Warrington, Mary Ann. “How Children Think about Division with Fractions.” Mathematics Teaching in the Middle School 2
(May 1997): 390–95.
How
Children
about
Division
with Fractions
390
Think
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
Copyright © 1997 by 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 other formats without written permission from NCTM.
MARY ANN WARRINGTON
W
HEN CHILDREN ARE ALLOWED TO cre-
ate and invent, their fertile minds
enable them to solve problems in a
variety of original and logical ways.
When their minds have not been shackled by
rules and conventions, children are free to invent procedures that reflect their natural
thought processes.
Much of the research and documentation concerning children’s inventions focuses on their
approaches to the addition, subtraction, multiplication, and division of whole numbers. Research and practice in these areas have shown
that children can develop sophisticated and
meaningful procedures in computation and
problem solving without explicit instruction in
the use of conventional algorithms. These invented procedures have been reported not only
in the United States (Kamii 1989, 1994; Madell
1985) but also in Brazil (Carraher, Carraher,
and Schliemann 1985), Holland (Heege 1978;
Treffers 1987), and South Africa (Olivier, Murray, and Human 1991). In the United States,
some leading educators, such as Burns (1994)
and Leinwand (1994), have renounced the
teaching of algorithms to young children, and
some researchers, including Kamii (1994), have
even shown the practice to be harmful. Despite
compelling evidence about children’s procedures
with fractions (Mack 1990; Streefland 1993),
most educators still believe that to handle the
more complex mathematics of the middle grades
and beyond, children need to learn specific procedures, or algorithms.
As a teacher of fifth and sixth grade, I am
passionately committed to reform in mathematics education and firmly believe in the merits of
children’s constructing their own knowledge.
Yet when I began working in the middle grades
four years ago, like many I was ambivalent
about whether students could continue to
progress in mathematics without instruction in
procedures, operations, and algorithms. Despite
my convictions about constructivist teaching in
the primary grades, the thought of tackling a
middle school curriculum involving fractions,
decimals, and percents with such an approach
MARY ANN WARRINGTON teaches at the Atrium
School, Watertown, MA 02172. Her interests in middle school
education center on children’s inventions in mathematics—how
they construct knowledge.
seemed overwhelming. However, the thought of
playing it safe and teaching in a traditional
manner contradicted everything I knew about
how children learn mathematics. Furthermore,
I was well aware of the alarming statistics regarding mathematics achievement in our
schools and the apparent ineffectiveness of
many currently used methods. Thus, I set forth
teaching mathematics to a class of fifth and
sixth graders using a constructivist approach in
which the students were encouraged to think
deeply about mathematics concepts and to invent their own methods of solving problems.
The purpose of this article is to let others
know that contrary to popular belief, children
can indeed construct knowledge about sophisticated and abstract concepts in mathematics
without the use of algorithms. I chose to focus
on the students’ division with fractions because
the rule of “invert and multiply” had always
puzzled me, and I was concerned about students’ ability to find meaning in an area that
few adults understand. Most adults “invert and
multiply” without any notion of why they are
doing so, and students usually cannot explain
the reasoning behind this frequently used and
widely accepted procedure.
The following account provides insight into
how children naturally think when they are encouraged to do their own thinking. The students
whose ideas are expressed in this article were
fifth and sixth graders in a self-contained,
mixed-age, mixed-ability classroom. The children had been exposed to a variety of teaching
practices in mathematics before entering my
class. Some had been taught algorithms, some
had worked extensively with manipulatives,
and some had had two years of a constructivist
approach to mathematics. The culture of the
classroom and school is one that values the
process of learning, and children are accustomed to sharing their ideas openly.
Children’s Thoughts on Dividing Fractions
BY THE TIME THIS TOPIC WAS INTRODUCED, THE students had constructed considerable knowledge
about fractions and were quite confident and
proficient with respect to equivalent fractions.
The work documented here began in February,
so I had been working with these students for at
least five months. The children had already
VOL. 2, NO. 6 . MAY 1997
391
proved capable of inventing ways to add, subtract, and multiply fractions without direct instruction on procedures or algorithms.
Because I believe that children construct
knowledge on the basis of what they already
know, I have always taught from this perspective;
when introducing a new concept, I begin with a familiar topic and move forward. Thus, the initial
discussion about dividing fractions began with a
general question about division. I asked the class
to think about the expression
4÷2
The proverbial “I don’t know how to do this” did not surface
and what it meant to them. Their responses
ranged from “It means if you have four things
and you divide them into two groups, how many
are in each group?” to the most common response, which was “It means how many times
does two fit into four or how many groups of two
fit into four?” This brief discussion of division
informed me about how the children were
thinking about this mathematical principle. It
is essential to learn what your students know
and how they are thinking before proceeding to
new territory. Their prior knowledge about division must be used as a base or a starting point.
Next, I presented them with the problem
392
2÷
1
.
2
Within seconds many children were eager to respond. When called upon, one student responded, “Four [pause] because one-half goes
into two four times.” Another child followed up
with “I think it’s four also [pause] because if you
had two candy bars and you divided them into
halves, you’d have four pieces.” I was pleased
with their thinking thus far and inspired by
their willingness to attempt to solve the problems using their own devices. The students
were used to relying on themselves for solutions, so the proverbial “I don’t know how to do
this” did not surface. The students seemed confident about their reasoning, and I was eager to
move into other problems, knowing that halves
tend to be easier for children than other fractions, such as thirds and fifths.
The next problem I wrote on the chalkboard
was
1÷
1
.
3
Without hesitation the children responded, almost in unison, “Three [pause] because onethird goes into one three times.”
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
I then wrote
1÷
2
.
3
Two responses were forthcoming. About onethird of the class said that the answer was 6,
and the rest believed the answer to be 1 1/2.
The reasoning of those claiming 1 1/2 was that 1
÷ 1/3 is 3, so 1 ÷ 2/3 must be 1 1/2 because 2/3 is
twice as big as 1/3 and so fits into 1 half as
many times. Since they had already determined
that 1 ÷ 1/3 is 3, l ÷ 2/3 must be half of that, or 1
1/2. This explanation seemed to convince others
that 6 was not feasible. Many of those who had
initially responded with 6 quickly retracted
their answer, whereas others took some time to
debate before noticing the flaw in their reasoning. The exchange of ideas is an important aspect of a constructivist classroom. Although
constructivists believe that children learn from
one another, they do not believe that children
acquire mathematical knowledge from other
people. Such knowledge has to be constructed
by each individual from the inside. Social interaction stimulates critical thinking, but it is not
the source of mathematical knowledge (Kamii
1994). In this case, the students who believed
that the answer was 6 had to think about their
own reasoning as well as that of their peers and
determine who was correct. In deciding that 6
was incorrect, they had to modify their thinking. The social interaction undoubtedly stimulated these children to question their thinking;
however, the actual construction of knowledge—determining the answer to be 1 1/2—was
internal.
From that point the students solved 3 ÷ l/3
with relative ease, and when asked to try
1
÷ 3,
3
they again constructed their reasoning from
what they knew. One child said, “It’s one-ninth
because one divided by three is a third, so if you
want to divide it [one-third] by three, you have
to take a third of a third, which is one-ninth.”
Another child’s explanation went like this: “I
think it’s one-ninth because if you had one-third
of a pie left and you were sharing it with three
people [two friends], each person would get oneninth.” What intrigued me about that argument
was that the child took a straight computation
problem and assigned meaning to it by creating
a word problem.
After working through several more prob-
lems, I gave the class the following:
I purchased 5 3/4 pounds of chocolate-covered
peanuts. I want to store the candy in 1/2-pound bags
so that I can freeze it and use it in smaller portions.
How many 1/2-pound bags can I make?
tions thus far, yet I was still curious about
whether they could continue this upward spiral
as the problems and computation became more
difficult. One day I asked them to solve the following problem:
4
By now the entire class could estimate that the
answer was “a little more than twelve.” After
estimating, they set out to calculate the exact
answer. As expected, this problem took longer
than previous ones, and more head scratching,
frowning, and exchanges of ideas ensued than
usual. After a while we gathered as a group to
discuss the outcome.
Several children remarked that they had an
answer that was close but were not sure if it
was exact. I assured them that I was more interested in hearing their strategies. Various
children volunteered answers, which I recorded
on the chalkboard. (All the answers were
slightly more than 13.) One child’s explanation
was the following: “I got thirteen and one-fifteenth [pause]. I started with four divided by
one-third and that’s twelve because one-third
goes into four [pause] twelve times. . . . Then I
changed two-fifths to six-fifteenths and onethird to five-fifteenths.” I interrupted her at
this point and asked her to explain why she did
that. She continued, “Because it is easier for me
to divide them now, and they are still the same
number [pause]. Then I figured five-fifteenths
goes into six-fifteenths one more time, which
makes thirteen, and there is one-fifteenth left
over, so it’s thirteen and one-fifteenth.” Several
children nodded with approval; some children
exclaimed, “That’s what I did!” Others asked to
have the thinking repeated.
After a lengthy discussion about this problem, everyone seemed convinced that the answer was indeed 13 1/15 except for the child
who had “doubled and divided by one” in the
previous example. She raised her hand and
claimed to agree with everything “except the
last part.” She said that “six-fifteenths divided
by five-fifteenths is one, and there is one-fifteenth left over, which still has to be divided by
five-fifteenths. One-fifteenth divided by five-fifteenths is one-fifth because five-fifteenths could
fit in [to one-fifteenth] one-fifth of a time, so the
answer is thirteen and one-fifth.” This response
was not only logical and mathematically correct
but also a shining example of the autonomy
VOL. 2, NO. 6 . MAY 1997
One of the glorious products of constructivism is intellectual autonomy
The students were used to estimating first,
so they quickly gave estimates ranging from 10
to 12. They then set out to find the exact answer. One child responded, “Eleven bags, and
you would have a quarter of a pound left over,
or half a bag.” When I asked how she obtained
that answer, she replied, “You get ten bags from
the five pounds because five divided by one-half
is ten, and then you get another bag from the
three-fourths, which makes eleven bags, and
there is one-fourth of a pound left over, which
makes half of a half-pound bag.” Another student solved the problem by changing the 5 3/4
to 6 pounds and then dividing that by 1/2 to get
12 bags. He then took the 1/4 he had added and
divided that by 1/2 to get the 1/2, which he subtracted from 12 to arrive finally at 11 1/2 bags.
Perhaps the invention that startled me the
most on this particular problem came from a
child who nonchalantly raised her hand and
said, “I just doubled it [five and three-fourths]
and divided by one.” Her peers responded, “Can
you do that?” She went on to explain that it did
not change the problem. She cleverly cited how
the answer remains the same when an equation
is doubled, as in 10 ÷ 5 = 2 and 20 ÷ 10 = 2. She
astutely used a mathematical relationship,
without direct instruction about proportions.
This sort of inventive thinking and intellectual
risk taking are simply not present in classrooms where teachers impose methods on children. I also found it fascinating, although not
surprising, that not one child considered converting the mixed numeral 5 3/4 to an improper
fraction of 23/4, which would be the first step in
a traditional algorithm. Although many advocates for teaching algorithms assert that children do not invent efficient methods to solve
complex computations, the evidence cited here
does not support such claims. What could be
more efficient than doubling 5 3/4 to make 11
1/2 and dividing by 1?
During the next week the children continued
to work with division of fractions. I was amazed
by the thinking that was taking place and
thrilled with how much I was learning from the
children. Their work had exceeded my expecta-
2 1
÷ .
5 3
393
(Kamii 1985, 1994) that children develop when
encouraged to think for themselves. Here was a
child standing alone, disagreeing with an entire
class of peers and perhaps even her teacher.
She was not willing to accept any thinking or
procedure that did not make sense. This is one
of the glorious products of constructivism: intellectual autonomy.
Needless to say, the debate over the answer
to this problem went on for some time, and for
many it carried over into recess. During the
heated debate, one child looked to me and said,
“Well, which answer is right?” When the class
realized that I was typically not giving answers,
the debate resumed. This sort of intellectual
bantering among children is a desirable and
typical occurrence in a constructivist classroom.
This type of social interaction or debate engages
children in critical thinking. It does not contribute to the sort of confusion that many people experienced in mathematics class, which resulted merely in frustration. It is a processing
of ideas that results in deeper understanding.
Piaget attributed great importance to social interaction. In his theory, social interaction is absolutely essential to the construction of knowledge, and it is indispensable in childhood for
the elaboration of logical thought (Kamii 1989).
We continued to work on dividing with fractions, and the problems became more involved
and had more complexity with respect to computation. The children continued to thrive, and
eventually almost all of them had resolved
their confusion about remainders (such as the
1/15 in the previous problem). Some students
continued to struggle with the more difficult
problems, yet they were able to give extremely
close estimates, which indicates a developing
understanding and excellent number sense.
Conclusion
WHEN I FIRST BEGAN TEACHING CHILDREN, I WAS
fascinated with their ability to think and reason. And their strategies for solving problems
have never ceased to amaze me. In many instances and with many different topics, such as
the one described in this article, students have
taught me what it means to be a teacher and
what it means to “think and communicate
mathematically.”
I want to stress to readers that children can
and do invent ways to do sophisticated mathematics; however, the culture of the classroom
must be one that truly values and encourages
394
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
thinking. Children must feel safe if they are to
take the intellectual risks necessary to construct knowledge. They must be given ample
time to think and reflect about numbers and to
exchange ideas with peers, and they must be
developmentally ready for the material being
presented. Furthermore, a child who has been
fed a strict diet of algorithms and has viewed
mathematics as simply calculating using a system of memorized rules cannot suddenly begin
to think deeply about numbers and to invent
procedures. Such children have learned to be
dependent on teachers for methods and solutions, and it is extremely difficult to change
such behavior.
As teachers it is our duty to provide learning
environments that allow children to be successful. This obligation means that we must look
carefully at the traits we value. Are we merely
interested in the correct answer, and is that all
we assess? Do we as educators truly value
mathematics, and if so, how do we communicate
that regard to students? Have we taken the
time and energy to learn the mathematics we
claim to teach? Do we really value and encourage intellectual autonomy?
Finally, one should not assume that the
teacher’s role in a constructivist classroom is
one of a passive observer who sits idly waiting
for children to construct knowledge. Setting up
a classroom environment in which the children
invent methods to solve problems is not an easy
task. The teacher must strive to understand
each child’s thinking and must carefully determine just when and how to guide a child to a
deeper and higher level of understanding. Creating such a classroom and formulating appropriate questions to probe children’s thinking
and lead them to new intellectual heights is
perhaps the subject of a subsequent article.
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