SUSAN B. EMPSON, DEBRA JUNK, HIGINIO DOMINGUEZ AND ERIN TURNER

FRACTIONS AS THE COORDINATION OF MULTIPLICATIVELYRELATED QUANTITIES: A CROSS-SECTIONAL STUDY

OF CHILDREN’S THINKING

ABSTRACT. Although equal sharing problems appear to support the development of frac-

tions as multiplicative structures, very little work has examined how children’s informal

solutions reflect this possibility. The primary goal of this study was to analyze children’s co-

ordination of two quantities (number of people sharing and number of things being shared)

in their solutions to equal sharing problems and to see to what extent this coordination

was multiplicative. A secondary goal was to document children’s solutions for equal shar-

ing problems in which the quantities had a common factor (other than 1). Data consisted

of problem-solving interviews with students in 1st, 3rd, 4th, and 5th grades (n = 112).

We found two major categories of strategies: (a) Parts Quantities strategies and (b) Ratio

Quantities strategies. Parts strategies involved children’s partitions of continuous units ex-

pressed in terms of the number of pieces that would be created. Ratio strategies involved

children’s creation of associated sets of discrete quantities. Within these strategies, children

drew upon a range of relationships among fractions, ratio, multiplication, and division to

mentally or physically manipulate quantities of sharers and things to produce exhaustive and

equal partitions of the items. Additionally, we observed that problems that included number

combinations with common factors elicited a wider range of whole-number knowledge and

operations in children’s strategies and therefore appeared to support richer interconnections

than problems with relatively prime or more basic number combinations.

KEY WORDS: fractions, children’s strategies, equal sharing, problem solving, elementary

K-8, distributed thinking, rational number, multiplicative structure

1. INTRODUCTION

Theoretical work on fractions (i.e., what it means to understand fractions)emphasizes their multiplicative structure (Kieren, 1988; Thompson andSaldanha, 2003; Vergnaud, 1988). Thompson and Saldanha, for example,described “mature” understanding of fractions as a synthesis of children’sunderstanding of multiplication, division, and ratio via measurement. Em-pirical research suggests that equal sharing – a form of partitive division(Greer, 1992) – may afford opportunities for children to construct this syn-thesis (Empson, 1999, 2003; Streefland, 1991). Yet, despite the prolifera-tion of studies on children’s strategies for partitioning (Lamon, 1996; Nunesand Bryant, 1996; Piaget et al., 1960; Pothier and Sawada, 1983), surpris-ingly little research has examined how children’s informal approaches to

Educational Studies in Mathematics (2005) 63: 1–28

DOI: 10.1007/s10649-005-9000-6 C© Springer 2005

2 S.B. EMPSON ET AL.

Figure 1. Partitive quotient fraction construct, based on Charles and Nason’s (2000)

definition.

partitioning support the construction of fractions as multiplicative struc-tures. We attribute this to a common focus in much of this prior work onchildren’s strategies for partitioning single units, which limits how fractionsas partitions and fractions as multiplicative structures can be developed andinterconnected.

One exception is recent work by Charles and Nason (2000), who pre-sented a taxonomy of children’s strategies for equal sharing problems sug-gesting a trajectory from basic partitioning strategies to the “partitive quo-tient fraction construct” (Kieren, 1988). Charles and Nason (2000) definedpartititve quotient fraction construct as a conceptual mapping in an equalsharing problem between the dividend and the numerator and between thedivisor and the denominator (Figure 1). Very little research, they noted,has studied “how young children’s intuitive partitioning strategies may fa-cilitate or possibly hinder the abstraction of the partitive quotient fractionconstruct” (p. 192). Although prior research on equal sharing (Empson,1999, 2003; Streefland, 1991, 1993), in particular, has documented the useof this type of problem in instruction to elicit a variety of informal butmathematically rich strategies, it has done so without an explicit analysisof the emergence of fractions as multiplicative structures.

Understanding fractions as multiplicative structures, we argue, involvesthe coordination of fractions with multiplication and division in a way thatemphasizes mathematical relationships (cf., Vergnaud, 1988), rather thanrelationships of association between dividend and numerator, divisor anddenominator, as suggested by Charles and Nason (2000). We present ev-idence from a cross-section of 112 children’s solutions to equal sharingproblems to show how these relationships may be constructed by children.Our primary goal was to analyze children’s coordination of two quantities(number of people sharing and number of things being shared) in their solu-tions to equal sharing problems and to see to what extent this coordinationwas multiplicative. A secondary goal was to document children’s solutionsfor equal sharing problems in which the quantities had a non-trivial com-mon factor. We conjectured that such problems would facilitate children’suse of multiplication and related concepts in the coordination of the sharingquantities. Prior research on children’s solutions of equal sharing problemswith a common factor is virtually nonexistent. Ultimately, our analysis isdesigned to assist teachers in planning instruction that elicits children’s

FRACTIONS AS MULTIPLICATIVE COORDINATION 3

informal knowledge of partitioning in problem-solving situations to de-velop fractions as multiplicative structures.

2. CONCEPTUAL FRAMEWORK

2.1. Partitioning multiple units to elicit fractions

Research on the role of partitioning in the development of fractions has fo-cused on fractions as quotients (Charles and Nason, 2000; Empson, 1999;Lamon, 1996; Streefland, 1991) and fractions as operators (Behr et al.,1992; Dienes, 1967; Schwartz et al., in press). In the latter set of stud-ies, fractions are mapped onto the action of partitioning a set; the unitof reference for the fraction is the set. For example, to divide a set into5 groups and consider 2 of those groups is to find two fifths of the set.In the former set of studies and in our framework for this study, frac-tions are mapped onto the state that results from the action of partitioninga set into a given number of groups; the unit of reference for the frac-tion is a single item in the set. For example, to divide a set of 5 itemsinto 4 groups is to find 1 and one fourth items per group. Although theoperator and the state are mathematically equivalent (Dienes, 1967; i.e.,1/4(1 + 1 + 1 + 1 + 1) = 1/4 + 1/4 + 1/4 + 1/4 + 1/4), they appear to bepsychologically distinct for young children (Kieren, 1995) who treat theirpartitioning actions and their descriptions of the results of partitioning asseparate entities (Empson, 1999). So, a child who partitions a candy intothirds may say that he or she “split in threes” but refer to each part as “a half.”

We define partitioning to admit mental as well as materially expressedactions (Steffe, 2002). Because one can only approximate equal shares intasks that require physically marking and cutting continuous items beyondrepeated halving, we posited that the depiction of equal shares of, forexample, sevenths in a part–whole representation is not a necessary stepto understanding the fraction 1/7 (for contrasting views, see Charles andNason, 2000; Lamon, 1996; Pothier and Sawada, 1983). What is necessary,however, is understanding that 1/7 is the amount one gets when 1 is dividedinto 7 same-sized parts.

In this study, we framed fractions in terms of children’s conceptions ofthe quantities that result when a set of one or more items is partitioned into agiven number of groups. There are numerous possible relationships amongquantities and actions on quantities within this framework (Figure 2). Ad-vanced understanding of fractions as multiplicative structures based on thisframework would consist of multiple coordinations between and within thetwo levels represented here and include the ability to reason flexibly aboutthe equivalence relationships from any starting point.

4 S.B. EMPSON ET AL.

Figure 2. Multiplicative structure of fractions: set of possible relationships between equal

sharing situations and mathematical notation.

The first level of this framework consists of the experientially-basedsituation of equal sharing. This level is the focus of our research and afoundation for fractions as multiplicative structures. The second level con-sists of the mathematical notation of the actions and outcomes at the firstlevel. Children may operate at each level somewhat independently of theother; Saxe et al. (in press) found that children’s understanding of fractionsand their use of standard notation to denote that understanding developsomewhat independently of each other.

Equivalence relationships within and between the two levels are repre-sented using dashed lines to indicate that children’s construction of theserelationships consists of several kinds of coordination of quantities andactions upon quantities. These strategies for coordinating quantites havebeen only partially explored by researchers. Understanding how childrencoordinate these quantities in equal sharing problems is key, we believe,to understanding how children’s thinking about fractions as multiplicativestructures develops.

Consider, for example, the path from b people share a things to a/bunit quantities. To figure out how much each sharer gets if 6 childrenshare 4 pounds of clay equally and exhaustively, for example, a child mustsomehow coordinate the sharing quantities. By coordination of quantities,

FRACTIONS AS MULTIPLICATIVE COORDINATION 5

we mean how a child physically or mentally manipulates the number ofitems to be shared in conjunction with the number of people sharing themto produce an exhaustive and equal partition of the items. Using a basicstrategy, a child may decide that all 6 people are to share the first pound,yielding 1 sixth per person; then the people share the second pound, yielding2 sixths per person; and so on, until each person has 1 sixth from eachpound of clay for a total of 4 sixths (Empson, 1999; Streefland, 1991). Thechild coordinated the number of people sharing with each pound of clayseparately. Alternatively, the child may know that the indicated divisionof quantities, 4 pounds of clay divided among 6 children, is equivalent tothe fractional quantity 4 sixths of a pound of clay per person. This typeof coordination suggests the child has synthesized the distribution of onequantity over the other into a single operation (Charles and Nason, 2000;Steffe, 1994). How children would coordinate bn people sharing an thingsto produce a/b, in contrast, has not been explored in prior research.

In the framework we include number combinations with a (non-trivial)common factor to make explicit the multiple routes for the construction ofequivalence (e.g., between two divisions, between two fractions, between adivision and a fraction). Although such number combinations make equalsharing tasks more complex and less likely to evoke a specific scheme, asin Steffe’s (2002) work, we suggest this complexity supports more connec-tions between number operations and fraction constructs than those sup-ported by partitions of single continuous units (e.g., Pothier and Sawada,1983) or even small sets of continuous units (e.g., Charles and Nason,2000; Empson 1999), and consequently provides a chance for children toconstruct mathematically rich structures.

In summary, the framework in Figure 2 represents a set of possibilitiesfor children’s reasoning about the relationships between or within the lev-els of multiplicatively related quantities and their mathematical extension.Although some of children’s reasoning has been documented in previousstudies as discussed above, with the current study we offer an analysis thatre-frames prior research by de-emphasizing partitions of single units andthe geometry of such partitions and focusing instead on children’s con-ceptualizations of multiplication, division, and ratios of quantities in thecontext of sharing multiple units among multiple sharers (cf., J. Schwartz,1988). Further, we include in this analysis problems with types of numbercombinations not previously studied, namely, those with a common fac-tor other than one (e.g., 8 people, 20 pancakes). Finally, although Charlesand Nason (2000) and others included in their analyses how children usedfractional terminology, we distinguish between children’s use of fractionterminology (“one fourth”) and their reasoning about quantities (“I’m go-ing to split this pancake into four equal parts and everyone will get one

6 S.B. EMPSON ET AL.

of those parts”), and therefore consider the conceptual sophistication ofchildren’s strategies independently of the use of conventional terms andnotation.

2.2. Children’s problem solving as distributed

We use distributed problem solving to refer to the interaction of a problemsolver with the resources available to represent and act upon problem-solving goals (Greeno et al., 1996). Prior research (Carpenter et al., 1993;Steffe, 1994) suggests that children’s problem solving may depend on ex-ternal resources (e.g., direct modeling) or internal resources (e.g., mentalschemes, number-fact recall) or some combination. Children’s initial solu-tions for a class of problems typically depend a great deal on the physicalsituation and problem-solving moves children make with materials andmarks on paper and for this reason are emergent. More advanced solutionsincorporate anticipatory thinking (Piaget et al., 1960). We used anticipa-tory thinking in the current study to refer to the planful act of incorporatingin advance some aspect of the sharing situation, such as the number of peo-ple, into the problem-solving goals (e.g., “I’m going to split this in fourthsbecause there are four people sharing”).

2.3. Questions

Our goal was to map how children coordinated quantities in equal sharingproblems as an index of the development of fractions as multiplicativestructures. Our questions were the following:

How do children coordinate the number of people sharing and the number ofthings shared in their equal-sharing strategies? In particular, how do children useconcepts of multiplication, division, and ratio to produce fractional quantities inthese strategies?

3. METHODS

3.1. Study participants

We analyzed two sets of problem-solving interviews with elementary-school students to maximize the spectrum of multiplicative thinking inour study. We sampled a range of achievement levels at each of the tar-get grades, either by purposive sampling based on achievement on a city-wide mathematics benchmark test (Sample 1) or by sampling exhaustively(Sample 2). Sample 1 consisted of 63 children: 15 first graders, 24 third

FRACTIONS AS MULTIPLICATIVE COORDINATION 7

graders, and 24 fifth graders (corresponding roughly to 6–7, 8–9, and 10–11year olds, respectively), from bilingual (Spanish and English) and mono-lingual (English) classrooms, all in the second semester of the school year.The children had had a small amount of exposure to equal-sharing prob-lems involving small quantities in the current or in prior years. Sample 2consisted of 49 fourth graders (children aged 9–10 years) from two hetero-geneously composed classrooms and who were interviewed after a 3-weekunit of problem-solving based instruction on equal sharing.

3.2. Interview protocol

Students in both samples were interviewed using a standardized clinicalinterview (Ginsburg et al., 1983). Students were read problems orally andcould choose to use paper and pencil, linking cubes, or no materials tosolve each problem. They were encouraged to solve problems in their ownways. Children were probed extensively for explanations of their solutionprocesses.

The first two problems in Table I included a number of items larger thanthe number of people sharing them to elicit fractional quantities greater than1, and also to provide opportunities for children to link partitions of discretequantities with partitions of continuous quantities. The remaining problemsin Tables I and II included a number of people larger than the number ofitems they shared to elicit fractional quantities less than 1. The numberof people sharing in each of these problems was chosen (a) to align witheasy partitions into unit fractions that could be compounded into non-unitfractions (problems a and b in Table I); (b) to be a multiple of the numberof things to be shared (problems c, d, and f in Table I); or (c) to have afactor other than itself in common with the number of things to be shared(problems c and e, Table I; problems b and c, Table II). Children weregiven problems with more difficult number combinations (e.g., Table I,problem e) only if they had solved easier versions (e.g., Table I, problemb). The order in which the quantities were presented in each problem (andwording of the problems) was varied to discourage rote approaches fromone problem to the next.

All third and fifth graders in Sample 1 were also assessed for their recallof 20 multiplication facts so that we could examine relationships betweenfact recall rates and the use of multiplication relationships in problem-solving strategies. This set of facts included most of those that applied insome way to solving the equal sharing problems as well as others up to12 × 12. Students were instructed to answer these facts verbally and fast.If a child delayed more than 2 seconds before answering, we did not countthat fact as recalled.

8 S.B. EMPSON ET AL.

TABLE I

Equal sharing problems given to children in Sample 1, in order given in interview

Indicated division Problems

a) 10 ÷ 4 (first grade)

23 ÷ 4 (third and

fifth grades)

4 children are sharing 10 (23) chocolate bars so that everyone

gets the same amount. The chocolate bars are all the same

size. How much can each person have?

b) 5 ÷ 3 (first grade)

32 ÷ 3 (third and

fifth grades)

Rosie the chef has 5 (32) pancakes for 3 hungry children to

share equally. How much can each child have?

c) 2 ÷ 6 6 children want to share 2 pounds of modeling clay so that

everyone gets exactly the same amount. How much clay

can each child have?

d) 3 ÷ 15 15 children want to share 3 pounds of modeling clay so that

everyone gets exactly the same amount. How much clay

can each child have?

e) 8 ÷ 12 On a field trip, 12 children get 8 apple pies to share among

themselves. If they share the pies equally, how much pie

can each child have?

f) 6 ÷ 24 Put out 6 unifix cubes (not linked). 24 alligators wanted to

share 6 key lime pies. One of the gators was just about to

split each pie into 24 equal pieces and give every gator 1

piece from each of the pies, when another gator

complained. He said these pieces would be way too small,

and he wanted the pies to be split into bigger pieces. How

can the gators share the pies equally without splitting each

pie into 24 pieces?

TABLE II

Equal sharing problems given to children in Sample 2, in order given in interview

Indicated

division Problems

a) 3 ÷ 10 10 children want to share 3 liters of soda so that each one gets the

same amount. How much can each child have?

b) 4 ÷ 6 6 children want to share 4 candy bars so that each one gets the same

amount. How much can each child have?

c) 9 ÷ 12 There are 9 giant peanut butter cookies that 12 children would like to

share. How much can each child have?

Interviews were conducted in English or Spanish, depending on studentpreference. Our goal was to facilitate children’s problem solving ratherthan analyze the role of language in children’s problem solving. All re-sponses were audiorecorded or videotaped then summarized and entered

FRACTIONS AS MULTIPLICATIVE COORDINATION 9

into a Filemaker Pro©R database. (The first-, third-, and fifth-grade childrenwere videotaped in order to compile videotapes for teachers’ professionaldevelopment.)

3.3. Analysis of findings

3.3.1. Strategy analysisAs our analysis progressed, the codes devised by the first author for priorresearch involving 6- and 7-year-old children (Empson, 1999) were aug-mented to accommodate the larger quantities and more varied numberrelationships in the problems. Six people (including the authors and twoadditional graduate students) coded the data and met periodically to refinethese codes. Each solution was coded for (a) type of strategy, (b) materialtools, and (c) fraction terminology.

3.4. Inter-rater reliability

Inter-rater reliability increased from 73% to 90% by (a) group discussion ofthe definitions and applications of codes, (b) resolution of disagreementson specific coded strategies, (c) discussion of these resolutions with theprincipal investigator, and (d) adjusting the coding of the data, based onthe refinements of the codes.

4. FINDINGS

We analyzed children’s thinking about fractions by examining how chil-dren’s strategies reflected efforts to coordinate the quantities in equal shar-ing problems. Apart from the most basic strategies that involved no co-ordination and hence did not exhaust the sharing material, we found twobroad categories of strategies: (a) Coordinating Parts Quantities and (b)Coordinating Ratio Quantities. Parts strategies involved children’s con-ceptualization of partitions of continuous units in terms of the numberof pieces that would be created (i.e., parts quantities). Ratio strategies in-volved children’s creation of associated sets of discrete quantities (i.e., ratioquantities), which avoided partitions of items. Within both categories ofstrategies, we identified a variety of additive and multiplicative coordina-tions of quantities.

In the following sections, we report children’s strategies distinguishedon the basis of how they coordinated the sharing quantities in a givenproblem. We use algebraic notation not to imply that children used thisnotation or that they should be introduced to it, but to emphasize the

10 S.B. EMPSON ET AL.

mathematical features of children’s strategies, when possible. Teachersmay find the notation useful for thinking about mathematical commonali-ties across strategies.

4.1. Precoordinating strategies

Children’s most basic strategies involved virtually no coordination betweenthe number of people sharing and the number of things shared. This lack ofcoordination took two forms: (a) distribute equal shares but not exhaust thesharing quantity or more commonly, (b) exhaust the sharing quantity butdistribute unequal shares. With both kinds of strategies children attended toonly one feature of the process of sharing, such as distribution or creationof shares, and used material tools of some sort. For example, to share 3pounds of clay among 15 people, a third grader drew three circles (3 poundsof clay), and partitioned the first into 3 parts and the second into 4 parts.Realizing that she needed 8 more parts for a total of 15, she partitioned thefinal pound of clay into 8 parts. Thus, each person got 1 unequally sizedpart as a share.

Overall in our data, 47 out of 59 invalid strategies were instances ofprecoordinating strategies. In these strategies, children used limited numberknowledge. Younger children tended to use precoordinating strategies tosolve problems with larger numbers of sharers, perhaps because they hadless knowledge of the multiplicative composition of numbers such as 12and 15 to use to determine an appropriate number of parts.

4.2. Strategies that involved coordinating parts quantities

All strategies in this category reflected the goal of creating a number of partsequal to or, rarely, a multiple of the number of people. Parts strategies did notnecessarily involve part-whole representations of the parts. Some childrenrepresented individual parts with tallies, some with drawings, some withhand gestures (especially for halves and fourths), and some with numerals.When materials or inscriptions were used in a strategy, the coordinationbetween number of people and number of things was distributed amongthe materials, and children partitioned the items to be shared one by one.Other children executed these strategies mentally, with minimal material orinscriptional support. More sophisticated strategies appeared to be basedon simultaneous partitions of the items to be shared, coordinated with thenumber of people.

We found three subcategories of Coordinating-Parts strategies, distin-guished by the structure of the coordination: (a) Progressive Coordinating

FRACTIONS AS MULTIPLICATIVE COORDINATION 11

strategies, (b) Single-Item coordinating strategies, and (c) Multiple-Itemcoordinating strategies. The third subcategory appeared only in problemswith number combinations that supported the use of multiplication (seeTable III).

4.2.1. Progressive parts strategiesProgressive parts strategies evidenced only basic number knowledge andwere distributed over material inscriptions and their manipulation. Mostcommonly children partitioned the items by using easy-to-make partitions,such as halves or fourths, and coordinated them only through a one-by-one distribution of parts to sharers. Children dealt with left-over parts bycreating new partitions of these parts into smaller parts. We described thiscoordination as Progressive because the number of parts is coordinatedwith the number of people as the parts are distributed and not in advanceof this distribution. We documented virtually no use of multiplication factsin children’s use of this strategy.

For example, a fourth grader solved 6 children sharing 4 candy bars(Table II, problem b) by partitioning 4 candy bars into halves (Figure 3).After distributing 6 of these halves by redrawing them under each child,he partitioned the last candy bar into eighths, and distributed these parts byredrawing them again. With 2 eighths left over, he finally coordinated thepartition of the candy bars with the number of people sharing by partitioningthe left-over fractional amount into 6 parts. Each share therefore consisted

Figure 3. Fourth grader’s progressive parts coordination for 6 children share 4 candy bars.

12 S.B. EMPSON ET AL.

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of 1 half of a candy bar, 1 eighth, and the difficult to name 1-sixth of2-eighths, which this student referred to as “one sixth.” Some children whoused a Progressive Parts strategy began with partitions other than halvesthat, fortuitously, worked. Their strategies were distinguished from moreanticipatory strategies by the reasoning they used to support the choice ofpartition. For example, to solve 24 alligators sharing 6 pies, a third graderused drawing to show that all 6 pies could be cut into 4 pieces, but onlyafter counting all 24 pieces did he know this partition would work. He saidhe cut each pie into 4 pieces because “si los hubiera cortado en pedacitosmas chicos, los cocodrilos se hubieran enojado [if I had cut the pies insmaller pieces, the crocodiles would get angry].”

Our data contained 58 instances of this strategy (Table III), which wasmore common among first and third graders. However, for more difficultnumber combinations (Table I, problems e and f; Table II, problem c), third,fourth, and fifth graders used a Progressive Parts strategy more frequentlythan on problems with easier number combinations.

4.2.2. Single-item coordinating strategiesThe most straightforward Parts strategy that involved anticipatory thinkingwas to partition each item into a number of parts (n) equal to the numberof people (P) sharing the items. We called this coordination Single Itembecause children coordinated the number of people with the number of partsin each individual item. Unlike the previously described strategies, thesestrategies required anticipatory thinking, which was reflected in children’splans to make a certain number of parts at the outset.

4.2.2.1. Single-item additive coordination. If children operated on eachitem individually, we called the coordination Single Item Additive. Thereare essentially two steps to an additive coordination: (a) coordinate thepartition of one thing with the number of people sharing, and (b) iterate thefirst partition on the rest of the things to be shared. If there is a total of T (forthings) items, then each person’s share is 1/n+1/n+· · ·+1/n[T times] =T/P . As in the previous strategy, there was virtually no use of multiplicationfacts in children’s coordination of the two quantities.

For example, to solve 12 people sharing 8 pies, a first grader used 8cubes (pies) and 12 cubes (people). Realizing the impossibility of an equaland exhaustive partition of the set by dealing the pies one by one to thepeople, he decided he would cut each pie “en doces [in twelves],” so eachperson would get one piece from each of the 8 pies for a total of 8 pieces.He initially called each share “8 mitades [8 halves]” but then explained thatthe pieces were less than halves, although he could not name the parts. To

FRACTIONS AS MULTIPLICATIVE COORDINATION 15

Figure 4. Fifth grader’s single item additive parts coordination for 12 children share 8 pies.

solve the same problem, a fifth grader drew 8 circles (8 pies) and partitionedone of them in 12 parts. It was difficult so the student decided to label theremaining circles with “12” to indicate similar partitions (Figure 4). Shewrote that each person would get “8/12.”

There were more instances (107) in our data of students using a SingleItem, Additive Coordination than any other strategy (Table III). This strat-egy was used with nearly equal frequency by first, third, and fifth gradersto solve problems with a small number of sharers, such as 3 or 4. Problemswith a number of sharers larger than 3 or 4 and that had a factor in commonwith the number of things being shared tended to elicit a wider variety ofstrategies, especially among fourth and fifth graders.

4.2.2.2. Single-item multiplicative coordination. If children executed aSingle-Item strategy by conceptualizing a partition of all items simulta-neously into P parts, we called it multiplicative. This mentally executedstrategy appeared to be an abbreviation of the additive Single-Item coor-dination. Each person’s share in this case can be denoted (T )(1/n) = T/P,where n = P.

For example, to solve 12 people sharing 8 pies, a first grader said shecould split all the pies into “twelves,” and because there were 8 pies eachperson would get 8 “pieces” or twelfths. Similarly, a fourth grader said thatif 12 children shared 9 cookies, each child would get 9 twelfths, explainingthat the 9 “goes on top” and then the 12 “on the bottom,” because each ofthe 9 cookies would be cut into twelfths.

This strategy appeared 54 times in our data (Table III). Children in Sam-ple 1 used it much less frequently (21 of 63 children) than in Sample 2 (33of 49 children), perhaps reflecting the fact that students in Sample 1 hadconsiderably less instruction on solving and discussing equal sharing tasks

16 S.B. EMPSON ET AL.

before participating in our problem-solving interviews. Further, amongthe children who had had comparatively little experience solving and dis-cussing similar problems, first and third graders used this strategy the least.The infrequent use of this strategy without instructional exposure may re-flect the non-trivial nature of the transition from iterative to simultaneouspartitioning of multiple units (cf., Steffe, 1994).

4.2.3. Multiple-item coordinating strategiesChildren who used these strategies operated on multiple items at a time,made up of either a subset of the items to be shared or all of the items.Although the strategies we report in this section differed in their relativesophistication, they were all made possible by number combinations thatpermitted the use of multiplication and related concepts. We distinguishin these strategies between the use of multiplication to coordinate quanti-ties in additive partitions and the multiplicative coordination of quantitiesin multiplicative partitions. The distinction between these two approachesto coordinating quantities is based on how children conceived of a sub-set of the items to be shared with respect to the entire set: in isolationfrom the set or in relationship to the set. This distinction is illustratedbelow.

4.2.3.1. Multiple-item strategies involving a subset of the set of items. Thesestrategies involved coordinating sets of two or more items with the numberof people sharing. In choosing to work with multiple items at once, childrenused doubling, trial-and-error skip counting, or multiplication to determinea number of parts that coordinated with (usually, equaled) the numberof people sharing. Like Single-Item strategies, we documented two waysin which children decided upon the number of items in the subsets. Thefirst involved an additive partition of the total set of items; the second amultiplicative partition.

4.2.3.1.1. Multiple-item strategies involving additive partition of numberof items. This strategy was possible whenever a problem included a non-prime number of people (P) sharing a number of items that was greater thanthe smallest non-trivial factor of P (e.g., Table I, problems e, f; Table II,problems a, b, c). Children chose a number of items, that created a subsetthat could be partitioned into a number of parts equal to the number ofsharers. They continued creating subsets that, when partitioned, yielded anumber of parts equal to the number of sharers. These subsets, which didnot need to be the same size, constituted an additive segmentation of theset of all things to be shared.

FRACTIONS AS MULTIPLICATIVE COORDINATION 17

Notationally, if ti represents the number of things in one of the subsets,and ni the number of parts into which each of ti things is partitioned, thenone person’s share is represented by 1/n1 +1/n2 +· · ·+1/ni + . . . where:t1+t2+· · ·+ti +· · · = T and each ti ni = P . Number knowledge, primarilyrelating to the number of sharers, facilitated the choice of number of itemsin a subset. For example, a fourth grader used multiplication facts for 12 tosolve 12 people sharing 9 things (Table II, problem c). She determined thatthe share per person could be 1/3 + 1/4 + 1/6, by taking 4 items (t1) anddividing each one into thirds (n1 = 3), then 3 items (t2) each divided intofourths (n2 = 4), and the last 2 items (t3) each divided into sixths (n3 = 6).Each subset was treated as a separate coordination with the number ofpeople, where ti ni = P .

Some children created only two subsets. For example, to share 8 piesamong 12 people, a fifth grader suggested splitting the first 2 pies into 6pieces each, giving 1 sixth to each sharer, and the remaining 6 pies intohalves, giving 1 half to each sharer, for a total share of 1 sixth and 1 halfper person (Figure 5). (Note the student’s use of the same fact, 6 × 2, tocreate two different partitions of the set.)

Our data contained 48 instances of this strategy (17 in Sample 1; 31 inSample 2; Table III). The low frequency in Sample 1 may be accounted forby the fact that the strategy depends on some fluency with multiplicationfacts not evidenced among the first and third graders, in particular. By far,the majority of the instances of this strategy occurred among the fourthgraders, underscoring again the potential value of instructional exposureto children’s ability to apply their knowledge of multiplication to solveproblems.

Figure 5. Fifth grader’s Multiple Item Parts Coordination for 12 children share 8 pies.

18 S.B. EMPSON ET AL.

4.2.3.1.2. Multiple-item strategies involving multiplicative partition ofnumber of items. If the number of people sharing had a non-trivial factor incommon with the number of items shared, some children used a MultipleItem, Multiplicative Partition strategy. The strategy requires coordinatingmultiple-item subsets with the number of sharers and, simultaneously, co-ordinating the number of subsets with the total number of things to beshared. This dual multiplicative coordination is based on the concept-in-action (Vergnaud, 1988) of common factor. That is, children did not neces-sarily indicate explicit knowledge of common factors but used the idea ofa common factor in their solution. Children with medium to high rates ofrecall of multiplication facts used this strategy, although having a mediumor high rate of recall did not necessarily lead to its use.

Like other strategies involving parts quantities, dual-coordination strate-gies were based on the goal of creating a number of parts equal to thenumber of sharers using a subset of the entire set of items to be shared.However, unlike previous strategies we have discussed, the number ofsubsets (g for groups) is multiplicatively coordinated with the number ofitems to be shared via a common factor. That is, the number of itemsin a subset (t) divides both the number of people sharing and the num-ber of items to be shared; it functions as a common factor. The resultingshare is 1/n + 1/n + · · · + 1/n (g times), or g(1/n), where nt = P andgt = T .

The only problems solvable using this strategy were problems e and f inTable I and problem c in Table II. For example, to solve 12 children sharing8 pies, a fifth grader reasoned that because 2 pies partitioned into 6 partsmade 12 parts (i.e., 1 part for each person, each part was a sixth of a pie),and because there were 4 two-pie groups in 8 pies, each person would get4 sixths. He coordinated an equi-partition of the 8 pies into 4 groups withthe number of parts he could make in each group. This strategy was usedonly twice in our study (Table III), perhaps because of its sophisticateddual-coordination structure.

4.2.3.2. Multiple-item strategies involving the entire set of items. We foundtwo strategies that involved operating on the entire set of items to be sharedthat were possible only with number combinations in which a commonfactor existed. If the number of items shared was a factor of the numberof sharers (e.g., 4 items, 12 people), children could use what we havecalled an Equal Quantities strategy. If the number of items shared wasnot a factor of the number of sharers, but a common factor existed (e.g., 8items, 12 people), some children used what we have called a CommensurateQuantities strategy.

FRACTIONS AS MULTIPLICATIVE COORDINATION 19

4.2.3.2.1. Multiple-item, equal quantities strategy. Many children used ap-proaches that suggested they had the goal of creating a number of parts equalto the number of people sharing using all of the items at once (T n = Pand each share is 1/n). However, this strategy was only successful whenthe number of things to be shared was a factor of the number of peoplesharing (problems c, d, and f in Table I). For example, to solve 15 studentssharing 3 pounds of clay, one student decided each pound of clay shouldbe partitioned into fifths, since 3 times 5 was 15. This strategy was used43 times by third and fifth graders (Table III). It admits a variety of entrypoints because it supports the application of multiplication facts as wellas precursors such as forming equal groups and skip counting. Thus, eventhird and fifth graders who did not have a high rate of recall of multiplica-tion facts used this strategy. However, no first grader used it, perhaps dueto few instructional experiences involving equal groups.

4.2.3.2.2. Multiple-item, commensurate quantities strategy. The most so-phisticated Parts strategy children used involved creating a number of partsthat was a (non-unit) multiple of both the number of sharers and the numberof items. To explain this strategy, we review the idea of commensurablequantities (Courant and Robbins, 1996, p. 58). Two quantities a and b arecommensurable if they “have as a common measure a/n which goes n timesinto a and m times into b” (for purposes of equal sharing, a, b, m, and n arenatural numbers). In terms of equal sharing problems, P and T are com-mensurable if there exists m and n such that T = P(n/m) or, equivalently,T m = Pn. (Put differently, n divides T and m divides P and T m = Pn.) Forexample, for 20 people sharing 8 things, this strategy would be enacted byfinding a number of parts, n, into which to partition each thing such that 8nequals some multiple, m, of 20 (that is, 8n = 20m). Because there is a totalof 8n parts to be distributed among 20 people, each person gets 8n/20 = mnumber of parts, for a fractional share of m/n (since each part is 1/n in size).

This strategy was used only twice (Table III). For example, a fifth graderused this strategy to solve 12 people sharing 8 pies. He began by settingup the problem with cubes, and after some thought, wrote 8 × 3 = 24and 2 × 12 = 24. At this point he exclaimed that he knew each personwould get 2 thirds. He explained that making thirds would yield 24 piecesand that the 24 pieces could be divided equally among the 12 people, for2 pieces (i.e., thirds) each. He spontaneously added that partitioning thepies so there were 48 pieces would also work, since 48 is a double of 24;each person would then get 4 sixths. Interestingly, this student had only amedium rate of recall of multiplication facts on our assessment. (Althoughrare, we include this strategy because we have seen it before in prior workwith children.)

20 S.B. EMPSON ET AL.

4.3. Strategies that involved coordinating ratio quantities

Children created ratio quantities by associating a group of sharers witha group of things (e.g., 3 people for 2 things). A minority of children inthe study created ratio quantities to coordinate sharers and shared items,although they did not necessarily use ratio terminology to communicateabout these quantities. In fact, most Ratio strategies were distributed overmaterials. As a group, these strategies appeared to be neither more norless sophisticated than Parts strategies, because both kinds of strategiesinvolved different combinations of distributed resources and anticipatorythinking.

Ratio strategies were characterized by attempts to act on both quantitiesin the sharing situation at once by coordinating a partition of both the setof items and the set of people sharing them. This is difficult to execute inan anticipatory way because both the number of groups and the size of thegroups are unknown. Without the use of multiplication facts in coordinationwith common factors, children typically used trial and error that weredistributed over materials and characterized by the progressive creation ofratio quantities in a series of steps that, when taken together, determined afinal share for each person. We termed these Progressive Ratio strategies.Other children used more anticipatory strategies and worked with the goalof splitting both sets of people and things into the same number of groups.We called these Partitive Ratio strategies. (The logic of division as splittingand as segmenting suggests the possibility of a Measurement Ratio strategy,in which a child would attempt to get a ratio-unit quantity that wouldexhaustively segment the sharing situation quantities. Only one child useda strategy that reflected this approach, with a great deal of trail and error.)

4.3.1. Progressive ratio strategies4.3.1.1. Generalized progressive ratio. These strategies were characterizedby children’s nonanticipatory creation of ratio-unit quantities. Studentsstarted by creating a collection of equal ratio quantities, such as 2 sharersfor 1 thing. If this partition of the sharing situation did not exhaust eitherthe number of sharers or things, students consolidated leftovers by re-associating groups of sharers or things.

For example, a first grader used cubes to solve 12 children share 8 pies.She first separated the 12 people into 3 groups of 4 and passed out 1 pie toeach group (for a 4 to 1 ratio quantity) with 5 pies left over (Figure 6a). Shethen passed out another pie to each of the groups which established a 4 to 2ratio (Figure 6b), and the last two pies were each given to the whole group(for a 12 to 1 ratio quantity) (Figure 6c). Based on these ratio quantities,she decided each person would get 2 “pieces” (fourths) plus 2 smaller

FRACTIONS AS MULTIPLICATIVE COORDINATION 21

Figure 6. First grader’s progressive ratio coordination for 12 children share 8 pies (using

cubes).

pieces (twelfths). The infrequency of the strategy (6 instances in our data,Table IV) may result from the difficulty of interpreting a configuration ofcubes that includes two or more ratio quantities in terms of an individualshare (e.g., Figure 4).

4.3.1.2. Special-case progressive ratio. This strategy was possible onlywhen the number of items shared was a factor of the number of sharers(problems c, d, and f in Table I). In this case, some children used an emergentcoordination in which they dealt sharers to things. For example, to solve 15people sharing 3 pounds of clay, a first grader used cubes to put 3 peoplewith each pound of clay. Realizing that this move did not exhaust the setof people, she re-adjusted the ratio quantity to 4 people for each item, andfinally 5 people for each item. This strategy was fairly common in our data(26 instances total, Table IV). Because children who used this strategy builtthe ratio quantities by trial and error, we classified it as a Progressive Ratiostrategy.

4.3.2. Partitive ratio strategiesChildren who used Partitive Ratio strategies worked with the goal of par-titioning T (number of things to be shared) and P (number of sharers) intothe same number of groups. This strategy was possible only when T and P

22 S.B. EMPSON ET AL.

TABLE IV

Valid ratio-quantities strategies by problem and grade level

Special-case Partitive ratio:

Progressive progressive Multiplicative

Problem ratio ratio (common factor)

First grade

10 ÷ 4 0 0 0

5 ÷ 3 0 0 0

3 ÷ 15 1 5 0

8 ÷ 12 1 0 0

Third grade

23 ÷ 4 0 0 0

32 ÷ 3 0 0 0

2 ÷ 6 0 4 0

3 ÷ 15 0 1 0

8 ÷ 12 3 0 2

6 ÷ 24 0 1 2

Fourth grade

4 ÷ 6 1 4 0

9 ÷ 12 0 1 4

3 ÷ 10 0 2 0

Fifth grade

23 ÷ 4 0 0 0

32 ÷ 3 0 0 0

2 ÷ 6 0 2 1

3 ÷ 15 0 4 1

8 ÷ 12 0 0 2

6 ÷24 0 2 0

Total 6 26 12

were both divisible by some number g (for number of groups, g > 1). Theresulting ratio, (T ÷ g)/(P ÷ g), determined the final share per person.

Although knowledge of multiplication facts facilitated finding g, somechildren who did not bring to bear this knowledge also took this approach byexecuting coordinations that were more recursive (e.g., repeated halvingof the entire sharing situation). For example, a fourth grader solved 12children sharing 9 cookies using what he called a “mirror image” strategy.He split both quantities in half so that on each side of the “mirror” 6 peoplewere sharing 4 and 1/2 cookies. He solved this reduced form of the problemusing a Parts strategy (Figure 7) and reasoned that the sharers on the otherside of the mirror would get the same amount.

FRACTIONS AS MULTIPLICATIVE COORDINATION 23

Figure 7. Fourth grader’s partitive ratio coordination for 12 children share 9 cookies.

Children who recognized the common factor and how it applied topartitioning the sharing situation (i.e., as a splitting partition rather than asegmenting partition) executed a simultaneous multiplicative coordinationof the sharing quantities. For example, to solve 12 people sharing 8 pies, afifth grader determined that both 12 and 8 were divisible by 4. This commonfactor meant the situation was equivalent to 3 people sharing 2 pies, andthat each person’s share would be 2 thirds of a pie. Like other strategiesinvolving multiplicative coordination, this strategy was used somewhatinfrequently (12 instances in our data, Table IV), perhaps reflecting thecomplexity of a double partition into a given number of groups. No firstgrader used this strategy, even in its more intuitive repeated halving form.

5. DISCUSSION AND CONCLUSIONS

Why undertake an analysis of children’s thinking about a single kind ofproblem situation? Solving and discussing equal sharing problems witha variety of number combinations generates a great deal of mathematics.Among the possible mathematics in children’s solutions to equal sharingwe have focused on the relationship between the two sharing quantities asan index of the development of fractions as multiplicative structures.

Our findings show that children’s attempts to make sense of equalsharing elicited relationships among fractions, ratio, multiplication, and

24 S.B. EMPSON ET AL.

division, evidenced in how children share things exhaustively and equallyamong sharers. Problems that included number combinations with one ormore (non-trivial) common factors elicited a wider range of strategies andappeared to support more connections than problems with relatively primeor more basic number combinations. We distinguished in our analysis be-tween children’s use of multiplication in strategies that involved coordi-nating quantities that were more additive in nature and the multiplicativecoordination of quantities. Table V summarizes the strategy analysis.

Children’s earliest attempts at coordinating equal sharing quantities,exemplified in Progressive Parts and Ratio strategies, were distributed overmaterials and had an additive, piece-meal character. These two types ofstrategies differed mainly in children’s use of material representations andhow they talked about those representations. Because these strategies weredistributed over various kinds of resources, it is possible that the primarydistinction between them is in their potential mathematical extension ratherthan in any underlying mental scheme.

Children who used more anticipatory Parts strategies, including Single-Item and Multiple-Item strategies, coordinated sharing quantities by cre-ating or conceptualizing a number of parts equal to the number of sharers,either by operating on single items or sets of items. In contrast to priorresearch on equal sharing problems (Charles and Nason, 2000; Lamon,1996; Streefland, 1991), the Single-Item and Multiple-Item strategies wedocumented distinguish between children’s operating on items taken singlyor as sets. Virtually no child produced a number of parts greater than thenumber of sharers as part of this coordination (the Commensurate Quan-tities strategy was the sole exception), in contrast to Lamon’s findings, inwhich a notable number of children partitioned individual items into num-bers of parts that were multiples of the number of sharers. Our findingssuggest that the number of sharers was a salient parameter for children indetermining partitions.

Some of the Multiple-Item Parts strategies that involved multiplicativecoordination were rare. In fact these may be used only temporarily by asubset of children who are proficient in multiplication but not yet equiv-alent fractions. Although these strategies were sophisticated in their useof multiplicative coordination, they were not necessarily efficient, and weanticipate that if followed developmentally we would see such strategiesgive way to strategies in which children move between equivalent divisionsand equivalent fractions with ease (as we suggested in Figure 2).

With the exception of the Partitive Ratio strategy, the coordination be-tween the sharers and shared quantities in children’s Ratio strategies wassituated in the physical association of sets of sharers with sets of itemsand thus was emergent. Within the Partitive Ratio strategies, halving the

FRACTIONS AS MULTIPLICATIVE COORDINATION 25

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26 S.B. EMPSON ET AL.

sharing situation quantities may function as a transitional strategy; the co-ordination is to some degree emergent, but may also serve as a bridge toa more anticipatory approach in which a number of groups into which topartition the sharing situation is planned in advance.

Conceptualizing partitioning in terms of numbers of parts appeared tocome naturally to the children in our study. Other researchers, in contrast,have recommended that children’s initial partitioning experiences shouldinvolve “analog objects . . . which are easy to partition such as length mod-els and long, thin rectangular region models” (Charles and Nason, 2000, p.193) or marking and cutting pieces on part–whole models (Lamon, 1996).The potential for multiplicative thinking embodied in the strategies docu-mented suggests that such recommendations may overemphasize the roleof activities that involve partitions of physical materials in children’s con-ceptualizations of fractions. Shifting the focus of instruction to the coordi-nation of two essentially composite quantities (items and sharers) allowschildren to express these relationships informally in terms of numbers ofparts while drawing on their developing understanding of multiplicationand division. This integration of partitioning with multiplication may bejust as important to the development of fractions as depicting partitionsinto equal parts. The strategies we have documented suggest it is a feasibleapproach.

ACKNOWLEDGMENTS

An earlier version of this paper was presented at the 10th InternationalCongress on Mathematical Education in Copenhagen, Denmark. We wouldlike to thank Kevin LoPresto, Luz Maldonado, and Stephanie Nichols forassistance in collecting and coding data for this research, Karen Heinz andTaylor Martin for critical reviews of this work, Jennifer Cook and ChrisBailor for editing, Norma Presmeg and the anonymous reviewers of themanuscript and, last but not at all least, Austin area schools for allowing usto work with their students. This work was supported in part by NationalScience Foundation grant no. 0138877 to Empson. The views expressedhere are the responsibility of the authors and do not necessarily reflect thoseof the funding agency.

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SUSAN B. EMPSON, DEBRA JUNK AND HIGINIO DOMINGUEZ

Science and Mathematics EducationUniversity Station, D5705The University of Texas at AustinAustin, TX 78712U.S.A.Tel: +1(512)232-9688Fax: +1(512)471-8460E-mail: empson@mail.utexas.edu

ERIN TURNER

University of Arizona