s, incze ight r J. P yr Elino t ar t cop sion y b e is ke of On ain. Jac by perm ed. r e d K d rv te ac ain Rem onnie M . Reprin ights rese A m in B r a by Fro All MacK any. ated illustr y Bonnie flin Comp b if 1995 ghton M u of Ho
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hildren’s literature can be a springboard for conversations about mathematical concepts. Austin (1998) suggests that good children’s literature with a mathematical theme provides a context for both exploring and extending mathematics problems embedded in stories. In the context of discussing a story, children connect their everyday experiences with mathematics and have opportunities to make conjectures about quantities, equalities, or other mathematical ideas; negotiate their understanding of mathematical concepts; and verbalize their thinking. Children’s books that prompt mathematical conversations also lead to rich, dynamic communication in the mathematics classroom and develop the use of mathematical symbols in the context of communicating. The National Council of Teachers of Mathematics (1989) emphasizes the importance of communication in helping children both construct mathematical knowledge and link their informal notions with the abstract symbols used to express mathematical ideas. This article relates how the book A Remainder of One (Pinczes 1995) was used in a fourth-grade classroom to teach the concept of partitive division. In partitive division, the student separates a group of objects into a given number of equivalent groups and finds the number in each group.
About the Book A Remainder of One is written in a lyrical style that makes reading the story melodic, and the colorful bugs illustrated by Bonnie MacKain are a visual delight for students. The book tells the story of Soldier Joe, a lovable bug who always seems to find himself labeled as the “remainder of one” by the other insects in the bug squadron.
Links to Literature
A Remainder of One: Exploring Partitive Division
In this story, Joe is a member of the twenty-fifth squadron, a group of twenty-five bugs that is marching in a parade before the queen. When the bugs divide themselves into two lines, Joe becomes the odd bug out. Because he is a determined little bug, Joe tries to find a way to divide the members of the bug squadron into even rows with no remainders so that he can participate in the parade. He divides the bug squadron into three rows for the next parade. When he is still the “remainder of one,” Joe does not give up but instead divides the bug troop into four rows. Children quickly relate to Soldier Joe’s feeling of wanting to belong and are motivated to work along Patricia Seray Moyer with him to solve the problem. Joe eventually solves the problem by dividing the bug Patricia Moyer,
[email protected], teaches mathematics education courses at the University of Alabama, Tuscaloosa, AL 35487. She is interested in preservice and in-service teacher education and teachers’ and students’ representations of mathematical ideas. The author extends special thanks to Keith Weaver and the mathematicians in his fourth-grade class at Stafford Global Studies Center, Tuscaloosa, Alabama. Edited by Jacqueline Harris,
[email protected], Toronto Catholic District School Board. Readers are encouraged to share experiences using children’s books with Teaching Children Mathematics. Please send such descriptions to “Links to Literature,” NCTM, 1906 Association Drive, Reston, VA 20191-9988.
APRIL 2000 Copyright © 2000 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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squadron into five rows, which evenly divides the twenty-five bugs in the squadron and eliminates the remainder of one.
Exploring Division Concepts in the Text
Fredneshia divides her “bugs” into three rows.
Photograph by Patricia S. Moyer; all rights reserved
FIGURE 1
The students were very receptive when I introduced A Remainder of One and told them that the star character was an insect. The illustrations, as well as the current popularity of animated movies that feature insects with human qualities, immediately captured the students’ interest. I began the lesson by asking students what they knew about the word remainder. Shanequa said, “It’s the last one there.” Brittany explained, “A remainder is like when you are playing Children relate to a game and you’re the last one left.” Then Laquisha said, Soldier Joe’s feeling “Like in division, a remainder of one.” This discussion was of wanting to belong the perfect lead-in to the book, which I read aloud to familiarand work to solve ize students with the story. I asked them to pay particular the problem attention to the mathematics that they saw in the book. When I finished reading, the students commented that they saw “lots of division” in the story. At this point, I told the students that we were going to read through the story again but that this time, we would model and record the division that we saw in the book. Each student was given a small plastic bag filled with twenty-five centimeter cubes to represent the bugs in the story. The students were delighted to have their own “bugs” to manipulate as the story was reread. Many of the students
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counted to be sure that they had exactly twenty-five bugs, and I allowed for exploration time, during which students stacked the cubes and made arrays. Once students had become familiar with the bugs, I reread the story up to the point at which the bugs divided into two lines. I asked, “How might we model the bugs’ dividing into two lines to find out how many bugs were in each line?” The students replied that we could group the cubes or make lines, as the bugs in the book had done. As they began making two lines, one student asked, “How long should the lines be?” I replied, “I guess we’ll have to see how long it turns out to be when we make the two lines. Try to make them even.” Shanequa remarked cleverly, “But we can’t if there are twenty-five.” As the students rearranged the bugs on their desks into two even lines, I heard a number of mathematical conversations throughout the room. “I’ve got one left over,” someone said, “a remainder of one!” After the children had the opportunity to divide their bugs into two even rows, we talked about the mathematics. I revisited the question asked earlier by a student, “How long should the lines be?” We talked as a class about how we might be sure that the same number of bugs was in each line. One student said that we could match the bugs in a one-toone correspondence to be sure that the same number was in each line. Another student explained a strategy of “one for you and one for me,” in which the bugs are placed in each line until they are all distributed. I modeled the one-for-you, one-for-me method for the students by starting with a group of twenty-five bugs on the back of my clipboard and partitioning the bugs into two lines until only one bug was left as the remainder. After this partitioning, we all agreed that each line had twelve bugs with a remainder of one. I wrote this number sentence on the chalkboard: 25 ÷ 2 = 12 R 1. I then continued reading the story up to the point at which the bug squadron divided into three lines. This time, I asked the students to model the bugs’ dividing, to find the number of bugs in each line, and to write the resulting division sentence. Dividing the bugs into three even rows with a remainder of one was easy for the children (see fig. 1), but it was not as easy for them to develop number sentences that captured the situation that they had modeled with bugs. When I asked students for their number sentences, they gave a variety of answers, including “eight divided by three,” “twenty-five divided by eight,” and “twenty-five divided by three.” Many students can solve mathematical exercises but have difficulty expressing their work using mathematical symbols when solving word problems (Witherspoon 1999). They also struggle when they are not TEACHING CHILDREN MATHEMATICS
sure what the problems require or how to translate them into symbolic equations. Developing the language of mathematics, including both the words that students use to communicate orally and the written symbol system, is essential for mathematical understanding yet lags behind the performance of mathematics. I helped students make these connections by showing them what each of the numbers in the division number sentence represented. The first number was the total number of bugs in the bug squadron, and the second number was the number of rows that we made with our bugs. The students then realized that the number that followed the equals sign told us how many bugs were in each of the lines after we had divided them. They agreed that “twenty-five divided by three” was the number sentence that represented dividing the bugs into three rows, and I wrote that sentence on the chalkboard (25 ÷ 3 = 8 R 1). I continued to reread the story in this manner, allowing the students to rearrange their centimetercube bugs to find the number of bugs in each row and to write the resulting mathematical sentences. When we agreed on a number sentence represented in the story, I wrote it on the chalkboard: 25 ÷ 4 = 6 R 1 and 25 ÷ 5 = 5. When the bugs divided evenly into five rows, I asked students about the remainder. They replied that it was “none,” or “zero.” To wrap up our lesson on the first day, I asked students what might happen if the pattern of dividing from the book had continued. The students said that the bugs would have divided into six rows, then seven. Before we manipulated the bugs, I asked students to predict whether they thought that a remainder might result when we divided by six, and if so, would the remainder be one? Most of the students thought that a remainder would occur, but they were not sure whether it would be a remainder of one. We investigated this problem by making six rows of bugs and found that, once again, we had a remainder of one. I wrote this division sentence on the chalkboard with the others: 25 ÷ 6 = 4 R 1. Students noticed that “the six and the four just changed places.” These connections are important for students to make as they build an understanding of the underlying patterns in mathematics. We finished that day by reviewing some of the patterns that we had noticed in our division and by speculating whether other numbers might result in a remainder of one as often as had the number of bugs in our story.
Extending the Lesson—Using Concrete and Pictorial Representations On the second day, I took the lesson one step further. I had two goals for the second part of this lesAPRIL 2000
son. I wanted students to transfer the problemsolving skills that they had used in the first part of the lesson to a new, yet similar, problem situation. I also wanted to demonstrate a method for creating a pictorial model of the mathematics. Instead of using the number of bugs from the story, each student was given a different number. I began the lesson by telling the students that they would each work with their own bug squadron that day. Each student received a bag containing from twenty-six to fifty centimeter cubes and a number card that showed how many cubes were in the bag. The students were excited about continuing our activities from the previous day. When the bags were distributed, they started counting and checking their bugs. “I’ve got thirty.” “I’ve got thirty-one.” “I have an Many students even number.” “I’ve got thirtyfour.” “I have forty-two.” I told struggle with how to the students that they would each be dividing their own bug translate word squadrons, so their answers and number sentences would be problems into different from those of everysymbolic equations one else in the class. The students also received sheets of twelve-by-eighteen-inch drawing paper that were divided into six equal sections, to be used as storyboards for their bug squadrons. We continued the lesson by reviewing the things that we had learned about division the day before. We talked about how we had used partitioning to divide the bugs into rows during the previous day’s lesson and how we might divide the new bugs during the lesson that day. The students remarked that they had received more than twenty-five bugs in their bags that day and so might not be able to do all the division in their heads as they had done the day before. Then I posed a two-part question: How many bugs will be in each line if your bug squadron divides into two lines for the parade, and will you have a remainder? I asked the students to write the number sentences that represented their individual bug squadrons divided into two lines. The children used the centimeter-cube bugs to make two even rows to find their answers (see fig. 2). As I walked around the room, I asked students to tell me what they were doing and to state the number sentences that represented their bug squadrons divided into two lines. Some students reported only parts of their number sentences: “I have forty-two divided by two,” “thirty-seven divided by two,” “forty divided by two . . . twenty.” Other students stated the entire number sentence. When I asked Jennifer about her number sentence, she replied, “Twenty-eight divided by two is fourteen.” Kayla said, “Thirty divided by two is fifteen.” I asked the 519
Photograph by Patricia S. Moyer; all rights reserved
FIGURE 2
Chaquetta lines up the forty-one bugs in two even rows of twenty with a remainder of one.
Jennifer adds legs and antennae to her bug illustrations.
Photograph by Patricia S. Moyer; all rights reserved
FIGURE 3
students to put these number sentences and any illustrations that would help others understand the division of their bug squadrons on the storyboard. Some students found that they had remainders, whereas others did not. Before she divided the bugs into two rows, Shanequa said, “I have an even number, so I’m not going to have a remainder.” As I walked around the room, the students shared their number sentences with me and showed me what they were writing on their storyboards. After the students finished dividing their bug squadrons into two lines and writing their corre-
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sponding number sentences, I asked the students how I might model what “divided by three” means. One student said, “Make three lines.” The students quickly began grouping their centimeter cubes and making three lines of bugs. Some students gave others mathematical suggestions or directions: “Make sure you divide them up.” “Make three lines.” “Make them even.” They also questioned one another: “Three lines of what—six?” Some students verbalized their thinking to others: “Thirty-three divided by three equals eleven.” After students divided the bugs into three lines, they wrote number sentences to show what they had found. Some students also drew on their storyboards illustrations showing a pictorial representation of the mathematics. A few of the students shared their division sentences and drawings with the rest of the class. Next, I told the students that the last division problem we were going to do would create four rows, and we were going to illustrate our last number sentence in a way that would make the bugs on our storyboards look like the ones in the book. Students need to see models of mathematical symbols as their mathematical language develops, and they need to see examples of how to create pictorial models. This activity gave students an example of using information from a problem-solving situation, manipulating concrete objects to model the situation, illustrating the solution route in the form of a picture, and writing the mathematical symbols. I demonstrated my method of partitioning my bugs into four equal rows. Then I counted aloud the number of bugs in each row. To create the pictorial model, I showed the students how to make the bugs by using their fingers and stamp pads to make fingerprints on the storyboards. I modeled this technique on a sheet of paper taped to the chalkboard. To make my fingerprints look more like the bugs in the story, I used a marker to add antennae, legs, eyes, and a mouth. Students quickly began dividing their individual bug squadrons into four groups so that they would know how many fingerprint bugs to place in each row on their storyboards. Once again, the mathematical talk in the classroom began. I heard several conversations in which students helped one another make sense of the mathematics. I overheard one child explain to another, “There’s eight in your row, so you put down thirty-three divided by four equals eight.” The students used the bottom half of their storyboards for their final number sentences and fingerprint illustrations (see fig. 3). At the end of class, the students shared their work. Students found patterns in the numbers that divided evenly. They noticed that sometimes TEACHING CHILDREN MATHEMATICS
no remainder occurred and that sometimes the remainder was more than one. Some of the students found that their numbers created an array. Students’ illustrations, their use of spoken language, and their written number sentences gave the teacher insights into the depth of their understanding of division in the context of this problem situation. Their manipulation of the centimeter cubes and their drawings and number sentences helped us interpret students’ developing understanding of what it means to divide. The book presented a problem-solving context for communicating, understanding, and exploring division concepts. Experiences such as these further students’ development of number sense and pattern recognition, foundational skills for the higher-level mathematics that they will encounter later in their educational careers.
Conclusion Children need many and varied experiences with mathematical concepts and symbols to develop fluency with them. It is important for teachers to promote conceptual understanding of what it means to divide before teaching the procedures for division. These experiences with division situations should include developing pictorial models and using both the spoken vocabulary and the written symbol systems of the language of mathematics. Opportunities to manipulate objects, illustrate mathematical thinking, and use the language of mathematics in oral and written form should be integral parts of the daily activities in an elementary mathematics classroom. This activity dealt with partitive division, in which a group is separated into a given number of equivalent groups and the students find the number of items in each group. Other examples of problem-solving contexts may also be used for measurement division, in which the number of items in each group is known and the number of groups must be determined. Developing an understanding of both partitive and measurement division in a variety of contexts gives students a foundation for transferring this skill to dividing fractions, decimals, integers, and algebraic expressions. It is equally important for students to develop the ability to translate the information embedded in a problem context into an equation to solve the problem. Students see many number sentences on worksheets throughout their school years. Yet the ability to extract the information needed to create number sentences from realworld situations promotes students’ problemsolving skills, which is an important developAPRIL 2000
mental step in students’ mathematical learning. Yet the ability to translate information in a problem context into a symbolic representation models the work that real mathematicians do. Experiencing many opportunities to read and interpret mathematThe book presented ics in a variety of contexts a problem-solving also enables students to become successful problem context for exploring solvers. Children’s books in which mathematics naturally division concepts emerges as a part of the story supply one meaningful context for modeling and exploring real problems.
References Austin, Patricia. “Math Books as Literature: Which Ones Measure Up?” New Advocate 11 (spring 1998): 119–33. National Council of Teachers of Mathematics (NCTM). Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: NCTM, 1989. Pinczes, Elinor J. A Remainder of One. New York: Houghton Mifflin Co., 1995. Witherspoon, Mary Lou. “And the Answer Is . . . Symbolic Literacy.” Teaching Children Mathematics 5 (March 1999): 396–99. ▲
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