Students’ whole number multiplicative concepts: A critical constructive resource for fraction...

$: Students’ whole number multiplicative concepts: A critical constructive resource for fraction composition schemes$
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Journal of Mathematical Behavior 28 (2009) 1–18

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The Journal of Mathematical Behavior

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Students’ whole number multiplicative concepts: A criticalconstructive resource for fraction composition schemes�

Amy J. Hackenberga,∗, Erik S. Tillemab

a Indiana University, 201 N. Rose Avenue, Wright Education Building 3060, Bloomington, IN 47405-1006, USAb IU School of Education at Indianapolis, 902 W. New York Street, Indianapolis, IN 46202, USA

a r t i c l e i n f o

Keywords:FractionsFraction multiplicationMultiplicative conceptsSchemesStudents’ mathematical learning

a b s t r a c t

This article reports on the activity of two pairs of sixth grade students who participated inan 8-month teaching experiment that investigated the students’ construction of fractioncomposition schemes. A fraction composition scheme consists of the operations and con-cepts used to determine, for example, the size of 1/3 of 1/5 of a whole in relation to thewhole. Students’ whole number multiplicative concepts were found to be critical construc-tive resources for students’ fraction composition schemes. Specifically, the interiorizationof two levels of units, a particular multiplicative concept, was found to be necessary for theconstruction of a unit fraction composition scheme, while the interiorization of three levelsof units was necessary for the construction of a general fraction composition scheme. Thesefindings contribute to previous research on students’ construction of fraction multiplica-tion that has emphasized partitioning and conceptualizing quantitative units. Implicationsof the findings for teaching are considered.

© 2009 Elsevier Inc. All rights reserved.

1. Introduction

Researchers have situated the relationship between students’ whole number multiplicative concepts and students’ con-struction of fractions in different ways. For instance, Streefland (1991), Mack (1995), and Behr, Wachsmuth, Post, and Lesh(1984) have argued that students’ whole number concepts can interfere with students’ reasoning with fractions. In contrast,Olive (1999) and Steffe (2003) have articulated a reorganization hypothesis for fraction knowledge, which states that stu-dents reorganize the quantitative operations they use to construct whole numbers in constructing fractions. Olive, Steffe,and colleagues have confirmed their hypothesis in analyses of data from a teaching experiment with students in their thirdthrough fifth grades (e.g., Biddlecomb, 2002; Olive & Steffe, 2002; Steffe, 2003, 2004; Tzur, 1999, 2004). Similar to Olive andSteffe, other researchers have found evidence for connections between students’ whole number multiplicative concepts andstudents’ fraction knowledge (e.g., Empson, Junk, Dominguez, & Turner, 2006; Hackenberg, 2007; Hunting, Davis, & Pearn,1996; Kieren, 1995; Thompson & Saldanha, 2003; Vergnaud, 1988). Thus there is considerable support for viewing students’whole number multiplicative concepts as critical constructive resources for building fraction knowledge.

Within the body of research on fractions, few studies have examined how students construct schemes for multiplyingtwo fractions (e.g., Armstrong & Bezuk, 1995; Mack, 2001; Olive, 1999; Steffe, 2003; Streefland, 1991), or fraction compositionschemes. A fraction composition scheme consists of the operations and concepts used to determine, for example, the size

� Portions of the research reported in this manuscript was part of the first author’s doctoral dissertation completed at the University of Georgia underthe direction of Leslie P. Steffe. All research reported in the manuscript was conducted at the University of Georgia.

∗ Corresponding author. Tel.: +1 812 856 8223; fax: +1 812 856 8116.E-mail addresses: [email protected] (A.J. Hackenberg), [email protected] (E.S. Tillema).

0732-3123/$ – see front matter © 2009 Elsevier Inc. All rights reserved.doi:10.1016/j.jmathb.2009.04.004


2 A.J. Hackenberg, E.S. Tillema / Journal of Mathematical Behavior 28 (2009) 1–18

of 1/3 of 1/5 of a whole in relation to the whole. The purpose of this article is to investigate the ways in which students’whole number multiplicative concepts are involved in the construction of fraction composition schemes. To implement thispurpose, we examine how four sixth grade students participating in an 8-month teaching experiment composed fractionsmultiplicatively. Our overarching hypothesis was that students’ whole number multiplicative concepts would be criticalconstructive resources for, and thus useful explanatory constructs for understanding, students’ construction of fractioncomposition schemes. This hypothesis was confirmed in our study and will be demonstrated in our analysis.

2. Schemes, whole number multiplicative concepts, and fraction multiplication

The conceptual framework for our study is grounded in scheme theory and in students’ whole number multiplicativeconcepts. These concepts are derived from prior research based on analysis of the schemes and operations students con-structed in the areas of whole number and multiplication of whole numbers (e.g., Steffe, 1992, 1994; Steffe & Cobb, 1988;Steffe, von Glasersfeld, Richards, & Cobb, 1983). So, in this section we provide a brief overview of scheme theory; review priorresearch on fraction multiplication; outline our conceptual framework for students’ whole number multiplicative concepts;and give a conceptual analysis of fraction composition, relating it to students’ whole number multiplicative concepts and toprior research.

2.1. Schemes as an analytic tool

A scheme is a “package” of operations (mental actions) that are organized toward accomplishing a goal. A scheme consistsof three parts—an assimilatory mechanism, an activity, and a result (von Glasersfeld, 1995). When a student encounters aproblem situation, she or he uses current schemes and operations to interpret it. Such interpretations may trigger recordsof prior operating, and the student may come to recognize the situation (i.e., assimilate it) as one that involves a particulartype of activity (e.g., partitioning). The student might then engage in this activity, mentally or also materially, and producea result (e.g., some number of parts of a “whole”). If the result is consistent with the student’s expectations (i.e., what thestudent anticipates the result will be), then the scheme is likely to close. However, if the result differs from the student’sexpectation, this event opens possibilities for the student to review the situation and modify any of the three parts of thescheme. We call this type of modification an accommodation, and we consider accommodations to be acts of learning. In thisstudy, we focused on two specific schemes—a unit fraction composition scheme (UFCS) and a general fraction compositionscheme (GFCS). We discuss these schemes in more detail below.

2.2. Previous approaches to studying fraction multiplication

2.2.1. The importance of partitioning and conceptualizing unitsAlthough researchers prior to Olive (1999) and Steffe (2003) have not necessarily used schemes in studying fraction

multiplication, they have emphasized partitioning – a particular mental operation – and conceptualizing units as criticalto students’ construction of fraction multiplication. In this research, partitioning refers to the process of dividing a unit intoequal-sized parts (Kieren, 1980), either solely mentally or also materially, and units refer to different quantitative units (e.g., 1candy bar, 1 yard, 2/3 of 1 foot, etc.) formed in the process of making a composition of fractions. For example, Armstrong andBezuk (1995) and Streefland (1991) drew upon students’ partitioning and conceptualizing of units in developing, respectively,a sequence of tasks for middle grades students involving fraction multiplication and fraction division, and computationalrules for fraction multiplication with upper elementary grades students.

In a similar vein, Mack (2001) found that fifth grade students’ informal knowledge of partitioning and conceptualizing unitscould be a basis for solving fraction multiplication problems. Initially the six students in her study conceived of partitioningonly in relation to a unitary whole; they did not apply partitioning to parts of wholes, and so finding 3/4 of 2/3 was problematicfor them. As the study progressed, Mack found that all students learned to solve this problem by partitioning parts—forexample, partitioning each third into two equal parts, and taking three of those four parts. In other words, to find three-fourths the students partitioned the given amount into four parts total so that they could take three of those parts. However,three of the six students had difficulty solving problems that involved conceiving of fractional parts as composite units. Thesestudents could not produce 2/3 of 9/10 or 3/4 of 7/8. Taking 2/3 of 9/10 was more problematic for these students than taking3/4 of 2/3 because the given amount (9/10) consisted of nine parts—and conceiving of those nine parts as three equal partswas a challenge. So, although these students learned to partition parts, they did not appear to unite fractional parts intocomposite units.1

Olive (1999) and Steffe (2003) found that the operation of recursive partitioning was a critical resource in fourth and fifthgrade students’ construction of a UFCS. Based on their research, we define recursive partitioning as partitioning a partitionin service of a non-partitioning goal, such as determining the size of 1/3 of 1/5 of one yard in relation to the whole yard. Inaddition, Olive (1999) analyzed the ways in which two of these students decomposed and recomposed quantitative units in

1 Note that taking 3/4 of 7/8 is even more complex than taking 2/3 of 9/10, because seven parts (eighths) cannot be made into four equal parts withoutfurther partitioning—e.g., partitioning each of the eighths into fourths.


A.J. Hackenberg, E.S. Tillema / Journal of Mathematical Behavior 28 (2009) 1–18 3

relation to non-unit fractions in the construction of a GFCS. In our study, we viewed partitioning and the conceptualizationof units as critical components of students’ work on fraction composition problems. However, in contrast with Armstrongand Bezuk, Streefland, and Mack, we conceived of students’ partitioning knowledge and conceptualization of units to beintricately tied to students’ multiplicative concepts. This approach is similar to that of Olive and Steffe, although we buildon their work (1) by coordinating older (sixth grade) students’ construction of fraction composition schemes (or lack ofconstruction of such schemes) with their different multiplicative concepts, and (2) by highlighting the distributive reasoningrequired for constructing a GFCS.

2.2.2. The focus on fractions as operatorsAnalyses of one component of fraction knowledge, fractions as multiplicative operators, are certainly related to how

students learn to compose fractions multiplicatively (e.g., Behr, Harel, Post, & Lesh, 1992, 1993; Davis, Hunting, & Pearn,1993; Hunting et al., 1996). Behr and colleagues have conceived of fractions as operators in terms of exchange func-tions, specifically a duplicator/partition-reducer model, a stretcher/shrinker model, and a multiplier/divisor model. Forexample, in the duplicator/partition-reducer model, taking two-thirds of a quantity could involve duplicating the entirequantity two times (applying a two-for-one exchange), and then taking one-third of the result, or applying a one-for-three“reducer.” The partition-reducer could occur first, or the exchanges could occur simultaneously (i.e., as a two-for-threeexchange), but Behr and colleagues have demonstrated that all cases require the formation and re-formation of quan-titative units.2 Hunting and colleagues have conducted their studies with “input–output machines,” whereby studentscan input a number of discrete items, observe the output for a given fraction, and make judgments about the sizes offractions. A pivotal finding of their research was how heavily students’ success with such tasks depended on their knowl-edge of whole number multiplicative relationships. Although we did not take an “operator” approach in this study, weconcur that students’ whole number multiplicative concepts are critical in their construction of schemes for multiplyingfractions.

2.3. Students’ whole number multiplicative concepts

2.3.1. The first multiplicative concept: the coordination of two levels of units in activityWe view students’ whole number multiplicative concepts in the context of their units-coordinating activity (Steffe, 1992,

1994). The most basic units coordination involves inserting a composite unit into the units of another composite unit. Thusit involves two ranks, or levels, of units. For example, a multiplicative coordination of 7 and 3 involves inserting 7 units intoeach of 3 units (cf. Kamii & Housman, 2000). Students may make this coordination in the activity of solving a problem, orthey may be able to bring it to a situation prior to operating. For example, consider the following problem:

Vacation Problem. Susan went on vacation for 3 weeks. How many days was she on vacation?

A student who solves this problem by counting on by ones from the first 7 days, monitoring the number of times she orhe has counted to seven (sometimes with the aid of physical markers like fingers or marks on paper), is likely making thiscoordination of two levels of units in activity, in that the student must carry out the units insertion. We identify operatingin this way as the first multiplicative concept3 used in our research. This student has constructed an operation of recursionthat she or he can use to repeatedly insert the same number of units into a composite unit.

2.3.2. The second multiplicative concept: the interiorization of two levels of unitsA student who brings the units coordination of inserting 7 units into each of 3 weeks to the Vacation Problem prior to

operating is likely to demonstrate some “strategic reasoning” in her or his solution. For example, the student may know that7 and 7 is 14, and then 6 more is 20, and 1 more is 21. One of the central differences between this solution and the onedescribed above is that this student took the 7s as material to operate on, which enabled her or him to disembed 6 from7, treating it both as part of 7 and independent from 7. When a student is able to disembed part of a whole and operatewith the part, the student has constructed part to whole operations. These operations are not available to students who areoperating with the first multiplicative concept because they cannot yet operate on the results of their units coordinatingactivity. Students who can operate on the results of their units coordinating activity have interiorized two levels of units. Wetake this interiorization (the result of a reflective abstraction, von Glasersfeld, 1995) to be the multiplicative concept of thestudents, and it is the second multiplicative concept used in our research.

With this second multiplicative concept students can make three levels of units in activity. For example, when thesestudents solve the Vacation Problem, they create 21 days as a unit that contains 3 units, each of which contain 7 units. If

2 We give an example of the stretcher/shrinker model when we demonstrate our conceptual analysis of fraction composition later in this section. Themultiplier/divisor model is less relevant to the work of our study, so we do not discuss it.

3 When we talk of students “with” or “having” a multiplicative concept, what we mean is that this concept is part of our current models of the students’thinking, because based on our interactions and experiences with those students, we attribute certain operations, schemes, and concepts to them. However,consistently stating this distinction – that a concept is really something that we have formulated and that we find useful in understanding and interactingwith students – is cumbersome. So we resort to less precise language with the intent for it to carry this meaning.



these students are told that Susan extended her vacation for 9 more weeks, they could determine that Susan was gone 63more days and would produce 63 days as a unit of 9 units each containing 7 units. Furthermore, if the students were askedto determine the total number of days that Susan was gone, they could unite 21 and 63 to find that Susan was gone for 84days. But it would be a separate problem for them to determine that Susan was gone for a total of 12 weeks (i.e., they mightsolve this problem by strategically counting by 7s up to 84). In other words, because they can make three levels of units inactivity, they can produce 21 as a unit of 3 units each containing 7 units, and they can produce 63 as a unit of 9 units eachcontaining 7 units. They can also determine that 84 must be a unit containing 21 and 63. But in doing so, 21 and 63 do notretain the status of being a unit of 3 units each containing 7 units and a unit of 9 units each containing 7 units, respectively.They lose that status because the students cannot take a three-levels-of-units structure as material to operate with further.So when they unite 21 and 63, each loses its status as a unit of units of units.

2.3.3. The third multiplicative concept: the interiorization of three levels of unitsStudents who have interiorized three levels of units, the third multiplicative concept used in our research, can solve the

above extension of the Vacation Problem by keeping track of all levels of units. These students might reason that 21 is a unitof 3 units each containing 7 units and unite this with 63, a unit of 9 units each containing 7, to produce 84 as a unit of 12units, each containing 7 units. The students operate in this way because they take the coordination of three levels of units asgiven, prior to operating, which allows them to use this “structure,” or concept, in further operating. We note that a primarydifference among students who have interiorized two versus three levels of units is the material on which the students areoperating. A student who has interiorized three levels of units uses similar operations to a student who has interiorizedtwo levels of units, but the operations are applied to different material. This point is illustrated in the example above—thestudents are able to reason that 3 sevens and 9 sevens are both part of 12 sevens, and they can disembed either from 12sevens to use in further reasoning. This way of operating is similar to students who have interiorized two levels of units:They can reason that 7 contains 6 and 1 and so can disembed either from 7 to use in further reasoning. This observationsuggests the recursive nature of students’ reasoning in constructing multiplicative concepts with whole numbers.

2.4. Conceptual analysis of fraction composition

In our study, we approached fractions as quantities—specifically, as measurable extensive quantities, and most oftenas lengths.4 This approach was facilitated by working with the students in two microworlds, TIMA: Sticks and JavaBars(Biddlecomb & Olive, 2000), which were designed to foster students’ construction of operations involved in building fractionschemes (Biddlecomb, 1994).5 In the teaching episodes, conceiving of one-third meant being able to make one-third of anidentified unit length (represented by a designated stick or bar). Taking one-seventh of one-third meant being able to makethat amount in the microworld and determine its length in relation to the unit length. We now provide a brief conceptualanalysis of a fraction composition to orient the reader to the data analysis later in the paper. We emphasize that this conceptualanalysis is our own and that how students constructed ways of operating to solve fraction composition situations was theprimary focus of our investigation in the study.

In making one-third, and in making one-seventh of one-third, fractions are both an operation on a quantity and a measur-able extent. One-third operates on the original unit length and the result is one-third of the length; one-seventh operates onone-third of the unit length, and the result is a length that is one-seventh of the one-third length (Fig. 1). Our conjecture thatstudents’ multiplicative concepts with whole numbers would be a fruitful basis for their construction of fraction compositionschemes was in part based on the observation that determining what one-seventh of one-third is in relation to the wholelength involves inserting seven units into each of the three units in the three-thirds bar. However, the goal of determiningthe size of one-seventh of one-third in relation to the whole does not explicitly involve partitioning all thirds into sevenths.A student solving this problem who sees a “need” to partition, mentally or also materially, all thirds into sevenths is engagedin recursive partitioning, from our point of view (cf. Steffe, 2003). That is, this student is partitioning a partition in service ofa non-partitioning goal.

In our conceptual analysis, we conjectured that more is involved in a situation like making one-seventh of four-thirds,given that the thirds marks are maintained.6 An immediate question facing the student grappling with this situation is: Howdo I take one-seventh of multiple equal parts? Or, how do I make four equal parts into seven equal parts? We conjectured thatsolving those questions requires distributive activity at some level. Solutions that we call “reasoning with distribution” involvetaking one-seventh of each part, or taking one-seventh of one part and then taking that amount four times. We conjecturedthat students who reason with distribution in these situations would use their UFCS to determine that one-seventh of onepart (one-third) is one twenty-first. Therefore, one-seventh of four-thirds must be four twenty-firsts (cf. Olive, 1999).

4 We acknowledge that working with areas is common in extant curricula addressing fraction multiplication (e.g., Lappan, Fey, Fitzgerald, Friel, & Phillips,2006). However, we chose to focus on lengths because fraction compositions with lengths can require more complex units coordinations than fractioncompositions involving the cross-partitioning of areas (cf. Izsak, 2008; Izsak, Tillema, & Tunc-Pekkan, 2008).

5 We discuss the microworlds further in Section 3.6 If the thirds marks are not maintained, determining the size of the result in relation to the whole is not possible based on the operations on the

representations of the quantities.



Fig. 1. Making one-seventh of one-third.

As we have mentioned, this brief conceptual analysis differs from previous analyses of fractions as operators by Behr etal. (1992, 1993) and approaches taken by Hunting and colleagues (Davis et al., 1993; Hunting et al., 1996). The main reasonthat we do not focus on exchange functions is that we aimed for the students with whom we work to produce results offraction compositions out of reasoning about fractions as extensive measures and operations on those measures. However,some of our analysis can be interpreted with previous analyses. For example, Behr and colleagues’ stretcher/shrinker modelsuggests that applying one-seventh to a quantity means stretching the quantity by a factor of one and shrinking it by a factorof seven, where this stretching and shrinking is applied to any partition of the quantity. Our example of taking one-seventhof four-thirds by taking one-seventh of each one-third could be interpreted as one instance of the stretcher/shrinker model.Another instance would be taking one-seventh of four-thirds by taking one-seventh of each sixth in four-thirds, or of eachsixtieth in four-thirds, or of two one-thirds and the remaining six-ninths. In other words, the stretcher/shrinker model is avery general and powerful adult analysis! We appreciate this and other analyses of fractions as operators, because they helpto demonstrate the complexity of fractions and rational number knowledge. But rather than start there, we chose to buildour work from the ways and means of operating that we could attribute to our students.

3. Methodology and methods

The data for this article is drawn from a constructivist teaching experiment (Confrey & Lachance, 2000; Steffe & Thompson,2000) with four pairs of sixth grade students at a rural middle school in north Georgia from October 2003 to May 2004. Inthis kind of teaching experiment, researchers seek to understand and explain how students operate mathematically and howtheir ways of operating change in the context of teaching. Thus teaching, as a tool for investigating learning, involves posingproblems and activities that are tailored to the mathematical thinking of the participating students and that are intendedto test out researchers’ hypotheses about students’ ways of operating. For this reason, teaching experiment methodologywas appropriate to test our overarching hypothesis that students’ whole number multiplicative concepts would be a criticalresource for the construction of fraction composition schemes. Teaching practices include presenting students with problemsituations, analyzing students’ responses, and determining new situations that might allow students to construct morepowerful schemes and operations.

The four pairs of sixth grade students were invited to participate after selection interviews in September and early Octoberof 2003, which we discuss later in this section. Each of the authors taught two pairs of students twice weekly in 30-minepisodes for 2–3 weeks, followed by a week off. So, over the 8 months each pair of students participated in approximately 35episodes. Most episodes included the use of TIMA: Sticks or JavaBars (Biddlecomb & Olive, 2000). All sessions were videotapedwith two cameras, one focused on the students’ work, and one focused on the interaction. The video files were subsequentlymixed into a single video file, which enabled us to correlate the student work with an analysis of the student–teacherinteraction.

The authors were witness-researchers when not teaching, and there was another witness-researcher present at nearlyall episodes; these three individuals formed the core research team for the project.7 The function of witness-researchersin this methodology is to assist in videotaping, ask a clarifying question or present an occasional task during an episode,offer feedback on the teaching activities between episodes, and provide triangulation of perspectives in retrospective dataanalysis. In this section, we provide details about the student participants in the experiment, the computer tools with whichwe worked, and our procedures for data analysis.

7 The project continued for two more school years, with additional researchers joining the team at various points.



3.1. Participants

In September and October of 2003 the research team conducted 20-min selection interviews with 20 sixth-grade studentsout of a pool of approximately 100 students, all of whom had the same classroom mathematics teacher. The first authorobserved four of this teacher’s five mathematics classes to identify students to interview. Based on these observations, theresearch team conducted interviews with the intention of selecting a pair of students who had constructed, as their toplevel of operating, each of the three whole number multiplicative concepts outlined in the previous section.8 During theone-on-one interviews, which were not recorded, we used tasks that were aimed at helping us assess the students’ currentmultiplicative concepts, and in this process, to identify if and how students used key mental operations (e.g., partitioningand units coordinating) that we have outlined above. Upon completion of the interviews, we consulted with the teacher, theschool counselor, and the principal in order to select students who attended school regularly and were likely to remain inthe district for the duration of the experiment.

This article reports on two of the pairs of students, who happened to be two pairs of girls, all Caucasian.9 Based on theirselection interviews, we conjectured that Sara and Amber had both constructed at least the first multiplicative concept, thecoordination of two levels of units in activity. Although we suspected that Amber’s ways of operating were somewhat moreadvanced than Sara’s, we paired them because of our conjecture that Sara would likely make rapid progress. We conjecturedthat the other pair of girls, Bridget and Deborah, had interiorized three levels of units, the third multiplicative concept. Overthe course of the experiment, we were to learn that some of our initial conjectures were inaccurate. In particular, Sara didnot make rapid progress and remained at the first multiplicative concept, while soon into the experiment we could attributethe second multiplicative concept, the interiorization of two levels of units, to Amber. In addition, we eventually found thatwe could not attribute the third multiplicative concept to Bridget. Although Bridget’s ways of operating were more advancedthan Amber’s, we had to conclude that Bridget had not interiorized three levels of units, even by the end of the experiment.That not all of our initial interpretations of the students’ ways of operating were viable illustrates the importance we placedon continually refining our models of the students’ ways of operating.

3.2. Sticks and bars

The computer programs that we used throughout the experiment, TIMA: Sticks and JavaBars (Biddlecomb & Olive, 2000),were developed to allow students to enact key mental actions such as partitioning, iterating, and disembedding, in estab-lishing fraction microworlds.10 A central difference between the two programs is that in Sticks students work only on linesegments (sticks), whereas in JavaBars students can draw rectangles of varying dimensions. In this paper, we present prob-lems where the students only focused on the length dimensions of the bars they drew, so we use “sticks” language to describethe operations in both programs. Students can partition sticks “by hand” or use a button (Parts) and menu dial to createsome number of equal partitions. Students can Break sticks into their constituent parts, Copy sticks or parts of sticks, and Jointhose copies together. Using Pull Out, students can pull out a part of a stick from a whole stick without destroying the wholestick, and they can Repeat that part to make a new stick some number of parts long. With the button Fill, sticks or parts ofsticks can be colored. Students can also partition partitions and Clear sticks of all marks. Finally, students canMeasure sticksin relation to a pre-designated unit stick.

3.3. Analytic procedures

In teaching experiment methodology, analysis occurs in two phases—on-going and retrospective analysis (Steffe &Thompson, 2000). The purpose of on-going analysis is to develop working models of students’ current schemes and opera-tions, and to plan tasks prior to conducting a teaching episode. In the actual teaching episodes, a researcher may deviate fromthese plans in order to be responsive to students’ emerging ways of operating. Thus, tasks are often modified or generated inthe act of teaching, as well as prior to the teaching episodes. During our on-going analysis, the research team met regularlybetween teaching episodes. To document this process, after each teaching day we completed a teaching episode log thatincluded a brief summary of each episode, as well as central observations and questions. In addition, we kept a researchjournal to record impressions of each episode with students and notes from research team discussions.

During retrospective analysis, we constructed second-order models of the students’ ways of operating, and changes inthose ways of operating (Steffe & Thompson, 2000). A second-order model is a researcher’s constellation of constructs todescribe and account for another person’s ways of operating. Some of our central constructs were the concepts discussedthus far, as well as schemes and operations from other research (e.g., Steffe, 2002). However, rather than simply “look for”our current understanding of a particular concept or scheme in the video files of a student’s activity, we aimed to set our

8 In addition, we aimed to select a pair of students who had not yet constructed the first multiplicative concept, but we do not report on these studentsin this paper.

9 The other two pairs of students in the teaching experiment were boys. There was no intention to pair by gender.10 Iterating is a mental operation that refers to repeatedly instantiating an amount to make another amount; disembedding refers to taking a part out of

a whole unit without mentally destroying the whole.



Fig. 2. Sara’s activity in TIMA: Sticks on the Cake Problem.

concepts to the side in order to see the students’ ways and means of operating as fully as possible, and to open possibilitiesfor re-conceiving prior conclusions about the nature of particular schemes and concepts. This orientation is critical so thatthe constructs we use to model students’ activity are adaptive, and so that we do our best to respect the mathematics ofstudents as a “living” subject (cf. Steffe, 1992).

Constructing second-order models involved us in repeated viewing of video files, creating notes, and then notes on notes(Cobb & Gravemeijer, 2008), in order to identify moments that captured (1) regularities in students’ ways of operating, (2)changes in those ways of operating, and (3) constraints that we experienced with the students. Our central goal was tocreate second-order models to account for these regularities, changes, and constraints that were coherent across all of thesemoments for a single student. We also aimed for our models to be consistent across different students, including those ofprior research (Clement, 2000).

4. Analysis: Sara and Amber

At the beginning of the teaching experiment, Sara and Amber had not constructed a UFCS. So in this section we examineSara and Amber’s activity in situations intended to engender this construction. By contrasting the two students’ ways ofoperating in these situations, we highlight the relationship between the students’ whole number multiplicative conceptsand their activity towards constructing a UFCS.

4.1. Sara’s activity on the Cake Problem

On 18 February, Sara had partitioned a unit “cake” (a stick) into 15 equal parts, and pulled out one of the parts. Ateacher–researcher then posed the Cake Problem to Sara. This was the first time she had encountered a problem intendedto provoke the construction of a UFCS.

Cake Problem, Task 1: You decide to share that piece (one-fifteenth) of cake between two people. How much of thecake would one person get?

Sara partitioned the piece (one-fifteenth of the cake) into two equal mini-parts and, at the request of the teacher-researcher, pulled out one of the mini-parts (Fig. 2). The teacher-researcher asked Sara if she could figure out the fractionalamount that one of the mini-parts was of the whole cake.

Protocol I: Sara works on the Cake Problem on 18 February.11

S: One fifteenth.T3: Well no, one person got half of one fifteenth.S: Yeah, half of one fifteenth.T3: How much of the whole cake is that?S: A half of a fifteenth.T3: That’s right. If you had to make a fraction for that, just one fraction—S [interrupts T3 and repeats]: A half a fifteenth![Amber says she wants to help Sara, but T3 asks her to wait until he is done asking his question to Sara.]T3 [to S]: You are exactly right, but I want to know if you had to give just one fraction, like a fiftieth or something likethat, what would a half of a fifteenth make? How much of the whole cake would that make? [Sara shakes her headand shrugs her shoulders, indicating that she is not sure. Amber wants to respond, but T3 asks her to wait to give Saraa chance to figure out the problem.] I bet Sara can find a way using the computer.S [works for 1 min in the computer microworld, trying several different buttons]: I have no idea.

The teacher-researcher continued to encourage Sara, asking her how many of the “mini-parts” it would take to make thewhole cake. Sara thought for 6 s and then said she could copy one of the mini-parts to see how many of them would make up

11 In the protocols, S stands for Sara, A for Amber, B for Bridget, and D for Deborah (all pseudonyms). T1, T2, and T3 stand for teacher-researchers in thisorder: the first author, second author, and a third member of the research team who was a witness-researcher (identified as W) most of the time. NormallyT2 taught Sara and Amber. On February 18th, T2 was ill, and so T3 taught the episode shown here. Comments enclosed in brackets describe students’nonverbal action or interaction from the teacher-researchers’ perspectives. Ellipses (. . .) indicate a sentence or idea that seems to trail off. Four periods (. . .)denote omitted dialogue.



the whole cake. She covered the original cake with copies of the mini-part but left spaces in between each copy. When shegot to the end of the whole cake, she had made 18 copies of the mini-part and concluded that 18 mini-parts would make thewhole cake. With prompting from her partner Amber, Sara joined the 18 parts together and found that the “new” cake wasshorter than the original cake. So she continued to copy and join mini-parts until she had made a cake the same size as theoriginal cake. Then she counted the number of mini-parts she had used and determined that, since there were 30 mini-parts,each mini-part would be one-thirtieth of the original cake.

In this situation, Sara did not initially seem aware that there would be a new fraction name for the smallest part of thecake she had produced, or that this situation might involve multiplication. The teacher-researcher tried to throw doubt onher answer of “one fifteenth” by suggesting that the amount was “half of one fifteenth,” not one-fifteenth. She agreed withthe teacher-researcher that the piece could be named half of one-fifteenth, but then seemed unsure about how she mightname her piece using only one fraction name (“I have no idea”). She eventually determined that she could make copies ofher piece to figure out how many of them would make up the whole cake, but she did not relate this activity to her activityof partitioning the first piece into two parts. In other words, Sara did not recursively partition in this situation. If Sara hadrecursively partitioned, she would have partitioned each of the 15 pieces into two equal parts. In doing so, she would haveinserted two units into each of 15 units, which would have involved the units-coordinating activity of her whole numbermultiplying scheme, to create a whole cake composed of 30 units. Instead, as described above, Sara found that her piece wasone-thirtieth of the cake by reconstituting the whole cake with a mini-part, not by recursively partitioning the 15/15-stickso that each fifteenth contained two equal parts.

4.2. Attempts to engender a recursive partitioning operation

Because Sara solved the Cake Problem without using her multiplying scheme, the teacher-researcher posed a new problemto her during the same teaching episode.

Sub Problem, Task 2: Can you share this sub sandwich fairly among 17 people? Now each person shares their piece withtwo other people (three people total share each piece). Could you figure out how much one little piece is of the wholesandwich?

Sara immediately partitioned each of the 17 parts into three equal parts. The teacher-researcher encouraged her to findout how many pieces there were, and she wrote 17 times 3 in a vertical format on a piece of paper, computing the resultwith her standard computational algorithm for whole number multiplication. She said there would be 51 pieces in the subsandwich and that one piece would be “one fifty oneth.”

In contrast to her activity in the Cake Problem, here Sara partitioned each seventeenth into three equal parts, and thisactivity in the computer microworld activated her whole number multiplying scheme. We infer that by partitioning each ofthe 17 parts of the sandwich into three equal mini-parts, Sara knew that she had made three mini-parts 17 times. After sheused her computational algorithm for multiplication, Sara established the sandwich as a unit of 51 units, and she used herfraction scheme to determine the size of one of the parts in relation to the whole. However, a clear difference between theSub and Cake Problems is that the former indicates that partitioning each part of the initial partition should occur. So, wecannot infer that Sara engaged in recursive partitioning in this situation. Posing the Sub Problem in this way was the teacherresearcher’s attempt to test whether this situation might be a situation involving multiplication for Sara, and whether thistype of problem might engender a recursive partitioning operation in future situations.

During the rest of the experiment (3 months), Sara correctly named a single unit fraction as the solution to problems likethe Sub Problem, but she did not do so in problems like the Cake Problem. In situations similar to the Cake Problem, Sarasometimes called the result one-half of one-fifteenth, or sometimes called it one-sixteenth, because the original cake wasbroken into sixteen parts and she had one (albeit the parts were not of equal size). Because we experienced this constraintrepeatedly with her, we infer that partitioning a partition in service of a non-partitioning goal was unavailable to Sara duringthis time. In other words, she had not constructed recursive partitioning, and so she did not construct a UFCS.

4.3. Explaining this constraint

We explain Sara’s way of operating in the Sub and Cake Problems based on her multiplicative concept. In the Cake Problem,we infer that Sara inserted two units into the first unit of the cake. However, she did not continue to insert two units into eachof the 15 units. She did not take this and similar situations as involving multiplication because she actually had to carry outthe insertion of units into other units, producing the results of the units coordination as part of the experiential situation. Inother words, Sara had to make two levels of units in activity—she had constructed the first multiplicative concept we outlinedpreviously, and she could not bring the insertion of units into units to a situation in service of another goal.

In the Sub Problem, which explicitly stated that Sara was to partition each of the 17 parts into three, Sara did insert threeunits into each of 17 units. Moreover, she was aware she had created a unit of three units 17 times based on her partitioningactivity, which is why she knew that she could multiply 17 times 3 to find the total number of pieces. Once she figured outthe total number of pieces in the whole cake, she could constitute the whole cake as a new unit of units structure (in thiscase, a unit of 51 units), which enabled her to name the fractional size of one of the mini-parts in relation to the whole cake.When she constituted the cake as a new unit of units structure, she no longer treated the first unit of units structure (17 units



of 3 units) as if it were part of the experiential situation.12 In short, because Sara had constructed the first multiplicativeconcept, she was able to solve problems like the Sub Problem, in which making the units coordination was an explicit goal.

This explanation suggests that the construction of a recursive partitioning operation requires the interiorization of at leasttwo levels of units (cf. Steffe, 2003). That is, it involves taking a partition of one unit (e.g., one-fifteenth) that creates a unitof units as indication that all of the units (e.g., all of the fifteenths) could be partitioned, without having to produce all of theunits of units by enacting the partitioning. So, in recursively partitioning, determining the number of parts in the whole andidentifying the fractional size of the part in relation to the whole result from the “same” units-coordinating activity, and seemto occur simultaneously. In contrast, for Sara, determining the number of parts in the whole and identifying the fractionalsize of the part in relation to the whole appeared to be separate but associated problems. To solve them, she seemed to useher multiplying and fraction schemes sequentially because each scheme involved her in making a new unit of units structure,which she had to produce in activity. This sequential use of schemes meant that taking one-third of one-seventeenth andconcluding that a mini-part was one fifty-first were not part of the same scheme. Therefore, constructing a UFCS seemedoutside of her zone of potential construction (Steffe, 1991) at this time.13

4.4. Amber’s solution of the Cake Problem

On 11 February, 1 week prior to Sara’s work on the Cake Problem, a teacher-researcher (the second author) posed anidentical problem to Amber. This was the first time Amber encountered a problem intended to provoke the construction ofa UFCS. Sara was sick on this day so Amber worked by herself (see Fig. 2 above).

Protocol II: Amber’s solution of the Cake Problem on 11 February.A: Okay mine was one fifteenth, so you would have half of one fifteenth. So it’d be. . .[continues to think for 15 s,indicating that she believes there may be another fraction name for the part]. I am not sure.T2: Could you use this piece [points to half of the 1/15-stick] to figure it out?A [drags the half of the 1/15-stick over to the first piece in the whole cake and checks that two can fit into the first1/15 of the cake]: Well you could fit two of his pieces in one, so. . .[Subvocally Amber says fifteen times two. Thewitness-researcher does not hear Amber and spends 15 s encouraging her. Then 15 s pass in silence.]T2: I think I heard you say it. Do you remember what you said?A [is not sure what T2 is referring to, and then seems to realize]: Oh, I said fifteen times two.T2: Oh, and why would it be fifteen times two?A: Because two of his pieces can fit into one-fifteenth and there’s fifteen pieces in the whole cake . . .soW: So how much would it be?A: It’d be. . . [stares at the ceiling in concentration] thirty.

Shortly after this point in the protocol, and in interaction with the teacher-researcher and witness-researcher, Amberstated that her mini-part would be one-thirtieth. Then at the request of the teacher-researcher, she used her mini-part tocheck how many times she had to copy it to make the whole bar. Although partitioning the first fifteenth into two parts didnot immediately call forth Amber’s units coordinating activity, using her piece to measure the first one-fifteenth did. Onceshe used her mini-part to measure the first fifteenth, she seemed to imagine that she could insert two units into each of the15 units, but she did not have to actually carry out this insertion in order to take it as part of the experiential situation. Thatis, she knew that multiplication was involved “because two of his pieces can fit into one-fifteenth and there’s fifteen piecesin the whole cake.”

Amber’s activity differed from Sara’s in two significant ways. First, Amber did not have to insert two units into each of the15 units in order to take this insertion as part of the experiential situation. So she could anticipate partitioning a partition inservice of a non-partitioning goal (finding out the fraction name of the mini-part) without actually having to carry out thispartition. For this reason, we attribute a recursive partitioning operation to Amber. Second, for Amber, copying the mini-part30 times served as a way to confirm her response of one-thirtieth, whereas for Sara, copying the mini-part was the way thatshe established the name for the mini-part. Amber’s copying activity appeared to be another way to establish the name forthe mini-part based on her concept of a unit fraction as one of so many equal parts of a whole. In contrast, Sara’s only wayto find the name of the fraction in the Cake Problem was to establish it as one of so many equal parts.

As noted, after Amber determined that 30 parts were in the whole cake in Protocol II, she did not immediately arrive ather final answer of one-thirtieth until she interacted further with the teacher-researcher and witness-researcher. However,because Amber appeared to initiate recursive partitioning in the situation, we infer that the situation had significant meaningfor her. We infer that Amber constructed recursive partitioning because she created a three-levels-of-units structure as aresult of her activity. That is, she treated the whole cake as a unit that was partitioned into 15 units, each of which she couldmentally partition into two units. Because she created a three-levels-of-units structure, she was able to compare half of the

12 This claim does not mean that Sara was not aware that she had just multiplied seventeen times three. However, it does mean she did not maintain thestructure that this multiplication implied.

13 The zone of potential construction (Steffe, 1991) is a time-sensitive concept for ways of operating that a student might construct based on the student’scurrent schemes and operations.



one-fifteenth piece to the whole cake by figuring out how many of these parts made up the whole cake. Amber’s coordinationof three levels of units in her activity in the Cake Problem is what we refer to as experientially creating three levels of units.We infer that her ability to conceive of the cake as a unit of 15 units each containing two units during her activity was anessential basis for her ability to view one-half of one-fifteenth and one-thirtieth as the same amount. Amber’s subsequentactivity with similar problems during February, March, and April indicates that her solution to the Cake Problem marked arelatively permanent way of operating with fraction composition—i.e., the initial construction of a UFCS.

4.5. Amber’s progress

4.5.1. Taking a non-unit fraction of a unit fractionDuring the rest of February and early March, the teacher-researcher investigated whether Amber could extend her UFCS

into taking a non-unit proper fractional amount of a unit fraction. Samples of the problems the teacher-researcher posed toAmber are the following:

Task 3: You are at a party and a cake is cut into nine pieces. Two people show up to the party late and you decide toshare your piece of cake with them. What fraction of the whole cake do the latecomers get together?

Task 4: Can you make 2/5 of 1/3 of the cake? How much is that of the whole cake?

Amber solved these problems by partitioning the cake into fractional parts, partitioning one fractional part into therequired number of mini-parts, pulling out the number of these mini-parts requested in the problem, and identifying thefractional size of the result in relation to the whole cake. For instance in Task 3, Amber partitioned the first ninth of the cakeinto three parts, pulled out two of them, determined that each one was one twenty-seventh of the original cake, and thereforeconcluded that the latecomers would get two twenty-sevenths of the whole cake. Although Tasks 3 and 4 do not requiresignificantly different ways of operating, we posed Task 4 to engender Amber’s awareness of making a fraction composition.We achieved this goal through the explicit use of fraction language in problems.

4.5.2. Taking a unit fraction of a non-unit fractionThe teacher-researcher wanted to investigate whether Amber could construct a more general fraction composition

scheme, so on 10 March he asked her to take a unit fraction of a non-unit fraction, Task 5.

Task 5: Can you make 1/3 of 2/5 of that cake and find out how much of the whole cake that is?

As we have outlined, such problems are more challenging than Tasks 3 and 4 because they require students to take a unitfractional part of a quantity consisting of multiple fractional parts.

On 10 March, Amber solved Task 5 by partitioning each fifth of the cake into three equal parts and pulling out one of theparts, stating that she got one-fifteenth of the cake. The teacher-researcher asked her if she had made one-third of one-fifth orone-third of two-fifths. Amber indicated that her solution was one-fifteenth and that she had taken one-third of two-fifths.Her solution suggests that she did not create two-fifths as a unit of two units. Rather, she seemed to establish the whole baras a unit of five units. So when she partitioned each fifth into three equal parts, we infer that she did not establish that therewere two one-fifth units, each of which contained three units. Instead, she appeared to reason that there were five one-fifthunits, each of which contained three units, and one of these parts was one-fifteenth of the whole bar.

During the rest of the experiment, the teacher-researcher posed more such problems to Amber, including one problemwhere he explicitly coached Amber through a solution of taking a unit fraction of a proper non-unit fraction. Then on 12May, the last day of the experiment, he again posed Task 5 to Amber. She solved the problem in the same way she had on 10March. Thus we conclude that constructing a GFCS was outside of her zone of potential construction at this time. Further, weconjecture that producing three levels of units in activity is not sufficient to construct a GFCS because constructing such ascheme involves operating on a three-levels-of-units structure. We articulate and test this conjecture by turning to the otherpair of students.

5. Analysis: Bridget and Deborah

At the start of the teaching experiment, both Bridget and Deborah could solve problems like the Cake Problem (Task1), as well as Tasks 3 and 4. That is, both had constructed at least a UFCS and could extend that scheme to taking non-unit proper fractions of unit fractions. We have already shown that a recursive partitioning operation is at the heart ofconstructing a UFCS, and that students’ multiplicative concepts are closely tied to the construction of such an operation. Inthis section we demonstrate that distributive reasoning is required to construct a GFCS, that distributive reasoning requiresthe interiorization of three levels of units, and that consistently reasoning with distribution is no easy feat.

5.1. Bridget’s multiplicative coordinations within a fractional unit

During the month of March, the girls and their teacher-researcher (the first author) worked on a variety of fractioncomposition problems. When solving these problems, Bridget tended to coordinate parts multiplicatively within a particular



Fig. 3. Each fourth of 3/4 of a unit yard partitioned into five equal parts.

fractional unit, but not in relation to the unit bar. Her solution to the Ribbon Problem near the end of the March 3rd episodeexemplifies these characteristics.

Ribbon Problem, Task 6: I have 3/4 of a yard of ribbon. My sister needs 2/5 of my piece. Can you make it and tell howmuch she needs?

Bridget said the problem was easy. She copied the unit yard and colored three-fourths of it. Then she expressed an ideaabout “dividing into fifteen.” After some discussion with Deborah and the teacher-researcher, she pulled out one-fourth ofthe unit yard, used it to make three-fourths, and partitioned each fourth into five equal parts (Fig. 3).

Protocol III: Bridget’s coordination of whole numbers of parts on 3 March.B: And then, that would be fifteen, three, she’d have six.T1: Six! Six what?B: Six—[pauses and laughs] six fifteenths.D [almost simultaneously with B]: Six twentieths.B: Or twentieths or something.D: Six twentieths of the whole bar, six fifteenths of yours.B: Yeah. Six fifteenths of yours, that’s what I mean.T1: Oh. Six fifteenths of mine. And why is it six twentieths of the whole bar?B: Because they’re all divided by five.D [simultaneously with B]: Because you have to add five more. Because you didn’t have the whole yard.

As shown in Protocol III, Bridget’s intention to make 15 parts total allowed her to make five equal parts out of a three-partbar because 15 could be divided by both 5 and 3. By intending to make 15 parts, we infer that she could insert 5 units intoeach of the 3 units of the 3/4-bar and then find one-fifth of the bar because 15 could be divided by 5. Thus she operatedwith three levels of units in her activity with respect to the 15-part bar: She created it as a unit of three units each containingfive units. Then she could reorganize the 15 parts to view it as a unit of five units each containing three units. So, Bridget’sactivity here contrasts with Amber’s activity in two ways: (1) Bridget established three-fourths as a unit separate from thewhole, while Amber did not; (2) Amber made one three-levels-of-units structure in activity, while Bridget made one suchstructure, followed by another.

However, in creating the second three-levels-of-units structure, Bridget did not seem to maintain the first structure. Thatis, Bridget’s naming of the bar as six-fifteenths is evidence that she did not retain the view of the 15 mini-part bar as a unitof three units each containing five units. Even though a 4/4-bar was in her visual field, it seemed as if, at the moment, her“mathematical world” was the 15 mini-part bar that she had made from the 3/4-bar. In a sense, the bar was no longer a 3/4-bar for her, but a 15/15-bar. Retaining a view of this bar as a unit of three units each containing five units would have openedthe possibility for her to name the resulting bar six-twentieths, precisely because she would have maintained awarenessof the three one-fourths of the unit yard into which she was inserting units of five. Bridget had a constructed a recursivepartitioning operation (Hackenberg, 2005), so she was well aware that partitioning some or all of the fourths of a bar intofive equal parts would yield twentieths. But in this situation that way of operating was not available to her—she could notseem to step “outside” of her mathematical world (the 15/15-bar) in order to view it in relation to the original whole. Thusit was as if the “top” level – the 15/15-bar as a unit – was a bound for her ways of operating, and she could not take it as agiven in relation to a still larger entity (the original whole).

Once Bridget had created the 15/15-bar, she used her fraction scheme to determine that one-fifth was 3 parts out of 15,and two-fifths was 6 parts out of 15. But for her these parts were not integrally related to the original whole, which indicatesshe did not take the original whole as a unit to which the 15/15-bar stood in relation as a unit. As a result, making the fractioncomposition did not include determining the length of the composition as an outgrowth of making it. She could interpretDeborah’s response of six-twentieths in reviewing the bar that she had made, because she could use her units coordinatingactivity to retrospectively organize the whole bar as a unit of four units each of which could contain five units. However,throughout the episodes in which we worked on fraction compositions, she did not ever independently produce such aresponse, in which she named the composition in relation to the unit bar.

5.2. Explaining this constraint

Our main explanation for Bridget’s way of operating to solve these problems involves the multiplicative concept we couldattribute to her. There was pervasive evidence throughout the teaching experiment that Bridget had not interiorized three



levels of units (Hackenberg, 2007, submitted for publication). We had consistent evidence that she had interiorized two levelsof units and could make three levels of units in activity. In fact, in working on fraction composition problems, Bridget madetwo different three-levels-of-units structures in activity, but when she made the second one (e.g., the 15/15-bar as a unit offive units each containing three), the first one (e.g., the 15/15-bar as a unit of three units each containing five) “dropped away.”In this sense Bridget operated in a way analogous to Sara, who made two different two-levels-of-units structures in activity,where the first one (17 units each containing 3 units) dropped away once she had made the second one (a unit of 51 units).

When students make three levels of units in activity, the “top” level is not a unit that can be used in further operating—itis not a unit they can take as given, but is produced in the process of acting. So, when Bridget made two different three-levels-of-units views in activity, she could not take that coordination as a unit in relation to the unit bar: Doing so requiresbeing able to take the three levels of units as given. Yet when Deborah said that the result was six-twentieths, Bridget madesense of this response by using her units coordinating activity. She focused on the unit bar as a unit of four units, each ofwhich could contain five mini-parts, and named the six mini-parts in relation to the twenty parts that would fill the entireunit bar. It is quite likely that at this point, the three-levels-of-units structures that had previously dominated her attentionalso dropped away; keeping track of all of these coordinations is a significant cognitive load for students who have not yetinteriorized three levels of units.

The ways of operating we have described here were corroborated in future episodes. For example, on 10 March Bridgetsolved the problem of determining 2/5 of 8/7 in much the same way as described here. Thus Bridget’s composition of twonon-unit fractions seemed to be a “two step” process: make the composition by multiplicatively coordinating parts within theinitial fractional quantity, and then, if she was prompted, she would determine the measure of the result in relation to the unitbar as a separate problem. So, at best we can claim that Bridget constructed a units-coordinating scheme with fractions, but nota GFCS. A units-coordinating scheme with fractions does not involve distributive reasoning to make fraction compositions.

5.3. An attempt to bring forth distribution

During the first two episodes of March (March 1st and 3rd), Deborah also did not seem to be reasoning with distributionto solve fraction composition problems. However, she repeatedly demonstrated distributive reasoning with whole numbermultiplication throughout the experiment,14 and we could clearly attribute the interiorization of three levels of units to her(Hackenberg, 2007, submitted for publication). Since both girls did not seem to be using distributive reasoning in solvingfraction composition problems, the research team discussed problems that might help provoke this possibility. As a result,during the 8 March episode the teacher-researcher posed problems in which the fractional quantity changed color (or flavor)every unit fractional amount (cf. Olive, 1999). For example:

Changing Color Ribbon Problem, Task 7: I have 7/9 of a yard of ribbon, but every ninth it changes colors. Make this ribbon.My friend needs 2/3 of what I have, and she wants all of the colors. Make a picture of my friend’s ribbon, and tell howmuch of a yard she has.

Initially Bridget seemed stumped by this question, and Deborah found a result of 14/27 by multiplying denominators andmultiplying numerators. To reorient them both, the teacher-researcher asked Bridget how she would make two-thirds of justthe first part (the first 1/9) and reiterated to Deborah that she had to make the new ribbon’s length, showing all of the colors.Both girls had the idea of partitioning each of the seven parts into three equal parts. When the teacher-researcher askedwhat one of those mini-parts was of the unit yard, Deborah instantly said one twenty-seventh. Bridget hesitated, then agreed.Deborah then made the new ribbon by pulling out one of the mini-parts and repeating it to make a 14 mini-part bar. She thencolored every two mini-parts of the 14 mini-part bar. When the teacher-researcher asked Bridget how she knew that Deborahneeded 14, Bridget said, “Twenty-one divided by three is seven, and seven is one. Seven times two is 14.” Her explanationis consistent with our analysis of how she used her units-coordinating activity within a fractional unit to solve fractioncomposition situations. Near the end of the work on this problem, Bridget still seemed uncertain about whether the resultwas 14/21 or 14/27, and Deborah still maintained that she had found her result by multiplying denominators and multiplyingnumerators. So it was not at all clear that using color provoked distributive reasoning: Bridget was still coordinating partsmultiplicatively within the given fractional amount, and Deborah seemed to be making the bars to illustrate her algorithmiccomputation, rather than to generate reasoning in the situation.

However, at this point a witness-researcher (the second author) asked what may have been a key question for Deborah:“How many thirds did you take of that first ninth? And the second ninth?” Deborah immediately said two, and then bothgirls said that they made two-thirds of each ninth. We moved onto a similar problem, starting this time with 5/7 of a yard ofribbon that changed color every seventh, with the goal of making 3/2 of it and showing all of the colors. Bridget solved theproblem by partitioning each of the five sevenths into two equal mini-parts and said “you would add five.”

Protocol IV: An attempt to bring forth reasoning with distribution on 8 March.B [after making a 5 mini-part bar, shown in lower right of Fig. 4]: You would add that [to the 10 mini-parts alreadyshown above].

14 For example, in February, to determine 104 divided by 8, Deborah reasoned that it had to be 13 because 10 times 8 was 80, and then there was only 24more, and 3 times 8 was 24.



Fig. 4. Five mini-parts to be added to the 10 mini-parts.

T1: Okay, can you make the whole new ribbon [sweeping a finger across the screen] down there?B: The whole ribbon?D: Yeah, just make it.[Bridget continues to repeat the mini-parts to make a 15 mini-part bar.]D: Now you just color it.B: Yeah, [clicking on the Fill button] this is going to take forever. Do I got to color it in order [of the colors on theoriginal 5/7-yard bar]?T1: Yeah, yeah, let’s color it in order.B: Okay. [She selects ochre to match the first color of the 5/7-yard bar and colors the first mini-part of the 15 mini-partbar ochre. Then she looks as if she is going to select Fill again.]D: No-no-no.B: But how do I make it even?D: But there’s three each, isn’t there? So there would be one, two, three [pointing at the first three mini-parts] is thatcolor; then one, two, three would be the next color.B [leaning in]: How many are there in all? [She counts the parts of the 15 mini-part bar subvocally by twos.] Fifteen.Okay, yeah, there’d be three. [She proceeds to color the bar accordingly.]

This protocol provides evidence that Deborah may have operated distributively in this situation, while Bridget clearlydid not. Bridget did not say how she determined that she needed to add five mini-parts to the original ribbon, but thefollowing interpretation is consistent with her previous ways of operating, as well as her conceptions of improper fractions(Hackenberg, 2007): Half of the 10 mini-parts in the 5/7-bar was five mini-parts, and to make three-halves required “addingon” one-half to the whole. When asked to color the 15 mini-part bar, Bridget had to count the total number of parts she hadmade and, we infer, divide 15 by the five colors in order to confirm that the color in the new ribbon should change everythree mini-parts.

In contrast, Deborah seemed to know – but we do not know for sure how she knew – that every three contiguous mini-parts in the new ribbon would be a different color. A potential explanation, which becomes quite plausible with furtherevidence below, is that she conceived of the new ribbon as three-halves of each of the sevenths in the old ribbon. Thus shehad no need to re-count the total number of mini-parts in the new ribbon and divide. When I asked about the size of themini-parts, as with most of the fraction compositions the girls made together, Deborah immediately stated that they wereone-fourteenths, while Bridget hesitated and then agreed.

5.4. Deborah reasoning with distribution

Although the teacher-researcher’s attempts to provoke reasoning with distribution on 8 March appeared to have littleimpact on Bridget’s ways of operating to solve fraction composition problems, it is possible that they opened possibilities forDeborah that she had not previously considered. For example, in the next episode on 10 March, Deborah solved the followingproblem in a rather sophisticated manner:

Complex Fraction Composition Problem, Task 8: Make 11/9 of a yard of ribbon. Your friend wants 4/3 of that piece. Makeit and tell how much your friend needs

Deborah made an 11/9-bar and divided the first one-ninth into three equal parts. Then she pulled out one small part andrepeated it to make 44 parts (Fig. 5). When the teacher-researcher asked how much one of those parts was of the unit yard,Bridget responded “One thirty-third.” Deborah insisted that one thirty-third was wrong. After 5 s, she stated that it was onetwenty-seventh, so the new piece of ribbon was 44/27 of a yard. In explanation of how she had made the composition, she



Fig. 5. Deborah making 4/3 of 11/9 of a yard.

said, “take one out of each of those boxes [the eleven one-ninths] and that equals eleven and that would be one-third, so Imultiplied that by four ‘cause she needs four.”15

This explanation is significant for two reasons. First, Deborah indicated that she had reasoned distributively to make aunit fractional amount of the given fractional quantity. That is, to make one-third of the entire eleven-ninths of a yard sheimagined taking one-third from each of the eleven ninths. So 11 small parts was one-third of the whole 11/9-yard bar. Thisway of operating is pivotal in the construction of a GFCS because it allows determination of the size of the small parts tobe embedded in the scheme for making the composition: One-third of each ninth is one twenty-seventh, so together the11 mini-parts is eleven twenty-sevenths. We conjecture that Deborah’s use of an explicit distributive operation in makingone-third of eleven-ninths of a yard is a central reason why she emphatically rejected Bridget’s idea that each part was onethirty-third and concluded that each part was one twenty-seventh.

Second, Deborah used her fraction scheme to make four-thirds by iterating one-third four times, but she did not use thetools of JavaBars to reflect this construction (i.e., she did not repeat 1 part to make 11 parts and then repeat the 11 parts fourtimes). In fact, she did not often independently use the tools of JavaBars to demonstrate her structural ways of operating withfraction composition problems, perhaps because she found making the bars somewhat tedious—or perhaps because makingthe bars remained largely an illustration of the end results of her mental activity. Nevertheless, from Deborah’s explanationof her solution of this complex fraction composition problem, we infer that she had made an initial construction of a GFCS.

Unfortunately, Deborah’s clarity in making fraction compositions was not consistent across the next few episodes. Weattribute this inconsistency in part to her personal preferences for using the standard fraction multiplication algorithm andto our own difficulty as a research team in adequately organizing fraction composition problems so that Deborah might havefound them more useful (Hackenberg, 2005). We also attribute it to her not being sufficiently aware of the pattern of herdistributive activity in a way that would make it possible for her to recognize and use the power of this pattern.

6. Discussion

6.1. Constructing a UFCS

In this study, we found that the interiorization of two levels of units (the second multiplicative concept) is necessaryfor a student to construct recursive partitioning, which in turn is the basis for constructing a UFCS. As demonstratedby our analysis of Sara and Amber’s activity, the second multiplicative concept is necessary because partitioning a par-tition in service of a non-partitioning goal, or anticipating the partitioning of a partition without actually carrying itout materially, means a student must be able to take the coordination of two levels of units as a given—as a tool forfurther reasoning. When students have constructed only the first multiplicative concept, they are “in” the activity ofmaking the units-coordination; they cannot “stand above it” in order to use it in further reasoning. For example, in theCake Problem, Sara could partition one of the fifteenths into two parts, but this activity did not help her anticipate thatpartitioning each of the other fifteenths might help her solve the problem situation. She needed to encounter prob-lems that explicitly stated that partitioning each of the units should be carried out for her multiplying scheme to beactivated.

Moreover, recursive partitioning seems to entail making three levels of units in activity (cf. Steffe, 2003), which is notpossible without the interiorization of two levels of units. Making three levels of units in activity is critical because, asSteffe notes, “producing a recursive partitioning implies that a child can engage in the operations that produce a unit ofunits of units, but in the reverse direction” (p. 240). For example, to solve the Cake Problem, Amber created the wholecake as a unit of fifteen units, each of which contained two units. In doing so, we claim that she learned to view prob-lems like the Cake Problem as situations of multiplication, which is supported by her solutions of similar problems duringthe rest of the experiment. However, Amber could not take a three-levels-of-units structure as a given prior to operating,nor could she operate further with a three-levels-of-units structure. So our analysis does not lead us to conclude that theinteriorization of three levels of units is necessary for the construction of recursive partitioning, or for the construction ofa UFCS.

15 Note the slight difference between Deborah’s distributive reasoning in this problem and the previous problem. In the previous problem Deborahappeared to take three halves of each one-seventh to make three-halves of five-sevenths. Here Deborah made one-third of each one-ninth and then tookthe resulting amount four times to make four-thirds of eleven-ninths.



6.2. Constructing a GFCS

However, we did find that the interiorization of three levels of units (the third multiplicative concept) is necessary fora student to construct a GFCS. As demonstrated by our analysis of Bridget and Deborah, the third multiplicative conceptis necessary for two reasons. First, students who are making the composition of a non-unit fraction, say 2/5, of a non-unitfraction, say 3/4, have to establish three-fourths as a unit of three units and then insert units into each of those units—whichat a minimum requires making three levels of units in activity. However, to maintain a view of 3/4 in relation to the wholeunit, 4/4, requires taking this three-levels-of-units coordination as material on which to operate, because a student mustrelate the three units, each of which contain five units, back to the unit bar. Bridget used her fraction scheme to operate on the15/15-bar that she had created from the 3/4-bar by making three levels of units in activity. But in doing so, the 15/15-bar didnot maintain its status as a three-levels-of-units structure. Taking such a structure as material for further operating opens theway to conceiving of the mini-parts as fifths of fourths, or twentieths, rather than fifths of thirds, or fifteenths, as Bridget did.

Second, taking the three-levels-of-units coordination within the 3/4 quantity as given opens the possibility for operatingfurther with it. For example, Deborah could take one-fifth of three-fourths by taking one mini-part from each of the threefourths, and then doubling that to make two-fifths of three-fourths.16 In fact, as shown by Deborah’s work on making 4/3 of11/9, Deborah did not have to take one mini-part from each of the given parts: She could disembed one mini-part from oneof the parts and take that as representative of doing so for all of the parts. Thus the interiorization of three levels of unitsopens the way to reasoning with distribution in fraction contexts because taking such a structure as given allows students tooperate on part of the structure as representative of operating on the entire structure—and in that process, the structure doesnot “drop away.” Moreover, reasoning with distribution in these contexts seems essential to “keeping track” of the fractionsthat are involved in the initial composition in order to name the mini-parts in relation to the whole. These conclusions areconsistent with Olive’s (1999) findings that a generalized number sequence is required to construct a GFCS. Students who haveconstructed a generalized number sequence have interiorized three levels of units.

6.3. Is the third multiplicative concept sufficient for constructing a GFCS?

As demonstrated by Deborah’s activity, the answer to this question is no. Knowing the standard computational algorithmfor fraction multiplication seemed to be a hindrance for Deborah in using her own quantitative reasoning to produce thecomposition of two fractions, even though she could do so in quite complex situations. In addition, Deborah’s inconsistentuse of distributive reasoning in solving fraction composition problems indicates that she was not aware of the distributivepatterns of her activity. So, constructing a GFCS appears to require the careful structuring of tasks in which students learn torecognize their distributive activity as a useful and powerful part of reasoning.

Developing awareness of the power of one’s ways of operating is in the realm of reflected abstraction (Piaget, 1977/2001;von Glasersfeld, 1995)—a retroactive thematization of one’s way of operating, which differs from the “vertical” learning thatseems necessary for students like Sara to construct a UFCS and for students like Amber and Bridget to construct a GFCS. Theresearch team made concerted efforts to design tasks to investigate Deborah’s ways of operating and open the way for herto make progress. However, we did not know ahead of conducting the experiment what sequencing and structuring of tasksmight be useful for Deborah to build awareness of the patterns of her fraction composition activity. In retrospect, we can putforth these conjectures: Asking Deborah to complete fraction composition tasks partially or fully in visualized imagination,as well as to describe how she would make a composition prior to making it in JavaBars, could help her build awareness ofher activity. A similar but even more advanced task would be to ask her to investigate and describe how to take one-third ofany non-unit fraction. Further research is needed to test these activities with students like Deborah.

6.4. Contributions to prior research

Our conclusions about what is required to construct a UFCS and GFCS contribute to the refinement of research constructsaimed at describing and accounting for students’ construction of the multiplication of fractions. This kind of refinementhappens over time, as researchers construct the mathematics of more students (cf. Steffe & Tzur, 1994), and in particular testout previous research constructs with students of different ages. For example, our analysis contributes to the discussionsby Olive (1999) and Steffe (2003) of the role recursive partitioning plays in the construction of a UFCS. Although we viewour conclusions to be compatible with those of Olive and Steffe, some differences remain. For example, we believe we areconsistent with how Steffe (2003) defines recursive partitioning (p. 240) as involving the production of three levels of unitsin activity. However, at times his application of recursive partitioning to account for the activity of two students in hisexperiment implies that the interiorization of three levels of units is required to construct recursive partitioning (p. 253). Wesee these differences as part of the nature of research aimed at formulating the mathematics of students, where researchersaim to learn mathematical ideas from students. Developing language for expressing such ideas is necessarily an evolvingprocess.

16 Alternately, as shown by Deborah’s solution to making 3/2 of 5/7, students who have interiorized three levels of units might take two mini-parts fromeach of the three fourths, or 2/5 of 1/4 three times.



More broadly, our study contributes to the development of research on the role of partitioning in students’ construction offraction multiplication. In 1980, Kieren proposed the importance of partitioning in the construction of fraction knowledge.Researchers who have investigated the multiplication of fractions, such as Mack (1995, 2001) and Streefland (1991), andthose who have analyzed fractions as operators, such as Behr et al. (1992, 1993), have carefully attended to this proposalin their research. Furthermore, they have expanded the emphasis on partitioning to include the importance of conceptual-izing (formulating and re-formulating) quantitative units. Olive (1999) and Steffe (2003) contributed further refinements,including the importance of recursive partitioning in constructing fraction composition schemes, and more generally, theview that students reorganize their whole number knowledge in their construction of fractions. Our study grows from thisbody of research by viewing partitioning and the conceptualization of units as intricately tied to students’ whole numbermultiplicative concepts, and by examining the construction of fraction composition schemes across students operating atdifferent levels of multiplicative reasoning.

The findings from our study extend the extant body of research in two ways. First, our findings indicate that partitioningand establishing different quantitative units in this partitioning activity is insufficient to claim that a student has constructeda fraction composition scheme. For example, students like Sara, who have constructed only the first multiplicative concept,can partition a partition and consider it to be a situation of multiplication, as Sara did in solving the Sub Problem. They canalso determine the size of a partition of a partition by repeating that part to re-create the whole, as Sara did in solving the CakeProblem. However, we could not conclude that Sara had constructed recursive partitioning or a UFCS. Similarly, students likeBridget, who have constructed the second multiplicative concept but have yet to construct the third multiplicative concept,can work on a variety of fraction composition problems by partitioning, recursive partitioning, and conceiving of quantitativeunits in the problem situation. However, in situations involving taking a unit or proper fraction of a proper fraction, thesestudents have difficulty keeping track of the work on the fractional quantity in relation to the whole because they are “in”the activity of making three levels of units. So reasoning with distribution, naming the results of their activity in relation tothe original whole, and the construction of a GFCS are outside of their immediate zone of potential construction. Thus ourstudy highlights new facets of what may constitute students’ fraction composition schemes, as well as persistent constraintsthat teachers and researchers may meet when interacting with students with different multiplicative concepts.

Second, our study indicates that analysis of students’ whole number multiplicative concepts may help explain, ratherthan primarily describe, students’ activity in fraction composition situations. So rather than situate students’ whole numberknowledge as a distractor in fraction contexts (cf. Mack, 1995; Streefland, 1991), we view students’ multiplicative concepts(constructed in whole number contexts) as an integral resource for students (cf. Hunting et al., 1996), as well as an explana-tory construct through which researchers can interpret students’ actions. For example, we cannot say for certain whetherthe students in Mack’s (2001) study who had difficulty conceiving of fractional parts as composite units had constructedmultiplicative concepts similar to Sara, Amber, or Bridget. However, we propose that their multiplicative concepts wererelevant in explaining their activity, in that they did not form units of composite units. In short, we propose that students’whole number multiplicative concepts are critical in accounting for these difficulties, as both researchers and teachers.

7. Implications for teaching

One implication for teaching from this research is that significantly different demands are placed on students’ reasoning bydifferent fraction compositions. So, a teacher needs to be sensitive to the different compositions she or he poses to students.Taking a unit fraction of a unit fraction is clearly the most basic fraction composition in which students can engage. However,what may be less obvious is that taking a proper fraction of a unit fractional amount is significantly different from taking aunit fraction of a proper fractional amount, in terms of reasoning demands. So, asking a student who has just constructed aUFCS to take 1/3 of 2/5 will likely be a significantly more difficult challenge than asking her to take 2/5 of 1/3. That does notmean that one should necessarily always ask students to take non-unit proper fractions of unit fractions before unit fractionsof non-unit proper fractions—sometimes posing a task that represents a significant “jump” in reasoning is warranted forstudents who the teacher hypothesizes may have constructed the operations to make progress on it. However, sensitivityto these differences is necessary. We see this sensitivity to be in the realm of mathematical knowledge for teaching (Ball &Bass, 2000; Ball, Lubienski, & Mewborn, 2001; cf. Izsak, 2008).

Indeed, we view understanding the differences in the demands on students’ reasoning in relation to these two typesof fraction composition to be similar to understanding the differences in the demands on students’ reasoning in relationto quotative versus partitive division problems in elementary school (Carpenter, Fennema, Franke, Levi, & Empson, 1999;Fosnot & Dolk, 2001; Greer, 1992). Students at young ages can solve both types of whole number division problems (Kamii &Housman, 2000). However, even for upper elementary school students, partitive division problems are often problems thatinvolve trying and then adjusting possible amounts, rather than problems that involve multiplicative reasoning. In contrast,such students may more readily conceive of quotative division problems as involving multiplicative reasoning becausestudents may solve them using strategies similar to the strategies they use for repeated groups multiplication problems. Inshort, even though from a certain point of view quotative and partitive division problems are mathematically identical, theyare not identical in students’ experiences or in the strategies students use to solve the problems.

Related to sensitivity to demands on students’ reasoning is expectation. As teachers interact with students over problems,they can form working models of students’ multiplicative concepts such as the ones we have described here. If a teacherhas some students who appear to have constructed only the coordination of two levels of units in activity, it is unlikely



that they will solve a problem like the Cake Problem via recursive partitioning. It is even more unlikely that they will solvea problem that involves taking 2/5 of 1/3, or 1/3 of 2/5. These comments are not intended to be deterministic: Students’ways of operating can be surprising, and we do not at all preclude the possibility that a student with the first multiplicativeconcept may construct the second multiplicative concept in the midst of working on fraction composition problems, forexample. However, getting a sense of the “boundaries” on students’ current ways of operating is critical in helping teacherspose problems that students find both challenging and workable, even though those boundaries are persistently in flux.

Acknowledgments

We thank Andy Norton for his helpful comments on an earlier draft, and we are indebted to Les Steffe for his guidanceduring the conduct of the research. We presented a version of parts of this paper at the Annual Conference of the AmericanEducational Research Association in April 2005.

References

Armstrong, B. E., & Bezuk, N. (1995). Multiplication and division of fractions: The search for meaning. In J. T. Sowder, & B. P. Schappelle (Eds.), Providing afoundation for teaching mathematics in the middle grades (pp. 85–119). Albany, NY: State University of New York Press.

Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.),Multiple perspectives on mathematics teaching and learning (pp. 81–104). Westport, CT: Ablex.

Ball, D., Lubienski, S., & Mewborn, D. (2001). Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge. In V. Richardson(Ed.), Handbook of research on teaching.. Washington, DC: American Educational Research Association.

Behr, M. J., Harel, G., Post, T. R., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), Handbook of research on mathematics teachingand learning (4th ed., pp. 296–333). New York: Macmillan.

Behr, M. J., Harel, G., Post, T. R., & Lesh, R. (1993). Rational numbers: Toward a semantic analysis–emphasis on the operator construct. In T. P. Carpenter, E.Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 13–47). Hillsdale, NJ: Lawrence Erlbaum Associates.

Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. (1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research inMathematics Education, 15(5), 323–341.

Biddlecomb, B. D. (1994). Theory-based development of computer microworlds. Journal of Research in Early Childhood Education, 8(2), 87–98.Biddlecomb, B. D. (2002). Numerical knowledge as enabling and constraining fraction knowledge: An example of the reorganization hypothesis. Journal of

Mathematical Behavior, 21, 167–190.Biddlecomb, B., Olive, J. (2000). JavaBars [Computer software]. Retrieved June 4, 2002 from http://jwilson.coe.uga.edu/olive/welcome.html.Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In R. Lesh, & A. E. Kelly (Eds.), Handbook of research design in mathematics

and science education (pp. 547–589). Hillsdale, NJ: Erlbaum.Cobb, P., & Gravemeijer, K. (2008). Experimenting to support and understand learning processes. In A. E. Kelly, R. A. Lesh, & J. Y. Baek (Eds.), Handbook of design

research methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching (pp. 68–95). New York: Routledge.Confrey, J., & Lachance, A. (2000). Transformative teaching experiments through conjecture-driven research design. In R. Lesh, & A. E. Kelly (Eds.), Handbook

of research design in mathematics and science education (pp. 231–265). Hillsdale, NJ: Erlbaum.Davis, G., Hunting, R. P., & Pearn, C. (1993). What might a fraction mean to a child and how would a teacher know? Journal of Mathematical Behavior, 12,

63–76.Empson, S. B., Junk, D., Dominguez, H., & Turner, E. (2006). Fractions as the coordination of multiplicatively related quantities: A cross-sectional study of

children’s thinking. Educational Studies in Mathematics, 63, 1–28.Fosnot, C. T., & Dolk, M. (2001). Young mathematicians at work: Constructing multiplication and division. Portsmouth, NH: Heinemann.Greer, B. (1992). Multiplication and division as models of situations. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp.

276–295). New York: Macmillan.Hackenberg, A. J. (2005). Construction of algebraic reasoning and mathematical caring relations. Unpublished doctoral dissertation, University of Georgia.Hackenberg, A. J. (2007). Units coordination and the construction of improper fractions: A revision of the splitting hypothesis. Journal of Mathematical

Behavior, 26, 27–47.Hackenberg, A. J. (submitted for publication). Students’ reversible multiplicative reasoning with fractions. Cognition and Instruction.Hunting, R. P., Davis, G., & Pearn, C. (1996). Engaging whole-number knowledge for rational-number learning using a computer-based tool. Journal for

Research in Mathematics Education, 27(3), 354–379.Izsak, A. (2008). Mathematical knowledge for teaching fraction multiplication. Cognition and Instruction, 26, 95–143.Izsak, A., Tillema, E., & Tunc-Pekkan, Z. (2008). Teaching and learning fraction addition on number lines. Journal for Research in Mathematics Education, 39(1),

33–62.Kamii, C., & Housman, L. B. (2000). Young children reinvent arithmetic (2nd ed.). New York: Teachers College Press.Kieren, T. E. (1980). The rational number construct: Its elements and mechanisms. In T. E. Kieren (Ed.), Recent research on number learning (pp. 125–150).

Columbus, OH: ERIC/SMEAC.Kieren, T. E. (1995). Creating spaces for learning fractions. In J. T. Sowder, & B. P. Schappelle (Eds.), Providing a foundation for teaching mathematics in the

middle grades (pp. 31–65). Albany, NY: State University of New York Press.Lappan, G., Fey, J. T., Fitzgerald, W., Friel, S. N., & Phillips, E. D. (2006). Connected mathematics 2. Boston, MA: Prentice Hall.Mack, N. K. (1995). Confounding whole-number and fraction concepts when building on informal knowledge. Journal for Research in Mathematics Education,

26(5), 422–441.Mack, N. K. (2001). Building on informal knowledge through instruction in a complex content domain: Partitioning, units, and understanding multiplication

of fractions. Journal for Research in Mathematics Education, 32(3), 267–295.Olive, J. (1999). From fractions to rational numbers of arithmetic: A reorganization hypothesis. Mathematical Thinking and Learning, 1(4), 279–314.Olive, J., & Steffe, L. P. (2002). The construction of an iterative fractional scheme: The case of Joe. Journal of Mathematical Behavior, 20, 413–437.Steffe, L. P. (1991). The constructivist teaching experiment: Illustrations and implications. In E. von Glasersfeld (Ed.), Radical constructivism in mathematics

education (pp. 177–194). Boston: Kluwer Academic.Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4(3), 259–309.Steffe, L. P. (1994). Children’s multiplying schemes. In G. Harel, & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics

(pp. 3–39). Albany, NY: State University of New York Press.Steffe, L. P. (2002). A new hypothesis concerning children’s fractional knowledge. Journal of Mathematical Behavior, 20, 267–307.Steffe, L. P. (2003). Fractional commensurate, composition, and adding schemes: Learning trajectories of Jason and Laura: Grade 5. Journal of Mathematical

Behavior, 22, 237–295.



Steffe, L. P. (2004). On the construction of learning trajectories of children: The case of commensurate fractions. Mathematical Thinking and Learning, 6(2),129–162.

Steffe, L. P., & Cobb, P. (1988). Construction of arithmetical meanings and strategies. New York: Springer-Verlag.Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh, & A. E. Kelly (Eds.),

Handbook of research design in mathematics and science education (pp. 267–306). Hillsdale, NJ: Erlbaum.Steffe, L. P., & Tzur, R. (1994). Interaction and children’s mathematics. In P. Ernest (Ed.), Constructing mathematical knowledge: Epistemology and mathematics

education (pp. 8–32). London: Falmer.Steffe, L. P., von Glasersfeld, E., Richards, J., & Cobb, P. (Eds.). (1983). Children’s counting types: Philosophy, theory, and application. New York: Praeger Scientific.Streefland, L. (1991). Fractions in realistic mathematics education. Dordrecht, The Netherlands: Kluwer Academic.Thompson, P. W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), Research companion to

the principles and standards for school mathematics (pp. 95–113). Reston, VA: National Council of Teachers of Mathematics.Tzur, R. (1999). An integrated study of children’s construction of improper fractions and the teacher’s role in promoting that learning. Journal for Research

in Mathematics Education, 30(4), 390–416.Tzur, R. (2004). Teachers’ and students’ joint production of a reversible fraction conception. Journal of Mathematical Behavior, 23, 93–114.Vergnaud, G. (1988). Multiplicative structures. In M. J. Behr, & J. Hiebert (Eds.), Number concepts and operations in the middle grades (pp. 141–161). Reston,

VA: National Council of Teachers of Mathematics.von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning London: Falmer.

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