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Harel, G. & Behr, M. (1990). The Construct Theory of Rational Numbers: Toward a Semantic Analysis. In G. Booker, P. Cobb, & T.N. de Merldicutti (Eds.) Proceedings of the PME XIV Conference (pp. 3-10). Mexico: Consejo Nacional de Ciencia y Technologia, Gobierno del Estado de Morelos.

 

THE CONSTRUCT THEORY OF RATIONAL NUMBERS: TOWARD A SEMANTIC ANALYSIS

 
Merlyn J. Behr, Northern Illinois Univ.
Guershon Harel, Purdue Univ.
 

A semantic analysis which is in progress will attempt to demonstrate the numerous interconnections among concepts in the multiplicative conceptual field. The analysis is based on two theoretical frameworks: mathematics of quantity and formation and reformation of units of quantity. The analysis employs two representational systems: noncontextualized drawings and the symbolism of the mathematics of quantity. The noncontextualized drawings are given in a notational system which might be a bridge between contextualized drawings or manipulative materials and mathematical symbolism. This paper presents, very briefly, new insights which have been gained through this type of analysis about subconstructs of rational numbers.

Numerous issues remain about how to facilitate children's construction of rational number knowledge. One of these issues is gaining deeper understanding of rational number subconstructs. We are gaining new insights into these subconstructs through an analysis based on the two perspectives of composition and recomposition of units (Steffe, Cobb and von Glasersfeld, 1988; Steffe, 1986, 1988) and mathematics of quantity (Kaput, 1985; Schwartz, 1988). The objective of our analysis is to provide a more firm theoretical foundation for research and development in this domain. An overview of our progress on this analysis is given in Figure 1.

A brief explanation of Figure 1 follows: The chart indicates that a rational number (3/4 is used to illustrate} has different interpretations depending on the rational number subconstruct considered. From the perspective of part-whole, a rational number is an extensive quantity and has a units formulation of either one composite 3/4-unit or three separate 1/4-units, independent of whether the analysis is based on a discrete or continuous quantity. Considering the quotient construct, a rational number is an extensive or intensive quantity according to whether the division of the numerator by the denominator is partitive or quotitive. For partitive division, 3/4 is the extensive quantity 3(1/4-unit)s per [1-unit] when the numerator, 3, has the units formulation of three singleton units of discrete quantity or one singleton unit of continuous quantity. When the units conceptualization of the numerator is one composite (3-unit) of discrete quantity it is 1/4(3-unit) per [1-unit]. For quotitive division, 3/4 is one composite [3/4-unit] when the numerator is conceptualized as 3(1-unit)s of discrete quantity (the analysis is incomplete for the numerator as one (3-unit) of discrete quantity and for continuous quantity). For the operator construct, a rational number is a mapping for which the domain and range varies according to the affect of several variables considered in the analysis: the interpretation given to the numerator and denominator in terms of type of operator, the order in which the numerator and denominator operators are applied, whether the operand of these operators is a discrete or continuous quantity, and according to the unitization of this quantity. We illustrate the operator interpretation of 3/4 in Figure 1 for the duplicator and partition-reducer interpretation of the numerator and denominator. When the operand is a discrete quantity, 3/4 is a function which maps quantities of the form 4n(1-unit)s to quantities of the form 3n(1-unit)s independent of the operator order; for continuous quantity, 3/4 is a function which maps a quantity of the form 1(1-unit): (a) onto a quantity of the form 1/4(3-unit) when the numerator operator (duplicator) is applied first and (b) onto a quantity of the form 3(1/4-unit)s when the denominator operator (partition-reducer) is applied first.

In the remainder of this communication we will briefly illustrate the procedures of the analysis. We employ two forms of analyses: diagrams to represent the physical manipulation of objects and the notation of mathematics of quantity. The diagrams provide a semantic analysis and the mathematics of quantity model a mathematical analysis; a very "close" step-by-step correspondence between the two representations suggests the mathematical accuracy embodied by the diagrams.

The Part-Whole Subconstruct. Our analysis of this subconstruct leads to two interpretations of rational number which we illustrate with 3/4. One interpretation is that 3/4 is three separate one-fourth parts of a whole and the second is that it is one composite three-fourths unit. Three-fourths as separate parts of a whole is 3(1/4-unit)s for a continuous quantity and is 3(1/4(4n-unit)-unit)s (i.e., three one-fourth-unit of units where each one-fourth-unit of units is a unitized one-fourth of a (4n-unit)) for a discrete quantity. We illustrate how our analysis leads to these interpretations in Figures 2 and 3. Three-fourths as a composite part of a whole is 1(3/4-unit); illustrations are omitted due to lack of space.

 


Figure 2. Three-fourths as parts of a discrete unit-whole leads to the interpretation that three-fourths is three one-fourth units.

1. (0 0 0 0 0 0 0 0)

One (8-unit). We could have started with any (4n-unit).

2. (0 0 / 0 0 / 0 0 / 0 0)

The (8-unit) is partition into 4 parts.

3. ((00) (00) (00) (00))

Each part is unitized as 1 (2-unit), and a unit-of-units is formed, (4(2-unit)s-unit).

4. ((0 0) (0 0) (0 0) (0 0))

Each (2-unit) is reconceptualized as (1/4(4(2-unit)s)-unit).

5. ((0 0) (0 0) (0 0) (0 0))

Three (1/4(4(2-unit)s)-unit)s are distinguished.


 

Figure 3 Three-fourths as parts of a continuous unit-whole leads to the interpretation that three-fourths is three one-fourth units.

1.

1(1-unit)

2.

The (1-unit) is partitioned into four parts.

3.

Each part is conceptualized as 1/4(1-unit).

4.

Three of the 4 parts are singled out to give 3 (1/4-unit)s.


The Quotient Construct. In this analysis we consider several variables: quotitive and partitive division, discrete and continuous quantity, and different unitizations of the numerator and denominator.

Partitive division gives two interpretations of a rational number which can be illustrated by 3/4 as the two intensive quantities: 3(1/4-unit)s per [1-unit] and 1/4(3-unit) per [1-unit]. We illustrate the first interpretation using discrete quantities.

 

Figure 4 Partitive division of 3 + 4, based on discrete quantities leads to three-fourths as three one-fourth units in one measure space per one unit of quantity from another measure space.

1.

(0 0 0 0) (0 0 0 0) (0 0 0 0)
     [*]    [*]      [*]       [*]

3(4-unit)s
4[1-unit]s
2.

(0/0/0/0) (0/0/0/0) (0/0/0/0)
     [*]    [*]      [*]       [*]

Each (4-unit) is partitioned into 4 parts.

3.

(ø ø ø ø) (0 0 0 0) (Ø Ø Ø Ø)
     [*]    [*]      [*]       [*]

Each of the 4 parts of each (4-unit) is reunitiezed as 1/4(4-unit).

4.

ø ø ø ø
[*][*][*][*]

Four 1/4(4-unit)s are distributed among the 4 [1-unit]s to give 1/4(4-unit)/[1-unit].

5.

0 0 0 0
ø ø ø ø

[*][*][*][*]

The third four 1/4(4-unit)s are distributed equally among the 4 [1-unit]s, another 1/4(4-unit)/[1-unit].

6.

Ø Ø Ø Ø
0 0 0 0

ø ø ø ø

[*][*][*][*]

The third four 1/4(4-unit)s are distributed equally among the 4 [1-unit]s, this gives a third 1/4(4-unit)/[1-unit].

7.

Ø Ø Ø Ø
0 0 0 0

ø ø ø ø

[*][*][*][*]

The three (1/4(4-unit))s are accumulated to 3(1/4(4-unit)-unit)s per [1-unit].

 
 

In each step of the mathematics of quantity model below, the step number from the preceding pictorial model is given in parentheses to demonstrate the closeness between the two models.

 

Figure 5 the partitive division of 3 + 4 which leads to the interpretation that three-fourths is three one-fourth units in one measure space per one unit of quantity in another measure space.

                 
1. 3/4 = 3(1-unit)s + 4[1-unit]s       (1)
                 
2.   = 3 x (4(1/4-unit)s) + 4[1-unit]s     (2)
                 
3.   = (4(1/4-u)s + 4(1/4-u)s + (4(1/4-u)s + (4[1-unit]s   (3)
                 
4.   = 4(1/4-u)s + 4[1/4-u]s + (4(1/4-u)s + 4(1/4-u)s) + 4[1-unit]s (4)
                 
5.   =
1(1/4-u)
4[1-u]s
+ (4(1/4-u)s + 4(1/4-u)s) + [4(1/4-u]s   (4)
                 
6.   =
1(1/4-u)
4[1-unit]s
+ 4(1/4-u)s + 4[1/4-u]s + 4(1/4-u)s + 4[1/4-u]s (5)
                 
7.   =
1(1/4-u)
4[1-unit]s
+ 1(1/4-u)
     4[1-unit]s
+ 4(1/4-u)s + 4[1/4-u]s   (5)
                 
8.   =
1(1/4-unit)
[1-unit]
+ 1(1/4-unit)
   [1-unit]
+ 1(1/4-unit)
   [1-unit]
    (6)
                 
9.   =
3(1/4-unit)s
[1-unit]
        (7)

 

Quotitive division, given the extent of our analysis, leads to the interpretation that 3/4 is the extensive quantity 1[3/4-unit], where 4(1-unit)s/[1-unit] is a measurement unit which is used to measure the numerator, 3, expressed as 3(1-unit)s.

 

Figure 6 Quotitive division of 3(1-units)/[1-unit] which leads to the interpretation that 3/4 is 1[3/4-unit]

     

(0) (0) (0)

[(*) (*) (*) (*)]

a.

3(1-unit)s

4(1-unit)s / [1-unit]

     

(0) (0) (0)

[(*) (*) (*) (*)]

b.
The measurement quantity is subunitized as 4[1/4-units]
     

c.
The "first" (1-unit) of the to-be-measured quantity is matched with the "first" sub measurement-unit (i.e. iteration of the measurement unit is begun).
     
d.
The "second" (1-unit) is matched with the "second" sub measurement-unit. Meanwhile the first matched pair is dropped from attention.
     
e.
The matchings are accumulated; i.e., both matched pairs are brought into cognitive attention.
     
f.
The "third" (1-unit) is matched with the "third" sub measurement-unit. Meanwhile, the first two matched pairs are dropped from attention.
     
g.
The matchings are accumulated.
     
h.
The 3(1-unit)s of the measured quantity are unitized to a 1(3-unit) and the 3 matched [1/4-unit of the measurement quantity are unitized as 1[3/4-unit]s. The measure of the quantity that was to be measured is 1[3/4-unit].
     
     
     
     

 

 

References

Kaput, J. (1985). Multiplicative word problems and intensive quantities: An integrated software response (Tech. Report). Harvard Graduate School of Education, Educational Technology Center.

Schwartz, J. L. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Heibert & M. Behr (Eds.), Research agenda for mathematics education: Number concepts and operations in the middle grades (pp. 41 -52). Reston, VA: The National Council of Teachers of Mathematics.

Steffe, L. (1986). Composite units and their constitutive operations. Paper presented at the Research Presession to the Annual Meeting of the National Council of Teachers of Mathematics, Washington, D.C.

Steffe, L. (1988). Children's construction of number sequences and multiplying schemes. In J. Heibert & M. Behr (Eds.), Research agenda for mathematics education: Number concepts and operations in the middle grades (pp. 119 -140). Reston, VA: The National Council of Teachers of Mathematics.

Steffe, L., Cobb, P., & von Glasersfeld, E. (1988). Construction of arithmetical meanings and strategies. New York, NY: Springer-Verlag.