8 ok_partitioning the emergence of rational number ideas in young children

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Partitioning: The Emergence of Rational Number Ideas in Young Children Author(s): Yvonne Pothier and Daiyo Sawada Source: Journal for Research in Mathematics Education, Vol. 14, No. 5 (Nov., 1983), pp. 307-317Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/748675Accessed: 15-10-2015 17:24 UTC

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Journal for Research in Mathematics Education 307 1983, Vol. 14, No. 4, 307-317

PARTITIONING: THE EMERGENCE OF RATIONAL NUMBER IDEAS

IN YOUNG CHILDREN

YVONNE POTHIER, Mount Saint Vincent University DAIYO SAWADA, University of Alberta

The study sought to trace the emergence and differentiation of partitioning as a process that leads young children to construct ideas of rational numbers. The method involved clinical sessions with children as they manipulated materials while solving tasks designed to reveal partitioning processes. The data analysis led to a theory concerning the emergence of partitioning capability, and conceptual structures that undergird the capability were identified. A five-level theory of the development of the partitioning process is presented.

Key position papers by Kieren (1976, 1980) have created a new theoretical context for inquiring into the child's acquisition of rational number concepts. Recent work within the new perspective (Kieren & Nelson, 1978; Kieren & Southwell, 1979) has lent support to Kieren's position.

Basic in Kieren's perspective is the process of dividing a whole into parts, a process variously referred to as partitioning, subdivision, or dissection. The purpose of the present study was to trace the emergence and differentiation of the process of partitioning as revealed in children's attempts to subdivide a continuous whole into equal parts.

METHOD The method involved a clinical interaction with a child as he or she

grappled with tasks specifically designed to reveal partitioning capabilities. As the child grappled with the tasks, the researchers grappled with the even more complex task of attempting to make sense of the child's behavior. We hoped to discover underlying patterns that would help us trace the emergence and differentiation of the partitioning process.

The method can be characterized as a clinical interaction technique set within a discovery paradigm. The interactive encounters were designed to yield insights rather than to verify hypotheses; validity, as opposed to generalizability, was our concern.

The interaction was characterized by flexibility in questioning. The initial question for each task was standard, but the subsequent questions, although following a general pattern, were varied, as were the numbers in the problem, depending on the behavior of the child. The method permitted probing,

This report is based on the first author's doctoral dissertation, completed at the University of Alberta in 1981 under the supervision of the second author.


308 Partitioning and the Emergence of Rational Number

following through a child's peculiar behavior, expanding on a question, or altering the problem. Sample

The 43 children who formed the sample were from kindergarten and the first three grades in a primary school in Alberta. The selection of respondents followed the method of theoretical sampling (Glaser & Strauss, 1967) that requires that no a priori decision be made regarding sample size; rather, the selection continues throughout the period of data collection. The children, all acquainted with the interviewer, were chosen on the basis of the interviewer's belief that they would respond freely to the task during a clinical session. The sampling continued until sufficient key insights had emerged and a satisfac- tory degree of conceptual integrity connected the various insights. At the close of the clinical sessions, the sample included 8 kindergarten, 8 Grade 1, 12 Grade 2, and 15 Grade 3 children. Their ages ranged from 4 years, 11 months to 9 years, 8 months. Task

Although five partitioning tasks were devised for the study, one task, the Cake Problem, proved to be highly effective both in enabling children to demonstrate their partitioning capabilities and techniques and in giving us key insights into the partitioning process. (For details of the other tasks, see Pothier, 1981.)

The materials for the Cake Problem consisted of one circular and four rectangular "cakes" of various dimensions, a circular "cookie," a supply of sticks, and a set of miniature dolls. The rectangular cakes were white Styrofoam models, 5 cm in height and topped with a piece of plush gold rug glued to the surface. The dimensions of the rectangular cakes were 10 cm by 25 cm, 15 cm by 20 cm, 20 cm by 25 cm, and 10 cm by 15 cm. The circular cake, 20 cm in diameter and 2.4 cm in height, was constructed of the same materials as the rectangular cakes. A circular piece of rug, 16 cm in diameter, was also used and was referred to as a "giant cookie." The sticks, which served to demonstrate cuts on the cakes and the cookie, were white plastic molding, 1 mm thick, cut in strips 0.5 cm wide and of lengths 10, 19, and 27 cm. The complete supply of sticks was available for the child's use when partitioning each cake or cookie for any number of dolls among whom it was to be shared. Procedure

The setting for the Cake Problem was a birthday party, and the child was asked to demonstrate how he or she would cut the cake so that each person at the party had the same amount. The question posed to the child was, "How would you cut the cake so that each one has as much?" After one way had been demonstrated, the child was requested to show another way to cut the cake so that each person had the same amount.


Yvonne Pothier and Daiyo Sawada 309

The child was first requested to partition the cake for 2 people. In most cases, the subsequent partitions were for 4, 3, and 5 people. The order of presentation varied, however, and not everyone was asked to partition the cakes for the same numbers of people. The older children were asked to make more partitions than the younger ones. Additional sequences for the older children were 4, 8, and 16 people; 5, 10, and 20 people; 3, 6, and 12 people.

The circular region was partitioned first; that is, the children were presented first with the giant cookie and subsequently with the rectangular cakes. The round cake was generally employed for further partitioning of a circular region following work with the rectangular cakes.

In addition to video and audio recording, any distinctive behavior observed or impressions made were recorded in written form during the sessions.

Analysis Scheme The analysis was based on a method suggested by Glaser & Strauss (1967):

Theoretical sampling is the process of data collection for generating theory whereby the analyst jointly collects, codes, and analyzes his data and decides what data to collect next and where to find them, in order to develop his theory as it emerges. This process of data collection is controlled by the emerging theory, whether substantive or formal. (p. 45)

Accordingly, a three-stage analysis scheme was used. Immersion stage. The first stage was a continuous process of analysis

characterized by an intense focus on the child's behavior. Its ongoing and flexible nature provided the direction for immediate probing and for varia- tions in the task.

Reflection stage. During the period of data collection, time was daily devoted to reflecting on the day's interactions and viewing, transcribing, and analyzing taped sessions. The interviewer kept a journal of insights gained and inferences made. The reflective process provided direction for further data collection.

Documentation stage. The third stage of the analysis was conducted after the data had been collected and comprised a systematic examination of the tapes and other records made during the first two stages. Charts, lists, and tables (some presented below) were constructed for the purpose of verifying the interpretations and for gaining further insights.

FINDINGS The theoretical account given below is a formalization of the insights that

emerged over the three stages of the data analysis. Realizing that the formalization is a rather severe distillation of the data, we first identify some constructs that play a key role in the theory, and we follow the account of the theory with excerpts from the transcripts.

To trace children's partitioning capabilities as precursors of rational number acquisition, the relevant mathematical constructs, contrary to expec- tation, are not those of rational number. Rather, we propose that the key



theoretical constructs come primarily from number theory and secondarily from geometry. Our description of the emergence and differentiation of the partitioning process hinged critically on the use of bipolar number-theoretic concepts: odd/even, prime/composite, factor/multiple. We synthesized the bipolar constructs into a final level of partitioning capability suggested by the fundamental theorem of arithmetic. We made refinements in each level of capability by using concepts from motion geometry: translation, reflection, rotation, similarity, congruence, and symmetry. The use of theoretical constructs derived from number theory and geometry does not imply that children are aware of these constructs. We simply find them valuable in making sense of the clinical interactions we had with the children.

We propose a five-level theory that describes the development of the partitioning process as we see it. Four levels are grounded in the data; the fifth level follows logically, but remains hypothetical.

Four Levels of Partitioning Capabilities The first four levels are outlined below in terms of three distinctive charac-

teristics: (a) the construct, or key concept, developing during the level; (b) the algorithm, or procedure, employed to produce the partitions; and (c) the domain, or extent, of the partitioning capabilities within the level. Level I: Sharing

"* Construct-breaking; sharing; halving "* Algorithm-allocating pieces ("a piece for you") "* Domain-social setting; counting numbers

Level II: Algorithmic halving "* Construct

-systematic partitioning in two -no notion of equality

"* Algorithm--repeated dichotomies "* Domain-one-half and other unit fractions whose denominators are

powers of 2 Level III: Evenness

* Construct-equality; congruence -repeated dichotomies becoming meaningful

* Algorithm--halving algorithm; geometric transformations -extension of algorithmic halving to doubling the number of

partitions and adding two parts * Domain-unit fractions with even denominators

Level IV: Oddness * Construct-even and odd

-search for a new first move -use of the new first move -geometric transformations



"* Algorithm--exploratory measures; trial and error -counting; one-by-one procedure

"* Domain-all unit fractions

Interpreting the Levels Level I: Sharing. The action of partitioning an object or a set of objects is

learned by a child in a social setting (Kieren, 1980). At an early age, the child witnesses situations in which things are shared and hears expressions such as "Break (or cut) it in half" and "Here's half." Later, the child participates in sharing activities, learns how to share, and eventually uses language to describe the sharing action and the result of sharing. But as far as rational number is concerned, this is rote learning. The child does not know the meaning of half in a number sense, and the necessary characteristics of evenness and one of two parts are not understood. This is shown by the child's use of the term half in expressions like break it in half in four pieces and split it in half in three pieces. Also, some children use the term half exclusively as an operation. An example is Lon, aged 5 years and 6 months (5; 6), who stated that the cookie was broken in half but called each piece a broken cookie. To the query "What could you call that piece?" he responded, "I don't really know."

At the Sharing level, the child begins to learn a halving mechanism, that is, how to construct a line through the middle of a region. Using the mechanism, the child can usually partition rectangular and circular regions in halves and fourths. However, some attempts result in uneven shares, in more parts than required, or in the region not being used up. Samples of these partitioning attempts are depicted in Figure 1. At this level, partitioning continuous quantities simply means allocating "pieces," and an equal number, regardless of size, is considered a "fair share."

Level II: Algorithmic halving. A second level is marked by the mastery of the doubling process that enables the child to partition rectangular and circular regions not only in halves and fourths but also in eighths and sixteenths. In other words, the child can double the number of parts to obtain fractional parts whose denominators are powers of 2. The child uses the acquired tool in an algorithmic manner, but there is no concern for equality. Successive partitions are performed systematically so that each part in turn is bisected, thus doubling the previous number of parts. Because the algorithm "works" so well, it is used without careful scrutiny of the partitions it produces. The procedure is often used in situations where it may be in- adequate or unsatisfactory. For example, a diagonal line can be used to partition a rectangular region evenly in two, so the child either performs repeated "dichotomies" in an effort to obtain fourths (see Figure 2a) or thirds (see Figure 2b) or mixes two halving lines (see Figure 2c). Vertical parallel lines that work in a rectangular region are used in a circular region to produce thirds, fourths, or fifths (see Figure 2d). In such instances, the parts are



Figure 1. Level I partitioning behavior.

(a) (b)

(c) (d)

Figure 2. Level II partitioning behavior: Thirds, fourths, and fifths.

declared to be fair shares. Examples of attempts at eighths and sixteenths are portrayed in Figure 3.

Level III: Evenness. A third level of achievement is attained when the child



Eighths

Sixteenths

Figure 3. Level II partitioning behavior: Eighths and sixteenths.

attends to the size and shape of the parts and critically examines them to determine if they are equal, or even. At this level, partitions are correctly classified as "fair" or "not fair," although some children may not be able to produce equal shares (e.g., thirds or fifths).

Level III is a major breakthrough in the child's thought. Algorithmic halving becomes more meaningful; equality (and evenness) is now a critical characteristic in designating fair shares. Diligent attempts are made to construct equal pieces. For many children partitions that do not look the same are designated "not fair." Moreover, the equality concept, together with geometric motions, enables the child to produce partitions for numbers other than powers of 2. Even numbers of parts can now be obtained. For example, to obtain sixths, the child typically partitions a circular region into fourths, bisects two sections, and then rotates two sticks to obtain six equal parts (see Figure 4a). In a rectangular region, sixths can be attained from fourths by translating a halving stick and placing an additional stick parallel to it (see Figure 4b). In a similar manner, tenths can be obtained from eighths, and twelfths from sixths.

(a) 2

1 3

1 3. "1

(b)

Figure 4. Level III partitioning behavior: Sixths.



This primitive use of geometric motions enhances algorithmic halving so that the child can obtain unit fractional parts whose denominators are even numbers. We refer to this more sophisticated capability as the halving algorithm. Although we did not test this conclusion, we consider the halving algorithm to be composed of two things: the doubling of a number of parts and the attainment of additional parts by adding another stick and making rotational moves.

In summary, the Evenness level is marked by two notions of evenness: (a) the equality, or evenness, of parts, and (b) the ability to partition unit fractions with even denominators. Children at this level are unable to obtain thirds and fifths. Typical attempts are depicted in Figure 5.

Figure 5. Level III partitioning behavior: Thirds and fifths.

Level IV: Oddness. Following the Evenness level, there is a period in which the child becomes increasingly aware of the inadequacy of the halving algorithm. The child learns that by beginning with a halving line, some fractional parts cannot be attained efficiently in either a rectangular or a circular region. The problematic numbers are three and five, or odd numbers as shown earlier in Figure 5. Some children label these as not even or the hard ones. When asked to make an odd number of parts, some children respond



with statements such as "I can't get it because it's not even" or "It can't be done."

Hof (8; 10) and Bro (8; 5) are children who had reached Level IV. When partitioning a rectangular cake for seven people, Hof stated, "This

one's an odd number.... I can do even numbers, but it's hard to do odds." He defined even and odd numbers: "It means like, if you're countin' by twos, they're even numbers. If you count by ones, like, I should say by threes-one, three, like that, they're odd numbers."

While partitioning a round cake, Bro stated that three and five parts were "hard to make." Asked, "What's hard about three and five?" she replied, "Well, if you try to make something out of 'em, you already made them." She explained further: "Well, you try to make five, but then you get six." In attempting thirds, she stated, "Like if you make three like this [a horizontal diameter cut and two vertical radius cuts], you'd get four, so, like if you go like this [moving the sticks], you still get four. If you go like this [one horizontal diameter cut], it's two halves."

Realizing that a different initial cut is necessary to partition into thirds and fifths, the child actively searches for a different move, thus freeing the partitioning from the dominance of the halving algorithm. For a circular shape, the first move is a cut along a radius of the circle, whereas for a rectangular shape, it is a cut in a vertical or horizontal line in a position other than the middle. The child uses this new move to obtain fractional parts whose denominators are odd.

In the case of an odd number of parts, the child uses a counting algorithm to guide the partitioning; that is, pieces are produced one by one, and counting is used to keep track of results. Such a strategy requires frequent readjustment of the partitioning lines to obtain equal shares. Having attained the oddness level, the child is able to partition rectangular and circular regions into any number of parts depending on his or her counting ability.

Level V: Composition. The counting algorithm used in Level IV is suitable for small odd prime numbers. However, it is an awkward algorithm for large numbers such as 9 and 15 because equal parts are not easily attained. In time, the child will probably question the one-by-one procedure and look for a different method. For composite numbers such as 9 and 15, there is clearly a more efficient strategy.

To obtain ninths, a region can first be partitioned into thirds and then each part trisected. In like manner, fifteenths can be composed by trisecting fifths. The possibility of alternative partitioning sequences demonstrates that the choice of a factor for a first move does not affect the result: an illustration of the fundamental theorem of arithmetic. A child who proceeds to partition composite numbers in this manner employs a multiplicative algorithm. Such a child has reached the Composition level and can now efficiently construct all unit fractionst

As stated earlier, the fifth level is not based on data; none of the children in



the study used a multiplicative algorithm when partitioning the regions. This level is strictly an implication of the theory-a level that would emerge, we hypothesize, with older children.

DISCUSSION We propose that the acquisition of the partitioning process is the culmina-

tion of a gradual progression through five levels. Each level is distinguished by certain conceptual characteristics, procedural behaviors, and partitioning capabilities. In particular, the partitioning process is mastered through the successive attainment of five subsets of the unit fractions: the fraction 1/2, fractions whose denominators are powers of 2, fractions with even denominators, fractions with odd denominators, and fractions with composite denominators.

A child first learns to partition in two; then, with the acquisition and eventual mastery of the halving algorithm, in powers of 2; then, with the use of geometric motions, in even numbers. Partitioning in odd numbers follows the learning of a first move other than a median cut. With the discovery of the new first move, children are able to partition in thirds, fifths, and other odd numbers; thus, thirds and fifths are achieved together. The algorithm involves counting, and equality of parts is usually achieved by rotational (for circular shapes) and translational (for rectangular shapes) moves.

For continued mastery of partitioning, ideas of odd and even appear to play a role. Children who constructed thirds or fifths but not both appeared not to be cognizant of the oddness quality. Their use of a nonhalving first move seemed accidental.

In attempts to partition rectangular and circular regions into thirds and fifths, the dominance of the halving algorithm was evidenced. Many children seemed incapable of deviating from employing a halving line as the initial "cut." A few who began with a line other than a median line did so only after several trials or with guidance. Since young children ordinarily master halves and fourths prior to thirds (Hiebert & Tonnessen, 1978; Piaget, Inhelder, & Szeminska, 1948/1960), the children in the sample were first asked to partition the regions into an even number of equal-sized parts. Would the results have been different if partitioning into an odd number of parts had been the first activity? Would the halving algorithm have been so dominant, or is the behavior a sequence effect? This issue is the focus of our current research.

CONCLUSION The goal of the study was to gain an understanding of the partitioning

process as it relates to the development of rational number ideas. From an interpretation of the partitioning behavior of young children, a theory has been generated wherein foundational constructs are identified and levels characterizing distinct partitioning capabilities are delineated. Within each level, developing concepts and algorithms, together with increasing domains



of partitioning capabilities, are described. Our results, like those of Hiebert and Tonnessen (1978), are in substantial agreement with the results of Piaget, Inhelder, and Szeminska (1948/1960). Our main departure has been in discovering the appropriateness of number-theoretic concepts in making sense of the emergence and differentiation of the partitioning process.

REFERENCES Glaser, B. G., & Strauss, A. L. The discovery of grounded theory: Strategies for qualitative

research. Chicago: Aldine, 1967. Hiebert, J., & Tonnessen, L. H. Development of the fraction concept in two physical contexts:

An exploratory investigation. Journal for Research in Mathematics Education, 1978, 9, 374-378.

Kieren, T. E. On the mathematical, cognitive, and instructional foundations of rational numbers. In R. A. Lesh & D. A. Bradbard (Eds.), Number and measurement: Papers from a research workshop. Columbus, OH: ERIC Information Analysis Center for Science, Mathematics and Environmental Education, 1976.

Kieren, T. E. The rational number construct-Its elements and mechanisms. In T. E. Kieren (Ed.), Recent research on number learning. Columbus, OH: ERIC Clearinghouse for Science, Mathematics and Environmental Education, 1980.

Kieren, T., & Nelson, D. The operator construct of rational numbers in childhood and adoles- cence-An exploratory study. Alberta Journal of Educational Research, 1978, 24 (1), 22-30.

Kieren, T., & Southwell, B. The development in children and adolescents of the construct of rational numbers as operators. Alberta Journal of Educational Research, 1979, 25 (6), 234- 247.

Piaget, J., Inhelder, B., & Szeminska, A. The child's conception of geometry (E. A. Lunzer, Trans.). New York: Basic Books, 1960. (Original French edition, 1948.)

Pothier, Y. Partitioning: Construction of rational number in young children. Unpublished doctoral dissertation, University of Alberta, 1981.

[Received May 1982; revised January 1983; revised June 1983]


Article Contentsp. 307p. 308p. 309p. 310p. 311p. 312p. 313p. 314p. 315p. 316p. 317

Issue Table of ContentsJournal for Research in Mathematics Education, Vol. 14, No. 5, Nov., 1983Volume Information [pp. 380 - 384]Front Matter [pp. 305 - 375]Editorial [p. 306]Partitioning: The Emergence of Rational Number Ideas in Young Children [pp. 307 - 317]The Effects of Test Language and Mathematical Skills Assessed on the Scores of Bilingual Hispanic Students [pp. 318 - 324]Sex Differences in Mathematical Errors: An Analysis of Distracter Choices [pp. 325 - 336]Vocabulary Instruction in Ratio and Proportion for Seventh Graders [pp. 337 - 343]Cognitive Style, Operativity, and Mathematics Achievement [pp. 344 - 353]Dissertations in Mathematics Education at the Higher School of Education in Cracow, 1968 to 1981: An Annotated Bibliography [pp. 354 - 360]Brief ReportThe Development of the Number-Word Sequence in the Counting of Three-Year-Olds [pp. 361 - 368]

ReviewSymbolism from A to W [p. 369]

Telegraphic Reviews [pp. 370 - 373]Letters to the Editor [p. 376]Back Matter [pp. 377 - 379]

8 ok_partitioning the emergence of rational number ideas in young children

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