Counting Isn’t Just for Primary Grade Students, Part 4
By Lynsey Gibbons and
Kendra Lomax, posted January 4, 2016 –
Learning about counting and number is foundational for young
children. In our three previous blog posts (Counting: Why is
it Important and How Do We Support Children? Part 1; Counting with Muna;
Activities to Try with Primary Students), we explored the complexity
of counting, and we shared activities for supporting young children in learning
to count. In this final post, we propose that older children also benefit from
opportunities to count and develop ideas about number and quantity. To better understand
why we think counting in the intermediate grades is important, let’s first look
at how Hamza, a fourth-grade student, worked through a division problem.
Ahmed has 146 pieces of candy. He
wants to put 10 candies in each box. How many boxes will he need? How many
candies will be left?
When we asked Hamza to explain his thinking about this
problem, he showed us the “boxes of ten” he had built out of Unifix® cubes (we recorded
what he built on his paper; see the image below). He said he wasn’t sure how
many boxes would be needed to fit all the candies; so we asked him to show us what
he had done so far. Hamza had made 12 sticks of ten with the Unifix cubes. He counted
the “boxes” aloud by tens: “10, 20, 30 . . . ” until he reached 100,
at which point he continued, “100, 102, 130.” He knew his model did not quite
match the story yet—he didn’t have enough boxes of tens—but he was not
confident about how to make 146.
Using a direct modeling strategy to represent the situation
in the story could have resulted in a correct answer, but Hamza did not have a solid
understanding of the counting sequence required to solve the problem. Fourth graders
could use a range of other strategies to solve this problem, such as finding
partial quotients (100 divided by 10; 40 divided by 10, and 6 left) or using
their knowledge of place value (there are 14 tens in 146). These strategies, as
well as counting and modeling strategies like Hamza’s, build on knowledge of
the base-ten number system.
In our experience, spotting counting struggles in the upper
grades can be difficult unless we get to closely observe students while they
work. By listening carefully to Hamza, we were able to better understand his
computational errors and their connection to counting. Once students start using
algorithms to solve problems, counting issues might go unseen. For example, when
solving a multidigit addition problem with the standard U.S. algorithm, one
need only be able to determine sums up to 9 + 9 and follow the
procedure. When students use this algorithm, however, it may remain unclear
whether they know the counting sequence with these larger numbers, understand
the magnitude of the sum of the numbers, could say which number is greater, and
so on. Thus, even in intermediate grades, continuing to work on ideas around
counting is important as students encounter larger quantities.
The counting activities below were introduced in the third
blog post about counting in the primary grades. Here we will offer suggestions
for how to use the same set of activities to provide older students with
opportunities to count with larger quantities and develop ideas about place
Connecting Quantity, Verbal Counting, and Symbolic Notation
Recall that in
post 3 we talked about counting
collections with primary students. Students in the intermediate grades can
benefit from this activity as well. By simply increasing the size of the
collections, we have found that older children will happily count a set of
objects with a partner to determine the total number and record how they counted.
Students that we work with have counted collections of more than 1000 items!
These larger quantities press students to find ways to group items and keep
track of what has already been counted. This process creates an opening to
consider ideas about regrouping tens into hundreds, switching between counting
from hundreds to tens to ones, and combining partial sums.
Another variation on this activity is to include packaged
items in a collection, such as boxes of paper clips, birthday candles, or
pencils. Including sets that cannot be broken apart as well as individual items
encourages children to develop ideas about multiplication, repeated addition, and
composition of number.
Developing Verbal Counting Sequences and Symbolic Notation
Counting is another great activity for children in the intermediate grades,
as well as in primary grades. Young children can explore counting forward and
backward by whole-number increments. We see the importance of these early
experiences when children call on their understanding of the number sequence to
solve problems, as Hamza did for the division problem. Students in the
intermediate grades elaborate on these earlier understandings to count larger quantities,
count by more challenging increments, and count by fraction or decimal amounts.
A video from
Teaching Channel shows how one teacher is using choral counting to
help students develop multiplication and division ideas in third grade.
Exploring Composition of Number
In intermediate grades, Quick
Images activities can be used to explore ideas of composing and decomposing
numbers as they relate to multiplication and division. Children can also
generate expressions or equations that match how they saw the number of objects
in a quick image, which can help them start to think about the order of
operations. When shown the image below, for example, a child may first count
the middle two groups of four and then the outer columns of three groups of
four. Students may generate a range of expressions to match this counting
strategy, including 8 + 3 × 4 + 3 × 4,
or 2(3 × 4) + 2 × 4, or (4 × 3) + 4 + 4 + (4 × 3).
Discussing how these each match the image and noticing that some people get
different answers when they compute each of these offers a great opportunity to
introduce and practice using parentheses and order of operations.
You can find planning tools and sample tasks for counting
collections, choral counting, and quick images at Teacher Education
by Design. For video of these activities, search for each activity
on the Teaching Channel.
We enjoyed digging into what’s important about counting and how
we can support children in learning to count. We hope you will find these ideas
and activities useful in your work with children. We are continually learning
and hope you will help us learn from you! Please let us know what you try out
and learn by tweeting us: @lynseymathed and @kendralomax. Happy counting!
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 about elementary school 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, area.