**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; and Counting 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 value.

**Counting Collections:
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.

**Choral Counting:
Developing Verbal Counting Sequences and Symbolic Notation**

Choral 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.

**Quick Images:
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.

**Resources**

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.

**Final Thoughts**

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.

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Sabrina Kuss- 9/2/2019 4:41:22 PMUsługa transportowa - Busy do Anglii.

Przewozy osób z Polski do Anglii

Busy Polska Anglia

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Mark Ludwig- 8/30/2019 12:59:26 PMPrzewozy osób z Polski do Holandii

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Busy Polska Holandia

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