POWERPOINT

Capturing Children's Multiplication and Division Stories

Author(s): Kelly K. McCormick and N. Kathryn Essex

Source: Teaching Children Mathematics , Vol. 24, No. 1 (September 2017), pp. 40-47

Published by: National Council of Teachers of Mathematics

Stable URL: https://www.jstor.org/stable/10.5951/teacchilmath.24.1.0040

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St r es Learn the value of having students create their own stories and pictures to represent number sentences as classroom assessments. Kelly K. McCormick and N. Kathryn Essex

and

Capturing Children’s

Multiplication Division

40 September 2017 • teaching children mathematics | Vol. 24, No. 1 www.nctm.org

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Copyright © 2017 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

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A third-grade student wrote the following repeated-addition story as part of an assessment given to 583 third graders near

the end of the school year (see fig. 1).

Once a group of five little marbles were walking. They ran into five more marbles. Now the five is ten. Five more marbles came by. Now a group of fifteen marbles are walking, and then they all bought lollipops.

We had asked the children to “make up a story and a picture about marbles for this number sentence: 3 × 5 = 15.” Students in this study came from pre- dominantly low- to average-income fami- lies living in three distinct geographical areas within the United States. We also collected work, which included a similar division task, from these students at the end of their fourth-grade year. In this article, we present findings describing the children’s multiplication and division stories and discuss the value of having students create their own stories and pic- tures as classroom assessments.

We wanted to capture and examine the children’s understanding of multiplica- tion and division. Research suggests that providing a foundational understanding of the meaning of an operation sup- ports students’ competence in problem solving and computation (Fuson 2003). Correspondingly, understanding multi- plication is a powerful tool; multiplica- tion is a primary operation that can be properly defined so it is fundamental for representing and solving many different situations (Otto et al. 2011). When solving word problems, children—

frequently choose an operation without making sense of the choice. . . . Know- ing why an operation is an appropri- ate choice for a solution strategy is an important part of establishing a robust understanding of mathematics. (Otto et al. 2011, p. 15)

Children’s

www.nctm.org Vol. 24, No. 1 | teaching children mathematics • September 2017 41

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42 September 2017 • teaching children mathematics | Vol. 24, No. 1 www.nctm.org

The Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) clearly emphasizes the importance of understanding the meaning of multiplication and division; CCSSM states that developing an understanding of multiplication and division is one of four criti- cal areas in third grade, when students are to—

develop an understanding of the mean- ings of multiplication and division of whole

numbers through activities and problems involving equal-sized groups, arrays, and area models; this includes understanding the meanings of whole number multiplica- tion and division. (CCSSI 2010, p. 21).

Third graders are to—

interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7. (CCSSI 2010, p. 23)

These standards state that third graders should be able to—

interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the num- ber of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are par- titioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. (p. 23)

CCSSM extends this focus to fourth grade, when students should be able to “interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5” (p. 29) and to the fifth, sixth, and seventh grades, when students should learn to “apply and extend previous understandings of multi- plication and division” to fractions and rational numbers” (pp. 34, 41, and 48).

Having children create their own multiplica- tion and division stories and pictures provided us with rich information about their under- standing of these operations. As educators, we could show students how to represent a prob- lem situation, but we learn more about their understanding of the operation and quantities involved when they create their own stories (Otto et al. 2011). In addition, because we rec- ognize the importance of children concurrently developing an understanding of multiplication and division and the relationship between the operations (CCSSI 2010; Fosnot and Dolk 2001; Mulligan and Mitchelmore 1997; Otto et al.

F IG

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E 1 As part of an end-of-the-year assessment, a third grader

wrote this repeated-addition story and drew a picture for the number sentence 3 × 5 = 15.

Story:

Picture of marbles:

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E 2 This child wrote a story in which the mathematical structure

was 5 + 5 + 5 = 15.

Story:

Picture of marbles:

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E 3 Here is an example of a third-grade student’s picture and

multiplicative story about equal groups for the number sentence 3 × 5 = 15.

Story:

Picture of marbles:

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www.nctm.org Vol. 24, No. 1 | teaching children mathematics • September 2017 43

groups of objects in two ways: each group repre- sents one thing at the same time it is a number of things. Before constructing the idea of unitizing, number is used to represent single units—six represents six marbles (Fosnot and Dolk 2001).

As Fosnot and Dolk (2001) note, children do not construct mathematical ideas in any set or ordered sequence. “They go off in many direc- tions as they explore, struggle to understand, and make sense of the world mathematically” (p. 18). We saw evidence of this when some of the children’s stories contained elements of both additive and multiplicative thinking. For exam- ple, the story in figure 4 starts with one group of marbles, and then two more groups are added.

Last, we also found multiplicative compari- son stories. Multiplicative-compare situations are about two sets; one set is a multiple of the other (Van De Walle et al. 2013). In these stories, a comparison is made between the amount in one group and the amount in a number of groups of the same size. For example, in the story shown in figure 5, the number of marbles in one group of three is compared to the num- ber in five groups of three.

Of the 583 students who were a part of this

2011; Russell 2010; Van De Walle et al. 2013), we collected work that represented the children’s understanding of both operations.

Children’s multiplication stories To better examine children’s understanding, we designed tasks that allowed us to explore the different types of multiplication and division stories that children compose. How children model a situation reflects their reasoning, and their explanation of their thinking guides their way of representing the situation and any expression or equation that they write (Otto et al. 2011). Thus, we decided to ask the children to create narratives and diagrams about marbles to model equations. Being provided with the context of marbles allowed the children to focus on the mathematics in the task; it made the task less time intensive by narrowing the pos- sibilities of the stories’ context and by reducing the time it took for children to draw a diagram, while still capturing their understandings of the meanings of multiplication and division. We then developed a coding scheme based on their work. We discovered that the children’s correct multiplication stories could be categorized into the following groups: stories—

• about repeated addition;

• about equal groups;

• with both multiplicative and additive aspects present; and

• about comparison situations.

We coded stories about adding three things five times or five things three times and multi- plication stories about repeated addition. With these types of stories, children wrote about having one group of five marbles, then hav- ing another, and then a third. For example, in figure 2, the child writes a story for which the mathematical structure is 5 + 5 + 5 = 15.

The multiplicative stories were about equal groups (see fig. 3): either three groups of five things or five groups of three things. The devel- opment from repeated addition to multiplica- tion requires children to understand a higher- order treatment of number, unitizing, in which groups are counted as well as the objects in the group (Fosnot and Dolk 2001). Children must be able to think about the numbers involved with

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E 4 A third-grade student’s story has both multiplicative and

additive aspects present as well as a picture for the number sentence 3 × 5 = 15.

Story:

Picture of marbles:

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E 5 For the number sentence 3 × 5 = 15, one third grader wrote

a multiplicative-compare story and picture.

Story:

Picture of marbles:

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44 September 2017 • teaching children mathematics | Vol. 24, No. 1 www.nctm.org

Let’s chat! On the second Wednesday of each month, TCM

hosts a lively discussion with authors and TCM readers about a topic important in our field.

You are invited to participate in the fun.

On Wednesday, September 13, 2017, 9:00 p.m. EDT, we will discuss “Capturing Children’s Multiplication and Division Stories”

by Kelly K. McCormick and N. Kathryn Essex. Follow along using #TCMchat.

You can also follow us on Twitter@TCM_at_NCTM and watch for a link to the recap.

study, 345, or 59 percent, wrote a correct story; 312 children also drew a correct picture. Fifty- two children wrote an incorrect story but drew a correct picture. Figure 6 shows an example of an incorrect multiplication story and pic- ture. Among the 345 correct stories, 262 were multiplicative stories, 28 were additive stories, 46 stories had both additive and multiplicative aspects, and 9 were comparison stories (see fig. 7). Thus, the majority of students who wrote correct stories exhibited the ability to think about equal groups; they had constructed the big idea of unitizing, which underlies the under- standing of place value, multiplication, and divi- sion (Fosnot and Dolk 2001).

Children’s division stories The fourth-grade division story task mirrored the multiplication task:

Make up a story and a picture about marbles for this number sentence: 18 ÷ 6 = 3.

Once again, we looked at not only the correctness of students’ responses but also the different types of stories composed. Division stories, like mul- tiplication stories, are about equal-size groups.

We coded stories about sharing items equally across a given number of groups as fair- sharing or partitive-division stories. The three examples of the fourth graders’ stories (see fig. 8) demonstrate that these children have a strong understanding that 18 ÷ 6 = 3 means that eighteen objects are divided equally into six groups and that there are three objects in each of the groups. Their stories show they understand that the groups must be equal, and their questions focus on the number of objects in each group.

Repeated subtraction or measurement division stories had contexts in which a given number of marbles were repeatedly subtracted from the whole group. We found fewer exam- ples of these stories in fourth graders’ work. Three examples of fourth graders’ stories (see fig. 9) demonstrate that these students have a strong understanding that 18 ÷ 6 = 3 means that eighteen objects are divided into groups of six and that three groups of six objects are in eighteen. With a repeated subtraction problem (see fig. 9a), the question is, “How many sixes are in eighteen?” However, because we had

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E 6 This child’s multiplicative story and picture for

the number sentence 3 × 5 = 15 is incorrect.

Story:

Picture of marbles:

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E 7 The majority of students who wrote correct stories had

constructed the big idea of unitizing, which underlies the understanding of place value, multiplication, and division.

Third graders’ correct multiplication stories

Multiplicative 262 Additive 28 Additive and multiplicative 46 Multiplicative-compare 9

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www.nctm.org Vol. 24, No. 1 | teaching children mathematics • September 2017 45

asked the children to “make up a story [not a problem] and a picture about marbles for this number sentence: 18 ÷ 6 = 3,” the story shown in figure 9b also demonstrates a correct under- standing of division. Even though the child does not directly pose the question, “How many sixes are in eighteen?” she explains how to determine the number of sixes that are in eighteen by repeatedly subtracting six from eighteen, twelve, and then six; and as the child then states, after that, “there were no marbles left, so the answer was 3.” If this were a class- room assessment, a follow-up question would be to ask this student what the three in her story means to ensure that she understands that the three is three groups of six.

Some children wrote correct stories for this task that we coded as multiplication stories (see fig. 10). In other words, the action in the story was about finding out how many marbles were in three groups of six objects or six groups of three. This suggested to us that these children understood that multiplication and division may be used to represent the same situation, that is, situations involving a given number of equal-size groups. Mulligan and Mitchelmore (1997) also found that children naturally relate these two operations and that when they do, they do not necessarily find one more difficult than the other, again emphasizing the impor- tance of providing children with opportunities to link the operations of multiplication and division more closely.

In the fourth grade, 356, or 61 percent of the 583 students, wrote a correct division story, and 309, or 53 percent of all the children, also drew a correct picture. An additional 42 students drew a correct picture but did not write a cor- rect story. Among the correct stories, 280, or 79 percent, were stories with fair-sharing con- texts. Another 46 students, or 13 percent, wrote stories with repeated-subtraction contexts. Eleven students wrote equal-group division stories that had neither a fair-sharing nor a repeated-subtraction context. One child wrote a comparison story. Eighteen students wrote multiplication stories.

In the classroom Research highlights students’ difficulty solv- ing story problems; they often guess at which operation to use to solve a problem if they do

not understand what the operations mean ( Verschaffel et al. 2007). For students to develop adaptive expertise in interpreting problems and carrying out appropriate com- putation to solve them, instruction—including assessment—must emphasize students’ under- standing of the action and the meanings of the operations in context (Russell 2010). Under- standing the multiple meanings of operations and the relationship between the meanings and operations is a critical part of establishing

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E 8 Three students’ fair-sharing stories and pictures show that

these fourth graders have a strong understanding that 18 ÷ 6 = 3 means that eighteen objects are divided equally into six groups with three objects in each.

(a)

(b)

(c)

Story:

Picture of marbles:

Story:

Picture of marbles:

Story:

Picture of marbles:

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46 September 2017 • teaching children mathematics | Vol. 24, No. 1 www.nctm.org

a strong, foundational understanding of math- ematics (Otto et al. 2011).

Having students create their own stories and pictures to represent number sentences gave us a snapshot of their understanding and ability to apply the meaning of the operations to the context of marbles. Likewise, having children

generate stories or story problems for a given equation or expression is a powerful way for classroom teachers to assess students’ knowl- edge of the action and meaning of the opera- tions (Drake and Barlow 2007; Van De Walle et al. 2013). It requires a higher level of thinking than simply solving a variety of story problems, which is how teachers typically assess students’ understanding of the meaning of the operations.

The stories and diagrams that children create offer a multitude of opportunities for teachers to facilitate rich classrooms discussions about the different meanings of the operations and how the meanings are related. For example, a teacher might frame a discussion around having stu- dents compare a child’s repeated-addition mul- tiplication story to another child’s equal-groups multiplication story. Similarly, another discus- sion could be built around having students compare a child’s fair-sharing division story to another child’s repeated-subtraction problem. When exploring part-whole relationships, ask- ing children, “What did you know?” and “What were you trying to find out in your problem?” is a powerful tool (Fosnot and Dolk 2001). Through the previous discussion, students should come to realize that with both types of division stories, the total number of marbles is known. However, in one story, they are trying to figure out how many marbles are in each group; and in the other, they know how many marbles are in each group and are trying to figure out how many groups. Building on the previous discussion, having the children determine which interpreta- tion of division is involved (how many groups or how many in each group) in other children’s sto- ries would deepen their operation sense. Other follow-up discussions might focus on the follow- ing questions: “Some of the how-many-groups stories say that the groups need to be equal or the marbles need to be shared evenly [see fig. 8a, b, and c]; is it important that the groups are equal and that the marbles were shared evenly? Why did you include that in your story?”

Using problem writing as an assessment reveals students’ understandings and mis- understandings of operations in a manner in which traditional assessments cannot (Drake and Barlow 2007–2008). The most powerful part of the learning experience described previously is that the stories come from the children. As students write and discuss their problems, they

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Three fourth graders’ stories demonstrate that these students have a strong understanding that 18 ÷ 6 = 3 means that eighteen objects are divided into groups of six and that three groups of six objects are in eighteen.

(a) With a repeated subtraction problem, the question is, “How many sixes are in eighteen?”

(b) Although this student never directly posed the problem, her example shows a correct understanding of division.

(c)

Story:

Picture of marbles:

Story:

Picture of marbles:

Story:

Picture of marbles:

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www.nctm.org Vol. 24, No. 1 | teaching children mathematics • September 2017 47

reveal their mathematical thinking; the value of student discussions, such as those previously described, is quite evident from the NCTM (2000) Standards and the Common Core’s (2010) Standards for Mathematical Practice.

Common Core Connections

3.OA.1 3.0A.2 4.OA.1

REFERENCES 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. http://www.corestandards .org/wp-content/uploads/MathStandards.pdf

Drake, Jill Mizell, and Angela T. Barlow. 2007– 2008. “Assessing Students’ Levels of Under- standing Multiplication through Problem Writing.” Teaching Children Mathematics 14, no. 5 (December–January): 272–77.

Fosnot, Catherine T., and Maarten Dolk. 2001. Young Mathematicians at Work: Construct- ing Multiplication and Division. Portsmouth, NH: Heinemann.

Fuson, Karen C. 2003. “Developing Mathemati- cal Power in Whole Number Operations.” In A Research Companion to Principles and Standards for School Mathematics, edited by Jeremy Kilpatrick, W. Gary Martin, and Deborah Schifter, pp. 68–94. Reston, VA: National Council of Teachers of Mathematics.

Mulligan, Joanne T., and Michael C. Mitchelmore. 1997. “Young Children’s Intuitive Models of Multiplication and Division.” Journal for Research in Mathematics Education 28, no. 3 (May): 309–30.

National Council of Teachers of Mathematics (NCTM). 2000. Principles and Standards for School Mathematics. Reston, VA: NCTM.

Otto, Albert D., Janet H. Caldwell, Cheryl A. Lubinski, and Sarah W. Hancock. 2011. Developing Essential Understanding of Mul- tiplication and Division for Teaching Math- ematics in Grades 3–5. Essential Understand- ing series. Reston, VA: National Council of

Teachers of Mathematics. Russell, Susan Jo. 2010. “Learning Whole-

Number Operations in Elementary School Classrooms.” In Teaching and Learning Math- ematics: Translating Research for Elementary School Teachers, edited by Diana V. Lambdin and Frank K. Lester Jr., pp. 1–8. Reston, VA: National Council of Teachers of Mathematics.

Van De Walle, John A., Karen S. Karp, Jennifer M. Bay-Williams, and Jonathon Wray. 2013. Elementary and Middle School Mathemat- ics: Teaching Developmentally. Boston, MA: Pearson.

Verschaffel, Lieven, Brian Greer, and Erik de Corte. 2007. “Whole-Number Concepts and Operations.” In Second Handbook of Research on Mathematics Teaching and Learning, edited by Frank K. Lester Jr., pp. 557–628. Charlotte, NC and Reston, VA: Information Age Publishing and National Council of Teachers of Mathematics.

Ed. note: For more on this topic, consult Multiplication and Division in Grades 3–5 in NCTM’s Essential Understanding series.

Kelly McCormick, kmccormick@ usm.maine.edu, is an associate professor of mathematics education at the University of Southern Maine in Portland. She is interested in how both children and preservice teachers make sense of mathematics. Kathryn Essex, [email protected], is a mathematics specialist at Indiana University in Bloomington. She is interested in

problem solving and how children and adults make sense of mathematics.

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0 Stories like this fourth grader’s multiplication story and picture for the number sentence 18 ÷ 6 = 3 show that some children naturally relate multiplication and division and do not necessarily find one more difficult than the other.

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