleagues, reading the mathematics
education literature, and interacting with children about counting. The
following colleagues have greatly informed their thinking about how to support
children in finding the joy in mathematics and in counting in particular: Ruth
Balf, Adrian Cunard, Megan Franke, Allison Hintz, Elham Kazemi, Becca Lewis,
Teresa Lind, Angela Chan Turrou, and many teachers in the Seattle, Washington,