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Behr, M., Harel, G., Post, T. & Lesh, R. (1991). The Operator Construct of Rational Number. In F. Furinghetti (Ed.) Proceedings of PME XV Conference (pp. 120-127). Assisi, Italy: PME.

 
THE OPERATOR CONSTRUCT OF RATIONAL NUMBER:
A REFINEMENT OF THE CONCEPT
 
Merlyn Behr
Northern Ill. Univ.
Guershon Harel
Purdue Univ.
Thomas Post
Univ. of Minn.
Richard Lesh
ETS
 

Some results of an ongoing mathematical/semantic analysis of the additive and multiplicative conceptual fields being conducted by the Rational Number Project are presented. These results suggest refinements of the operator construct of rational number, one of which presented herein, is called duplicator/partition-reducer (D/PR). The analysis uses two notations: a generic manipulative aid and a generalized notation for mathematics of quantity. As D/PR, a rational number such as 3/4 is a 3-for-4 exchange function. Implications of this for computational procedures and problem solving are suggested and presented.

A mathematical/semantic analysis of the subconstructs of rational number (Kieren 1976) conducted by the Rational Number Project (Behr, Harel, Post, & Lesh, 1990, in press) has led to a refinement and a deeper understanding of these constructs. Behr et al. (in press) imposed different combinations of interpretations on the numerator and denominator and hypothesized 5 distinct interpretations of the operator construct: Duplicator/Partition-Reducer, Stretcher/Shrinker, Multiplier/Divisor, Stretcher/Divisor, and Multiplier/Shrinker. Extensive analysis of the first two has identified interesting similarities and differences. The purpose of this paper is to report some findings about the Duplicator/Partition- Reducer (D/PR) interpretation.

The numerator as a duplicator suggests that its effect on an operand quantity is to take it as a single entity and make a number of copies of that quantity so the total number equals the numerator. The denominator as a partition-reducer suggests that its effect on an operand is to partition that whole quantity into a number of parts equal to the denominator and then to adjust the quantity to the size of one of its parts. Both the duplicator and partition-reducer operate on the operand quantity as a whole (a unit), not on the points or objects that make up that whole. On the other hand, stretcher and a shrinker interpretations do operate on the points or objects that comprise the whole (Behr et. al., in press).

We have developed two notational systems to conduct, and communicate results of, these analyses: a generic manipulative aid notation and a generalized mathematics of quantity notation. Both indicate the type of units we hypothesize children would need to form in order to understand the particular concept under analysis.

Representations using the generic manipulative aid suggest object manipulations through which a child should be guided to provide an experiential base for understanding. We present the notation needed to interpret some of the analysis of the D/PR interpretation:

 

  1. From a single object, 0, we can conceptualize a singleton unit, denoted in the manipulative aid and mathematics of quantity notational systems respectively as: (0), 1(1-unit); or, we can conceptualize several singleton units: (0) (0) (0), 3(1-unit)s.

  2. From several objects, 0 0 0, we can conceptualize a composite unit: (000), 1(3-unit).

  3. From several singleton units, (0) (0) (0) (0), or several composite (3-unit)s, (000) (000) (000) (000), we can conceptualize a unit-of-units: ((0) (0) (0) (0)), 1(4(1-unit)s-unit), or ((000) (000) (000) (000)), 1(4(3-unit)s-unit).

  4. From several composite units-of-units, ((00)(00)(00)(00)) ((00)(00)(00)(00)) ((00)(00)(00)(00)), we can conceptualize a composite unit-of-units-of-units: (((00)(00)(00)(00)) ((00)(00)(00)(00)) ((00)(00)(00))(00))), 1(3(4(2-unit)s-unit)s-unit).
 

We will use 3/4 applied to 8 in all illustrations. In Figure 1 a manipulative aid representation of 3/4 as a composite of the operators 3/1 and 1/4 is given. It suggests that 3/1 and 1/4 are 3-for-1 and 1-for-4 exchange functions; and consequently, that 3/4 is a 3-for-4 exchange function.

A manipulative representation of 3/4 as a composite of 3/1 and 1/4 varies in complexity depending on the order in which 3/1 and 1/4 are applied. The partition to go from step d to e (Figure 1) is more complex with 3/1 applied first because of perceptual distractors, reunitizing the 3(8-unit)s to (24-unit)s and then to 4(6-unit)s might be less complex. A demonstration of this, one with 1/4 applied first, and a mathematics of quantity representation which matches Figure 1 in a step-for-step manner appears in (Behr et al., in press).

In Figure 2 a manipulative aid representation of 3/4 of 8 with 3/4 as a direct 3-for-4 exchange function rather than as a 3/1-and-1/4 composite function is given.

 

Application of the D/PR Interpretation to Computation Algorithms.

The D/PR exchange function interpretation of rational number is very powerful. It provides a close relationship between a manipulative level interpretation of a rational number and the syntax of the corresponding mathematics-of-quantity symbolic representation (Behr et al., in press). The manipulative representations suggest algorithmic computation procedures for the arithmetic-of-numbers. At the manipulative level a 3-for-4 exchange of objects in an application of the D/PR concept of 3/4 to some operand can be accomplished as follows: (a) arrange the objects of the operand into 4 groups and then (b) replace these 4 groups by 3 groups of the same size. This generalizes to all rational numbers in the obvious way. The procedure carries over to a symbolic representation in exactly the same way; that is, to apply 3/4 as a D/PR to 8, first rewrite 8 as 4 x 2, then replace the 4 in this expression with a 3, to give 3 x 2. For example:

 

3/4(8)

= 3/4(4 x 2) (partition 1 group of 8 into 4 groups.)

= 3 x 2 (Replace 4 groups of 2 with 3 groups of 2.)

= 6 (Reunitize 3 groups of 2 to 6 groups of 1.)

 

The manipulative display (Figure 2) which matches the symbolic form above suggests the 3/4 function rule, exchange 4 groups with 3 groups. What the adult reader might think of as cancellation of the 4 in 3/4 with the 4 in 4 x 2 is, as suggested in Figure 2, is an application of the concept of 3/4 as a 3 groups-for-4 groups exchange function. The 3/4 in the notation 3/4 (4 x 2) is a function rule. The rule is to exchange the 4 in 4 x 2 with a 3. Thus, application of the function 3/4 to the preimage number 8 (after being expressed as (4 x 2) gives the image value 6 (expressed as 3 x 2).

The notation 3/4(8) is ordinary function notation. More precisely, 3/4(8) means the value of the function 3/4 evaluated at 8. With this interpretation of the rational number, the expression 3/4 times 8 really does mean 3/4 of 8.

With 3/4 interpreted as a composition of a 3-for-l and a 1-for-4 exchange, as in Figure 1, or as the a 1-for-4 and 3-for-1 (not demonstrated with the manipulative aid in this paper), arithmetic-of-number computation algorithms are suggested as shown in Figure 3. The symbolic representation in Figure 3a corresponds almost step-for-step to the manipulative representation in Figure 1; Figure 3b shows an additional step in going from 3 x 8 to 4 x 6 which reflects an alternate unitization to that shown in Figure 1; Figure 3c shows the computation algorithm for 3/4 interpreted as a 3-for-l exchange applied to the result of a 1-for-4.

Curricular implications. This analysis of rational number as a D/PR exchange operator suggests several prerequisite knowledge structures: (a) ability to partition quantities, (b) flexibility in formation and re-formation of units, (c) understanding of and ability to perform partitive division, (d) an understanding of the concept of function as a mapping, and (e) skill with and understanding of multiplication as repeated addition.

 

Application of the D/PR construct to problem solving.

A major issue in developing problem-solving skill rests with the ability of students to form a representation that accurately reflects the quantities in the problem and the relationships among these quantities. Two matters concerning word problems relate to so-called extraneous-data and multiple-step problems, both make problems difficult for children to solve. To illustrate, consider the following problem situation and two questions.

Problem Situation. Many brands of gum are sold in the form of packages with 5 sticks in a package. Jane has 8 packages of gum. Mary has 3/4 times as much gum as Jane. Question 1. How many packages of gum does Mary have? Question 2. How many sticks of gum does Mary have?

In traditional problem-solving instruction, the information that there are 5 sticks in each package would be considered extraneous data for Question 1 because this question could be answered without that information. Nevertheless, the presence of this data causes difficulty for children. One reason for this might be that the model that is used in traditional instruction to answer Question 1 is not an accurate model of the problem situation. That there are 5 sticks in each package is part of the situation. Would problem-solving performance be improved if symbolic models to answer Question 1 could more accurately model the situation? When concern is for an answer to Question 2, traditional instruction classifies the problem as a multistep problem. A difficulty for children in solving multistep problems is that carrying out the first step (in this case multiplying 8 times 5 or 3/4 times 8) introduces still another quantity into the situation and the relationship of this new quantity to the existing quantities must be established.

We will interpret the problem situation in the generalized mathematics of quantity notation and show that the same initial representation can be used to answer both questions. Differences in the solution process will be seen to depend on a different re-formation of units of quantity. The quantities are as follows: each 5 sticks of gum is 1(5-unit); 8 packages of 5 sticks, a unit-of-units, is 1(8(5-unit)s-unit), which is also 1(8-unit); 3/4 is initially taken in the generic sense of a 3 (1-unit)s for 4 (1-unit)s exchange, as the solution progresses we think of it as a 3 (2(5-unit)s-unit)s for 4 (2(5-unit)s-unit)s exchange (See Steps 2 and 3.). The important issue for 3/4 as a D/PR is that 4 units are exchanged for 3 units of the same type.

 

References

Behr, M., Harel, G., Post, T., & Lesh, R. (in press). Toward a semantic analysis: Emphasis on the operator construct. In T. Carpenter & E. Fennema (Eds.), Learning, Teaching and Assessing Rational Number Concepts: Multiple Research Perspectives.

Behr, M., Harel, G., Post, T., & Lesh, R. (1990). The construct theory of rational numbers: Toward a semantic analysis. In G. Booker, P. Cobb, & T. N. de Mendicuti (Eds.), Proceedings of the 14th Annual Conference of the International Group for the Psychology of Mathematics Education, Vol III (pp. 3-11), Mexico.

Kieren, T. (1976). On the mathematical, cognitive, and instructional foundations of rational numbers. In R. Lesh (Ed.), Number and Measurement: Papers from a research workshop (pp. 101-144). Columbus, OH: ERlC/SMEAC.