RELATIONSHIPS BETWEEN
FRACTIONAL KNOWLEDGE AND
ALGEBRAIC REASONING: THE CASE
OF WILLA
Mi Yeon Lee & Amy J. Hackenberg
International Journal of Science and
Mathematics Education
ISSN 1571-0068
Volume 12
Number 4
Int J of Sci and Math Educ (2014)
12:975-1000
DOI 10.1007/s10763-013-9442-8
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MI YEON LEE and AMY J. HACKENBERG
RELATIONSHIPS BETWEEN FRACTIONAL KNOWLEDGE AND
ALGEBRAIC REASONING: THE CASE OF WILLA
Received: 5 September 2011; Accepted: 5 June 2013
ABSTRACT. To investigate relationships between students’ quantitative reasoning with
fractions and their algebraic reasoning, a clinical interview study was conducted with 18
middle and high school students. The students were interviewed twice, once to explore
their quantitative reasoning with fractions and once to explore their solutions of problems
that required explicit use of unknowns to write equations. As a part of the larger study, the
first author conducted a case study of a seventh grade student, Willa. Willa’s fractional
knowledge—specifically her reversible iterative fraction scheme and use of fractions as
multipliers—influenced how she wrote equations to represent multiplicative relationships
between two unknown quantities. The finding indicates that implicit use of powerful
fractional knowledge can lead to more explicit use of structures and relationships in
algebraic situations. Curricular and instructional implications are explored.
KEY WORDS: algebraic reasoning, fractional knowledge, fractions as multipliers,
reciprocal reasoning, reversible iterative fraction scheme
According to the National Mathematics Advisory Panel (2008), fractional
knowledge is critical for learning algebra: “By the nature of algebra, the
most important foundational skill is proficiency with fractions. The
teaching of fractions must be acknowledged as critically important and
improved before an increase in student achievement in Algebra can be
expected” (p. 1-xiv). However, research on exactly how students’
fractional knowledge and algebraic reasoning are related has been
relatively undeveloped (Lamon, 2007) compared to research on
relationships between students’ whole number knowledge and algebraic
reasoning (e.g. Kaput, Carraher & Blanton, 2008). Although some research
has been conducted (e.g. Ellis, 2007; Empson, Levi & Carpenter, 2011;
Steffe, 2001), much is unknown about precisely how fractional knowledge is
critical in learning algebra.
To address this issue, we conducted an interview study on the
relationships between middle and high school students’ quantitative
reasoning with fractions and their algebraic reasoning in the domain of
writing and solving linear equations. As a part of the larger study, the
authors conducted a case study of Willa, who was one of the seventh
grade students. The purpose of this paper is to investigate how Willa’s
International Journal of Science and Mathematics Education (2014) 12: 975Y1000
# National Science Council, Taiwan 2013
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fractional knowledge was related to her writing and solving of linear
equations. The research questions for the case study are:
1. How does Willa reason with fractions as quantities?
2. How does Willa develop equations to solve algebra problems that
involve unknowns?
3. What are the relationships between Willa’s quantitative reasoning with
fractions and her algebraic reasoning in the domain of writing and
solving equations?
PERSPECTIVES
ON
RELATIONSHIPS BETWEEN FRACTIONAL KNOWLEDGE
AND ALGEBRAIC REASONING
The aim of this section is to indicate how we view fractional knowledge
and algebraic reasoning, as well as how we approached the study based
on prior research on relationships between students’ fractional knowledge
and algebraic reasoning. We view students’ fractional knowledge in terms
of students’ operations and schemes, which are discussed in the
conceptual framework of the paper. Following Kaput (2008) and other
early algebra researchers (e.g. Carraher, Schliemann & Scwartz, 2008;
Russell, Schifter & Bastable, 2011; Smith & Thompson, 2008), we view
early algebraic reasoning to be about generalizing and abstracting
arithmetical and quantitative relationships and systematically representing
those generalizations and abstractions in some way, but not necessarily
with standard algebraic notation. As students make progress, they learn to
reason with standard algebraic notation in lieu of reasoning with numbers
and quantities. This perspective can be motivating for students because
use of algebraic symbols is based on experience with meaningful referents
and relationships—specifically, quantitative relationships (Kaput, 2008;
Smith & Thompson, 2008).
Relational Thinking
The research of Empson et al. (2011) is based on this perspective. They
propose that children’s strategies for solving problems involving fractions
are based on fundamental properties of operations and equality, which
form the foundations of algebra. For them, relational thinking refers to
children’s explicit or implicit use of the fundamental properties of
operations and equality (e.g. the distributive property) as children reason
through problems involving fractions (as well as whole numbers). According
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to the authors, “relational thinking is a critical precursor—perhaps the most
critical—to learning algebra with understanding, because if children
understand arithmetic that they learn, then they are better prepared to solve
problems and generate new ideas in the domain of algebra” (p. 426).
To construct a relational understanding of fractions, Empson et al.
(2011) state that students need to understand fractions as quantities, and
they emphasize that Equal Sharing Problems are useful. For example, in a
quest to share two identical candy bars equally among five students, a
student could decide to share each candy bar equally among five students.
This student’s idea can be expressed in notation as follows: 1/5 × 2 = 1/5
(1 + 1) = 1/5 × 1 + 1/5 × 1 = 1/5 + 1/5 = 2/5. So, the student’s strategy
could implicitly involve the distributive property. Empson et al. propose
that helping students develop their implicit use of fundamental properties
of operations and equality in problem solving will support students in
explicitly using standard algebraic notation to represent generalizations of
their relational thinking. That is, the perspective of Empson et al. is that
regular implicit use of algebraic ideas in quantitative situations can lead to
explicit use and awareness of these ideas.
Linear Co-variational Reasoning
Other researchers also point out the importance of students developing
awareness of patterns and structures in their ways of thinking with fractions
(e.g. Ellis, 2007; Steffe, 2001). For example, in Ellis’s study, seven middle
school students in a teaching experiment reasoned about two quantitative
situations, one involving gear ratios and the other involving speed. These
students created emergent ratios as quantities, which meant that they used two
quantities that were directly measured, such as distance and time, to create a
third, intensive quantity, such as speed (Schwartz, 1988). For example, if a
character walking at a constant speed went 15 cm in 12 s, students reasoned
that for every 5 cm he walked, 4 s would elapse, and ultimately that 1 cm
corresponded to 4/5 s. Students used these emergent ratios to make explicit
statements about slopes of lines in cycles of generalizing and justifying
activity. Ellis’s research indicates that creating equivalent ratios with
fractional quantities is required for constructing ideas of slope.
CONCEPTUAL FRAMEWORK
Based on these ideas from the literature, we used a quantitative approach
in our study. In this section, we present this approach and other key
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constructs in our framework for the study: operations and schemes, as
well as multipliers on unknowns and reciprocal reasoning.
A Quantitative Approach
A quantitative approach places emphasis on operating with quantities and
their relationships (Thompson, 1994, 1995). Thompson (1994) defined a
quantity as a people’s conception of the measurable attribute of an object
or phenomena, along with a unit, and a process for assigning a numerical
value to the attribute. For example, weight is a measurable attribute of
people’s concepts of a bag of rice; together with a unit such as a kilogram
and a scale, one can develop a process for assigning a numerical value to
the weight of a bag of rice. In our study, we aimed to approach fractions
as lengths, where a fractional length was always considered in relation to
a length that was identified as the unit. Lengths could also represent other
quantities such as weight or amount of money. In our study, we also
considered unknowns as quantities that have yet to be measured.
Quantitative reasoning can promote developing images of quantities
and relationships between them. Broadly, an image is produced by a regeneration of a prior sensory experience such as sound, motor, or visual
image—i.e. an image is a result of a re-presentation (von Glasersfeld,
1995). von Glasersfeld defined re-presentation as “a mental act that brings
a prior experience to an individual’s consciousness” (p. 95). Thus, to
reconstruct experience, it is important to recollect the figurative materials
that comprised the experience. Furthermore, to use the reconstructed
experience productively in another situation, it is necessary to abstract
features of those materials across situations. In keeping with these ideas,
Thompson & Saldanha (2003) maintained that quantitative reasoning
entails “the development of operative imagery—the ability to envision the
result of acting prior to acting” (p. 29). In our study the words “drawing”
and “picture” refer to pictorial representations of quantities, and we were
interested in understanding the operative imagery of our participants.
Operations and Schemes for Fractional Knowledge
Along with a quantitative approach, in this study we viewed fractional
knowledge as consisting of operations and schemes. An operation is a
mental action. Schemes are composed of operations, and they are a way
of describing people’s goal-directed behavior. A scheme has three parts:
an assimilated situation, an activity that involves implementation of
operations, and a result, which a person assimilates to his or her
expectations (von Glasersfeld, 1995).
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From our perspective, to imagine dividing a length into equal parts
requires engaging in some operations. Critical operations for fractional
knowledge include partitioning, iterating, and disembedding. Partitioning
involves dividing a whole into some number of equal parts (Lamon,
1996; Steffe, 1991), while iterating refers to repeating a unit or a part of a
unit in order to make a larger amount (Steffe, 1991). For example,
iterating 1/3 of a sandwich three times will produce a partitioned
sandwich equal to the initial sandwich. Disembedding involves taking a
part out of a whole unit without mentally destroying the whole (Steffe &
Olive, 2010). That is, the part and the whole are distinguished as separate,
yet related. For example, if a student with a disembedding operation ate 1/3
of the above sandwich, the student would still think of the remaining two
parts as thirds because the student would keep the whole, 3/3, mentally intact
despite the fact it was no longer all physically present.
The Splitting Operation. Steffe (2002) has also found that the splitting
operation is critical for the construction of fractional knowledge. The
splitting operation is more advanced than just partitioning or iterating
alone; it is a composition, or unification, of partitioning and iterating
(Hackenberg, 2010; Norton, 2008; Steffe & Olive, 2010). For example,
consider this problem, which is a basic task to assess whether a student
has constructed a splitting operation: “The given bar (a rectangle) is five
times the length of another bar; draw the other bar.” To solve this
problem, students need to envision the other candy bar as independent
from the given bar but related to it. The other candy bar can be iterated
five times to make the given candy bar, and simultaneously the other
candy bar is made out of partitioning the given bar into five equal parts.
Solving the problem in this way means students think of partitioning and
iterating as unified.
The Iterative Fraction Scheme. In particular, splitting is necessary for the
construction of an advanced fraction scheme, an iterative fraction scheme
(IFS) (Steffe, 2002). A situation of an IFS is a request to create a fraction
from a given unit—for example, to make another bar that is eight-fifths of
a given bar. The activity of the IFS involves partitioning the given bar
into equal parts, disembedding one of the parts, and iterating that part
some number of times. For example, to make eight-fifths of a given bar
students with an IFS would partition the bar into five equal parts and
iterate one part eight times. Students who have constructed an IFS regard
the result as eight times one-fifth, as well as one whole (five-fifths) and
three more fifths. More generally, students who have constructed an IFS
regard any fraction (proper or improper) as a multiple of a unit fraction
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(Steffe & Olive, 2010), and they also maintain the relationship between
an improper fraction and the whole.
Students who have constructed an IFS can learn to reverse their
scheme. For example, these students can start with a fractional amount,
such as 7/5 of a candy bar (an unmarked rectangle), and make the whole
bar. To do so they typically partition the 7/5-bar into seven equal parts
and then iterate one of those parts five times. These students can operate
in this way because they think of 7/5 as seven times what they need to
iterate to make the whole (1/5). Students can use their splitting operation
to split 7/5 into seven equal parts, creating a unit fractional amount of the
whole bar (Hackenberg, 2010). Solving such problems means that a
student has constructed a reversible iterative fraction scheme (RIFS).
In addition, researchers have found that constructing an IFS requires a
particular multiplicative concept, the third multiplicative concept
(Hackenberg, 2007; Steffe & Olive, 2010). Students’ multiplicative concepts
are based on how they produce and coordinate composite units (Steffe,
1992). Students who have constructed the third multiplicative concept (MC3
students) can coordinate three levels of units prior to operating and flexibly
switch between different three levels of units structures (Hackenberg, 2010).
For example, prior to acting in a situation, MC3 students can view a number
like 15 as a unit of three units each containing five units, a three-levels-ofunits structure, and they can switch to viewing 15 as a unit of five units each
containing three units. This kind of coordination is important for conceiving
of improper fractions as described above because eight-fifths is a composite
unit of eight units (1/5 s), in which any one of those 1/5-units indicates a
composite unit of five units (5/5, the whole to which 8/5 refers and from
which it is also independent).
Whole Numbers and Fractions as Multipliers on Unknowns
Hackenberg (2010) has suggested that coordinating three levels of units prior
to operating is necessary but not sufficient for conceiving of fractions as
multiplicative operators, or multipliers, on known quantities. In this study, we
considered how both whole numbers and fractions may be used as multipliers
on unknowns in equation writing. For example, consider this problem:
Cord Problem. Stephan has a cord that is some number of feet long and his cord is five
times the length of Rebecca’s cord. Draw a picture of the situation. Then write an equation
for the situation. Can you write another, different equation for the situation?
By using the multiplicative relationship between the two unknowns,
students can write equations such as S = 5R or R = (1/5)S, in which S
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represents the length of Stephan’s cord and R refers to the length of
Rebecca’s cord. In this example, students who represent “five times the
length of Rebecca’s cord” as “5R” may have developed a concept of a
whole number as a multiplier on an unknown. Similarly, students who
represent “one-fifth the length of Stephan’s cord” as “(1/5)S” may have
developed a concept of a fraction as a multiplier on an unknown.
Researchers have analyzed fractions as multipliers in terms of scaling
(Behr, Harel, Post & Lesh, 1993; Kieren, 1995).1 For example, to conceive
of 3 as a multiplicative operator on 5 ft, each foot could be exchanged for
3 times that amount, to produce 15 ft. So each foot could be “scaled” by
three. However, this scaling could be applied to any partition of the 5 ft,
such as each 1/2-ft, or each 1/3-ft, etc. So, although the result of the
scaling is a quantity of 15 ft, the nature of the transformation between
quantities is complex.
The Cord Problem can certainly be interpreted in terms of scale
factors: The length of Rebecca’s cord can be scaled up by a factor of 5 to
produce the length of Stephan’s cord. Yet, this concept does not seem
necessary to start work on the problem because one can interpret the
relationship between the quantities in terms of repeated lengths (i.e. five
repeated lengths, each of which represent Rebecca’s cord length, will
make Stephan’s cord length). However, this idea of repeating lengths is
not sufficient for conceiving of what to multiply Stephan’s length by to
make Rebecca’s length. Hackenberg (2010) has proposed that abstracting
a fraction like 1/5 as a concept is necessary to conceive of 1/5 as a
multiplier on a known. In this interpretation, a student’s concept of 1/5
contains all the ways of making it (e.g. partition a whole into five equal
parts and disembed one part) and unmaking it (e.g. iterate 1/5 five times
to produce the whole). If, for the student, one-fifth of a whole amount
includes the idea of taking that part five times to produce the whole, then
the whole multiplied by one-fifth must produce one-fifth of the whole.
Conceiving of fractions as multipliers on knowns and unknowns is
important for reciprocal reasoning (Hackenberg, 2010), which is in turn is
important for writing and solving linear equations (Driscoll, 1999).
Reciprocal reasoning involves flexibly switching between taking either of
two quantities as the referent for “measuring” the other (Hackenberg,
2010). For example, students who can reason reciprocally know that if the
quantity A is two-fifths times quantity B, then the quantity B is fivehalves of quantity A. In this study, we aimed to further develop this
interpretation by investigating Willa’s use of both whole numbers and
fractions as multipliers on unknowns and by exploring her reciprocal
reasoning in the domain of solving linear equations.
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METHODS
To investigate students’ fractional knowledge and algebraic reasoning
in the domain of writing and solving equations, we conducted a taskbased clinical interview study. Clinical interviews are a way to collect
and analyze data to reveal students’ hidden thinking process (Clement,
2000). Clinical interviews can draw out students’ naturalistic thinking
and provide evidence for building models of students’ understanding.
Within the larger study, the first author conducted a case study of two
students, one of whom was Willa. A case study is an analysis of people
or events that are studied holistically (Thomas, 2011). Outlier cases that
are extreme or atypical tend to be selected as subjects more often than
representative cases because outliers can produce novel insights
(Thomas, 2011; Yin, 2009). Case study methods usually involve an
in-depth, longitudinal exploration of a single individual, a collective
(e.g. classroom) or an event.
Participants and Procedures in the Larger Study
Students were invited to participate in the larger study after the researchers
observed six middle school mathematics classrooms and conducted 20-min
unrecorded selection interviews with 24 students. In these interviews, the
researchers used tasks to assess students’ whole number multiplicative
concepts. Seventeen students from the seventh and eighth grades at a middle
school and one tenth grade student from a high school2 were selected for the
large study. They represented a range of three distinct multiplicative
concepts (Hackenberg & Tillema, 2009; Steffe, 1992) and six students were
selected with each of these concepts.
Each student participated in two 45-min, semi-structured, video-recorded
interviews, a fractions interview and an algebra interview. All students
completed the fractions interview prior to the algebra interview. To
triangulate conclusions about students’ fractional knowledge, students also
completed a written fractions assessment (Norton & Wilkins, 2009) that we
present analysis of elsewhere (Hackenberg & Lee, under review).
At each interview, one researcher was an interviewer and the other was
a witness-researcher who observed and occasionally suggested a question
that the first author, in the midst of interaction, may not have considered
(cf. Steffe & Thompson, 2000). The interviewers aimed to elicit students’
ideas and to capitalize on them in order to encourage the students to
demonstrate their most advanced ways of thinking. The interviewers
followed a protocol, which involved regularly asking students to draw
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pictures of quantitative situations, to solve problems and write equations,
and to describe their solutions and equations based on their pictures.
However, the interviewers were also free to depart from the protocol to
investigate interesting or unusual student responses. Two digital video
cameras were used to video-record each interview: One camera recorded
interaction between the interviewer and student, and the other recorded
the student’s written work. These videos were electronically mixed into
one file for analysis.
The interview protocols had been refined in a prior pilot study
(Hackenberg, 2009). The protocols were related in that reasoning with
fractions in the fractions interview could be drawn upon for solving
problems in the algebra interview (Appendixes 1 and 2). In the fractions
interview, unknowns could remain implicit in a student’s thinking. In the
algebra interview, students were asked to explicitly reason with
unknowns and to use algebraic notation to represent situations. For
example, in the fractions interview students were asked to draw a picture
of the situation and figure out the amount of money involved when Tanya
had $84, which was 4/7 of David’s money. In the algebra interview,
students were asked to draw a picture of and write equations for a similar
situation in which both quantities were unknown.
A Case Study of Willa
Willa3 was a student who had constructed the third multiplicative
concept (MC3). She was taking an advanced seventh grade
mathematics class. When we observed her class, we saw that she
used a standard algebra textbook and that classroom activities focused
on manipulations of standard algebraic notation. The first author
interviewed Willa, and the second author was a witness-researcher.
During the interviews, Willa showed some characteristics that were
different from the other MC3 students in the larger study. Like other
MC3 students, she had constructed powerful fraction schemes and
operations. Yet, unlike these other students, she sometimes appeared
to operate in ways that were less powerful than her potential. Thus,
we chose to investigate Willa’s fractional and algebraic knowledge in
a case study in order to illuminate possible reasons for the differences
that she demonstrated in relation to her peers. Although the case
study is based on just 2-hour-long interviews, these interviews
revealed detailed snapshots of her thinking. Based on these
interviews, we do not claim to know everything about Willa’s
fractional knowledge and algebraic reasoning—we claim to have
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learned a small but important subset of her ideas. In particular, we
sought to assess whether Willa had constructed an IFS and RIFS, to
understand how Willa wrote and solved basic linear expressions and
equations, and to generate a hypothesis about the relationship
between the two.
Data Analysis
Analysis occurred in two phases. In the first phase the authors built a model
of Willa’s fraction operations, schemes, and equation writing and solving. To
do this, the researchers repeatedly viewed Willa’s video files and created
detailed analytical notes, which included making transcriptions, writing
summaries of Willa’s work on each problem, and recording interpretations of
and conjectures about Willa’s reasoning via memos (Corbin & Strauss,
2008, p. 117). In order to write the memos, the researchers developed
probing questions about Willa’s data and made theoretical comparisons to
models from prior research (e.g. Hackenberg, 2007, 2010; Steffe, 2002).
After the researchers had developed the detailed analytical notes, the first
author wrote a narrative summary of the model of Willa’s fractional
knowledge and algebraic reasoning. The second author read the summary
and the authors discussed it, viewing segments of video to reconcile any
differences in interpretations and refine conjectures.
In the second phase of analysis, the researchers identified Willa’s
RIFS and the use of whole numbers and fractions as multipliers on
unknowns, as well as reciprocal reasoning with unknowns, to articulate a
relationship between Willa’s fractional knowledge and algebraic
knowledge. The first author generated a written synthesis of Willa’s
fractional knowledge and algebraic reasoning, which was developed from
the narrative summary. Then, the first author discussed this synthesis
with the second author to determine its trustworthiness. Based on this
synthesis, the researchers developed a written account of the relationship
between Willa’s fractional knowledge and algebraic knowledge in the
domain of writing and solving equations.
ANALYSIS
AND
FINDINGS
In this section, we first describe Willa’s fractional knowledge. In analyzing
Willa’s fractional knowledge, we focused whether she had constructed a
splitting operation, an IFS, and an RIFS. Then, we describe her algebraic
reasoning, focusing on the extent to which she had constructed whole
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numbers and fractions as multipliers on unknowns and whether she engaged
in reciprocal reasoning.
Willa’s Fractional Knowledge
Splitting Operation. At the time of the study, Willa had constructed a
splitting operation. The evidence for this conclusion is her solution to the
following task:
String Problem. My string is 7 times longer than yours. Can you draw your string?
To solve this problem, Willa partitioned the given string (a length) into
seven fairly equal parts and drew out one of those parts below the given
string. She stated that her piece was one-seventh because her piece was
one of seven equal parts.
Willa used her splitting operation to split composite units, as shown by
her response to the second interview problem:
Sara-Roberto Stack Problem. Sara has a stack of CDs that is 65 cm tall. That’s 5 times the
height of Roberto’s stack. Draw a picture of this situation. How tall is Roberto’s stack?
To solve this problem, Willa first divided 65 by 5 to get 13. Then she
drew a long rectangle and partitioned it into five equal parts to represent
Sara’s stack. She drew another rectangle that was the same size as one of
the five parts to represent Roberto’s stack. When asked how she got the
picture, she said, “Well I don’t know really how I draw it.” Since Willa
had drawn an accurate picture, we infer that she had some images for the
quantitative situation. However, she might not have been aware of her
imagery: Stating that she did not know how to draw the picture may
indicate that she was not aware of how the drawing and numerical results
were related, or that she was not accustomed to articulating her imagery.
Iterative Fraction Scheme. We can also attribute an IFS to Willa. When
asked to draw a separate candy bar that was 9/7 of a given bar (a
rectangle), Willa first partitioned the given bar into seven equal parts.
Then she drew a new bar that was longer than the original bar. The first
author asked her to show all the parts in the new bar, and she partitioned
it into nine equal parts, where the first seven parts aligned with the seven
parts in the given bar. The first author colored one piece of her new bar
and asked how much that piece was. Willa said, “one-seventh—wait. I
think it’s one-ninth. Hold on. I think one-seventh or one-ninth. But I don’t
know which one because it was one-seventh.” The first author colored
another piece of the original bar and asked if those two colored pieces
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were the same. Willa replied, “This [a piece from the new bar] is oneseventh on the top [original] bar and this one is one-ninth on the bottom
[new] bar.” When asked if it was possible for same piece to have different
names, she said, “Yes, the piece can be both one-seventh and one-ninth.”
This data excerpt shows that Willa had likely constructed an IFS but was
not immediately aware of the relationships between the various parts and
wholes. She may have become more aware through our interaction.
However, since this interview did not assess learning, we cannot make
that claim with certainty.
Reversible Iterative Fraction Scheme. In addition, we can conclude that
Willa could reverse her IFS. Willa was shown a candy bar (rectangle) and
told that this bar was 3/5 of another bar. To draw the other bar, Willa split
the given bar into three parts. Then, she drew a new bar the same length
as the original bar and added two more of these parts to make five equal
parts. The interviewer followed up by asking her to draw another candy
bar in the case that the given candy bar was 4/3 of another one. Willa
partitioned the given bar into four equal parts and then drew another bar
consisting of three of those parts. Her solutions to these problems
indicated that Willa had constructed an RIFS.
However, Willa did not immediately use her RIFS with composite
units. The interviewer posed the following problem to her:
Tanya-David Money Problem. Tanya has $84, which is 4/7 of David’s money. Can you
draw a picture of this situation? How much money does David have?
To solve this problem, Willa first drew a large rectangle and partitioned it
into seven equal parts to represent David’s money. Then she drew another
rectangle consisting of four of these parts to represent Tanya’s money
(Fig. 1). The first author asked her who had more money, and Willa said that
David had three-sevenths more than Tanya had.
When the first author asked her how much money David had, Willa
divided 84 by 7 to get 12. She initially said that each piece was $12 in the
bar representing Tanya’s money. However, she quickly said that was
wrong and then wrote $12 in each part on David’s bar. Here, she
appeared to calculate seven $12 s to determine David’s money, but she
knew that David did not have $84. When the first author asked if one of
Tanya’s pieces represented $12, Willa divided $84 by 4. She said, “Hold
on, there is $21 [in each of Tanya’s pieces]. So that equals $84 and then
he has three more pieces. David has to have more money because she has
just four-sevenths of his money.” After a short pause, Willa computed
$21 times 3, added $63 to the $84, and arrived at an answer of $147.
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Figure 1. Willa’s work on the Tanya-David Money Problem
In her solution to the Tanya-David Money Problem, Willa immediately
drew a picture of seven-sevenths to represent David’s money, and then
she identified four of those sevenths to represent Tanya’s money. Doing
so meant she did not use her RIFS. That is, she did not split four-sevenths
into four equal parts and iterate one of these parts seven times to make
seven-sevenths. Instead, Willa began with seven-sevenths and identified
four of those parts as Tanya’s money. Approaching the problem in this
way means her splitting operation and RIFS were not activated, even
though we assess that Willa had constructed this operation and scheme.
As a result, Willa had some difficulty determining David’s money. To
figure out how much David had, Willa, immediately divided $84 by 7.
This work is sensible since she began with seven-sevenths, and it is
further indication that she did not split four-sevenths. However, Willa
appeared to experience doubt about this solution. That doubt, in
conjunction with the interviewer’s questions, resulted in her dividing
$84 by 4 and viewing each of the four equal parts of Tanya’s money as
$21. Although it is likely Willa used her splitting operation at this point,
she did not iterate the result of splitting, $21, seven times to create
David’s money.
This situation may be partly explained by Willa’s imagery. Willa
demonstrated that she had constructed a splitting operation and RIFS and
so likely had, or was developing, operative images of splitting and of her
RIFS. However, when composite units were included in a problem, her
splitting operation and RIFS were not activated. One reason is that she
may not have had an operative image—an ability to envision the result of
acting on a quantity prior to acting (Thompson & Saldanha, 2003)—when
she had to keep track of several unit structures. To solve the Tanya-David
Money Problem with an RIFS, students need to conceive of the $84 as a
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unit of four units each containing some number of units (a three-levels-ofunits structure), and also see these four units as embedded in a unit of
seven units each containing some number of units (another three-levelsof-units structure). This kind of structure is complex to make, let alone
envision or anticipate (Hackenberg, 2010). So, when faced with that
complexity in the Tanya-David Money Problem, Willa may not have
known right away how to use or adapt her operative images. This aspect
of her ways of operating might also be used to explain why Willa was not
able to articulate how she arrived at the picture in the Sara-Roberto Stack
Problem, although she easily drew a picture to represent Sara’s stack
height and Roberto’s stack height.
Willa’s Algebraic Knowledge
Whole Numbers and Fractions as Multipliers on Unknowns. All
problems in the algebra interview included opportunities to use whole
numbers and fractions as multipliers of unknowns. For example:
Multiple Identical Candy Bars Problem. Here are 5 identical candy bars (rectangles) and
each candy bar weighs some number of ounces. If h represents the weight of one bar, how
much does 1/7 of all the candy weigh?
When Willa worked on this problem, she immediately wrote 5h
7 . When
asked to draw the situation, she was quiet for 15 s and said that she did not
know how to draw it. So the first author asked her to draw a modified
version of the problem involving smaller quantities: “There are three candy
bars. How much does 1/5 of all the candy bars weigh?” Willa said that she
could write an equation but did not know how to draw it. The second author
rephrased the question, asking her to show how to share the three candy bars
Figure 2. Willa’s work on the Multiple Identical Candy Bars Problem
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fairly among five people. Willa partitioned each of the three bars into five
equal parts, and she colored three parts out of the 15 parts (see Fig. 2). When
asked to write an expression for the weight, Willa wrote 3h
5 . The second author
asked again her to explain how she got the expression. Willa answered that
every person would get three-fifths of a candy bar and so it would be 35 h . The
3
first author asked whether the two expressions (3h
5 and 5 h ) were the same or
not. Willa said they were not because she thought the first one represented 3 times h
divided by 5 and the other indicated 3 divided by 5 times h.
Then, the second author asked Willa whether the two expressions
involved the same picture or different pictures, and Willa answered that they
were different. The second author asked her which expression was shown in
her picture. Willa indicated the 35 h. Then, the second author asked her to draw
the other expression, 3h
5 . She said, “Wait, that’s [the picture she drew] this one
3h
[ 5 ] because three candy bars are divided by five people. This [35 h] is three
divided by five h. Umm …
3 I don’t know. Maybe these are same. I don’t
know how to draw this one 5 h . I think this is the same. I am confused. They
might be the same. They are not the same equation, but I think they are the
same.” So, in working on this problem, Willa determined another expression,
, that was different from her initial expression, 3h
5 , and she was not sure
whether the two expressions were quantitatively identical.
By articulating that the two expressions came from different pictures
and thinking processes, Willa may have been indicating some initial
understanding that the two expressions demonstrated different parts of a
process of finding 1/5 of the weight of three identical bars (the first one is
the sum of the three bars’ weights divided by 5, while the second could be
seen as the result of taking one-fifth of each of the three bar’s weights to
yield three one-fifths of one bar’s weight). However, Willa was not
certain whether operating multiplicatively on an unknown with whole
numbers (e.g. multiplied by 3 and divided by 5) was equivalent to an
unknown multiplied by a fraction (3/5). That is, she seemed to understand
that the two expressions came from different thinking process, but she
seemed uncertain about whether the two expressions represented the same
quantity. Her response implies that she had not fully developed her
images of how algebraic expressions were related to the transformations
of quantities in the problem.
In addition, based on this data, we can say that it appeared more natural
to Willa to operate multiplicatively with whole numbers on unknowns,
rather than with fractions. In this problem, Willa’s main meaning for 3/5
appeared to be a quotient meaning, 3 ÷ 5, which is a perfectly fine—and
advanced—meaning. When she drew her picture (Fig. 2), she seemed to
view the shaded part as 3/5 of h, but that may have been primarily a part-
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whole meaning for her, even though she was capable of more complex
meanings (three-fifths as three times one-fifth). In sum, Willa seemed to
be creating the idea of using 3/5 as a multiplier on an unknown in the
process of solving this problem, since she began the problem by operating
multiplicatively on the unknown solely with whole numbers and because
she was not certain about the equivalence of her expressions. Thus, this
data provides initial evidence that fractions were not necessarily
multipliers of unknowns for Willa.
Reciprocal Reasoning. Further light is shed on Willa’s construction of
fractions as multipliers on unknowns by her work on the Sam-Theo CD
Problem.
Sam-Theo CD Problem. Theo has a stack of CDs some number of cm tall. Sam’s stack is
two-fifths of that height. Can you draw a picture of this situation? Can you write an
equation for how tall the height of Sam’s Stack is?
Willa drew a picture of a rectangle partitioned into five equal parts and
then another rectangle that spanned the first two of those parts (Fig. 3).
Then, she wrote an equation related to their relationship: 25 t = s, where t
represented the height of Theo’s stack height and s represented Sam’s
stack height. To investigate her reciprocal reasoning, the first author
asked her to write another equation. She first wrote 2 5t = s. The first
author made the question more specific by asking Willa to write an
equation for the height of Sam’s stack in terms of the height of Theo’s
stack. After sitting in silence for 15 s, Willa wrote 5s2 = t. When asked to
explain the equation with her picture, Willa responded that she got 10
pieces by taking Sam’s two-part stack height five times, and then she
divided by two in order to find Theo’s five-part stack height.
Figure 3. Willa’s work on the Sam-Theo CD Problem
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However, when asked to use her picture to find Theo’s stack
height in the case that Sam’s stack height was 10 cm, Willa pointed
to each part of Theo’s stack in the picture, quietly saying “5, 10, 15,
20, 25.” Then, she said that the Theo’s height was 25 cm. So, in the
process of finding the larger unknown quantity when the smaller
quantity was known, Willa took one-half of Sam’s stack five times.
This process was different from her formulation of the equation
of 5s2 = t, in which she first iterated Sam’s stack five times and then
divided by two.
Then, Willa used this pair of numbers (10 and 25 cm) to check her
equation by writing 5ð210Þ = t. She made a computational error, getting 20/
2 = t and thus, 10 = t. At first she did not realize her error, and so she tried
to revise the equation from 5s2 = t to 5t2 = s in order to adjust the equation to
her incorrect numerical result. When the first author asked her to review
her computation, she realized her mistake and corrected the result to t =
25. Then the second author asked her which equation of 5s2 = t and 5t2 = s
was right. Willa erased the equation 5t2 = s.
This excerpt provides further evidence that Willa did not always use
fractions as multipliers on unknowns. In this problem she used 2/5 as a
multiplier initially and without hesitation. However, she did not use 5/2 as
an operator to write the second equation. Instead, Willa used ideas about
multiplication and division of whole numbers as inverses of each other.
So, although she developed two correct equations of 25 t = s and 5s2 = t, her
work on this problem does not indicate that she viewed fractions like 5/2
as multipliers on unknown.
In addition, in the process of checking her equation of 25 t = s with s =10,
Willa first multiplied 2/5 on each side to produce the other equation. This
kind of error (not multiplying by the reciprocal) may stem from solving
an equation based on memorization of a procedure, rather than based on
an understanding of reciprocal relationships. Some of the MC3 students
in the larger study took Sam’s stack height partitioned into two equal
parts, as a unit. Then, they developed the idea that five of those halves of
Sam’s stack would produce Theo’s stack height. Alternately, other MC3
students took Sam’s stack as a unit and thought of Theo’s stack height as
two of Sam’s stack heights and one-half more. MC3 students who viewed
Theo’s stack height as two and a half times Sam’s stack height, under
questioning from the interviewer, produced 5/2 as a multiplier on Sam’s
stack height. In contrast, we did not see evidence that Willa was
reasoning in either of these ways, using five-halves or two and half as a
multiplier of an unknown, although she regularly used whole numbers
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and sometimes proper fractions as multipliers. Thus, we conclude that
Willa had not yet constructed reciprocal reasoning.
DISCUSSION
In this study, we described Willa’s quantitative reasoning with fractions
and algebraic reasoning in the domain of writing and solving equations.
In terms of Willa’s fractional knowledge, we assessed that she had
constructed a splitting operation, an IFS, and a RIFS. However, she did
not initially use her RIFS with composite units. That is, in the TanyaDavid Money Problem Willa did not split four-sevenths into four equal
parts and iterate one of those parts seven times, although she did
eventually use splitting to solve the problem. In addition, Willa did not
appear to have developed imagery for some of her reasoning with
fractions prior to working explicitly with quantitative relationships via
drawings.
When investigating Willa’s algebraic reasoning, we found that she used
whole numbers as multipliers of unknowns, but she did not always use
fractions as multipliers of unknowns. There were some indications that she
was constructing or had constructed fractions as multipliers, such as when
she wrote “35h” in the Multiple Identical Candy Bars Problem after discussion
with the researchers and when she used 2/5 as a multiplier on an unknown in
the Sam-Theo CD Problem. However, the evidence was not consistent
across all of her work: In that same problem, Willa did not appear to conceive
of 5/2 as a multiplier on an unknown. Instead, in order to generate a second
equation for that problem she operated multiplicatively on unknowns solely
with whole numbers, as she had initially done in the Multiple Candy Bars
Problem. Willa’s work on the Sam-Theo CD Problem allowed us to propose
that Willa had yet to construct reciprocal reasoning. We note that this lack of
reciprocal reasoning was somewhat unusual in that the other MC3 students
demonstrated reciprocal reasoning, although it was often elicited through
supporting questions in the course of the algebra interview (Hackenberg &
Lee, under review).
To explain Willa’s lack of reciprocal reasoning, we appeal to a
connection between Willa’s fractional knowledge and her algebraic
knowledge with regard to the use of her RIFS. This connection surfaced
between her solution of the Tanya-David Money Problem in the fractions
interview and her work on the Sam-Theo CD Problem during the algebra
interview. In the Tanya-David Money Problem, splitting four-sevenths
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into four parts (fourths) and iterating one of those parts (one-fourth of
the known quantity) seven times are the operations necessary to make
seven-fourths of the known quantity. That is, in using an RIFS in this
problem a student produces, at least implicitly, the result as sevenfourths of the known, ¼ of $84 taken 7 times is $147, so $147 is sevenfourths of $84. Students may not be aware of this structuring of the
result—they may not see that they have produced seven-fourths of the
known (Hackenberg, 2010). But the reciprocal fractional relationship
between the two quantities is implicit in this way of operating. Since
Willa did not use her RIFS in solving this problem, it is unlikely that
she would even implicitly consider the result as seven one-fourths of the
known quantity.
Willa’s solution to the Tanya-David Money Problem is relevant for
understanding how she reasoned with a fractional relationship between
two unknown quantities in the Sam-Theo CD Problem. In solving that
problem, Willa used two-fifths as a multiplier on an unknown and
clearly viewed Sam’s stack height as two one-fifths of Theo’s stack
height. However, she never used five-halves as an operator on an
unknown: She never explicitly viewed Theo’s stack height as five onehalves of Sam’s stack height. We propose that Willa not initially using
her RIFS to solve the Tanya-David Money Problem, and then not
being aware of the structuring of the result of the problem (seven onefourths of Tanya’s money is David’s money) had an impact on her
abilities to be more explicit about reciprocal relationships in the SamTheo CD Problem. That is, since we do not have evidence that Willa
experienced implicit reciprocal relationships in how she reasoned with
fractions as quantities in the first problem, it is not surprising that she
was not ever explicit about them in writing equations to represent
relationships in the second problem.
Implications
This analysis has three implications. First, this study corroborates findings
from prior research that students’ fractional knowledge is related to their
algebraic knowledge (e.g. Ellis, 2007; Empson et al., 2011; Steffe, 2001).
In this study, Willa did not use her RIFS to solve the Tanya-David Money
Problem, and so it was quite unlikely that she was aware of the reciprocal
structuring of quantities in that problem. We have argued that her not
experiencing implicit reciprocal relationships in reasoning with fractions
as quantities was linked to her lack of reciprocal reasoning in writing
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equations. In addition, Willa only sometimes used fractions as multipliers,
which impacted her expression and equation writing. So, Willa’s
fractional knowledge—specifically her RIFS and use of fractions as
multipliers—influenced how she wrote equations to represent
multiplicative relationships between two unknown quantities.
Second, Willa’s case study shows that students’ operations and
schemes require curricular tasks and activities to support the use of
them. Otherwise, such operations and schemes—even powerful
ones—are unlikely to develop and will unlikely to be “material” for
generalization in algebraic contexts. Although we assessed that Willa
had constructed an advanced fraction scheme, the RIFS, this scheme
was not necessarily activated for her by problems in which she might
use it. We surmise that her RIFS was probably not regularly activated
outside of our study, and so she may not have had many
opportunities to use this scheme to develop her fractional knowledge,
let alone to draw upon it when faced with algebraic problems where
it could be useful. Thus, we suggest that students like Willa can
benefit from experiencing instruction that regularly supports the use of
their fraction schemes. Teachers of students like Willa can help these
students by posing rich quantitative problems that help students use the
fraction schemes they have constructed, as well as to construct new schemes.
This implication directly supports the view of Empson et al. (2011) that
development of powerful fractional knowledge that involves implicit
mathematical relationships can lead to more explicit use of these
relationships in algebraic situations.
Third, the interviews in this study demonstrate one way to elicit students’
fractional knowledge so that it can be a basis for building algebraic knowledge
in writing algebraic expressions and equations. In our interviews, we proposed
that students consider fractions and unknowns as quantities, and we provided
tasks in which students were asked to draw pictures in order to help them represent their constructed schemes and operations. In addition, we encouraged
our students to reflect on the quantitative meaning of their reasoning and
algebraic notation by using probing questions. Regularly encouraging the
use of imagery and supporting questions in order to help students think
quantitatively in working on fractions problems such as Tanya-David Money
Problem has the potential to allow students to create abstractions and
generalizations of quantitative structure when students solve algebraic
problems such as Sam-Theo CD Problem. This is because, as Thompson
(1995) mentioned, students who have operative images when dealing with
quantities have a higher probability of using their constructed operations and
schemes in advanced problem situations.
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APPENDIX 1: FRACTION INTERVIEW PROTOCOL
F1. The drawing (a line segment) below shows my piece of string. Think
of your piece of string so that mine is five times longer than yours.
Can you draw what you’re thinking of? Can you show for sure that
mine is 5 times longer than yours?
F2. Sara has a stack of CDs that is 65 cm tall. That’s 5 times the height of
Roberto’s stack. Draw a picture of this situation. How tall is
Roberto’s stack of CDs? How did you solve this problem?
F3. The drawing (a rectangle) is a picture of a candy bar. Draw a separate
candy bar that is 9/7 of that bar. If the student completes this
problem, ask her or him to shade one piece and tell how much it is of
the original bar. (After the interviewer shading one piece from the
student’s picture) “How much is that piece? How much of the top bar
is it? How much of the bottom bar is it? Is it possible for the same
piece to have different names?
F4. This candy bar (a rectangle) is 3/5 the size of another candy bar.
Make a separate drawing of the other candy bar. How did you make
your drawing? What is the fraction name of another bar? Which bar
represents the unit?
a. If difficult, try: This candy bar (a rectangle) is 1/5 of another
candy bar. Make a separate drawing of the other candy bar.
b. If easy, try: This candy bar (a rectangle) is 4/3 of another candy
bar. Make a separate drawing of the other candy bar.
F5. Tanya has $84, which is 4/7 the size of David’s amount of money.
Draw a picture of this situation. How much does David have?
a. If difficult, try: Tanya has $15, which is 1/5 the size of David’s amount
of money. Draw a picture of this situation. How much does David have?
b. If very easy, try: Cassie earned $48 babysitting. That’s 4/3 the
amount of money Serena earned. Draw a picture of this situation.
How much money did Serena earn?
APPENDIX 2: ALGEBRA INTERVIEW PROTOCOL
A1. Do you have a cord with earplugs for listening to music? How long
do you think it is? It’s not a value that we know exactly, right? But
we could measure it to find the exact value. Okay, so Stephen has a
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cord for his iPod that is some number of feet long. His cord is five
times the length of Rebecca’s cord.
a. Could you draw a picture of this situation? Describe to me what
your picture represents.
b. Can you write an equation for this situation? Elicit what the letters
represent to them. Can you tell me in words what your equation
means?
c. As necessary. Can you check your equation with your picture?
d. As necessary. Check your equation using this question: Who has a
longer cord, Stephen or Rebecca?
e. Can you write more than one equation? As necessary (if they have
only written something like t = 5*q, where t represents Stephen’s
cord length and q represents Rebecca’s cord length): Can you
write an equation to express Rebecca’s cord length in terms of
Stephen’s?
f. As necessary (if they have written something like t = q÷5): Can
you write this equation using multiplication?
g. Let’s say Stephen’s cord is 15 ft long. Explain how to find the
length of Rebecca’s cord.
A2. There are 5 identical candy bars (show picture) and each candy bar
weighs some number of ounces. Let’s say that h = the weight of
one bar. How much does 1/7 of all the candy weigh?
a. If this question is hard, start with 2 or 3 bars and ask about 1/3 or 1/5.
b. If still hard, use sharing language to find out about whether the
student can make fair shares.
c. Could you draw a picture of 1/7 of all the candy?
d. Can you write down an expression for the weight of 1/7 of all the
candy?
e. If part (d) is hard, then drop back to notating quantitative situation
of sharing 5 bars fairly among 7 people.
f. If okay, move onto how much 3/7 of all the candy weighs. What
would a picture of that look like? Can you write down an
expression for the weight?
g. To test out improper fractions, could try asking for 9/7 of all the candy.
A3. Theo has a stack of CDs some number of cm tall. Sam’s stack is
two-fifths of that height.
a. Draw a picture to represent this situation.
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b. Can you write an expression for how tall the height of Sam’s
stack is? Elicit what the letter(s) represent(s) to them. Can you tell
me in words what your expression(s) mean(s)?
c. Can you write an equation based on your expression in (b)? Can
you tell me in words what your equation means?
d. Can you write another equation for the situation? Can you tell me
in words what your equation means?
e. (If students write the second equation using inverse of 2/5), Can
we switch the numbers like that? What relationship do you see in
these two equations?
f. Ask them to test their equations with particular numbers, such as
10 cm as the height of Theo’s stack.
g. If this is hard, start with: Sam’s stack is 1/5 of that height.
h. If this is easy, start with: Michael’s stack of CDs is 7/5 of the
height of Theo’s.
A4. Christina earned some money babysitting. That’s 4/3 of what Serena
earned.
a. Draw a picture to represent this situation.
b. Can you write an equation that relates the amount of money
Christina earned to the amount of money Serena earned? Can you
tell me in words what your equation means?
c. Can you write another equation? Can you tell me in words what
your equation mean(s)?
d. Ask them to test their equations with particular numbers, such as
$36 for Christina’s money.
NOTES
1
This meaning of fractions has often been referred to as its operator meaning (Behr et
al., 1993), but we use the term multiplier to distinguish between operating multiplicatively
versus additively with fractions.
2
The original intent of the study was to include both middle and high school students.
Due to scheduling difficulties, high school students were unavailable to participate. Thus
more middle school students were interviewed to fulfill the targeted number of
participants in the study, 18.
3
Willa is a pseudonym. We do not give information beyond Willa’s grade level and
her class because we did not study her knowledge in relation to her mathematics
classroom; we were only studying her knowledge in relation to models of other MC3
students.
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Author's personal copy
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MI YEON LEE AND AMY J. HACKENBERG
Mi Yeon Lee
Mathematics Education
Arizona State University
Farmer Education Building, 1050 S Forest Mall, Tempe, AZ, 85282, USA
E-mail: [email protected]
Amy J. Hackenberg
Indiana University
Bloomington, IN, USA
E-mail: [email protected]