ann's fraction schemes

$: Ann's fraction schemes$
ADALIRASAENZ-LUDLOW

ANN'S FRACTION SCHEMES

ABSTRACT. A longitudinal constructivist teaching experiment that lasted approximately one academic year was conducted with six third graders. The purpose of the teaching experiment was to analyze the itinerary of children's ways of operating while solving fraction tasks. Ann was one of the third graders who participated in the teaching experiment, and her case study presents the author's interpretation of the generation and evolution of Ann's fraction schemes.

1. INTRODUCTION

The difficulties that children and adolescents experience when operating with fractions have been well-documented by an extensive body of research (among others, Behr, Lesh, Post and Silver, 1983; Behr, Wachsmuth, Post and Lesh, 1984; Behr, Wachsmuth and Post, 1985; Berg- eron and Herscovics, 1987; Green, 1969/1970; Hunting, 1980/1981, 1983, 1986; Kerslake, 1986; Kieren, 1980, 1988; Muangnapoe, 1975; and Nik Pa, 1987/1988). The results of the National Assessment of Educational Progress (NAEP) analyzed by Brown, Carpenter, Kouba, Lindquist, Silver and Swafford (1988a, 1988b) indicated that nine-year-olds, thirteen-year- olds, and seventeen-year-olds have low performance in computation with fractions and little conceptual understanding. These findings indicate the students' need to conceptualize fractions as quantities before they are intro- duced to conventional symbolic algorithms. After all, the algorithms that we use today to operate with fractions are sophisticated procedures generated and refined in the course of the development of mathematics (Waismann, 1959; Smith, vol. 2, 1953; Dantzig, 1954; Courant and Robbins, 1969). In the absence of the conceptual structure behind these algorithms, children memorize them without attached meaning.

For a conventional arithmetical algorithm to become meaningful to a child, it must represent the coordination of mental operations and conventional notations. The premature introduction of conventional notations may represent an obstacle to children's learning of mathematics (Piaget, 1973) by having them use symbols foreign to them to represent their mental operations. Initially, it could be necessary to accept children's informal written representations as transitional notations before conventional notations are negotiated with them. Piaget (1967) views operations with quantities as

Educational Studies in Mathematics 28: 101-132, 1995. (~) 1995 Kluwer Academic Publishers. Printed in the Netherlands.

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mental actions that first take place in experimental contexts under the necessary, but insufficient, influence of language that gives these operations "extension, mobility, and universality" (p. 92). That is, children's use of natural language to verbalize their mental activity could serve as a first step toward symbolization.

Behr et al. (1983, 1985); Kieren (1980, 1988); Kieren, Nelson, and Smith (1985); Piaget, Inhelder, and Szeminska (1960); Peck and Jencks (1981); Pothier and Sawada (1983, 1990); and Streefland (1979, 1984, 1990), among others, have documented the importance of partitioning in the understanding of fractional numbers. McLellan and Dewey (1908) also recognized the importance of "the mental operation of rhythmic parting and wholing" (p. 83) in the generation of units of measurement and fractional units. "Parting and Wholing" seem to be essential mental operations in the development of children's fractional meanings. In the context of natural numbers, the "wholing" operation or unitizing operation has been documeted by Steffe (1986, 1988, 1990, 1991) and Steffe and Cobb (1988). Although the importance of partition and its concomitant emerging units have been suggested in several studies, there have not been studies that focus on the importance of the unitizing operation in the generation of fractional-number units and the relationship between children's ways of operating with natural numbers and their ways of generating fractional numbers and operations with them.

2. METHODOLOGY

2.1. Teaching Experiment

During approximately one academic year, a constructivist teaching experiment was conducted with six third-graders. The objective was to for- mulate explanations and models of these children's conceptualizations of fractions.

The methodology of the teaching experiment is based on two apparently opposite, but nonetheless complementary, perspectives - the Vygotskian and the Piagetian theoretical frameworks. Under the Vygotskian perspective, social interactions are essential for the internalization of external activity into inner activity. That is, the individual's knowledge is seen as the product of social interaction through which the individual's intellectu- al growth is determined (Wertsch and Stone, 1985). However, under the Piagetian perspective, social interactions are viewed as a source of cognitive conflict that facilitates the individual's cognitive development. That is, the cognitive subject constructs knowledge by adapting his or her prior knowledge to remain viable in a social or physical world (yon Glasersfeld,

FRACTIONS SCHEME 103

1980). The main difference between these two theoretical frameworks concerns the genesis of knowledge as a transformation of external into internal activity or as an internal self-regulated process. The main similari- ty is that socio-cultural factors have a shaping influence on the individual's knowledge. The constructivist teaching experiment methodology (Steffe, 1983) reconciles the dualism between individual cognition and implicit socio-cultural negotiation of meanings.

Piaget's (1970, 1970/1972) epistemological position assumes that knowledge arises from progressive interactions that actively take place between the subject and the exterior world; but in the end, it is only the subject's mental activity at every level (perception, intuition, and thought) that gives rise to concepts. This mental activity follows a pattern of mental coordinations of actions that he calls a scheme of action. A scheme is the generalizable aspect of specific actions of knowing. Thus a scheme must conserve itself under repetition and consolidate itself through application to situations which vary because of modification of context (Piaget, 1972).

This paper discusses an attempt to assist and orient Ann's construction of fraction schemes. Her constructions were supported by her collaborative interaction with the teacher. Even though the focus of this paper is the conceptualization of Ann's fraction schemes, it is worth noting that such schemes were the result of Ann's cognitive activity facilitated by the collaborative interaction with the teacher/researcher. The role of social interaction in the development and evolution of Ann's fraction schemes is not analyzed in this paper, though it is freely recognized as a basic element of the teaching experiment methodology.

Four phases were taken into consideration in this teaching experiment: an observation phase, a clinical-interview phase, a teaching phase, and an analysis phase.

2.1.1. Observation phase For two months, the researcher observed and sometimes taught arithmetic to the participating third grade class. Based on the children's verbal and written mathematical performance, six children of different mathematical ability were chosen to participate in the teaching experiment. This phase also provided an opportunity to establish a collaborative relationship with the teacher and the whole class.

2.1.2. Clinical-interview phase During four consecutive weeks, each child was interviewed on a week- ly basis. Each interview lasted 60 minutes. The objective of the clinical interviews was to ascertain the children's current ways of operating with natural numbers as well as their current understanding of fractions. These

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interviews were not intended to advance these children's actual knowledge. The purpose of the first three of the four clinical interviews was to probe these children's unitizing operation (i.e., the conceptual act of con- ferring unity to a collection of elements, Steffe and Cobb, 1988) and their ability to work simultaneously with different composite units. Counting, missing items, division, and multiple-units tasks were verbally posed to the children. All children were given the same tasks and they were asked the same questions. The fourth clinical interview probed children's current understanding of fractions.

2.1.3. Teaching phase Teaching episodes are the hallmark of the teaching experiment methodology. A teaching episode is characterized by: (a) the teacher's explicit goal to foster the generation and evolution of the child's conceptual constructions by judiciously selecting tasks; and (b) the teacher-student interdependent activity through verbal and nonverbal communication in which language is used as a tool to achieve and express cognitive processes.

At the outset, each teaching episode is bounded by conceptual analysis done prior to instruction; however, the essence of a teaching episode is that instruction is modified continuously, according to the student's ways of thinking as evidenced by his/her answers and explanations.

Once a week, each child participated in a 60-minute teaching episode. On the average, each child participated in 19 teaching episodes over the school year. All children were given similar tasks, but questions had vary- ing degrees of difficulty depending on each child's ways of thinking as inferred by the teacher from the analysis of his/her prior teaching episodes and the child's current explanations.

Whether in an interview or teaching episode, it was important for the teacher/researcher to view each posed task and the child's solution to that task from the child's perspective. (Vygotsky, 1934/1986, observed that the child and the adult look at the same object from two fundamentally different perspectives: the child's framework is concrete and situated, whereas the adult's framework is conceptual.)

2.1.4. Analysis phase Every clinical interview and teaching episode was videotaped and analyzed first daily and then retrospectively in an integrated and cumulative manner. The daily analysis of the first three clinical interviews used the conceptual framework of children's mental operations when dealing with natural numbers as developed by Steffe and Cobb (1988). This daily analysis was followed by cumulative analyses (i.e., interviews 1, 2, 3, of each child, were analyzed as a unit). The fourth clinical interview was analyzed


separately, and its retrospective analysis was included in the cumulative analyses of the teaching episodes.

The daily analysis of each teaching episode provided an opportunity to revisit each child both retrospectively and prospectively. Retrospective- ly because the teacher/researcher could put into perspective the child's answers and explanations. Prospectively because the teacher/researcher could decide when to use previously conceptualized tasks or to generate new ones as well as to prepare questions for the following session. The cumulative analyses of the teaching episodes of each child were performed each four weeks in cumulative sequences (i.e., 4, 5, 6, 7; 4, 5, 6, ... 11; 4, 5, 6 .... 15; 4, 5, 6 .... 19) to gain inductive insight into the child's ways of operating.

3. TASKS

Tasks were designed to integrate the use of children's prior natural-number knowledge in the generation of their concepts of fractions. The importance of the liaison between mental or physical measuring acts and the conceptualization of units in the development of fractions was noted by McLellan and Dewey (1908),

the logical and psychological relation between the number that defines the measuring unit and the number that defines the measured quantity; or as it is sometimes expressed, the "relation between the size of the parts" (the measuring units) and the number of parts composing or equalling the measured quantity ... cannot be ignored in the teaching of fractions because [it] cannot be ignored in the teaching of whole numbers (pp. 242-243). (Emphases in the original)

Division and multiplication seem to be essential when considering part- whole relations and fractions as a means of quantifying such relations. Fractions are intimately related to division and multiplication, namely "analysis of a whole into exact units, and synthesis of these units into a defined whole" (McLellan and Dewey, 1908, p. 242).

Fraction tasks were embodied in contextual situations familiar to children to facilitate the construction of meanings and to elicit the use of their natural-number units. Some tasks were conceived within the context of the U.S. monetary system because of its well-structured units symbolized on bills and coins of different denominations (the same could be said of any other monetary system). The variety of exchanges of a given amount of money offers the potential of conceptualizing exchanges as partitions, and the recomposition of the total amount of money as a multiple of bills or coins of smaller denomination. In general, money seems to facilitate the conceptual abstraction of fractions due to the child's focus of attention on

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the value of a bill or coin that could be mentally disembedded from a larger bill or coin that represents the whole or unity, instead of focusing only on the physical extension of a part of a continuous or discrete whole.

Sets of blocks of the same and different sizes, and patterns that simultaneously contained several implicit partitions were also used to generate fraction tasks. The simultaneity of different partitions of the same whole fostered the cognitive need to correlate them and the generation of equivalent fractions emerged. The structure of these patterns was not, in essence, different from the structure of the money system, since a bill can also have several implicit exchanges (or partitions).

Tasks were posed verbally to provide opportunity to conceptualize fractions in the absence of symbolic notation. Using contextual situations, the children reconstituted a whole as a multiple of one of its parts, conceptualizing the whole as part-dependent and giving rise to part-to-whole relations. Then, based on such multiplicity, the children conceptualized the part as whole-dependent and the whole-to-part relations emerged. This last relation was verbalized in terms of the prior multiplicity of parts in the whole. For example, when a part was used as a measuring unit three times to exhaust the whole, then one part (or one measuring unit) was referred to by the children as "one out of three" or "one third." Both ways of express- ing whole-to-part relations were always encouraged. It is worth noting that fraction words such as "one half" and "one third" were already part of the children's vocabulary. Fractional number-words instead of fractional symbols are used in reporting this case study because only number words were used by the children and the teacher in the course of the teaching experiment.

4. ANN'S MEANS OF OPERATING WITH NATURAL NUMBERS

Ann, a nine-year-old girl when the teaching experiment began, was a third-grader perceived by her teacher as a very capable student. She was a talkative and reflective child, willing to overcome the challenge of difficult questions and to recapture her mental processes to verbalize explanations for her solutions. Ann had some classroom instruction on fractions in the prior grades, but had no instruction on fractions during the time of the teaching experiment.

The paper presents significant episodes of Ann's constructive development of fractions. Two points about Ann's conceptualizations of fractions are in order. First, although tasks were conceptualized to help these children bridge the gap between natural-number and fractional-number knowledge, the interpretations of and solutions to these tasks were their own. Second,


Ann's ways of operating were not, in essence, different from the ways of operating of other participants in the study (S~ienz-Ludlow, 1990).

It was necessary to analyze Ann's ways of operating with natural numbers because one of the main conjectures of the study was that children would use and modify their ways of operating to generate natural number units to generate unit fractions as generalizations of part-whole relationships. The analysis of children's ways of operating with natural numbers was based on Steffe's (1986, 1988, 1990, 1991) theory of children's conceptualizations of natural numbers as composite units.

This section presents an interpretation of excerpts from the first three clinical interviews which dealt with natural numbers. Contextual situations, manipulatives, and verbal interaction provided the means of communication between Ann and the teacher.

Using the context of the back-to-school party thrown for the class by the third-grade teacher, Ann was presented with the following task.

T: If there are twenty-four students in your class, and each group of four students eats one pizza, how many pizzas will be needed?

A: [Quickly] Six.

T: How do you know?

A: Because 4-8-12-16-20-24 [while lifting up one finger for each number word].

To find her answer quickly, Ann had to see twenty-four as a composite unit from which she could exhaustively disembed units of four prior to any counting activity. The act of disembedding (Steffe and Cobb, 1988) presupposes the mental act of establishing part-whole numerical relations in which the whole is conserved as a mental representation even after the mental act of disembedding smaller composite units from it. Ann's last explanation "because 4-8-12-16-20-24" and her lifting up one finger for each counting of four indicated that counting came after segmenting twenty-four into composite units of four. That is, four was for Ann an iterable unit. A composite unit is iterable (Steffe, 1991) for the child when he/she uses it to segment a larger unit; nonetheless, reconstitution of the larger unit as a multiple of the segmenting unit must be anticipated prior to the act of segmentation.

The following dialogue illustrates Ann's strategy to solve a similar, but structurally different problem.

T: If the twenty-four students in your class go camping, and there are eight tents to be equally shared by all the students, how many students will be assigned to each tent?

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A: Well ... it could be said that is twenty-four divided by eight.

T: O.K. How many students in each tent?

A: [Quickly] Three.

T: How do you know?

A: Because eight plus eight is sixteen plus eight is twenty-four.

Ann's counting activity indicated that her expression "well ... it could be said that is twenty-four divided by eight" meant, for her, twenty-four segmented into units of eight. However, her answer of three students per tent was the result of counting the three anticipated units of eight to generate eight units of three.

Ann also used her iterable units and her disembedding operation to solve tasks that an adult would solve using the division and multiplication algorithms.

T: If two liters of juice cost ten dollars, how much will one liter cost?

A: [Quickly] Five dollars.

T: How did you find your answer so fast?

A: Because five and five is ten.

To find the cost of one liter, Ann segmented ten into two composite units of five and implicitly made a one-to-one correspondence between the two liters and the two composite units of five. Her explanation "because five and five is ten," after she found the cost of five dollars per liter, was the result of anticipating five as a unit that could segment ten exhaustively. Therefore, it could be said that five also was, for Ann, an iterable unit.

The continuation of the above dialogue shows how Ann used her numerical part-whole relations to solve a task that could be solved using division and multiplication.

T: If two liters of juice cost six dollars, how much will four liters cost?

A: [Quickly] Twelve dollars.

T: How did you figure out your answer so fast?

A: Because when you said two liters for six dollars, I figured out three dollars per liter. 3 -6-9-12 [lifting up one finger for each number word and finally showing four fingers].


Ann's answer indicated that six was, for her, a composite unit from which she could disembed two composite units of three using her part-whole numerical relation. Then she used three as an iterating unit four times to find the cost of four liters of juice.

Ann's ability to deal with different units simultaneously was indicated in the following dialogue. She was asked to put five one-dollar bills into one cup, and sixteen quarters (25-cent coins) into another. The cups were covered and the teacher posed questions about the total number of dollars and quarters, respectively.

T: How many dollars altogether?

A: [Quickly] Nine.

T: Why?

A: Let's see. 4 -8 -12 -16 [lifting up a finger for each four that she counts], that's four. Four and five is nine.

T: How many quarters altogether?

A: [Immediately] Thirty-six.

T: Why?

A: Nine dollars. That's sixteen [looking at the cover under which the quarters are located]. That's twenty [looking at the cover under which the five one-dollars bills are located]. Twenty plus sixteen is thirty-six.

Since Ann's units of four were iterable, it was natural for her to take four as a segmenting unit to find the number of dollars in sixteen quarters. Her lifting up a finger for each count of four was indicative of the application of her unitizing operation to reconstitute four quarters as one dollar. To find the number of quarters, she mentally decomposed again four of the nine dollars into sixteen quarters, as initially given, and five dollars were decomposed into twenty quarters. Then Ann formed a composite unit of thirty-six quarters by unitizing twenty and sixteen quarters.

Because of the speed with which Ann was able to operate with two different units, it was inferred that she might be able to operate numerically with coins and bills of three different denominations. Ann was asked to put thirteen dimes (10-cent coins) into a new cup, cover it, and put it with the others. Then she was questioned about the number of dollars, quarters, and dimes in all the cups.

T: How many dollars altogether?

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A: [Quickly] Ten dollars and thirty cents.

T: How many quarters altogether?

A: [Quickly] Forty-two.

T: How did you find that out so fast?

A: One hundred and thirty dimes ... wait, one hundred and thirty cents, that's one dollar and thirty cents. Four quarters in a dollar. There are ten dollars. Forty quarters and one quarter, forty-six [she adds one quarter and five cents]. No p, forty-one and five cents.

T: How many dimes altogether?

A: [Quickly] O.K. One hundred and three.

T: How did you find that answer so fast?

A: Because ten times ten is a hundred. One hundred dimes is ten dollars, and then thirty cents is 1-2-3.

Her answer "ten dollars and thirty cents" indicated that she used the result of nine dollars, from the above tasks, and added to it "one dollar and thirty cents". One dollar and thirty cents was the representation that she made of thirteen dimes after unitizing ten dimes into one dollar and decomposing three dimes into thirty cents. To find the total number of quarters, she represented ten dollars as forty quarters, and thirty cents as one quarter and five cents while monitoring herself in the process of explaining her answer. Similarly, she represented ten dollars and thirty cents as one hundred dimes and three more dimes.

In the above dialogues, the teacher gave Ann no prompts to guide her solutions; therefore her answers seem to be the product of her own cognitive activity. Ann's solutions to these and other natural-number tasks, posed to her during the course of the first three clinical interviews, were based on her conceptualizations of natural numbers as composite units and her flexibility to operate with them.

The question that needs to be addressed is whether or not Ann's well- grounded conceptualizations of natural-number units could indeed facilitate her constructions of fractions. The following sections are dedicated to the interpretive analysis of Ann's solutions to fraction tasks using continuous and discrete wholes as well as wholes that were kept verbally constant but not specifically defined.

FRACTIONS SCHEME l l l

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Fig. 1.

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5 . A N N ' S M E T R I C P A R T - W H O L E S C H E M E : C O N T I N U O U S C O N T E X T

This section presents excerpts of the fourth clinical interview. This last clinical interview was to infer Ann's current understanding of fractions. During this interview, Ann consistently used the part as a measuring unit to segment a given whole, and then she recast the unity of the whole in a composite unit. Her awareness of the necessity to view the whole as reconstituted into parts of equal size was indicated in the following dialogues. Figure 1 shows a square subdivided into parts of different sizes to facilitate the conceptualization of different partitions. In this article, the term "partition" is taken to mean the process of segmenting the whole exhaustively (either physically or mentally) by using the same part as a segmenting unit.

T" This [showing A] is what part of the square?

A: [Counting A-like pieces one by one] One sixteenth.

T: Can you show me one fourth of the figure?

A: This one [showing part B].

T: Can you find another name for one fourth?

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, ,

bl / Fig. 2.

A: One fourth. Oh! One quarter!

T: Is one fourth the same as four sixteenths?

A: Well, sometimes it is and sometimes it isn't. Because you could have twenty of this [showing A], and one fourth would be five twentieths. But in this case, one fourth is the same as four sixteenths.

Ann's initiative to count A-like squares, even though the pattern had squares of different sizes that could be considered, indicated her awareness of decomposing the whole into parts equal in size to that of part A. This decomposition led her to give A a fractional quantification with respect to the larger square. When she was asked to show one fourth of the larger square, she used B as a segmenting unit to exhaustively segment the whole and recast it as a unit of four B-parts. Furthermore, her last answer indicated her anticipation of another partition of the whole into twenty parts. That is, multiplicity of equal-sized parts in the whole seems to be, for Ann, an essential prerequisite to generate a fractional quantification for a part.

Figure 2 shows a hexagon given to Ann. Several copies of the parts bl, b2, b3 of the hexagon were available to her in case she wanted to use them. She was to find fractional quantifications for the pieces bl, b2, and b3 with respect to the hexagon. Letters are used here for descriptive purposes, but they were not used when posing the tasks.

T: This piece [referring to bl] is what part of the whole figure?

A: One half because two of this make the whole figure.

FRACTIONS SCHEME l 13

T: One of this [referring to b2] is what part of the whole figure?

A" [Takes two pieces b2 and juxtaposes them over the hexagon] One third because three of this [referring to bz] make the whole figure.

T: This piece [referring to b3] is what part of the whole figure?

A: [Takes three pieces b3, juxtaposes them over the hexagon and thinks for a while] One sixth because six of this [referring to b3] make the whole figure.

Ann's answers indicated that her fractional quantifications for the parts bl, b2, and b3 (with respect to the whole) were linked to the numerosity of these parts in the segmentation of the whole (when each of these parts was taken as a measuring unit) as well as the multiplicative reconstitution of the whole. In other words, the unity of the whole was preserved regardless of its partitioning. Nonetheless, fractional quantifications were not meaningful to her (when a given part of the whole could not segment the whole an integral number of times) as the following dialogue shows.

T: This piece [referring to b2] is what part of this one [referring to bl]?

A: Well ... this is not really anything, because you cannot fit one or two or three of this in one half.

T" If I give you this piece [referring to b3], can you find a relationship between bl and b2?

A: [Quickly] Well ... in that case, this piece [referring to b2] is two thirds of one half.

Ann's comment "Well ... this is not really anything because you cannot fit one, or two, or three of this in one half" indicated her conscious awareness of the need to reconstitute the whole as a multiple of equal-sized parts in order to find a fractional quantification for one of the parts. She could not generate a partition of bl using the part b2 as a segmenting unit. Once the piece b3 was given to her, she reconstituted bl as three b3-pieces and b2 as two b3-pieces. Then, she generated the fractional quantification of two thirds for b2 with respect to bl. She said, "two thirds of one half" because she considered bl as the new whole that she had to deal with even though it was only half of the hexagon.

In summary, this clinical interview indicated that at the beginning of the teaching experiment Ann already had underway the construction of what I call metric part-whole scheme.This scheme was the result of physical or

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mental measuring of the whole with a part and reconstituting the whole as a composite unit that had a multiplicative relation with respect to the part. This multiplicative relation was then used to generate a fractional quantification for such a part. This scheme synthesizes the complexity of part-whole relations. First, the conceptualization of the whole as a composite unit of equal-sized units and recognition of the part-dependency of the whole by establishing part-to,whole relations. Second, after establishing the part-to-whole relation (conceptualization of the whole as part- dependent), she inverted this relation to generate the whole-to-part relation (conceptualization of the part as whole-dependent). Ann's inversion of the part-to-whole relation into the whole-to-part relation allowed her to confer a particular part a quantification (fractional in nature) with respect to the whole.

In other words, the bases of the metric part-whole scheme are the multiplicative recomposition of the whole, and the disembedding of a part from the whole while mentally conserving the unity of the whole. Then, the part becomes "one out of many" and is granted a fractional quantification with respect to the whole.

It is worth mentioning that up to this point, the teacher made no inten- tional effort to facilitate Ann's generation of fractions. Her interpretations of and solutions to the posed tasks could be considered the result of her own mental activity, though encouraged by her verbal interaction with the teacher.

6. ANN'S MODIFICATION OF HER METRIC PART-WHOLE SCHEME: DISCRETE CONTEXT

In the above section, it was observed that in the context of continuous wholes, Ann could generate fractions as a result of her metric part-whole scheme. The teacher's inference was that it could be plausible for Ann to modify her scheme to solve fraction tasks with discrete wholes. Most of the time, the dollar monetary system was used as a discrete whole. The value of coins and bills in this system resembles the value of different units used in the decimal notational system, and children were at ease working with money even when using large numbers; furthermore, units can be conceived in a concrete or abstract way depending on the child's ability to abstract.

Ann's initial conceptualizations of fractions within the context of discrete wholes seem to be a consequence of the exchange of a given quantity of money into bills and coins of smaller denomination. Such exchange allowed her to reconstitute the given amount of money as a multiple of


bills and coins of smaller denominations. For example, Ann exchanged a fifty-cent coin into quarters (25-cent coins), dimes (10-cent coins), nickels (5-cent coins), and pennies (1-cent coins) and then the following dialogue took place.

T: A dime is what part of a fifty-cent coin?

A: [Quickly] One fifth because it takes five of this [showing a dime] to make one of those [pointing at the fifty-cent coin].

T: A nickel is what part of a fifty-cent coin?

A: That is a tenth [pause]. Let's see, if this is one fifth [showing a dime], five more of this [showing a nickel] will make fifty. [Ann puts 10 nickels on the table and counts them by fives lifting up a finger for each count of five] 5-10-15-20-25-30-35-40-45-50. Yeah! One tenth.

Focusing exclusively on the number of coins, she generated a fractional quantification for a dime (10-cent coin) upon reconstituting fifty as a composite unit of 5 dimes (or units of ten) and she gave the dime a fractional quantification of "one fifth because it takes five [dimes] to make one of those [a fifty-cent coin]." Moreover, she was able to anticipate a fractional quantification of one-tenth for a nickel (5-cent coin) before she used her first answer to generate the second by representing fifty as 5 dimes and each dime as 2 nickels. That is, Ann conceived fifty as a unit of units of five to generate a fractional quantification for the nickel.

The above dialogue indicated that in each exchange of the fifty-cent piece, Ann was aware of two numerosities: the numerosity of the composite units (5 dimes or 10 nickels) in fifty, and the numerosity linked to each composite unit (ten or five). Nonetheless, to generate fractional quantifications for a dime and a nickel, Ann decided to use only the numerosity of the composite units and not the numerosity corresponding to the composite unit in itself.

Several questions could be asked. Why Ann used only the number that indicated the numerosity of composite units and not the number that indicated the numerosity or magnitude of the composite unit in itself? Was the second numerosity left implicit on purpose? To locate indicators that guided the elucidation of these questions, Ann was provided with tasks that involved different fractional quantifications of the same part. Ann was asked to find as many exchanges as possible for a one hundred dollar bill in bills of smaller denomination. She found that it could be exchanged into one-, five-, ten-, twenty-, and fifty-dollar bills. For example, she exchanged

l 16 ADALIRA S,~ENZ-LUDLOW

it into 20 five-dollar bills and she said, "there are 50, no wait, 5 -10 -15 - ... - 45 -50 [showing her ten fingers and assigning one finger to each count of five]. Yes, that's twenty because 5 -10 -15-20- . . . - 85 -90 -95-100 [using her hands twice]." Similarly, she quickly found the exchange of the one hundred dollar bill into 2 fifty-dollar bills, 5 twenty-dollar bills, 10 ten- dollar bills, and 100 one-dollar bills. Then, the following dialogue~took place.

T: One half of a hundred-dollar bill is the same as how many hundredths?

A: [Quickly] As fifty hundredths.

T: One fifth of a hundred-dollar bill is the same as how many hundredths?

A: [Quickly] As twenty hundredths.

T: Can you give me another name for one half?

A: [Quickly] One half is the same as ten twentieths.

T: Can you give me another name for one fifth using hundredths, twentieths, and tenths?

A: [Quickly] Twenty hundredths, four twentieths, two tenths.

T: Which is bigger, one tenth or one twentieth?

A: [Looking at the bills] One tenth.

T: Which is bigger, one twentieth or one half?

A: [Looking at the bills] Onehalf.

The ease and speed with which Ann answered the questions indicated that her comparisons of fractions were based on her conceptualization of one hundred as a unit of composite units of fifty, twenty, ten, five, and one and each of the composite units was decomposed again into more elementary units. That is, Ann conceived one hundred as a unit of units of units. For example, her conceptualization of one hundred as 2 fifties and each fifty as 50 ones allowed her to see each fifty as one half, each one as one hundredth, and therefore, each fifty as 50 hundredths. Given that the same quantity (fifty) had two fractional quantifications, she concluded that the two fractional quantifications were the same. Similarly, conceptualizing one hundred as 5 twenties, 20 fives, 10 tens, and 100 ones, she generated for twenty, five, ten, and one the fractional quantifications


one fifth, one twentieth, one tenth, and one hundredth, respectively. Her further conceptualization of twenty as 20 ones, 2 tens, and 4 fives allowed her to recognize one fifth as "twenty hundredths, four twentieths, [and] two tenths." Her looking at the bills to find her answers indicated her implicit coordination of natural-number units and fractional-number units. Ann's coordination put in correspondence two fractional units with two natural-number units. Then, the fractional units were ordered according to the order of the natural-number units.

To confirm the hypothesis that Ann was coordinating natural-number units and fractional-number units, the teacher gave her another fraction task with money, to provide her with an opportunity to express such coordination explicitly. Ann was given a box with fake bills of several denominations (one-, five-, ten-, twenty-, hundred-, five hundred-, and thousand- dollar bills) ordered by increasing value; there were at least twenty bills of each denomination. Ann was asked to put one thousand-dollar bill on the table and to choose from the box quantities of money that corresponded to fractional parts of one thousand. This activity is illustrated in the following dialogue.

T: Would you please give me one fiftieth of one thousand-dollar bill?

A: [Quickly] I think it is twenty.

T: Why?

A: Because 5 -10-15-20-25 /30-3540-45-50 [Showing her ten fingers and simultaneously assigning one finger per hundred and five twenties per hundred]. That's a thousand.

T: Would you please give me one twentieth of one thousand-dollar bill?

A: [Quickly] It is either two hundred or fifty. I think it is fifty.

T: Why?

A: Because two fifties make one hundred. Here is a thousand [showing the fingers of her two hands], 100-200-300-400-500-600-700-800- 900-1000 [lifting one finger per each hundred]. [Counting again on her fingers] 2-4-6-8-10-12-14-16-18-20.

Ann did not use the bills in the box, instead she resorted to counting as a way of operating. To find one fiftieth of one thousand dollars, Ann seemed to have anticipated twenty as a segmenting unit to partition one thousand into fifty parts. Upon request, she verified her answer by finger-counting.

118 ADALIRA S.~ENZ-LUDLOW

This counting activity indicated that she conceptualized one thousand as 10 hundreds and each hundred as 5 twenties. To find one twentieth, Ann anticipated fifty as a segmenting unit and to verify and/or explain such anticipation she again resorted to counting. Her counting indicated that she had conceptualized one thousand as 10 hundreds and each hundred as 2 fifties. In other words, Ann conceived one thousand as a unit of units by interpreting it as a multiple of units of twenty and fifty. Then she used this multiplicity to fractionally re-interpret twenty and fifty, with relation to one thousand, as one fiftieth and one twentieth, respectively.

To generate fractional quantifications for parts of discrete wholes, Ann modified her initial part-whole scheme. The modification was indicated by her awareness not only of the numerosity of the parts in the whole, but also of the numerosity within each part (the content of it). The awareness of these two numerosities prompted her coordination of natural-number units and fractional-number units.

Ann's coordination of natural-number and fractional-number units guided her construction of the inverse relationship between the number of parts and the size of each part in a given whole. In the following dialogue, Ann showed her idiosyncratic way of reasoning.

T: Can you give me one hundredth of a thousand dollar bill?

A: [Quickly] Yes, one ten-dollar bill.

T: Why?

A: Because if one dollar is one hundredth of this [showing a hundred- dollar bill], then ten dollars must be one hundredth of this [showing a thousand-dollar bill].

T: Very good way of thinking. Can you verify that in another way?

A: Well, the only other way Would be 10-20-30-40-50/60-70-80-90- 100 [showing the fingers of her two hands and implicitly assigning one finger to each hundred and 10 tens to each finger].

T: Five dollars is what part of one thousand-dollar bill?

A: [Quickly] One two-hundredth.

T: Why?

A: Because if ten is one hundredth, then five is one two-hundredth. [Then Ann counts by twenties on her fingers and implicitly assigns one finger to each hundred and 20 fives to each finger] 20-40-60-80- 100/20-40-60-80200. That's one thousand.

FRACTIONS SCHEME l 19

Ann found that ten was one hundredth of one thousand dollar bill "because if one dollar is one hundredth of this [one hundred], then ten dollars must be one hundredth of [one thousand]." An interpretation of her answer is that Ann conceptualized the task multiplicatively, and her mental reasoning went as follows: Knowing that

one hundred is 100 times as large as one, if

I take a quantity that is 10 times as large as one hundred (i.e., one thousand) and another quantity that is 10 times as large as one (i.e., one ten), then

the multiplicative relationship between one thousand and ten must be the same as the multiplicative relationship between one hundred and one since I enlarge the content of each part leaving unchanged the number of parts. This answer, graphically interpreted is shown in Figure 3.

The same answer could also be interpreted numerically, as in Figure 4.

In fact, Ann's counting by tens to verify her answer indicated that she zould also conceive one thousand as 10 hundreds and each hundred as 10 tens.

The second answer could be interpreted graphically in a similar manner, see Figure 5.

This answer numerically interpreted is shown in Figure 6. When Ann used counting to verify her answer, she re-interpreted the

unit one thousand as 10 hundreds and each hundred as 20 fives. This re- conceptualization of one thousand from a unit of units to a unit of units of units was another path that Ann used to find the multiplicative relationship between five and one thousand. She used, then, this multiplicative relationship to give five the fractional quantification "one two-hundredth."

Ann's answers indicated that her conceptualizations of numerical quantities as units of units, in different ways, were essential to find part-to-whole multiplicative relationships that in turn induced fractional quantifications for the subunits embedded in the containing unit. Furthermore, these conceptualizations of a quantity as a unit of units in different ways induced the construction of the inverse relation between the number of parts and the content of each part. The more subunits she could disembed from the given unit, the smaller the subunits were, and the smaller the fractional quantification of these subunits.

1 2 0 A D A L I R A S ~ E N Z - L U D L O W

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Ann's realization of the above relationship enabled her to deal with the ordering of fractions quite readily. The following dialogue indicated that Ann's comparisons of fractions were based on the inverse relation

122 ADALIRA S~,ENZ-LUDLOW

between the number of parts and the content (size) of each part, as well as the coordination of natural-number units and fractional-number units.

T: Which is bigger, one half or one hundred ninety-nine two-hundredths of one thousand-dollar bill?

A: One hundred ninety-nine two-hundredths. Oh] No.

T: [Repeats the question]

A: One hundred ninety-nine two-hundredths is less than one half? Wouldn't it be? I don't know why.

T: Try to figure it out.

A: O.K. It takes two hundred two-hundredths to make one thousand. One hundred ninety-nine two-hundredths is one less [meaning one two- hundredth less] than one thousand. But one half, I need two of this [meaning one half] to make one thousand and I have only one. One hundred ninety-nine two-hundredths is bigger.

T: Which is bigger, ninety-nine hundredths or nine tenths of one thousand dollar bill?

A: Ninety-nine hundredths because you need ten more dollars to make one thousand. Nine tenths, you need one hundred dollars more to make one thousand. Ninety-nine hundredths is bigger.

Ann's successful struggle to compare the fractions one half and one hundred ninety-nine two-hundredths of one thousand dollars seems to be due to three different, but sequential, mental processes. First, she conceptualized one thousand dollars as a composite unit of fractional units in two different ways - two halves and two hundred two-hundredths. Second, she implicitly compared one half and one two-hundredth using the inverse relation between the number of parts and the size (content) of each part. Third, she used the comparison between the unit fractions (one half and one two-hundredth) to compare their complementary fractions (one half and one hundred ninety-nine two-hundredths). My interpretation is that these three processes synthesized a mental and implicit sophisticated way of reasoning that could go as follows: if one half is larger than one two- hundredth, then the complementary fraction of one half must be smaller than the complementary fraction of one two-hundredth.

To find her second answer, Ann reconstituted one thousand into a composite unit of fractional units in two different ways - ten tenths and one


hundred hundredths. Then, she used the multiplicative part-to-whole relations (one thousand is the same as 10 hundreds or 100 tens) to find the content of one tenth and one hundredth (one hundred and one ten, respectively) and coordinated these natural-number units with the given fractional units. Finally, because of her conservation of the unity of the whole as a unit of units (using different units), she proceeded to complete the whole (one thousand) to compare the given fractions.

In solving two similar tasks, Ann used two different ways of reasoning. Her first answer was the result of her initial awareness of the inverse relation between the number of parts and the size of each part. To find her second answer, she resorted to the coordination between natural-number units and fractional-number units. These two different quantitative strategies, con- secutively performed, indicated that Ann's metric part-whole scheme was evolving into an operative scheme adaptable to more general situations.

In summary, given a discrete whole, Ann was able to anticipate a segmenting unit to partition it exhaustively as a result of a mental measuring act. In doing so, she remained aware of the unity of the given whole and conceived the whole as a part-dependent entity by establishing a part-to- whole relationship. Subsequently, she inverted this relation by conceiving the part as a whole-dependent entity and establishing a whole-to-part relationship. Her part-to-whole relation was embodied in the reconstitution of the whole as a multiple of the part or as a unit of units, but her whole-to- part relation was embodied in the fractional quantification for a part with respect to the whole.

The question that needed to be answered was if Ann's coordination of natural-number units and fractional-number units fostered the constructive evolution of the inverse relation between the number of parts and the size (content) of each part. The following section is indicative of her developing reasoning when wholes were not specified and when the specific content (numerosity) of each part could not be numerically represented to operate with it.

7. ANN'S C ONSOLIDATION OF HER METRIC PART-WHOLE SCHEME

Ann's metric part-whole scheme for continuous and discrete wholes was an essential mental construct that supported her development of fractions. The fraction tasks given to Ann, up to this point, used well-specified discrete and continuous wholes. To facilitate her abstracting activity, tasks were posed using nonspecific wholes. For example, in the following dialogues, Ann had to deal with fractions of a "certain amount" of money.

124 ADALIRA S,~ENZ-LUDLOW

T: I have in my pocket a certain amount of money and I am going to give to you twenty-five fortieths and to Michael fifteen fortieths. What part of the money will I have left over?

A: I don't know because I don't know the amount of money that you have in your pocket.

T: Even if you don't know the amount of money in my pocket, I 'm giving you all the clues.

A: What is one fortieth?

T: One out of forty.

A: None.

T: None what?

A: None fortieths. But I have to figure out how much is one fortieth!

The lack of specificity of the amount of money made Ann think that she was unable to solve the task. However, once she realized that one fortieth was the fractional quantification for one part out of forty parts (whether she knew the total amount of money or not), she was able to find the answer "none fortieths." To find her answer, she had to mentally represent the "certain amount of money" as a unit of forty fortieths from which she disembedded two composite fractional units (twenty-five fortieths and fifteen fortieths) that were complementary in the sense that these units exhausted the whole. The last two lines of the dialogue indicated that she knew what she meant by none, but she still wanted to know the exact content (size) of one fortieth.

The above task used fractions that were the result of one partition of a "certain amount of money". However, the following task used fractions that were implicitly determined by two partitions of the same amount of money.

T: Suppose that you give me one seventh of your money and then one fourteenth of your money. What part of your money did you give me?

A: But what is my money? If I don't know it, I cannot do it.

T: Well it doesn't matter what amount of money you have, I will be happy with the part that you give me.


A" Give me a clue!

T: What is bigger, one seventh or one fourteenth?

A: Well, fourteen doubles seven ...

T: So ...

A: One seventh equals two fourteenths.

T: [Repeats the initial question]

A: [She takes some seconds] Three fourteenths.

T: Why?

A: Well, one seventh is two fourteenths and one fourteenth more is three fourteenths.

Ann's request to have a clue indicated her willingness to go on with the task, regardless of not knowing what the total amount of money was. The teacher's hint to compare one seventh and one fourteenth was interpreted by Ann in terms of partitions of the same whole that she coordinated by embedding one into the other. Her expression "fourteen doubles seven" indicated that she generated a partition with fourteen parts out of the partition of a whole into seven parts by segmenting each of the seven parts into two equal parts. Using her mental representation of these two partitions, she generated a relationship between the two given fractions: "one seventh equals two fourteenths." Then, she added two fourteenths and one fourteenth to find three fourteenths.

Ann's coordination of these partitions was a product of her metric part- whole scheme which allowed her to see the whole (her money) as a unit of 7 units of two. Ann's interpretation of the whole as a unit of units was an indication that her solution was'linked to the conceptualization of different partitions of the same whole.

Ann's sophisticated reasoning with fractions was indicated when she had to deal with a fraction of a fraction as in the following dialogue.

T: If I ask you to give me one fourth of one half of your money, what part of your money would you give me?

A: [Taking some seconds to think] That question, I 'm not sure I can do it without the amount of money.

T: [Gives Ann a sheet of paper] Suppose this is your money.

126 ADALIRA SJ~ENZ-LUDLOW

A: [Ann folds the paper in two parts, folds it again in four parts and shows one piece] Here is one fourth of one half of my money.

T: Perfect! But this is what part of the whole amount of money?

A: This is hard! [She takes some seconds to think] This is one eighth of my money.

T: Why?

A: Because four in this half and four in this half.

T: Now, can you give me one half of one fourth of your money?

A: [Ann uses another piece of paper, folds it in four parts, and again in two parts] Oh! One eighth.

T: When do you give me more money, when you give me one fourth of one half, or when you give me one half of one fourth?

A: I give you the same amount of money.

To think about a fraction of a fraction as the result of a partition of a partition of a nonspecified whole was a challenging task for Ann without any visual aid, so the teacher suggested the use of a sheet of paper to represent the total amount of money. From then on, Ann's interpretation of the expression "one fourth of one half ' guided the folding of the paper in two parts and then in four parts as if she had anticipated that she had to make first two parts and then four parts. However, to find the fractional quantification for the part that was the result of a partition of a partition was more difficult for Ann. Once she realized that the result of these consecutive foldings was a partition of the whole into eight parts, she gave to each part the fractional quantification of one eighth.

After solving the first task, the second task of finding "one half of one fourth" was easy for Ann. Nonetheless, she was surprised, as indicated by her "Oh!" expression, when she found that the result of this new task was also "one eighth." Then she used this finding as a stepping stone to establish that one fourth of one half and one half of one fourth were the same. On her own, Ann seemed to have used transitivity and commutativity perhaps without being aware. Ann's solutions to these tasks were linked to the reconstitution of the whole as a unit of units in two different ways: a unit of 4 units of two, and a unit of 2 units of four. These multiplicative conceptualizations of the whole (her money represented in a sheet of paper) and the fractional quantification of one eighth for one of the parts were clearly the result of the application of her metric part-whole scheme.


In the following dialogue, Ann demonstrated a sophisticated use of her metric part-whole scheme by coordinating two implicit partitions of the same whole.

T: Suppose that you are given one third of a certain amount of money. From the money that you receive, you decide to give me as much as one twelfth of the initial amount of money. What part of the money do you have left?

A: Give me a clue. One fifth or one ninth. Wait, wait. Five thirds. No, sixteen twelfths. Wait, one third is four twelfths...

T: Can you use your relation "one third is four twelfths" to solve the problem?

A: [Takes some time] One and one-third twelfths. Give me a clue.

T: You already found the main clue, but if you want to, you can assume that the top of the desk is the total amount of money that you have.

A: [Ann partitions the desk in three parts and each of the new parts in four parts, see Figure 7] One third minus one twelfth is three twelfths.

T: Perfect!

Without realizing that the fractions one third and one twelfth were determined by different partitions of the same whole (certain amount of money) that could be coordinated, it would have been impossible for Ann to solve this fraction task without knowing the algorithm to subtract fractions. On her own initiative, Ann proceeded with some trials that led her to find the relation "one third is four twelfths." However, she was unaware of being just one step away from the solution. The teacher's suggestion to take the desktop as a representation of the total amount of money prompted Ann to express the relationship between the given fractions using partitions. She partitioned the desk into three parts and, subsequently each part into four parts; then she disembedded one twelfth from one third to find the complementary fraction three twelfths.

In summary, Ann's sophisticated solutions to more difficult fraction tasks were the result of the evolution of her metric part-whole scheme through its application to situations in which the wholes were not explicitly given. Ann's cognitive efforts in the pursuit of a solution were an autonomous decision that facilitated the evolution of her scheme.

128 ADALIRA S,~ENZ-LUDLOW

i/3

-----1 --------1

1/12 ~

Fig. 7.

8. DISCUSSION

Ann's case study indicates that the evolution of her fraction scheme was the product of her cognitive activity and her collaborative interaction with the teacher. In this interactive process, the tasks posed by the teacher facilitated Ann's constructions, but Ann's answers guided the teacher's questions and inferences.

Ann's fraction units were the result of her metric part-whole scheme. The construction of this scheme was first indicated, during the fourth clinical interview, by Ann's solutions to fraction tasks within continuous contexts, and this scheme evolved as a result of its application to solve fraction tasks within discrete contexts. The consolidation of this mental construct was indicated when Ann solved fraction tasks using wholes that, while not specified, were kept constant.

The metric part-whole scheme seems to be a synthesis of her act of measuring and the concomitant and cyclical establishment of part-to-whole and whole-to-part relationships. A result of her physical or mental measuring acts was the conceptualization of a whole as a composite unit while establishing the one-to-many (part-to-whole) relationship that was part- dependent and multiplicative in nature. Disembedding a part (or subunit) from a whole (composite unit) while conserving its unity generated a many-to-one (whole-to-part) relationship that was whole-dependent and


fractional in nature. Ann's fractions were the quantification of these cyclical relationships between the whole and its parts. In other words, once the part-to-whole relationship was complemented with its inverse (the whole- to-part relationship), this inverse relationship induced the generation of fractional quantifications for a part with respect to the whole. For some time, Ann compared fractions that were the result of different partitions of a known discrete whole, based on the actual content of each fraction. More- over, the coordination of natural-number units and fractional-number units guided the conceptualization of the inverse relation between the number of parts and the content (size) of each part. This inverse relation allowed her to compare unit fractions of known and unknown wholes, without depending on their content, as well as the generation of equivalent fractions for solving addition tasks with no formal algorithm.

Ann refined her initial fraction conceptualizations to the point of generating sophisticated ways to operate with fractions when wholes were not explicitly defined. Her conceptualization of continuous or discrete wholes as a unit of units and the verbal explanation of her answers were essential factors in the development and evolution of her fraction scheme over a period of 7 months. Furthermore, her fraction concepts were rooted in her sophisticated ways of counting and her already established part-whole numerical operations (Steffe and Cobb, 1988) within the context of natural numbers.

Several aspects of the teaching-learning process are exemplified in this case study. First, Ann was able to generate her fraction conceptualizations in the absence of numerical notation and in the midst of using natural language and fraction number-words; it appears as if verbalization of her mental activity fostered the development of her fraction concepts. Second, Ann's fraction concepts evolved over time and her collaborative interaction with the teacher supported such evolution. Third, Ann used her prior knowledge of natural numbers and her strong conceptualization of units to generate and develop her fraction concepts.

ACKNOWLEDGEMENTS

This article is based on portions of the author's doctoral dissertation com- pleted at the University of Georgia-Athens in 1990 under the direction of Leslie R Steffe.

130 ADALIRA S./~ENZ-LUDLOW

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ADALIRA S~ENZ-LUDLOW,

Assistant Professor of Mathematics Education, Department of Curriculum and Instruction, Purdue University, West Lafayette, Indiana 47907-1442, USA

ann's fraction schemes

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