A Remainder of One: Exploring Partitive Division
Moyer, Patricia Seray, Teaching Children Mathematics
Children'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.
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 with him to solve the problem. Joe eventually solves the problem by dividing the bug 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
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 a game and you're the last one left." Then Laquisha said, "Like in division, a remainder of one." This discussion was the perfect lead-in to the book, which I read aloud to familiarize students with the story. I asked them to pay particular 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 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. …