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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.
517
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
518
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-
520
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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