Reasoning about the Multiplication and Division of Fractions
Statement of the Problem
As an elementary school teacher I have encountered certain math concepts that are difficult to
teach. By difficult I mean, without the procedure I learned as a child and use as an adult, I am
unsure of how to explain the problem to my students. Two ideas that I understand procedurally,
but not conceptually, are multiplying and dividing of fractions. For example, I was taught that to
multiply two fractions you multiply across and then simplify. Likewise, for division of two
fractions, you invert the second fraction and then multiply across. I find that my teaching of
fractions is limited to repeating and remodeling the same procedures over and over again. I
don’t know how to engage my students in reasoning around this subject. It is unclear to me how
to connect these concepts to my students’ lives and therefore make them worthy of learning. I
would like to increase both my own and my students’ understanding of multiplying and dividing
fractions through improved instruction, student participation, and class discussion.
In order to make these improvements in my classroom instructional practices I need to know
what types of conversations I should be having with my students about fractions? I need to
know what prior knowledge is necessary for students to have in order to enable them to engage
in mathematical reasoning about fractions. I am curious as to what mathematical reasoning looks
likes in regards to multiplication and division of fractions? Lastly, I feel that it is important for
me to determine how other teachers are approaching multiplication and division of fractions with
their students? To begin to examine these questions I have looked at examples of textbooks, read
dialogue in which children are reasoning about multiplication and division of fractions, and have
researched different literature regarding mathematical reasoning and the teaching of fractions.
Review of the Literature
In, Multiplication of Fractions: Teaching for Understanding, Cramer and Bezuk (1991) suggest
using the Lesh Translation Model, (1979), as a way to increase conceptual understanding of
multiplying fractions in students. The Lesh Translational Model suggests 5 modes of
representation including: real-world, manipulatives, pictures, spoken symbols, and written
symbols. (Cramer and Bezuk, 1991, p. 35) “Relationships between and within these modes of
representation are called translations. A translation is the reinterpretation of a concept from one
representation mode to another or within a single representational mode.” (Cramer and Bezuk,
1991, p. 35) For example, if students are presented with a story problem this is considered to be
a representation of real-world. If the problem is then worked out using a geoboard or counters
the problem has been translated from real-world to manipulatives. Cramer and Bezuk (1991)
suggest that “organizing instruction around these translations enhances students’ understanding
of fractions and improves their ability to transfer and apply this understanding to unfamiliar
problems. Below is a diagram of the Lesh Translation Model:
Reasoning about the Multiplication and Division of Fractions
The Lesh Translation
Model (1979)
Manipulatives
Real-World
Situations
Pictures
Written
Symbols
Spoken
Symbols
Cramer and Bezuk emphasize that even though students can perform the algorithm for
multiplying fractions, it doesn’t mean that they have a conceptual understanding of why they do
the steps or why the answer makes sense. In this article they suggest asking quantitative
questions involving estimation of the answer and estimation of size of answer in comparison to
the size of each factor. Students are encouraged to engage in mathematical reasoning by judging
the reasonableness of the answer they get. (1991, p. 36)
I think that this is a good model to use in both multiplication and division of fractions.
Presenting problems in one of the modes and asking kids to translate them to another mode
seems like a really effective way to strengthen the conceptual understanding of the problem and
to increase student flexibility in solving different problems involving similar concepts.
In, Understanding Division of Fractions, Bezuk and Armstrong guide teachers through a short, 5
day unit of real-world division problems that increases in difficulty as the days proceed.
Although it is not specifically stated, they use the same method of translating between different
modes of representation to build students’ conceptual understanding of dividing fractions. In the
form of spoken symbols, they state, “dividing fractions can be viewed as counting the number of
parts of a certain size that it takes to cover another part of a certain size.” (1993, p. 43) They then
Reasoning about the Multiplication and Division of Fractions
use real-world problems to illustrate this concept to students. To solve the real-world problems
they use the manipulative mode of representation.
For example, one of the problems they present the children with on the first day is as follows:
Paula and her road crew can resurface 1/8 of a kilometer of highway in one day.
1/8 of a kilometer is represented by this strip:
__________________
-------------------------__________________
1.
In May, Paula and her road crew resurfaced the ½ kilometer section of highway
represented below. How many days did it take to finish the work? First try to estimate
how many times the 1/8 kilometer strip can be measured out of the ½ kilometer section of
highway. Then use the strip to check.
_____________________________________________________________________________
-------------------------
-------------------------
---------------------------
--------------------
_____________________________________________________________________________
You can think of this problem as ½ of a kilometer divided by 1/8 of a kilometer per day is
_______.
On the fourth day they have evolved to questions such as:
List five ‘division of fraction’ problems that equal 3. Describe a pattern that you see in these
equations (other than that they are all equal to 3). How would you convince someone else that
your problems are equal to 3?
You can see that the level of cognitive demand increases as students’ conceptual understanding
of dividing fractions increases. The unit that Bezuk and Armstrong present begins as
“procedures with connections” as described by Stein and Smith, in that there is a suggested
pathway to follow that has a close connection to underlying conceptual ideas. There are multiple
representations such as visual diagrams, manipulatives and problem situations. (1993, p. 348)
Later in the unit problems develop into “doing mathematics” as described by Stein and Smith.
There is no predictable pathway and the problems require students to explore the nature of the
mathematical concept. Furthermore, later problems “require students to access relevant
knowledge and experiences and make appropriate use of them in working through the task.”
(1993, p. 348) They emphasize the importance of having students work collaboratively on the
problems and encourage discussion as a mechanism to further strengthen conceptual
understanding. Note that the algorithm for dividing fractions is never introduced to students.
Reasoning about the Multiplication and Division of Fractions
This short teacher’s guide demonstrates that the Lesh Translation Model is versatile and can be
used in teaching mathematical reasoning about multiplying and dividing fractions.
These two readings helped me to understand ways in which I can strengthen my students’
conceptual understanding of multiplying and dividing fractions. I did get a couple of ideas about
when multiplication and division of fractions might be used in real life, however, would still like
to explore this question further. These articles touched a bit on mathematical reasoning by
mentioning the idea of judging the reasonableness of answers. However, I feel like the readings
provided a pretty structured way of teaching and I am curious to see what it looks like when the
students are allowed to explore and openly discuss these concepts among themselves, in a more
student-centered atmosphere. I originally wanted to know why the procedures we traditionally
teach work, however am now wondering if that is even important. Some of the readings suggest
avoiding teaching those procedures until high school!
Both Chazan, in Beyond Formulas in Mathematics and Teaching and Lampert, in Teaching
Problems and the Problems of Teaching, have ideas about how to create good whole class
discussion about mathematics. In Chapter 4, Chazan describes some impressive techniques used
by Lampert. According to Chazan, Lampert has a very unique way of approaching teaching
mathematics. Instead of assigning problems for students to do after an algorithm has been
taught, she assigns the problems before any algorithm has been taught. This forces students to
explore and reason their way through the problem. She has found that this allows students to
approach math problems in more innovative ways. (Chazan, p. 121) Furthermore, Chazan states
that “Lampert portrays school mathematic like the discipline itself, as a living and growing field
in which developments occur when people create solutions to problems. She encourages her
students to see mathematics as a field in which one makes hypotheses and revises them.”
(Chazan, p. 122) This makes so much sense to me but it is completely opposite of how I was
taught mathematics and of how I was taught to teach mathematics!
Chazan states that we must “avoid categorizing mathematical statements as right or wrong.” It
places the teacher as the authority and forces students to take our word for it. It also causes
students to devalue their own reasoning and logic. Instead, he cites an example in which two of
his students spend a week arguing a misconception (zero is not a number) and he lets them hang
on to this argument even though the class disagreed. He discussed the idea that the mathematical
community considers zero a number and was able to introduce the idea of accepted views in the
field of mathematics. (Chazan, p. 137) Through this process there resulted a lot of mathematical
conversation in which the teacher wasn’t the center of instruction. When teaching math, Chazan
and Lampert do not focus on obtaining an answer to questions they propose to their students.
Instead, they value the rationale for student responses. They want students to discuss the logic
and evidence that they use to solve a problem in order to convince their classmates that their
responses make sense. (Chazan, p. 127)
Reasoning about the Multiplication and Division of Fractions
In, Teaching Problems and the Problems of Teaching, Lampert walks the reader through the
steps she takes in engaging students in a conversation about a specific problem she has proposed.
There are several things that she does in order to inspire conversation. First and foremost, she
aims to teach students that “they are responsible for reasoning through a piece of mathematics.”
(2001, p. 159) She practices asking questions in a way that provokes student responses, whether
or not they think they have solved the problem. For example, she might ask, “Who has
something to say about A?” (2001, p. 145) Another technique she uses to guide the conversation
is strategically choosing who to call on. She watches their independent work and knows what
each has to offer. She can then choose the individual whose information is particularly useful for
consideration at that time. Then she can direct the class to consider what the speaker has said.
(2001, p. 146)
In terms of classroom conversation, Herbel-Eisenmann suggests using a focusing-interaction
pattern. This pattern, as Herbel-Eisenmann suggests, serves many purposes including, “allowing
the teacher to see more clearly what the students were thinking or requiring the students to make
their thinking clear and articulate so that others can understand what they are saying.” (2005, p.
486) This method is less teacher-centered and puts the responsibility on the student to explain
their thinking. The teacher’s role is more to ask clarifying questions that they anticipate other
students might have.
In her planning, Jill Lester tries to make the problems that she presents to the students,
“challenging but accessible.” (1996, p. 89) She is patient at the beginning of the year and begins
teaching the process of mathematical reasoning by asking, “how they had arrived at the answer”.
(1996, p. 90) She frequently provides manipulatives for children to use to validate their answers.
Lester gradually moves from a teacher-centered learning environment to a student-centered
learning environment. By the first month of school, during one of her math classes, she notes,
“It was also clear that they were paying absolutely no attention to me.” (1996, p 100) The kids
were interacting with one another and listening closely to each other’s explanations.
Chazan struggled with the question, “Why should I learn this?” when teaching Algebra to his
middle school students. He believed that the key was to think of Algebra as a way, “to capture
the relationships between quantities – where quantities are qualities of experience that you can
quantify – that change. When you begin to think that way, functions become things that you see
all around you.” (p. 13) He realized that to teach middle school Algebra successfully he would
have to make it relevant to his learners. He decided to ditch the text book and create meaningful
projects.
Chazan realized that he didn’t know half of what went on in his students’ lives. This disconnect
was a barrier to creating meaningful Algebra problems for his kids to work on. He tried things
like asking them to write stories about measurement. “I have to find ways for them to teach me
about them - instead of me making assumptions about their lives.” (p. 14) He also boosted
student efficacy by letting them go out and interview local businessmen about “the role that
Reasoning about the Multiplication and Division of Fractions
measurement and quantity play in their work.” (p. 14) It turned out that many of the
businessmen didn’t even realize that the calculations they do could be written down. The
students actually taught their interviewees something through this project. (p. 14)
Lester and Chazan have similar strategies to approaching mathematics with their students. Each
teacher demonstrates questioning as a strategy to encourage thinking in their students. When a
wrong answer is given these teachers allow the students to discuss what makes sense and what
does not. Each teacher has made the effort to connect the math that the kids are learning to their
lives in some way. I notice that more time is spent on one problem; the goal being to develop
and demonstrate a conceptual understanding of the math involved. In each class the student’s
show enthusiasm and begin to interact with each other instead of relying on the teacher for all the
answers.
In chapter 8, Lampert discusses helping children to make connections across lessons. She
chooses to uses problem context to connect ideas across lessons. “More pedagogically
challenging is the problem of figuring out how to use a context, both to bring students in contact
with the connected universe of important mathematical ideas that the problems posed make
available and to make the contact that they have with those ideas productive of learning.” (2001,
p. 211) In the lessons she describes, she uses time-speed-distance relationships relating to the
story, The Voyage of the Mimi, to connect ideas about fractions, division, and ratio over time.
By using a story that the kids are reading it makes the math meaningful and “worthy of
investigation”. (2001, p. 181)
She also encourages students to communicate about mathematical ideas by using graphic
representations. Along the way she mentions the difficulty deciding how organize the work of
her class without creating a dependence on her. She decided to provide them with a diagram
(journey line) for consistency and to make communication about the mathematics easier. Most
of her students were creating variations of this diagram in their personal notebooks and were not
yet using it to “figure out the relationships in the problem.” She expresses her uncertainty as to
how much she should direct her students verses letting them come up with as many variations on
the correct diagram as they could think of. The dilemma being that if she directs them entirely
she will become the expert but if she lets them go in different directions it will make
communication about the mathematics very difficult. (2001, p. 196)
Student work is often a tool she uses in this process to help her create a foundation on which
future lessons can be built. By looking through their notebooks she can determine where their
understanding lies and can tailor her instruction to best meet their needs. Another part of her
work in helping students to see the important relationships among mathematical concepts she
calls anticipation. She notes the importance of developing the foreknowledge that she needs in
order to take full advantage of the opportunities that could arise as students work in a particular
problem context. (Lampert, 2001, p. 184)
Reasoning about the Multiplication and Division of Fractions
In chapter 9 Lampert discusses connecting the required topics of the curriculum to the student’s
work and creating a situation in the classroom in which these topics seem worthwhile of
investigation to the students. With experience on her side, she notes that by using a mental
checklist of all the topics in the curriculum, she is able to take advantage of opportunities in
which these topics might come up through the independent investigations of her students. (2001,
p. 213)
What she does in her classroom by using problems is allow the students to uncover the math as
they approach a problem in which the math is necessary. She documented her student’s work
over a period of 6 class periods and found that they “covered” the math in an order completely
different from that of a curriculum framework or textbook index. She found that in order to
connect the math she couldn’t teach one topic at a time, one after another. Instead, students must
be “simultaneously engaged with several topics in each lesson.” (Lampert, 2001, p. 217) Put
another way; students are, “moving back and forth between big ideas and the facts and
procedures that logically flow from them.” (Lampert, 2001, p. 217)
“A real-world problem is one that someone, perhaps even the student, might encounter outside of
school. The notion is that students will be attracted to problems that implicitly suggest that
mathematics is a useful body of knowledge that allows people to solve problems they face in
their lives.” (Chazan, D., Beyond Formulas in Mathematics and Teaching, p. 40)
Mode of Inquiry
In order to investigate my questions I focused on collecting classroom data from experienced
teachers. I looked at both their approach to the mathematics and their approach to encouraging
children to become actively involved in mathematical reasoning.
First, I collected data in the form of teacher-suggested strategies from different experts on
teaching methods. I was observing their approach to the math.
From these resources focused on teaching methods I recorded the following:
1. What prior knowledge is suggested?
2. What strategies does the author suggest using when introducing multiplication and
division of fractions?
3. Does the author suggest using pictures and/or manipulatives?
Second, I looked at actual classroom conversations in which students are engaging in
mathematical reasoning. In these samples I am focusing on what mathematical reasoning looks
like in a classroom. I am paying particular attention to the role the teacher plays and what types
of questions they ask to facilitate mathematical reasoning among the students.
Results
Reasoning about the Multiplication and Division of Fractions
Approaching the Content:
SAMPLE 1:
Marilyn Burns begins her lesson on multiplying fractions by reviewing six statements about
multiplication that the students had created together in a previous lesson.
Six statements about multiplication: (Burns, 2003, p. 13)
1. Multiplication is the same as repeated addition when you add the same number again and
again.
2. Times means “groups of”.
3. A multiplication problem can be shown as a rectangle
4. You can reverse the order of the factors and the product stays the same.
5. You can break numbers apart to make multiplying easier.
6. When you multiply two numbers, the product is larger than the factors unless one of the
factors is zero or one. (or a fraction smaller than one)
She continues this lesson by posing each of the above statements to her students by asking them
to think of them in terms of fractions.
She begins with statement one and asks, “Let’s think about this statement together with this
problem – six times one-half,” (Burns, 2003, p.15) Burns encourages students to, “Talk with
your neighbor about how you might make sense of this problem.” (Burns, 2003, p. 15) This leads
students to discuss repeated addition. They decided that:
6x½=½+½+½+½+½+½=3
To facilitate discussion, Burns asked questions like, “Does it make sense to you?” and “Can you
explain what Juanita did?”
In regards to statement two Burns asks her students, “Does it make sense to read ‘six times onehalf’ as ‘six groups of one-half’?
Going on to the 3rd statement she asks: “Do you think we can use rectangles to show
multiplication problems when we’re multiplying fractions? Can we draw a rectangle to show six
times one-half?” (Burns, 1994, p. 15)
She used the rectangle for 6 x 1 to help students visualize what a rectangle might look like for 6
x ½.
The students came up with the following rectangle:
6
Reasoning about the Multiplication and Division of Fractions
½
½
This leads a student to ask the question, “What about if both the numbers are fractions?”
Burns replies, “Let’s think about drawing a rectangle for the problem one-half times one-half.”
Burns models the drawing of this rectangle when her students are quiet and unresponsive:
1
1/2
1/2
1
Burns uses statement 4 to address the language that can be used when discussing multiplication
of fractions. She states, “If we think about the times sign as ‘group of’ then one-half times six
should be ‘on-half groups of six.’ But that doesn’t sound quite right. It does make sense,
however, to say ‘one-half of six’, and leave off the groups part. This sounds better, and it’s still
the same idea. What do you think ‘one halve of six’ could mean?” She is able to help her
students make sense of the language of fractions and in turn give them another way to think
about what it means to multiply fractions. (Burns, 1994, p. 17)
With statement number 5 Burns engages students in a discussion about the problem 6 x ½ again.
“Talk with your neighbor about how you could apply this statement to the problem six times
one-half.” She wants them to decide whether you could break apart one of the numbers to help
with the multiplication.
One student decides that you could break the six into three twos.
6 x ½ = (2 x ½) + (2 x ½) + (2 x ½) = 1 + 1 + 1 = 3
Another suggests breaking the six into two and four.
Reasoning about the Multiplication and Division of Fractions
When the last statement is addressed the students decide it cannot be true for fractions because
the answer of 3 is less than the factor 6 above. They modify it to state “…unless one of the
factors is zero or one or a fraction smaller than one.”
This particular sample illustrates a method used to introduce multiplying fractions to a group of
students. The teacher’s questions help to encourage the students to create their own
understanding and strategies to solve the problems. She connects this new information to what
the students have already come to understand about multiplication and helps them to revise their
previous understanding. She models the use of rectangles to illustrate certain problems. She
also addresses language used in multiplying so that students will gain a better understanding of
what it means to multiply fractions.
SAMPLE 2:
Burn’s students have a strong understanding of division of whole numbers already. They have
created the following statements about division that they revisit during their introduction to
division of fractions: (Burns, 1994, p. 76)
1. You can solve a division problem by subtracting.
2. To divide two numbers, a ÷ b, you can think, “How many b's are in a?”
3. You can check a division problem by multiplying.
4. The division sign (÷) means “into groups of.”
5. The quotient tells “how many groups” there are.
6. You can break the dividend apart to make dividing easier.
7. Remainders can be represented as whole numbers or fractions.
8. If you divide a number by itself, the answer is one.
9. If you divide a number by one, the answer is the number itself.
10. If you reverse the order of the dividend and the divisor, and the quotient will change.
The first problem that Burns presents to her students is 3 ÷ ½. The first statement is used to
solve the problem through repeated addition until zero is reached. The second statement causes
Burns to ask her students, “How many ½’s are in 3? One student drew 3 circles, cutting each in
½ to illustrate the 6 ½’s within the three circles. The third statement is used to check the answer
of 6 that the class had come up with. Through this discussion, one student noticed that doubling
the dividend and the divisor made the problem even easier:
6÷1=6
When the problem 4 ½ ÷ ½ was proposed to students, one student decided that statement six
from the above list would help make the problem easier to solve. She changed 4 ½ into 4 + ½.
4½=4+½
4÷½=8
Reasoning about the Multiplication and Division of Fractions
½÷½=1
8+1=9
So, once again, Burns draws on her student’s prior knowledge. She is helping them make
connections to both division of whole numbers and their new understanding of multiplication of
fractions.
SAMPLE
X or ÷
Prior
knowledge used
Questions posed
by teacher
1
X
Six statements
about
multiplication of
whole numbers
Do you think that
all of these
statements are true
when we think
about multiplying
fractions?
2
÷
Rules for
division of
whole numbers
Do you think that
all of these
statements are true
when we think
Multiplication of about dividing
fractions?
Fractions
Strategies
used by
students
Repeated
addition
Pictures or
maniputatives
used
Rectangles
Breaking
numbers
apart
Repeated
subtraction
Circles
Breaking
apart
numbers
I think these two data samples help to answer my question about what type of prior knowledge
students need in order to demonstrate mathematical reasoning. They need a deep understanding
of both multiplication and division of whole numbers. I think that mathematical reasoning was
evident in the way that students drew upon common knowledge that the class had obtained
earlier in the year. I think it was also demonstrated in the way that students took risks to use
their understanding of multiplication and division to create strategies to solve new types of
problems. Students used tools that they were familiar with, such as rectangles and circles, to
illustrate their thinking to other students. They also applied familiar strategies such as repeated
addition and repeated subtraction to show others that their answer made sense. Students even
simplified problems using strategies such as breaking apart numbers, another method that
demonstrates their ability to connect their prior knowledge to the new problems they are trying to
solve.
SAMPLE 3:
Reasoning about the Multiplication and Division of Fractions
In, Elementary and Middle School Mathematics, John A. Van de Walle makes several
suggestions of the teaching of multiplication and division of fractions.
Van de Walle suggests using the concept of equal shares to begin multiplying fractions. He
suggests that asking students to first find a fractional part of a whole number is a good place to
start.
For students who are ready to approach multiplication of two fractions here is a sample problem
he has provided and his suggestion on how to think about the multiplication necessary.
“You have ¾ of a pizza left. If you give 1/3 of the leftover pizza to your brother, how much of a
whole pizza will your brother get?” (Van de Walle, 1990, p. 323)
He remarks that the problem is 1/3 of 3 things. “The focus remains on the number of unit parts
in all, and then the size of the parts determines the number of wholes.” (Van de Walle, 1990, p.
323)
Another example of this is as follows: (Van de Walle, 1990, p. 323)
¾ x 4/5
¾ of 4 things is 3 things
¾ x 4/5 = 3/5
In more difficult problems such as the following:
The zookeeper had a huge bottle of the animals’ favorite liquid treat, Zoo Cola. The monkey
drank 1/5 of the bottle. The zebra drank 2/3 of what was left. How much of the bottle of Zoo
Cola did the zebra drink?
Van de Walle maintains the importance of the different roles that the top and bottom number
play. “The top number counting and the bottom number naming what is counted play an
important role.” (Van de Walle, 1990, p. 324)
He notes that in the problem above you are finding thirds of four things, the 4 fifths of the cola
that are left.
One interesting problem that he uses in his book is as follows:
Reasoning about the Multiplication and Division of Fractions
3/5 x 2/3
One strategy he suggests is counters. (Van de Walle, 1990, Figure 17.8, p. 324)
2/3 is 10 counters
1/5 of 10 is 2 counters
3/5 of 10 counters is 6 counters
3/5 x 2/3
6/15 or 2/5
Another strategy he suggests is using the Communitive Property.
Switching the problem around to read 2/3 of 3/5 is easier to immediately visualize using the
same method he suggests earlier. Finding 2 of the 3 things is simple.
Van de Walle does suggest introducing a task that could perhaps lead students to the traditional
algorithm used for multiplying fractions. By using grids students can illustrate problems such as
3/5 x ¾ as follows: (Van de Walle, 1990, p. 325)
By first drawing all the lines in one direction students are representing the ¾. Next, students can
be instructed to divide that region into fifths. Then, by extending the lines it becomes evident
what fractional part each little square is of the whole.
He suggests that the teacher should avoid pushing the students to formalize any rule. Let
students notice on their own that the number or rows and columns are actually the two
numerators and the two denominators.
SAMPLE 4:
Van de Walle also suggests certain strategies for introducing division of fractions.
In the problem,
Reasoning about the Multiplication and Division of Fractions
Cassie has 5 ¼ yards of ribbon to make three bows for birthday packages. How much ribbon
should she use for each bow if she wants to use the same length of ribbon for each?
Van de Walle suggest thinking about 5 ¼ as 21 fourths. Then it is easy to divide 21 by 3 to
determine that there will be 7/4 yard of ribbon for each bow.
In more difficult problems in which the numbers must be split apart as follows, Van de Walle has
another strategy he suggests.
Mark has 1 ¼ hours to finish his three household chores. If he divides his time evenly, how
many hours can he give to each?
Because the 5 fourths Mark has for his chores doesn’t split evenly into 3 parts, this problem is
more difficult. For this type of problem he suggests using pictures such as number lines and
counters to illustrate dividing each of the fourths into three equal parts, creating twelfths.
Because there are a total of 15 twelfths there is obviously 5/12 for each chore.
There are two algorithms associated with division of fractions. One is of finding the common
denominator and the other is the algorithm for the actual division.
To help students develop the common denominator algorithm he suggests using pie pieces and
fraction strips to model converting each fraction into the same fractional part.
To help students develop the invert and multiply algorithm he uses the following example:
A small pail can be filled to 7/8 full using 2/3 of a gallon of water. How much will the pail hold
if filled completely? (Van de Walle, 1990, p. 330)
The task is to find the whole. A full pail is 8/8. The water in the pail is 7 of the 8 parts needed to
fill the pail. Therefore, dividing the water by 7 and multiplying by 8 solves the problem or fills
the pail. (Van de Walle, 1990, p. 330)
Sample #
3
X or ÷
Prior Knowledge
Needed
X
Multiplication of
whole numbers
Finding fractions
of a whole
Strategies
Suggested to use
in teaching
The focus should
remain on the
number of unit
parts in all.
Use the word
“of” in place of
Pictures or
manipulatives
used
Rectangles
Counters
Grids
Reasoning about the Multiplication and Division of Fractions
“times”.
Use the
Communitive
Property if it
makes the
problem easier.
4
÷
Division with
whole numbers
Use pictures to
illustrate the
problem
Start with wholenumber divisors.
2 meanings of
division:
partition and
measurement
Think about
mixed numbers
as fractional
parts.
Counters
Number-lines
Keep in mind
that the question
for partition
problems is
“How much is
one?”
Again, I think that this resource does emphasize the need for a deep understanding of
multiplication and division of whole numbers. Therefore, it is important that the teacher is
connecting these new ideas to what the students have already learned. Van de Walle stresses the
importance of choosing the numbers used in story problems carefully so that they work to
illustrate the concept that you are trying to teach in that particular lesson. He uses a lot of
drawing and illustrations and suggests the use of counters in working to create a common
denominator. I think that his methods could be applied in a classroom in which the goal was to
encourage mathematical reasoning.
Observing Mathematical Reasoning in the Classroom
SAMPLE 1:
In, Keeping Out Right Answers, Marty Schnepp, a once traditional teacher of mathematics, visits
Chazan and Bethel’s classroom to witness a new way of teaching math. “Kids in her class were
thinking, and finding ways to represent that thinking…These kids who would have to write, or
make tables and graphs to represent ideas, they were able to draw the mathematics out of
Reasoning about the Multiplication and Division of Fractions
situations…discussions led to new inquiries, and ideas came from students.” (Chazan, p. 15) He
decided to adopt their methods and to use their materials in both of his Algebra classes.
One example of mathematical reasoning in Schnepp’s class was when they were presented with
the problem: 4x-5
a.
b.
c.
d.
Write the rule in words
Make a table of six inputs and outputs
Find the input(s) that make zero
Describe all inputs that make the output negative
When kids began sharing their answers for part (a) one student wrote, “Take the input. Multiply
it by 4 and subtract 5.” When Schnepp asked his class, “Anyone have anything at all different?”
Another student spoke up and said, “I have ‘Take 4. Multiply it by the input and subtract it by
5’” (Chazan, p. 16) Schnepp pushed his students to continue thinking about these two statements
by asking, “What’s different about that?” This led the kids into a discussion about when order
matters. The students began thinking about subtraction, addition and then division and discussed
whether order matters in each of those situations. They are examining concepts such as the
commutative property without memorizing a definition. “…students think through and explain
their solutions instead of seeking or trying to recollect the "right" answer or method.” (NCTM
Reasoning and Proof Grades 3-5, Cobb et al. 1988) The other thing that this example illustrates
is how important it is to articulate mathematical descriptions accurately. This became a lesson in
the language of math and what difference one word can make. This is a good example of
mathematical reasoning because the kids are comparing two different answers and are working
to evaluate which one makes sense. They are using discussion to clarify a mathematical concept.
They are learning about how to talk about math.
SAMPLE 2:
One example of children doing math reasoning in Jill Lester’s class was when they were working
on the following problem: If each of 15 6th graders and each of 25 second graders need
paintbrushes, how many shall I buy?
Lester accepts the answers 52 and 40 and asks the kids, “Are there any answers here that bother
you? One child offers that 52 is too big because, “There are only three 10s and two 5s.” (Lester,
1996, p.99) Another agrees, “There’s one 10 in 15 and there are two in 25. That can’t be 52.”
(Lester, 1996, p.99) By accepting all of the proposed answers, she encouraged kids to, “assess
their answers for reasonableness”. (Lester, 1996, p. 96) She frequently asks, “Are there any
answers here that bother you?” (Lester, 1996, p. 99) This is a great way to provoke healthy
debate without pointing the finger at any individual. Kids in her classroom did not seem to mind
when others challenged their answer. It had simply become an accepted process as students
continued to build their understanding of math.
Reasoning about the Multiplication and Division of Fractions
She allowed the kids to then work in small groups. Some groups proved the answer was 40 in
several ways. They shared their justification for their answer with the whole group by stating
such ideas as, “We took the two 5s – one from the 15 and one from the 25. We added them
together. That made 10. 20 is two 10s. There’s a 10 from the 15. That made 40. (Lester,
1996, p. 100) She challenges her second graders to think about, “how it was possible that three
seemingly unrelated solution processes could produce the same answer”. (Lester, 1996, p. 92)
This is a good example of mathematical reasoning because both kids are considering the
reasonableness of the answer. They are also solving the problem in multiple ways and providing
justification to the group for their answers.
It is one month into the school year and Lester makes several observations of her 2nd grade
students as she watches their ability to reason about math develop. The children present right
and wrong answers, all of which are written on the board by Lester. Students have learned not to
be upset when their answer is disputed by a peer. Groups are working together to solve this
problem in several different ways. They are dying to share their ideas with their classmates. The
kids are no longer simply being quiet while others share to wait their turn to share. Now they are
listening to each other and are commenting on each other’s responses. Lester sums up their
progress by stating, “The children had listened to one another in order to figure out ideas and
how they related to their own solutions. And they had validated each other’s solutions without
looking to me for direction or support.” (Lester, 1996, p. 102)
The kids were interacting with one another. They were responding directly to the comments
other students were making. Lester makes the observation, “…no one looked to me for
recognition.” (Lester, 1996, p. 100) This means that they had moved past appeal to authority, as
Carpenter, Franke, Levi (2003), describe it. They were exuding confidence and the energy in the
room was high. Lester observes the kids while they volunteer to be called on, “Jack is bouncing
up and down in his place. His voice almost too loud. Keith is up on his knees and pulling on his
earlobe” (Lester, 1996, p. 100) Even the quietest kids are participating. “Everyone seemed to be
involved in the process.” (Lester, 1996, p. 101)
Conclusions and Limitations
From this research several things have become clear to me with regards to mathematical
reasoning about the multiplication and division of fractions. Most importantly, I have determined
that through creating an environment in which children are taught to reason about math, a teacher
can make any subject in math reasonable. So, while I started by focusing on the multiplication
and division of fractions, I have decided that my biggest gains in understanding have been in the
area of making math, in general, reasonable.
Mathematical reasoning in school is imperative in teaching students that math is logical, can be
tested, and can be proved. If students are engaged in mathematical reasoning in school they are
making connections between the math they are doing in school and real life. When students are
Reasoning about the Multiplication and Division of Fractions
reasoning about math they are using their prior knowledge to try to solve a new problem. They
are being innovative and using creativity. They can demonstrate why their strategies and
answers make sense to other students in the classroom. They aren’t asking the teacher whether
an answer is right or wrong. Students are using language that is mathematical and familiar to
others in the class in discussions about math.
With regards to the multiplication and division of fractions, students need to have prior
knowledge of the multiplication and division of whole numbers. Real-world problems help to
create sense of “worthy-of-learning” among the students. Using manipulatives and number lines
help students to visualize the math that is happening. Choosing alternative language and
familiarizing students with the different way people can say “times” will help over all
understanding to evolve as well.
To support mathematical reasoning about the multiplication and division of fractions, a teacher
must approach math in a non-traditional way. It is important to allow students to work on new
problems without first teaching the algorithm. They must force students to use their own
creativity and understanding of math. Problem-based mathematics which connects the math
from lesson to lesson over time and which connects the math to real world problems can help
engage students and can help them to make important connections to the math that they are
doing. Providing different ways of illustrating the mathematics is important; use of diagrams,
manipulatives, and pictures can help students to gain a stronger understanding about math.
A classroom culture is important in facilitating mathematical reasoning. One that is student
centered, allows both independent and group work, encourages mathematical discourse between
students and requires that ideas be explained is most supportive of that goal. The discussions
that students engage in should not be teacher-centered but the teacher must be there to ask
students for clarification when needed. Student responses should be focused on explaining their
logic and why the math they did makes sense. The teacher is not after a right answer but instead
sound reasoning and evidence presented by students. The responsibility to reason through
problems is placed on the students. A classroom culture that is flexible in allowing students to
discover math through a natural process instead of rigidly planned curriculum is also important.
I realize that the biggest limitation to my research is lack of classroom access. I would have
found it beneficial to gather my own data throughout the year as my students reasoned their way
through math. I think it would have been helpful to experience some of the benefits to the
methods that the literature on mathematical reasoning suggests.
Next Steps
Now that I have learned about what it takes to make math reasonable I think there are several
“next steps” I could take. One of the most interesting ideas I came across was the idea of a
Reasoning about the Multiplication and Division of Fractions
problem based curriculum. The connections across lessons and between the math and real life
seem very valuable in the effort to make math reasonable. This leads me to ask the following
questions:
1. How can teachers who are bound by a structured curriculum and regular, standardized
assessments create and maintain a problem based curriculum?
2. Have other teachers found ways to create a problem based curriculum in which the
multiplication and division of fractions is covered?
3. What resources are available for a problem based curriculum in which the multiplication
and division of fractions is the mathematical focus?
Reasoning about the Multiplication and Division of Fractions
Resources:
Burns, Marilyn (1944). Teaching Arithmetic: Lessons for Multiplying and Dividing Fractions,
Grades 5-6. Sausalito, California: Math Solutions Publications, c2003.
Bezuk N.S., Armstrong B. E., Understanding division of fractions. (1993). The Mathematics
Teacher, 86(1), 43. Retrieved October 22, 2009, from Research Library Core. (Document
ID: 5244277).
Chazan, D. (year) Keeping out right answers. In H. Featherstone (Ed). Changing Minds.
Michigan Extension Service: Michigan Department of Education.
Cramer, Kathleen, Bezuk, Nadine. (1991). Multiplication of Fractions: Teaching for
Understanding. The Arithmetic Teacher, 39(3), 34. Retrieved October 22, 2009, from Education
Module. (Document ID: 1862133).
Herbel-Eisenmann, Beth and M. Lynn Breyfogle. "Questioning Our Patterns of
Questioning." Mathematics Teaching in the Middle School 10 (May
2005): 284- 289.
Lampert, M. (2001). Teaching Problems and the Problems of Teaching. Yale University Press:
New Haven and London.
Lester, J. (1996). In Schifter, D. (Ed): What's happening in math class?
Vol. 1: Envisioning new practices through teacher narratives. New York: Teachers
College Press
Van de Walle, J. (1990). Elementary and Middle School Mathematics. New York : Longman.