Relationships between Students' Fraction Knowledge and Equation Solving

Fraction Knowledge and Equation Solving, 1 NCTM Research Pre-session 2009

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Relationships between Students’ Fraction Knowledge and Equation Solving

Amy J. Hackenberg

Indiana University

Correspondence concerning this manuscript should be addressed to Amy J. Hackenberg, Department of Curriculum and Instruction, Wright Education Building 3060, Indiana University, Bloomington, IN 47402. E-mail: [email protected]


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Relationships between Students’ Fraction Knowledge and Equation Solving

In the United States, the word “algebra” can refer to a course taken in middle or high

school, as well as to a web of knowledge and skills consisting of at least five major areas (Kaput,

1998, 2008). Both algebra as a course and algebra as a way of thinking continue to be

challenging for students (Blume & Heckman, 2000; Chazan, et al., 2007; RAND Mathematics

Study Panel, 2003), and to function as a gatekeeper for further educational opportunities and full

participation in society (Kaput, 2008; Moses & Cobb Jr., 2001). Adequate preparation of

teachers of algebra is also a pressing concern (National Commission on Mathematics and

Science Teaching for the 21st Century, 2000; RAND Mathematics Study Panel, 2003).

In the past 15 years researchers have responded to these issues by studying students’

early algebraic reasoning (e.g., Carpenter, Franke, & Levi, 2003; Carraher, Schliemann,

Brizuela, & Earnest, 2006) and the teacher preparation necessary to support it (e.g., Blanton &

Kaput, 2005; Jacobs, Franke, Carpenter, Levi, & Battey, 2007; Schifter, Monk, Russell, &

Bastable, 2008). This research has focused primarily on drawing out algebraic aspects of

elementary grades students’ reasoning with whole numbers. Thus far little attention has been

paid to connections between students’ algebraic reasoning and fractional knowledge. However,

researchers have suggested that fractional knowledge is critical for learning algebra (e.g.,

Kilpatrick & Izsak, 2008; National Mathematics Advisory Panel [NMAP], 2008b; Wu, 2001).

Moreover, in a recent national survey, rational number knowledge was one of three areas in

which algebra teachers reported that their students were most poorly prepared (NMAP, 2008a).

Perhaps partly as a result, the Report of the Task Group on Conceptual Skills and Knowledge

from NMAP (2008c) has emphasized that for algebra, “…the most important foundational skill

is proficiency with fractions…The teaching of fractions must be acknowledged as critically


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important and improved before an increase in student achievement in Algebra1 can be expected”

(p. 1-xiv).

The clinical interview study reported on in this paper aimed to address the open question

of what it means to consider reasoning with fractions as critical to learning algebra. More

specifically, the purpose of this study was to assess relationships between five ninth grade

students’ quantitative reasoning with fractions and their algebraic reasoning in the area of

equation solving. The research questions for the study were:

(1) How are the students reasoning with fractions as quantities? More specifically, what

fraction schemes can be attributed to the students?

(2) How do the students solve algebra problems that involve unknowns and equations?

(3) What are the relationships between students’ quantitative reasoning with fractions and

their algebraic reasoning in the area of equation solving?

Fraction Knowledge: Fractions as Quantities and Fraction Schemes

A fraction can be conceived of as either an extensive or intensive quantity—e.g., as a

quantity that can be measured “directly,” such as length, or as a quantity that is made from a

multiplicative comparison of extensive quantities, such as a rate (Schwartz, 1988). Fractions as

extensive quantities have been found to be critical in the conceptual analysis of the construction

of rational number (Behr, Harel, Post, & Lesh, 1993; Kieren, 1980, 1988; Pitkethly & Hunting,

1996), and in learning to compare and operate with fractions (Behr, Wachsmuth, Post, & Lesh,

1984; Saenz-Ludlow, 1994). Some researchers have emphasized fractions as extensive quantities

(e.g., Olive & Steffe, 2002; Saenz-Ludlow, 1994; Steffe, 2002; Tzur, 1999) in order to

understand how, in constructing fractions, students reorganize the schemes and operations that 1 The authors of the NMAP report use “Algebra” to refer to algebra as a course.


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they have used to produce whole numbers. In this research, an operation is a mental action, and a

scheme is goal-directed way of operating that consists of an assimilated situation, activity

triggered by the perceived situation, and a result of the activity that is also assimilated to a

person’s expectations (Piaget, 1970; von Glasersfeld, 1995).

A student who has constructed a partitive fractional scheme (Steffe, 2002) conceives of

one-fifth of a candy bar as a part that, when iterated five times, will produce the “whole” bar.

Many situations might be the trigger for the student to use the activity of her partitive fractional

scheme, one of the most obvious being a situation in which the student has a goal of making

three-fifths of a given candy bar (a rectangle). To do so, a student who has constructed a partitive

fractional scheme can partition the rectangle into five equal parts, take out one of those parts, and

iterate it to make three parts.2 Thus the activity of the partitive fractional scheme involves

operations of partitioning, disembedding, and iterating.3 In particular, the student has transcended

part-whole conceptions of fractions in that the student can disembed a unit fractional part from

the partitioned whole and iterate it to make another fractional part of the partitioned whole.

However, the meaning for three-fifths for the student (the result of the scheme) comes from it

being part of a whole, not from it being a fraction that is three times one-fifth (cf. Norton, 2008).

In this sense students who have constructed only partitive fractional schemes remain tied to part-

whole conceptions. They cannot use their partitive fractional schemes to make, for example,

seven-fifths of a candy bar by partitioning the bar into five equal parts, disembedding one of the 2 They can use iterating in this way because their units of one are iterable. This implies that they have interiorized two levels of units in their construction of number (Steffe, 1992, 1994). 3 I use partitioning to refer to equi-partitioning: the mental action of making some number of equal parts in an unpartitioned whole. Disembedding refers to the mental action of taking a part out of the whole without (mentally) destroying the whole. Iterating refers to the mental action of repeatedly instantiating a part to produce another amount. Partitioning, disembedding, and iterating are often accompanied by physical or material actions, if a child is drawing or using a computer tool. However, physical actions do not always indicate the construction of a particular mental action.


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parts, and iterating that one-fifth of the bar seven times. If these students do iterate one-fifth

seven times, they will often call what they have made “seven-sevenths.”

Constructing a fractional scheme that includes improper fractions, an iterative fractional

scheme, is a major advance in students’ fractional knowledge (Hackenberg, 2007; Olive &

Steffe, 2002; Steffe, 2002; Tzur, 1999). Students can use their iterative fractional schemes in

situations such as making seven-fifths of a given candy bar (rectangle). To do so students must

posit a bar that stands in relation to the given bar yet is freed from relying on being part of a

whole for meaning (i.e., seven parts out of five does not make sense, cf. Thompson & Saldanha,

2003). So, students who have constructed an iterative fractional scheme can fulfill making this

posited bar by partitioning the given bar into five equal parts, disembedding one of the parts, and

iterating it seven times.4 Most critically, the result of this activity is, for these students, a bar that

consists of seven one-fifths, not seven-sevenths. So they conceive of the result of their scheme in

relation to the given whole, yet simultaneously as independent of the whole.

Steffe (2002) has conjectured that the splitting operation is necessary to construct an

iterative fraction scheme. The splitting operation, a composition of partitioning and iterating, is

the fundamental operation involved in solving this problem: “This candy bar (rectangle) is five

times the length of your bar. Draw your bar.” To solve this problem, students need to posit their

bar, which is made from a partition of the given bar, and simultaneously can be iterated five

times to create the given bar. Positing this bar that is, essentially, multiplicatively related to the

given whole is one basis for positing an improper fraction that is both independent of and related

to a given whole.

4 For these students, unit fractions are iterable units. When a unit fraction is an iterable unit, it implies the composite unit that can be made from it in iteration.


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In addition, in prior research (Hackenberg, 2007) I have found that students’

multiplicative concepts are critical in the construction of an iterative fractional scheme. In

particular, the interiorization of three levels of units seems to be necessary. Students who have

interiorized three levels of units bring to a situation a three-levels-of-units “view” of number, and

they can operate further with that view. For example, they “automatically” view 35 units as a

unit of five units each containing seven units. Being able to take this view as given is important

in making improper fractions because three levels of units are involved in any improper fraction:

the improper fraction itself, the whole to which the improper fraction refers (but from which it is

also independent), and the part that was used in iteration to create the improper fraction. For

example, to conceive of seven-fifths as a number “in its own right” means to take it as a unit of

seven units (fifths), any of which could be iterated five times to create the “whole,” with respect

to which seven-fifths is named (Figure 1). In my prior research, students who had not yet

interiorized three levels of units did not operate in a way that allowed me to attribute an iterative

fractional scheme to them.

Figure 1, The three levels of units involved in seven-fifths.

Quantitative Reasoning as a Basis for Algebraic Reasoning

Unit Bar

seven-fifths

five-fifths

one-fifth


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As portrayed in the previous section, researchers have found treating fractions as

extensive quantities to be fruitful in students’ construction of fraction knowledge. In addition,

researchers have proposed that quantitative reasoning is a fruitful basis for developing algebraic

reasoning because quantitative reasoning and algebraic reasoning have some significant

similarities, and because reasoning with quantities can be meaningful for students (Chazan,

2000; Confrey, 1998; Dossey, 1998; Thompson, 1993). Researchers who highlight generalized

quantitative reasoning as a basis for early algebraic reasoning (e.g., Carraher, et al., 2006;

Dougherty, 2004; Smith & Thompson, 2008) capitalize on students reasoning about relationships

among quantities as an avenue for conceiving of unknowns and variables, writing equations and

inequalities, and tracking changes in quantities that co-vary.

For example, emphasizing quantitative reasoning as a basis for algebraic reasoning means

that graphic items (numbers as well as letters) refer to known and unknown quantities: A known

is a fixed quantity that has been measured and the measurement is known, while an unknown is a

fixed quantity that is unmeasured but the measurement has the potential to be known. Reasoning

about relationships among quantities is the foundation for constructing and solving equations.

Thus equation solving can be “about something” other than manipulation of formalisms (cf.

Kaput, 1998), which may appeal to students by connecting with their experiences.

Although fractions may be implicit or disguised in solving equations like 4x = 28, they

soon become explicit in solving equations like 3x = 7. First, the solution to an equation like 3x =

7 is an improper fraction. Arriving at such a solution requires determining how to divide seven

identical units into three equal parts—an extension of whole number division that requires

sophisticated multiplicative coordinations (Empson, Junk, Dominguez, & Turner, 2006;

Hackenberg, in review; Toluk, 1999). Furthermore, to reason with reciprocals in the solution of


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either of these two equations requires unit fractions like 1/4 and 1/3 to be multiplicative

operations for students—one facet of fractional knowledge that has been found to be particularly

complex (cf. Behr, Harel, Post, & Lesh, 1993; Davis, Hunting, & Pearn, 1993; Kieren, 1995).

Methodology and Methods

As a small-scale, exploratory interview study, this project did not have the power to

assess learning. However, a strength of clinical interviewing is “the ability to collect and analyze

data on mental processes at the level of a subject’s authentic ideas and meanings, and to expose

hidden structures and processes in the subject’s thinking that could not be detected by less open-

ended techniques” (Clement, 2000, p. 547). Interview studies are an important tool to generate

scientific knowledge because science involves dynamic, on-going discussion (Kvale, 1996) in

attempts to formulate explanatory models (Clement, 2000; Steffe & Thompson, 2000) for

observed and experienced phenomenon.

Each of five 9th grade students participated in two hour-long, semi-structured interviews,

a fraction interview and an algebra interview. The interview protocols were designed so that the

reasoning involved in the fraction interview formed a central basis for solving problems in the

algebra interview. For example, the fraction interview consisted of problems in which unknowns

could remain implicit in a student’s thinking and solution processes. In the algebra interview,

students needed to become more explicit about using unknowns, as well as algebraic notation, in

order to solve the problems.

Each interview was videotaped with two cameras, one focused on the interaction between

the researcher and student, and one focused on the students’ written work. These videos were

mixed into one file for purposes of analysis and all files were transcribed. During analysis, I

engaged in repeated viewing of the videofiles and in recursive rounds of note-taking (Cobb &


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Gravemeijer, 2008). In this process I formulated a second-order model of each student’s

fractional schemes (Steffe & Thompson, 2000) to the extent that was possible over two

interactions. A second-order model is a researcher’s constellation of constructs to describe and

account for another person’s ways and means of operating. I used and built on constructs from

previous research (Hackenberg, 2007, in review), as well as from previous second-order models

of students (e.g., Nabors, 2003; Steffe, 2003, 2004). Formulating these models provided a

consistent “portrait” of each student’s fraction schemes and concepts (first research question), as

well as her or his ways of solving problems involving unknowns and equations (second research

question). In addition, I looked across the two interviews of each student to articulate

relationships between each student’s quantitative reasoning with fractions and her or his

algebraic reasoning in the area of equation solving (third research question).

Analysis and Findings

I now describe the three main findings of the study, which respond to the first two

research questions. First, I demonstrate that no student had constructed an iterative fraction

scheme and that three of the students did not conceive of fractions as multiplicative operations

(research question 1). Second, I describe how the students reasoned with quantitative situations

that involved multiplicative relationships, and how they represented these situations with

unknowns and equations (research question 2).

Improper Fractions

No iterative fraction schemes. I could not attribute an iterative fraction scheme to any of

the five students. When I asked each student to draw seven-fifths of a pictured candy bar (a

rectangle), three of the five students stated that it could not be done. One student, Frank, stated

definitively that “the top number can’t be higher than the bottom number.” Another student,


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Sheila, offered that she could make five-sevenths instead. A third student, Melanie, said “I keep

thinking that’s [seven-fifths] over a candy bar—not possible!” Melanie and a fourth student,

Eliza, expressed concern that more candy would be needed than what they had. I countered by

telling them the candy bar was magic—when you take a piece out, the candy fills right back in.

This intervention did not seem to fully alleviate their difficulties, but both drew a bar.

Eliza drew a 5/5-bar and then added on two more pieces the same size as the fifths

(Figure 2a). In explanation, she said, “to get 7 out of 5 you have to add two more.” When I

shaded the last piece of her bar and asked what fraction that was of the original, she said it was

one-seventh. She gave the same response when I shaded the first piece. Then I asked what the

size of the first piece was before adding on the two pieces, and she said it was one-fifth. Melanie

drew a 5/5-bar and another 5/5-bar on which she shaded two parts (Figure 2b). Because I was

uncertain whether Melanie believed she had to draw the parts separately, or why she had placed

the bars the way she did, I decided to investigate further.

Figure 2a (above) and 2b (below), Eliza and Melanie’s initial drawings of seven-fifths.


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In fact, with Sheila, Eliza, and Melanie I used an intervention to test whether or not their

difficulty in making seven-fifths was a significant cognitive constraint.5 The intervention consists

of presenting students with a candy bar (rectangle) and asking them to make a separate candy bar

that’s three-fifths of that bar. Students who have constructed partitive fraction schemes can do

this by, for example, partitioning the given bar into fifths, disembedding one-fifth, and taking

that three times. All three students drew a separate unmarked 3/5-bar. 6 The next part of the

intervention involves asking students to “show all the parts in their bar,” to shade any one of the

three parts, and to state how much of the original candy bar that part is. All three students

identified one of the three parts of their bar to be one-fifth of the original candy bar. Because all

three students found this task unproblematic, I can conclude that they had likely constructed a

partitive fraction scheme.7

Then I asked each student to make a bar that consisted of seven of those parts. When

Sheila drew her bar, I asked her how many fifths she had made; she said seven. So it is possible

that she made some progress based on this intervention—i.e., she may have seen that it was

possible to make a bar that consisted of seven one-fifth parts. Note that this alone is not enough

to attribute an iterative fraction scheme to a student, because I would have to be convinced that

the student was consistently coordinating all three levels of units involved in making improper

fractions over multiple instances. When Eliza drew her seven-part bar, I asked her to shade one

part of it. She was still not sure whether that part was one-fifth or one-seventh of the original

5 Due to my assessment that Frank had interiorized at most two levels of units and struggled with whole number multiplication and division, I did not pursue this intervention with him. 6 Drawing an unmarked bar means that the students’ iterating operations were not clearly evident. 7 Some caveats are in order here: At times Sheila did not appear to have constructed, or have activated, her disembedding operation, which meant she did not always make separate bars (even when asked). Similarly, at times Melanie did not appear to disembed or iterate, even though I believe she had likely constructed both operations since she showed evidence of them in some situations.


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candy bar. Finally, when I asked Melanie to draw this bar, she again drew a 5/5-bar and then

another, smaller bar consisting of two of those fifths (Figure 3). She said that if you joined the

two bars, it became seven-sevenths.

Figure 3, Melanie’s post-intervention drawing of seven-fifths.

I did not do the above intervention with the fifth student, Rebecca, in part because she

appeared to sometimes use improper fractions. However, in the second interview with her I

asked her to draw a picture of five-halves of a given candy bar. She first drew a bar and

partitioned it into fifths. Then she paused and appeared to change her mind. She then drew an

additional candy bar and a half candy bar, partitioning the two whole bars to show all the halves.

So Rebecca seemed to make five-halves accurately, although like Melanie she seemed to think of

it as two whole bars and one half of a whole bar. So, based on the evidence from the interviews, I

could not attribute an iterative fraction scheme to any of the students, although in our interactions

both Sheila and Rebecca may have made some progress toward constructing such a scheme.

Further activity with all four girls confirmed that improper fractions continued to be problematic.

Reasoning with improper fractions as the solution to a problem. For example, late in the

first interview I posed the following problem to all students except Frank:

Peppermint Stick Problem. The 7-inch peppermint stick shown (Figure 4) is three times the length of another stick’s length; draw the other stick and tell how long it is.


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Figure 4, A seven-inch long peppermint stick.

Sheila solved this problem by partitioning the first 6 inches into three equal parts consisting of 2

inches each, and then by partitioning the seventh inch into thirds.8 So she drew a stick that she

said was 2 and 1/3 inches, but did not know an improper fraction for her solution. When I asked

her if she could show how many thirds of an inch were in her solution, she partitioned each of

the two inches into three equal parts and, upon further questioning, stated that there were seven

(thirds).

Eliza determined two and one-third inches for her solution to the Peppermint Stick

Problem by dividing seven by three in her head, arriving at “two point three three repeating.” She

said she knew that was two and one-third, so she marked off two and one-third inches on the 7-

inch stick, and then counted off another two and one-third inches along the stick to make another

mark. She drew out an unmarked stick that was two and one-third inches long (Figure 5). Since

Figure 5, Eliza’s initial solution of two and one-third inches.

8 Shelia justified this solution by saying that if she “timesed” her stick by three, it would make the full 7-inch stick. This justification shows evidence of an iterating operation.


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she had explained that she divided seven by three to find her solution, I asked if she could show

on her picture how to divide seven inches into three equal parts. She (spontaneously) partitioned

each inch of the 7-inch stick into thirds. I asked how many of those pieces would be in her

solution, and she counted seven. However, when I asked for the length of the stick, she said

seven twenty-firsts. After I asked about its length in inches, she said it was two and one-third. I

shaded one of the seven parts of her solution, and she said that was one-third. So, when I asked

how many thirds were in her solution, she said seven. We talked about whether seven one-thirds

of an inch was a valid name for her solution, and whether it was the same amount as two inches

and one-third of an inch. She thought they did name the same amount. Then Eliza and I looked

back at making seven-fifths of a bar. This time she said that seven-fifths was “a whole and two

more.” But like Melanie, she claimed that you could not make the parts all together, but had to

keep them separate (i.e., as a whole and two more fifths).

When I posed the Peppermint Stick Problem to Melanie, she partitioned each inch into

three equal parts. Then she calculated that she had made 21 parts total, divided by 21 by 3 to get

7, and concluded that 7 parts was the size of the other stick. When I asked about the length of

that stick, like Eliza, she said it was seven “twenty-oneths” or one-third of the original. After I

asked about the length in inches, she grouped every three parts and determined the length to be

two and one-third inches. I then asked her how many thirds of an inch that was. She answered

correctly and promptly, seven-thirds of an inch, and she affirmed that that was the same amount

as two and one-third. Rebecca solved the Peppermint Stick Problem in a nearly identical way. So

on this problem both Melanie and Rebecca stated improper fractions swiftly and

unproblematically when prompted. Nevertheless, for most of the work over the two interviews,

both girls used mixed numbers and decimals, not improper fractions.


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Algebraic Representations of Quantitative Situations Involving Multiplicative Relationships

No fractions as multiplicative operations. Three of the students—Frank, Sheila, and

Eliza—had not constructed fractions as multiplicative operations. Evidence for this claim

includes their work on the following problem, which I posed to all five students in their second

interview:

Money Problem. Javier has some money, and so does Melissa. Javier’s money is five times Melissa’s money. Can you draw a picture of that situation? Could you write an equation for this situation? Could you write another equation for the situation? Could you write the equation with division using multiplication (if one of their equations involved division)?

Frank approached the problem at first by writing $1 for Melissa’s money and $6 for

Javier’s. He confirmed, when I asked, that he thought $6 was five times more than $1. Even after

I prompted using a drawing with segments, Frank continued to use the relationship of five more

rather than five times. I asked Frank to find out how much money Melissa had if Javier had $15.

He determined that Melissa had $3. In explanation, he said that he counted by 3s to 15. He

explained that tried 3 and miscalculated, went to 4 and that was too much, then thought of 3 and

1/2, and then checked 3 again and it worked. In all of his pictures of the situation, he included

Melissa’s money in with Javier’s money (Figure 6a), commenting that he’d take one of the 3s off

for Melissa. For the equation, he wrote P ÷ 5 (vertically) and drew a horizontal line (P

represented Javier’s amount of money). When I asked for a name for Melissa’s amount of

money, he wrote an M below the line (Figure 6b). For the second equation, he also wrote in a

vertical format: M x 5 (line) P. Then I asked for horizontal equations, and he wrote them. When I


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Figure 6a (above) and 6b (below), Blake’s picture and initial equations for the Money Problem.

asked if he could use multiplication for the division equation, he did not know how to do it.

Upon probing, he did articulate that Melissa’s money was one-fifth of Javier’s, but he did not

have a way to write that.

In solving the Money Problem, all four girls initially wrote equations like H = 5*M and H

÷ 5 = M, where H represented the amount of money Javier had and M the amount of money

Melissa had. When I asked about whether the division equation could be represented

multiplicatively, Sheila said she didn’t see how you could multiply a bigger number to get a

smaller number, unless it was a negative. I asked whether she could use a fraction. She

eventually wrote M = 1/5 H. In an attempt to find out what that meant to her, I asked whether

what she wrote was 1/5 times H. She said no, it just meant that Melissa’s money would be equal

to one-fifth of Javier’s money. This response seems to contraindicate awareness of one-fifth as a

multiplicative operation, even if she could write down an equation that appeared to involve one-

fifth as a multiplicative operation.

Eliza’s first equation was x * 5 = y, where x represented Melissa’s money and y

represented Javier’s money. When I asked for another equation besides that one, she said she

couldn’t think of one. Then I asked whether she could write an equation for finding Melissa’s


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money, and she wrote 5 ÷ y. She said that expression meant dividing Javier’s money by five to

get Melissa’s money, and she completed her equation: 5 ÷ y = x. I asked if she could determine

Melissa’s money if Javier had $35. Eliza wrote 5 ÷ 35 = 7. When I probed about this division

statement, she said you write it 35 ÷ 5 = 7 “or else it would be negative.” So she then adjusted

her equation to x ÷ 5 = y. When I asked about the meanings of x and y, she changed the equation

to y ÷ 5 = x. To write the equation with multiplication, she wondered aloud about -5 and wrote y

* -5. I asked about whether she could use a fraction, and she offered 1/5. She then wrote y ÷ 1/5

= x. We discussed an example with Javier having $25. I asked her whether $25 divided by 5 was

the same as $25 divided by 1/5. Eliza continued to think that division by one-fifth was correct.

So for Eliza at least, conceptions of division may have been significant in her work on the

Money Problem. In any case, her work does not indicate that she conceived of fractions as

multiplicative operations.

Fractions as multiplicative operations, at least sometimes. In contrast, Melanie and

Rebecca, when prompted, produced H*(1/5) = M as another way to state their equation involving

division (H ÷ 5 = M). From that point onward in the second interview, both girls sometimes used

fractions as multiplicative operations in equations.

For example, in the second interview I posed the following problem to both girls:

CD Problem. Theo has a stack of CDs some number of inches tall. Sam’s stack is 2/5 the height of Theo’s. Can you draw a picture? Could you write an equation for this situation? Could you write another equation for this situation?

Rebecca wrote the equation T*(2/5) = S, then T*(.4) = S, where T represented the height of

Theo’s stack and S represented the height of Sam’s stack. However, to write an equation to

produce the height of Theo’s stack from Sam’s stack height, Rebecca said T was 1 and 1/2 times

greater. Most likely she meant that to get Theo’s height she needed to add on 1 and 1/2 of Sam’s


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height to Sam’s height. She then changed her mind to 2 and 1/2 times, which she wrote as 5/2 in

a specific example when Theo’s stack height was 16 cm. I asked her how she could verify that

what she had written (16 x 5/2 = 40) was a true statement, and she hesitated. Then I asked about

the number of “one-half’s” in five-halves, and she was unsure. This comment led to me asking

her to draw five-halves of a candy bar, as I have already described.

On the same problem, Melanie wrote T*(2/5) = S, where T represented the height of

Theo’s stack and S represented the height of Sam’s stack. Instead of asking her about an

equation to get Theo’s stack height from Sam’s, I posed this extension of the CD problem to

Melanie:

Extension of CD Problem. Michael’s stack is 7/5 the height of Theo’s. Could you draw the height of Michael’s stack in relation to the other two? Could you write an equation to find the height of Michael’s stack?

To solve it, Melanie drew another segment that was two-fifths longer than the segment

representing Theo’s stack (Figure 7). She then wrote T*(1 2/5) = M. To investigate her ideas

Figure 7, Melanie’s picture for the Extension of the CD Problem.


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about improper fractions a little more, I shaded in one part of the segment representing Michael’s

stack, and I asked her how much that was of Theo’s stack. She responded that it was one-fifth. I

did the same for another part, and she responded similarly. She confirmed that she had made

seven-fifths for Michael’s stack. When I noted that she had not used seven-fifths in her equation,

she could have written T*(7/5) = M. So, both her work and Rebecca’s work on this problem

indicate that they used at least proper fractions as multiplicative operations in writing equations

to represent quantitative situations. Thus they may have conceived of at least proper fractions as

multiplicative operations.

However, counterindications of this conclusion were also evident for both Rebecca and

Melanie. For example, in the second interview Rebecca worked on this problem:

Weight of Candy Bar Problem. Each of 5 identical candy bars has weight B ounces. How much would 1/7 of all the candy weigh?

Rebecca created the expression (B*1/7)*5, but she did not know what fraction of B that was. An

example with Melanie occurred following the Extension of the CD Problem. In working on a

similar problem in which the given multiplicative relationship was an improper fraction, Melanie

encountered difficulty reasoning reversibly. Such difficulty demonstrated that improper fractions

as quantities or multiplicative operations were not unproblematic for her, as I will explore in the

next section.

Algebraic equations that reflect additive reasoning. None of the students consistently

wrote multiplicative equations when working on problem situations involving multiplicative

relationships. In other words, all five students reasoned additively when using algebraic notation

to represent multiplicative relationships among quantities, such as the CD Problem, the

Extension of the CD Problem, and the following problem:


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Babysitting Problem. Serena earns $48 babysitting, and that’s 4/3 the amount of money that Christina earns. Could you draw a picture? How much does Christina earn? Then: We don’t actually know how much Serena earned, but the relationship between the girls’ earnings is the same. Could you write an equation for the relationship between the two girls’ earnings?

Frank was the only student who did not work on any of these three problems. As I have already

noted, his drawing on the Money Problem (a similar problem to the three above but with a whole

number multiplicative relationship between quantities) indicates that he saw the relationship in

that problem as additive. Because of the challenges he experienced on the Money Problem, as

well as my conjecture that he had interiorized at most two levels of units, during the remainder of

his second interview I posed problems to him that I thought were more closely within his zone of

potential construction (Steffe, 1991).

However, all four girls worked on the CD problem. I have already described Rebecca’s

and Melanie’s solutions. The problem proved to be revealing for Sheila and Eliza as well. Recall

that the CD Problem is:

CD Problem. Theo has a stack of CDs some number of inches tall. Sam’s stack is 2/5 the height of Theo’s. Can you draw a picture? Could you write an equation for this situation? Could you write another equation for this situation?

Sheila drew a picture of a segment partitioned into five equal parts and two other segments to

highlight two-fifths of that segment (Figure 8).9 For an equation, she first wrote S = 2/5. Then T

= S + 3/5, where T represented the height of Theo’s stack and S represented the height of Sam’s

stack. I asked her to check her equations if Sam’s stack was 20 cm. She determined each fifth of

Sam’s stack would be 4 cm. Then she wrote T = S + 12. I asked her more about what 3/5 meant

in her initial equation. She wrote “of T’s stack” on her equation, so it looked like this: T = S +

3/5 of T’s stack. For another equation, she wrote S = T + 2/5 of S’s stack. Then on her own 9 Note that her picture for this situation is one example of not disembedding a fractional part from a whole.


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Figure 8, Sheila’s picture for the CD Problem.

initiative, she changed it to T = S – 2/5 of T’s stack. Then, again on her own initiative, she

changed that to T = S – 3/5 of T’s stack. She stated that Sam’s stack was 2/5 of Theo’s stack, and

I encouraged her to write that in words. At the end of our exchanges, she wrote S = 2/5T of T’s

stack. Note that for Sheila, the letters that represented the unknowns were not entirely

quantitative because she used them to indicate names. That is, she wrote “of T’s stack” or “of S’s

stack” as well as using T and S to ostensibly represent the heights of the stacks.

When I posed the CD Problem to Eliza, she drew a picture similar to Sheila’s, except that

she clearly represented both stacks (Figure 9). Her initial equation was T – 3 = S, where S

represented Sam’s stack height and T represented Theo’s stack height. When we talked through

what her equation meant, she said “no” and crossed it out. Then she wrote T – 3/5 = S. I

suggested a numerical example of 25 cm as Theo’s stack height; could she find Sam’s stack

height? She explained that you have to divide 25 by 5, then times 5 by 3 and subtract that off

from 25, and the result would be two-fifths. I asked if she could write an equation with 2/5 in it.

She wrote S + 2/5 = T. I again posed a numerical example with Sam’s stack height equal to 16

cm. Eliza tried to divide 16 by 5. I directed her back to the picture, and she wrote S x 2.5 = T.


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Figure 9, Eliza’s picture of the CD Problem.

She explained that Theo’s height was two (of Sam’s heights) and a half. She then crossed out S +

2/5 = T. For an equation to get Sam’s height, she wrote T ÷ 2.5 = S. When I asked about whether

she could use multiplication for that equation, she said she didn’t know. I asked again about how

to make Theo’s height from Sam’s, referencing the 16 cm. She wrote S + S + (1/2S). In referring

back to her equation S x 2.5 = T, I asked her if she knew what fraction was the same as 2.5. She

asked whether it was three-fifths. Then she said one-half, because 2.5 over 5 is one-half. I probed

again and suggested she could also think about what to multiply Theo’s height by to make Sam’s

height. She said she didn’t know.

As I have already noted, Melanie solved the CD problem and its Extension by writing

equations in which she used fractions as multiplicative operations. However, she had more

difficulty with the Babysitting Problem.

Babysitting Problem. Serena earns $48 babysitting, and that’s 4/3 the amount of money that Christina earns. Could you draw a picture? How much does Christina earn? Then: We don’t actually know how much Serena earned, but the relationship between the girls’ earnings is the same. Could you write an equation for the relationship between the two girls’ earnings?


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When I posed that problem, Melanie immediately exclaimed, “so Serena got more or less!?” She

concluded that Serena earned more because four-thirds is “over one.” I asked for a picture of her

thinking. She drew Christina’s money as one segment and Serena’s money as a segment four

times longer than Christina’s segment. Her explanation was that Serena has four-thirds more than

Christina. I asked: What if Christina had $48 and that was 2/3 of Serena’s money? What would

the picture look like then? Melanie asked if Serena had two-thirds less. Then she drew a picture

where the segments representing Serena’s money was two-thirds of the segment representing

Christina’s money.

Because reversible reasoning seemed to have become problematic for Melanie, I stepped

back and posed the Second Babysitting Problem, in which the given relationship was a proper

fraction:

Second Babysitting Problem. This time Christina earned $24. That $24 is 3/8 of what Serena earned. Could you draw a picture of this situation? Could you write an equation to show the relationship between the two amounts of money?

Melanie said: “so Serena earned 3/8 of $24.” I repeated the situation, and Melanie said that

Serena had five-eighths more. She drew a picture of two segments, a segment partitioned into

Figure 10, Melanie’s picture for the Second Babysitting Problem.


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eight equal parts, and another segment that was three of those parts long (Figure 10). However,

at first she labeled the 8/8-segment as C, and then she appeared to change her mind. For an

equation, she wrote S*3/8 = C, where S represented Serena’s earnings and C represented

Christina’s earnings. But then she seemed not sure about whether the equation was right. I

suggested checking it out with numbers—how much money should Serena have? Melanie found

three-eights of 24: She divided 24 by 8 to get 3, and then took 3 times 3 to get 9. She did not

think this was right and focused on Serena having 5/8 more. She said that 3*5 = 15, so Serena

had $24 + $15 = $39. I asked about whether this worked with her picture—and I wondered in

particular what the littlest segments in her picture represented. Melanie seemed torn between

whether each segment represented $8 or $3. We did not get beyond this for the rest of the

interview.

Although in working on the CD Problem Melanie had written equations for Theo’s and

Sam’s stack heights using fractions and mixed numbers as multiplicative operations, her work on

the Babysitting Problems leads me to be suspicious about what this meant to her—particularly

the equation T*(1 2/5) = M. The main reason for this suspicion is the “disruption” of her

reversible reasoning once I posed the Babysitting Problem, in which the multiplicative

relationship was an improper fraction. We worked on problems to address her reversible

reasoning for the rest of the interview. Although she still wrote some multiplicative equations,

she reasoned additively from point onward.

When I posed the Babysitting Problem to Rebecca, she drew a reasonable picture (Figure

11): A segment to represent Serena’s money that she split into four equal parts, and then a

segment next to it that spanned three of those parts. For the equations, initially she wrote C + 1/3

= S, and S – 1/3 = C, where C represented the earnings of Christina and S represented the


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Figure 11, Rebecca’s picture for the Babysitting Problem.

earnings of Serena. In checking the equations with the initial scenario where Christina earned

$48, she wrote 36 + 12 = 48 and 48 – 12 = 36. She indicated that she was suspicious of the

second equation because one-third of 48 is not 12. She articulated that in the equation C + 1/3 =

S, the 1/3 refers to 1/3 of Christina’s money. So, she then wrote C ÷ 1/3 = 12 x 4 = S or 48.

I then posed The Height of the Room Problem, so I could test out how Rebecca was

thinking about taking an improper fractional amount of an unknown quantity:

The Height of the Room. This room is some number of feet tall. How would you write an expression for four-thirds of the height of the room?

Rebecca stated that she needed to divide the height of the room, R, by 3. She wrote R ÷ 1/3 = ?.

Then she said, “it’d be question mark times four,” writing “x 4 = 4/3.” Thus her statement looked

like this: R ÷ 1/3 = ? x 4 = 4/3. When I asked about whether she intended to divide by three or

divide by one-third, she said you need one-third and then you times that by four. So she seemed

to intend to take one-third of the height and use that to make four-thirds of the height. Indeed,

later she changed her notation to reflect that she was dividing R by three. After some other

examples with unit fractions (e.g., can you write one-fourth of the height of the room), she wrote

R x 4/3 to represent four-thirds of the height of the room. Still, in determining specific examples


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(e.g., the height of the room is 12 feet), she always added on one-third of the whole, rather than

taking one-third of the whole four times. When we returned to the Babysitting Problem, Rebecca

wrote C x 4/3 = S as an alternative to her first equation. To try to “fix” the second equation, she

said “S x 3/3, but that’s just one.” I infer that she meant she knew that expression wasn’t

satisfactory because it would just yield S. With further probing, she did eventually write S x 3/4.

Discussion and Conclusions:

The Relationship Between Fraction Knowledge and Equation-Solving

In this study, none of the students demonstrated strong and consistent reciprocal

reasoning in solving equations, and none of them consistently conceived of fractions as

multiplicative operations. So, the study suggests that not having constructed fractions as

multiplicative operations precludes students from reasoning with reciprocal relationships in, and

using reciprocal relationships to represent, situations involving a multiplicative relationship

between two quantities. This conclusion is significant because it suggests a close connection

between students’ fraction knowledge and their basic algebraic reasoning, and it implies that

working to help students conceive of fractions as multiplicative operations could ameliorate

some difficulties that students experience early in learning algebra.

In prior research (Hackenberg, in review) I have found an iterative fraction scheme to be

necessary but not sufficient for developing a conception of fractions (even proper fractions) as

multiplicative operations. The general reason for this conclusion is that students need to be able

to take the result of a scheme as given before they can use it in further operating, such as using a

fraction as a multiplicative operation in another situation. Students who have constructed only

partitive fractional schemes can sometimes take the results of their schemes as material for


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further operating,10 but (1) the meaning of the results of their schemes still rests on the

relationship of the result to the whole, and (2) they certainly cannot take any fraction as a number

in its own right.

More specifically, conceiving of a fraction as a multiplicative operation appears to

require abstracting the operations of one’s iterative fraction scheme and the operations of one’s

reversible iterative fraction scheme into one “program” constituting the conceptual meaning for

any particular fraction (Hackenberg, in review). So, for example, to conceive of one-fourth as a

multiplicative operation means students take the operations of the scheme they have for making

it (e.g., partition a quantity into four parts and take out one part), as well as the operations of the

scheme that reverse the making of one-fourth—that made back the whole quantity from one-

fourth of it (e.g., iterate that part four times)—as a program of operations. When one-fourth

consists of this program of operations, reciprocal reasoning with one-fourth comes right out of

the program—one-fourth “contains” the notion of being able to be iterated four times to make

back a whole because the program includes reversing the making of the fraction. If one-fourth

“automatically” entails taking a part of a length four times to make back the whole length, then

the whole length is automatically four times one-fourth of itself. So, since four times one-fourth

is one (whole length), one-fourth times the whole length must be one-fourth of the length.

This conclusion is not contradicted by the data in this study in that none of the five

students had constructed an iterative fraction scheme, and none reasoned seamlessly with

fractions as multiplicative operations. Upon prompting, three of the girls (Sheila, Melanie, and

Rebecca) produced the notation that x ÷ 5 was the same as x*(1/5). However, Sheila explicitly

10 My current conjecture is that students have to have constructed a splitting operation and be able to use it with their partitive fraction scheme in order to do so.


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rejected the idea that Javier’s money was being multiplied by one-fifth. Melanie and Rebecca

seemed to accept this multiplicative notation and use it in working on the CD problem. Yet

neither used it unproblematically in working on the Babysitting Problem. For both of these

students—who were quite strong multiplicative reasoners—fractions as multiplicative operations

seemed to be something they had to work hard to produce in the activity of solving a problem,

not something that they could read into a situation and use at will. My current conjecture is that

these two students had accepted conventions for multiplying quantities by fractions—in fact, it

may have even made intuitive sense to them. But that does not mean that they had constructed

fractions as multiplicative operations, and therefore that they had constructed reciprocal

reasoning.

Not having constructed reciprocal reasoning meant that all five students relied on additive

reasoning when representing multiplicative situations. In some cases the students built

expressions or equations that faithfully represented the multiplicative relationships in a certain

situation. For example, Eliza wrote T + T + (1/2T) to produce Sam’s stack height from Theo’s

stack height in the CD Problem. This expression accurately captures the relationships between

the two heights and uses one-half as a multiplicative operation, although it rests on additive

accumulation. However, in many cases, the equations students wrote did not capture all of the

multiplicative relationships in the problem. For example, Rebecca wrote C + 1/3 = S and S – 1/3

= C in the Babysitting Problem. She herself thought at least the second equation was

problematic, to her since “1/3” referred to one-third of Christina’s earnings. Yet neither equation

captures that multiplicative relationship, and both equations indicate that representing the

quantity “one-third of Christina’s earnings” (let alone one-fourth of Serena’ earnings) was


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problematic. Rebecca’s equations demonstrate the close relationship between conceiving of

fractions as multiplicative operations and the construction of reciprocal reasoning.

An even more basic concern is that without an iterative fraction scheme, solutions to

problems like the Peppermint Stick Problem do not make sense. Students may be able to view

the result in terms of mixed numbers, but they then miss building and generalizing the idea that

one-third of seven units is seven-thirds of one unit. This idea is important for reasoning with

ratios: If 7 gallons of gas cost 3 dollars, how much flour can you buy for 1 dollar? To distribute 7

gallons across each of the 3 dollars can involve distributing one-third of each of the 7 gallons

across each of the 3 dollars. Thus each dollar corresponds to seven 1/3-gallons, or seven-thirds of

one gallon. Of course this solution is not the only one possible, but it has considerable power for

generalizing “ratio-taking” to any number of gallons and any number of dollars.

An implication of this study is that to reason reciprocally even in “basic” algebraic

problems requires (at least) the interiorization of three levels of units, since this multiplicative

concept is required to construct an iterative fraction scheme. This multiplicative concept is a

significant achievement for students (cf. Olive, 1999), and in fact students can be powerful

reasoners prior to the construction of this concept (Hackenberg, 2005, in review). So, this

research begs further investigation into at least two areas: (1) What work can be done with

students in schools to more explicitly open opportunities for the interiorization of three levels of

units and the construction of iterative fraction schemes? (2) For students who have not yet

interiorized three levels of units in their ninth grade year, what progress can they make toward

reasoning algebraically?


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Relationships between Students' Fraction Knowledge and Equation Solving

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