Units of Quantity; A Conceptual Basis Common to Additive and Multiplicative Structures
Traditionally we have taught children the arithmetic of numbers with the ultimate objective of their being able to use this arithmetic to model and solve real-world problems. The order of this approach has been first to teach the arithmetic of numbers, essentially divorced from a social context, and then to teach problem solving by making attachments between numbers and operations on numbers with measurements of and operations on quantities to model relationships between quantities. What is proposed by the arithmetic of quantity is that arithmetic should grow out of social contexts. That is, by beginning with the observation of how quantities behave and by attaching numbers to attributes of quantity through the process of measurement, the arithmetic of quantities and ultimately the arithmetic of numbers should be suggested. An important distinction to be made here is that arithmetic of numbers is apparently based on the assumption that all the numbers represent quantities of the same unit of one, a quantity of singleton units. The arithmetic of quantity involves composite conceptual units of various composition. The arithmetic of quantity requires special attention to measure units and different types of composite units. The arithmetic of numbers apparently assumes only singleton units.
Research in the Soviet Union has given some attention to children's ability to deal with different types of quantity units. Davydov (1982), describing experimental work conducted in the Soviet Union found that first through third graders in a traditional program master procedures for addition and subtraction of single and multidigit numbers and can easily determine that 3 + 4 is equal to 7, for example. Many of the children were able to describe contexts for which the number sentence was a mathematical model. These same children were immediately perplexed when asked what possible sense they could make out of the unexpected sentence 3 + 4 = 5. In a teaching experiment he found some first grade children were successful with a problem that could be modeled with this sentence when represented with physical objects, three containers of the same size filled to the brim with water and four containers half the size also filled with water and the question posed was of how many containers of the larger size would be needed to contain the water. This situation is easily represented as follows: 3 units of size 2 + 4 units of size 1 = how many units of size 2? Through a reformation of units, 4 units of size 1 can be thought of as 2 units of size 2 (Behr, Harel, Post, & Lesh, 1992a); then, the preceding can be rewritten as 3 units of size 2 + 2 units of size 2 = 5 units of size 2.
This discussion illustrates that a hidden assumption in the arithmetic of numbers is that all quantities are represented in terms of units of 1. This hidden assumption has a negative impact on the elementary and middle school curriculum. The current curriculum on whole number arithmetic gives problem situations in which children deal with quantities essentially expressed only in singleton units, rather than providing problem situations in which they represent quantity in various unit types (Steffe, 1988). Even in so-called multistep problems such as "John has four bags with six marbles in each and three bags with four in each. How many bags with two marbles in a bag can he make?" in which some of the given problem quantities are in composite units, the traditional approach is to change to units of one in the solution of the problem. The traditional approach is to change the six-marble and four-marble units to twenty-four and twelve one-marble units, add to find that there are thirty-six one-marble units, and then divide by 2 to find that there are eighteen two-marble units. Am alternate approach is to reformulate the six-marble and four-marble units to two-marble units, a unit common to the two given and the unit referred to in the problem question. Representation and solution of the so-called multistep problems do not require a conversion to units of 1 in every case. Indeed there are problem situations in which it is more efficient to find a common unit other than a unit of 1 and in some cases, using a unit of I is actually contrary to the constraints of the problem situation. Freudenthal (1983) and more recently Lamon (1989) refer to the process of conceptualizing a situation in terms of a common unit other than 1 as norming.
Galperin and Georgiev (1969) suggest that the concept of unit has a special place in the formulation of elementary mathematical notions and, indeed, indicate that all mathematical concepts assume the notion of a unit. The work of Steffe and his colleagues (Steffe, Cobb, and von Glasersfeld (1988) strongly point out the relationship between formation of units and the development of concepts of number, addition, and subtraction of whole numbers. More recently, they have suggested that concepts of multiplication and division and rational numbers depend on the formation of certain units.
We assert that giving children situations of whole number arithmetic that involve a variety of unit types and units of units and experience in representing and manipulating quantities that can be represented in these unit types will provide a more adequate foundation for learning and understanding whole number arithmetic and a cognitive bridge to learning and understanding rational number concepts and operations. Therefore the thrust of our analysis is to exhibit the units structure of these mathematical constructs.
CONCLUSIONS AND DISCUSSION
This analytical work represents an attempt to accomplish two goals. One goal is to hypothesize the cognitive structures that develop, or need to be developed, in acquiring an understanding of the concepts discussed. A second goal is to consider these hypothesized cognitive structures to suggest kinds of learning activities that children ought to experience so that they have an opportunity to develop those structures. A conjecture that these unit structures that we hypothesize correspond to mental structures learners develop has some support from cognitive science (S. Ohlsson, personal communication, August 7. 1991).
We wish to emphasize that the notational systems we have developed and. communicated in the chapter were developed for theoretical analysis and communication within the research community. We neither advocate that the notational systems be used with children nor disavow the possibility that particular instantiations of the notational systems might be used with children. ' However, the development of these instantiations goes beyond the scope of this chapter. We do argue, however, that the generic manipulative aid unit analyses do provide a template for the construction of such instantiations to provide appropriate manipulative experiences for children.
Whether or not the cognitive structures we hypothesize develop during, or are necessary for, the learning and understanding of the concepts under discussion remains to be determined by research. That the analysis given in this chapter has validity is supported by work in progress by Simon and Blume (1991) In their research with preservice teachers, they have investigated the development of the understanding of the area of a rectangular region as a multiplicative relationship between the lengths of the sides. Their work supports, first of all, that insights into how the area of a rectangle is related to the product of the length-measures of the two sides is essentially nonexistent prior to instructional intervention. Through some insightful instructional moves on the part of these researchers, preservice teachers developed their understanding of this relationship to the point where their insight into area was essentially that of the cross product of units of width and units of length. The units used were long and short sticks. This understanding was displayed in the context of the following problem: Two people work together to measure the size of a rectangular region, one measures the length and the other the width. They each use a stick to measure with. The sticks, however, are of different lengths. Louisa says, "The length is four of my sticks." Ruiz says, "The width is five of my sticks." What have they found out about the area of the rectangular region?
Students worked on this problem in groups of three. A response by Tonie is particularly appropriate to the analysis presented in this paper: "I think that if you had enough sticks to build an entire rectangle, they would fall naturally into miniature rectangles to fill it. If you had all the sticks ... it would go four across, four across, four across, and then it would naturally form rectangles inside the rectangle." It seems apparent that this student had in mind that successive right-angle pairs of sticks juxtaposed to the edges of the rectangle or previously placed pairs would enclose rectangular regions. Further protocol analysis would be needed to determine the closeness of the match between this student's cognitive structures and those hypothesized in Figures 5.9 or 5.10.
Connection between mathematical concepts is a notion that gets considerable attention in current discussions about mathematics learning (e.g., Curriculum and evaluation standards for school mathematics, 198% Clear specifications of what constitutes a connection appears less frequently. We argue that attention to units of quantity does point to such connections and suggests both the cognitive and mathematical connection. Using a problem involving addition of problem quantities given in "unlike" units (other than units of 1) we were able to show connections between such a problem structure and addition of multidigit numbers and also between such a problem structure and addition of rational numbers. In the section Multiplication of Rational Numbers, we showed how interpretation of the same problem in whole number units and in fractional number units provides a connection between a whole number problem structure and a rational number problem structure. Exemplification of these connections in instructional situations, we hypothesize, will enrich children's understanding of problem and computational procedures and facilitate the extension of children's knowledge about whole number situations to rational number situations.
The development of this chapter was in part supported with funds from the National Science Foundation under Grant No. DPE 84-70077 (The Rational Number Project). Any opinions, findings, or conclusions expressed are those of the authors and do not necessarily reflect the view of the National Science Foundation.
1. Although this notation represents knowledge structures, a more in-depth investigation of the mental operations and actions that constitute those knowledge structures is given by other researchers in this volume, e.g., Confrey, Steffe, and Thompson.
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