Mixing
Strategie
to Compare Fractions
s
Strictly teaching algorithms or procedural computations can “encourage
children to give up their own thinking” (Kamii and Dominick 1998).
Although such procedures are valid
for finding solutions, students often
know only how to use them, and they
lack the understanding of why a procedure works. For example, consider
the following prospective teacher’s
method for comparing 1/6 and 1/8
(Tobias 2009):
With common denominators, the
bottom numbers have to be the same.
So I knew that 6 and 8 both go into
24. So whatever it [6] takes to get to
24, which is 4, I multiply by the top.
And whatever it [8] takes to get to
24, which is 3, I multiply by the top.
And since this one is obviously larger,
4 instead of 3 things, 1/6 is bigger.
376
Jennifer M. Tobias
Although the procedure was valid
for comparing 1/6 and 1/8, its use
altered the problem so that it was a
comparison between 3 and 4. As a
result, 1/6 and 1/8 were no longer part
of the problem, and reasoning about
these two fractions was nonexistent.
In 2001, the National Research
Council (NRC) published a report suggesting that mathematically proficient
students have five strands of knowledge.
These include strategic competence,
or the ability to problem solve; adaptive reasoning, or the ability to explain
and justify; conceptual understanding,
which is understanding why an idea is
important and how it connects to other
ideas; procedural fluency, or knowing
how to solve problems efficiently; and
productive disposition, which is seeing
mathematics as a worthwhile activity
(NRC 2001). Only one strand pertains
MatheMatics teaching in the Middle school
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Vol. 19, No. 6, February 2014
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This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
to students’ ability to use a
procedure to solve problems;
in other words, it alone will not
be enough to allow students to
be successful with mathematics
(NRC 2001).
This article describes teaching
strategies to support students in
becoming mathematically proficient.
The examples originate from a class
of prospective teachers. The instruction focused on having the prospective
teachers understand more than just
a procedure for comparing fractions
through experiences similar to what
they will then have to implement in
their own classroom, as outlined by the
Common Core State Standards for
Mathematics. Because the prospective teachers were the students in an
education course, they are referred to
throughout this article as students.
WAVEBREAKMEDIA LTD/THINKSTOCK
When students have many tools,
including context and common
numerators, they are more apt to
develop mathematical proficiency.
ies
Provide a Context
Problems that are placed in a context
can help students’ problem-solving
abilities or strategic competence
(NRC 2001). Word problems allow
students to make sense of numbers
before they are formally asked to work
with them. In addition, providing
contextual word problems gives students the opportunity to develop reasoning strategies on their own without
being directly told what the strategies
are and how to use them. For example,
solve the following problem:
fter a party, 1/6 of a large mushA
room pizza remained, and 1/8 of
a large sausage pizza was leftover.
Which pizza had more leftover?
How did you solve the problem?
Chances are the context provided you
with a way to visualize what the two
fractions represent, and your thinking may have sounded similar to the
student’s work shown in figure 1.
The student found that 1/6 is greater
because an object cut into 6 will create
larger pieces than the same object cut
into 8. Rather than just finding a common denominator, a context can give
students ways to think about what the
numbers in the problem represent.
The intent of this problem was
to have the class develop reasoning
strategies related to comparing fractions with common numerators. With
common numerators, each fraction
will have the sa me number of pieces,
and the comparison is in terms of
the size of each piece. As seen in the
student’s work, when given a context,
he or she was able to reason through
the problem and discover that the
larger the denominator, the smaller
the piece. Without a context, a student
may not develop these understandings
and instead may revert to computing
numbers, as evidenced by the explanation at the beginning of this article.
Ask Students to Explain
and Justify
Asking students to explain how they
solve a problem and justify why their
methods work can aid in their development of adaptive reasoning (NRC
2001). This is important to determine
if they are using valid methods to construct their answer and in what ways
they are finding a solution. For example, consider the following problem:
birthday party took up two
A
tables. One table had 9 large pizzas
for 18 people. The other table had
2 large pizzas for 4 people. Each
table shared the pizzas equally. At
which table would you want to sit
to get the most pizza?
Note that this problem was written
with ratios instead of fractions. This
decision was purposefully made to see
if students would relate the quantities of 9/18 and 2/4 to determine that
each table would have 1 pizza for
every 2 people.
Some students reasoned about
the numbers as ratios to compare the
two tables, as seen in the following
student’s explanation:
Fig. 1 Reasoning with a context will allow a student to figure out the larger piece that
remained.
378
Mathematics Teaching in the Middle School
●
Vol. 19, No. 6, February 2014
It doesn’t matter which table you
sit at because they are equal. Both
tables have half as many pizzas as
people, which is 1 pizza for every
2 people.
By understanding the relationship
between the number of pizzas and
people at each table, this student was
able to determine that there would
be 1 pizza for 2 people at each table.
Thus, everyone would receive the
same amount. Other students reasoned about how much pizza each
person would receive:
Either, because each person, regardless of which table they are at, is getting the same amount of pizza: 1/2
of one pizza. At the table of 18, we
can divide each of the pizzas in half
because 9 goes into 18 twice. At the
table of 4 people, both of the 2 pizzas
can also be split in half, giving each
person half of a pizza.
This student correctly determined
that everyone at both tables receives
the same amount of pizza. This thinking was related to how much pizza
each person would receive. By understanding that 18 is twice as much as 9,
he or she reasoned through what the
numbers meant and determined that
each table received the same amount
because each person would get 1/2 of
a pizza.
As seen with both students’ responses, they determined that people at
each table received the same amount.
However, some students found this
by working with fractions and finding
that each person would receive 1/2 of
a pizza. Others worked with ratios and
determined that 1 pizza would feed
2 people. Although students drew the
same conclusion, without an explanation and justification it may not be
evident that this answer was derived
from using different meanings of rational numbers.
Connect Strategies
Presenting students with situations
in which they can connect strategies
with one another can support their
conceptual understanding and ability
to develop other reasoning strategies
(NRC 2001). For example, the class
was presented with the problem of
comparing the leftover 1/6 pizza to
the leftover 1/8 pizza immediately
before solving the following problem
in which both fractions had the same
number of pieces missing.
Fig. 2 Students needed to coordinate
multiple fractions at the same time
when ordering five fractions by size.
Pete held a pizza-eating contest. The
following table shows how much of
a large pizza each contestant ate.
Rank the five contestants in order
from first place to fifth place.
Contestant
Pizza Eaten
Michael
7/8
Marie
7/13
At the party, the trapezoidal table
Joseph
9/20
was decorated with 5/6 spool of
ribbon. The rectangular table was
decorated with 9/10 spool of ribbon. If both spools had the same
amount of ribbon, on which table
was more ribbon used?
Walker
23/24
Nicole
3/20
Evidenced from one student’s explanation, members of the class were
able to connect the two problems. The
student explained:
But what we first thought of was kind
of in comparison to the last question
[leftover 1/6 versus leftover 1/8 pizza],
now we are looking for the largest
piece. In this case, we are looking for
the smallest piece leftover.
From this explanation, the class was
able to start relating this problem to
the previous example comparing 1/6
with 1/8. In that situation, students
needed to understand that they had
the same number of pieces. With this
problem, they have the same number
of pieces missing. By comparing 1/6
and 1/10, students had to use common
numerator reasoning to determine that
1/6 is greater, meaning that 5/6 has
more missing than 9/10, and subsequently 5/6 is smaller than 9/10.
The missing pieces strategy is the
most difficult for students to understand. This strategy requires thinking
backward from the common numerator
strategy. Rather than having the same
number of pieces, the pizzas now have
the same number of pieces missing. By
presenting a missing pieces problem
directly after a common numerator
situation, students can start to make
connections between the two.
Have Students Use
Multiple Strategies
Providing students with situations
that require them to use multiple
strategies at once can be a way to
develop their procedural fluency
(NRC 2001). According to the NRC,
students who are procedurally fluent have a “knowledge of procedures,
knowledge of when and how to use
them appropriately, and skill in performing them flexibly, accurately, and
efficiently” (NRC 2001, p. 121). In
other words, when students are given
multiple comparisons at once, they
need the skills to be able to distinguish between which strategy best
fits the given situation. For example,
consider the problem in figure 2.
In this problem, students had to
coordinate five fractions at the same
time. Many started by finding the
smallest fraction first. Because 3/20
Vol. 19, No. 6, February 2014
●
and 9/20 were the only fractions less
than half, students saw that the 3/20
was smaller because both fractions
had the same denominators. One
student stated, “They’re both out of
the same portion.”
When comparing 7/8 and 23/24,
students used missing pieces to determine that 23/24 is greater.
There are more pieces in the first one
[23/24]. So, therefore, if the pizzas are
the same size, those pieces are going to
be smaller. And then since they’re each
missing 1 piece, the piece is the smallest one leftover, so it has the most.
Then when comparing 7/8 and
7/13, students saw that the fractions
had the same numerator, thus the
pieces were larger in 7/8 of a pizza,
making 7/8 greater.
Seven is closer to 8 than 7 to 13. And
13 would have more slices, and they’d
be smaller.
Presenting an ordering situation
where students had to coordinate
more than two fractions at the same
time allowed them to think about
multiple strategies at once. This
type of situation would have been
tedious if students only knew how to
find common denominators. With
multiple reasoning strategies at their
disposal, students were able to solve
the problem efficiently without the
use of an algorithm.
Address Incorrect
Student Thinking
Although students develop the reasoning strategies described, many errors may be prevalent in their thinking
as well. A common error in students’
fraction thinking includes reverting
to whole-number ideas (Behr et al.
1984). Using whole numbers in fraction problems can provide a way for
incorrect student thinking to come to
Mathematics Teaching in the Middle School
379
the forefront of classroom conversations. These misconceptions can then
be addressed and used as a foundation
for developing correct reasoning.
Take a few moments to solve the
following problem before reading on:
The 22 people at the party sat in the
party section of the restaurant, which
holds 42 people. At the same time,
16 people were in the nonparty section, which holds 36 people. Which
section was closer to capacity?
Fig. 3 Looking at the number of filled seats resulted in a correct answer for an
incorrect reason.
Fig. 4 Examining the number of empty seats using an illustration (a) and computation
(b) resulted in an incorrect answer for the question asked.
The intent of this problem was
to have students compare 22/42 and
16/36 to a benchmark of 1/2 to determine that the party section is closer to
capacity because it is more than 1/2
full. This problem was written without fractions to determine if students
had incorrect conceptions of fractions
based on whole numbers.
Students used three whole-number
methods to solve the problem. The
first method looked at the number of
people in each section. When only
looking at the number of filled seats,
students determined that the party
section was closer to capacity, as seen
in figure 3. This would be analogous
to students comparing fractions by
only looking at the numerator. Some
students wrote, “Since 22 > 16, 22/42
is larger.”
The second method compared
the number of seats in each section,
or the denominators. Students using this reasoning determined that
the nonparty section was closer to
capacity because the size of each of
the 36 pieces is larger than the size
of each of the 42 pieces. One student
explained that “36 pieces are larger
than 42 pieces, making 16/36 larger.”
Although students understood the
idea that the larger the denominator,
the smaller the piece, they only looked
at how large each piece was without
coordinating this number with the
number of filled seats.
380
(a)
(b)
The third method found the number of empty seats in each section (see
fig. 4). Students using this reasoning
concluded that both sections were
equal. By using a picture (see fig. 4a),
or by setting up the fractions 16/36
and 22/42 and subtracting the numerator and denominator (see fig. 4b),
students found that both sections had
20 empty seats. From here, they concluded that both sections were equal.
One student stated:
. . . when I subtracted, I know
in fractions they both have
20 seats leftover. So wouldn’t
they be equal?
Mathematics Teaching in the Middle School
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Vol. 19, No. 6, February 2014
Posing problems with whole numbers
instead of fractions can provide insight
into the types of understandings
students have with respect to fractions.
This situation brought out several misconceptions that included only comparing numerators, only comparing
denominators, or finding how many
pieces were missing in each. If students
were instead only given a procedure for
how to solve the problem, these misunderstandings may not have surfaced.
A productive disposition
toward mathematics
According to the NRC, students’
development of strategic competence,
adaptive expertise, conceptual understanding, and procedural fluency
will inherently lead them to develop a
productive disposition toward mathematics (2001). Supporting students’
understanding by providing contexts,
asking them to explain and justify, and
connecting and using multiple strategies will give them the opportunity to
develop and gain a deeper understanding of how to reason when comparing
fractions. In addition, they can engage
in mathematics in ways similar to
those emphasized in the Common
Core Standards for Mathematical
Practice. Supporting ways for students
to develop their own reasoning strategies can provide them with a foundation for thinking about the numbers,
not just calculating them.
References
Behr, Merlyn J., Ipke Wachsmuth,
Thomas R. Post, and Richard Lesh.
1984. “Order and Equivalence of Rational Numbers: A Clinical Teaching
Experiment.” Journal for Research in
Mathematics Education 15 (5): 323–41.
doi:http://dx.doi.org/10.2307/748423
Common Core State Standards Initiative
(CCSSI). 2010. Common Core State
Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the
Council of Chief State School Officers.
http://www.corestandards.org/assets/
CCSSI_Math%20Standards.pdf
Kamii, Constance, and Ann Dominick.
1998. “The Harmful Effects of Algorithms in Grades 1–4.” In The Teaching
and Learning of Algorithms in School
Mathematics, 1998 Yearbook of the
National Council of Teachers of Mathematics (NCTM), edited by Lorna J.
Morrow and Margaret J. Kenney,
pp. 130-40. Reston, VA: NCTM.
National Research Council (NRC). 2001.
Adding It Up: Helping Children Learn
Mathematics, edited by Jeremy
Kilpatrick, Jane Swafford, and
Bradford Findell. Washington, DC:
National Academies Press.
Tobias, Jennifer M. 2009. “Preservice
Elementary Teachers’’ Development
of Rational Number Understanding
through the Social Perspective and the
Relationship among Social and Individual Environments.” PhD diss. Orlando,
FL: University of Central Florida.
Any thoughts on this article? Send an
e-mail to [email protected].—Ed.
Jennifer M. Tobias,
[email protected], is an
assistant professor of
mathematics education at
Illinois State University in
Normal. She is interested in how students
develop an understanding of fraction concepts and operations and the preparation
of teachers.
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