UNDERSTANDING THE MULTIPLICATIVE STRUCTURE
Problem situations in the multiplicative conceptual field (MCF) are analyzed with respect to a wide range of task variables from five structures: numeric, semantic, propositional, contextual, and mathematical. This paper focuses on the Invariance Structure a substructure of the Semantic Structure and the mathematical structure which consists of principles underlying the Invariance structure. These principles are organized into two main classes order determinability principles class and order determination principles class corresponding to two solution actions the action of declaring whether an order relation between two quantities is determinable and the action of specifying what that order relation is. The analysis is believed to (a) reflect the structure of the MCF; (b) contributes to our understanding of what constitutes a conceptual knowledge of the MCF and solutions of multiplicative problems; (c) guide researchers to design activities for children to construct MCF knowledge; and (d) aid to investigation of the effect of various multiplicative variables on childrens performance.
In the last few years, we have focused our research on theoretical analyses of the multiplicative conceptual field (MCF), based on our own and others previous work on the acquisition of multiplicative concepts such as, multiplication, division, fraction, ratio, and proportion and relationships among them. The goal of this research is to better understand the mathematical, cognitive, and instructional aspects of the MCF structure. Results from this research indicate that the mathematical structure of the MCF is very complex and cognitively very demanding (see Harel and Behr, 1989; Harel, Behr, Post, and Lesh, in press; Behr, Harel, Post. and Lesh, in press; Behr and Harel, this volume). Thus, because of the importance and ubiquity of the MCF in mathematics, it is an important challenge for mathematics educators, instructional psychologists, and curriculum developer, to study this field and understand childrens acquisition of the knowledge it involves, so that instruction that facilitates childrens construction of this knowledge can be devised.
Vergnaud (1988) introduced the notion of conceptual field "as a set of situations, the mastery of which requires mastery of several concepts of different natures" (p. 141). In particular, Vergnaud characterized the multiplicative conceptual field (MCF) as consisting of all problem situations whose solutions involve multiplication or division, and classified these situations into three categories: simple proportion, product of measure, and multiple proportion (Vergnaud, 1983, 1988). In our recent work on the structure of the MCF, we analyzed multiplicative situations with respect to a wide range of task variables, some of which are known to the research on multiplicative concepts, the others are new. This analysis, integrates Vergnauds analysis and others analyses (e.g., Nesher, 1988; Thompson, 1990), and results in a system of structures which are believed to reflect the nature of the MCF, both mathematically and cognitively.
The analysis is organized around five structures: Numeric, Semantic, Propositional, Contextual, and Mathematical, with several substructures (Figure 1). Because of space restrictions, we limit our discussion to two strongly interrelated structures: the Invariance Structure (one of the substructure of the Semantic Structure) and the Mathematical Structure. The latter consists of mathematical principles classes underlying situations in the Invariance Structure.
Multiplicative Invariance: Problem Situations and Solution Principles
Variation is one of the most important and ubiquitous actions in mathematical reasoning. In many problem situations in mathematics, quantities are varied by applying to them a certain transformation. The goal in these problems is to identify the type of transformation applied in order to determine whether the relation between the initial quantities is invariant under the transformation applied, and, if not, to determine the new relation between the quantities that resulted from the transformation, and/or to search for an appropriate compensation for the change that results. For example, in problems such as "divide 163 by 0.3," children are taught to change the decimal divisor 0.3 to the whole-number divisor 3 by multiplying by 10, divide 163 by 3, and, multiply the result by 10. What constitutes the understanding of this procedure is the awareness that the equality relation between the dividend 163, the divisor 0.3, and their quotient is not invariant under the change of the divisor 0.3 to 3, and that the "multiply by 10" transformation applied to the quotient of 163÷3 is an appropriate compensation for this change. Moreover, many of childrens inventions of computations can be explained in terms of reasoning on variation and invariance. We bring two examples of such inventions. The first is form the domain of basic addition. Cobb and Merkel (1989) reported that children solve the problem 8+9 = ? by changing the nine into 8 + 1, to get the well-remembered basic fact 8+8 = 16, and then add 1. We hypothesize that when children use this strategy, they are aware (at least implicitly) of the fact that the equality relation in this problem is invariant under the change of 9 to (8+1). The second example is from the domain of division. We asked an eight-year-old child to solve the problem 156÷2.75, before he was taught the standard algorithm for dividing decimals. In responding to this problem, he multiplied 2.75 by 2 and got 6.5, then he multiplied the 6.5 again by two and got 13. He then divided 156 by 13, using the standard algorithm, and multiplied the result, 12, by 4. We believe that the conceptual base for this solution is the awareness that changing the problem quantities is an allowable action provided that the appropriate compensations are applied. This child applied consecutively the "multiply by 2 transformation" to change the decimal divisor 2.75 into the whole number divisor 13; he did so to change the given problem, 156÷2.75, into a familiar one, 156÷13. He knew that this quantity change is allowable but it must be followed by an appropriate compensation with respect to the equality relation between the dividend and divisor on the one hand and their quotient on the other. He chose a multiplicative compensation by applying the "multiply by 4" transformation to the quotient of 165÷13. This solution as well as many other taught to children are based on basic invariance principles which will be specified below.
Problem situations. Additive reasoning and multiplicative or proportional reasoning can be characterized in terms of the type of compensation, additive or multiplicative, children employ for the transformation(s) applied to the problem quantities. Children who are additive reasoners interpret changes to the values of the quantities as additive transformations, and, therefore, employ additive compensations even in situations in which multiplicative compensations are required. Proportional reasoners, on the other hand, differentiate between situations in which quantity changes must be interpreted as additive transformations and those which must be interpreted as multiplicative transformations, and, accordingly, employ the appropriate compensations. When the transformation "add k," for example, is applied to the quantity a in the inequality a < b, children have little difficulties finding an appropriate compensation that keeps the direction of the order relation unchanged (e.g., add k to the quantity b). On the other hand, children have great difficulty solving problems in which multiplicative compensations are involved; for example they do not easily understand that the "smaller than" relation in the inequality a / b < c / d is invariant under the transformation "add k" (where k is positive) to the quantity b, and offer the transformation "add k" to the quantity d as a compensation for this transformation. It is the awareness of the type of transformation (additive or multiplicative) applied and the ability to find the appropriate compensation for that transformation that constitutes proportional reasoning.
An important goal for mathematics education is to identify the knowledge that constitutes adequate reasoning for additive invariance and multiplicative invariance tasks, and to offer learning activities that can help children to develop such knowledge. In the rest of this paper, we focus on categories of multiplicative situations, and present classes of mathematical principles that constitute their solutions. From a pure mathematical point of view, these principles are easily derived from the axioms of the ordered field, and thus, they are self-evidence, at least for mathematically sophisticated people. From a cognitive point of view, however, as research on the concept of proportion imply, these principle are not obvious.
Multiplicative and proportion tasks are instances of two general categories: Invariance-of-ratio and invariance-of-product (see Figure 1). The classification of tasks into these categories was made according to the type of solutions whether are based on ratio comparison or on product comparison - commonly used by subjects. For example, the orange concentrate task used by Noelting (1980) belongs to the invariance-of-ratio category, because the correct solutions used by children to solve this task always involve comparison of two ratios. On the other hand, the balance scale task used by Siegler (1976) belongs to the invariance-of-product category, because the correct solutions used by children in Sieglers study involve comparison of the products of the distance-weight values for each side of the fulcrum to determine which side goes down.
Quite different mathematical principles, and thus different reasoning patterns, are involved in the solution of these types of tasks. To describe these principles, a refinement of each category is needed (see Figure 1). There are two subcategories of the invariance-of-product category: The Find-product-order Subcategory consists of problems which give two order relations between corresponding factors in two products, and asks about the order relation between the two products (e.g., given that a > b and c = d and the question is about the order relation between a X c and b X d); and the Find-factor-order Subcategory consists of problems which give an order relation between two corresponding factors in the two products, and asks about the order relation between the other two factors (e.g., given that a X c < b X d, and a > b, and the question is about the order relation between c and d). Likewise, there are two subcategories of the invariance-of-ratio category: The Find-rate-order Subcategory which consists of problems which give two order relations between corresponding quantities in the two rate pairs and asks about the order relation between the two rates (e.g., given that a > b and c = d and the question is about the order relation between a/c and b/d); and the Find-rate-quantity-order Subcategory which consists of problems which give an order relation between two rates and an order relation between the other two quantities in the two rate pairs (e.g., given that a / c = b /d, and a > b, and the question is about the order relation between c and d).
Solution Principles. Each task from either one of these four subcategories involves three number pairs a and b, c and d, and either a pair of product a X c and b X d or a pair of ratios a / c and b / d and the order relation between quantities within two of the three pairs are given and the problem is to determine, if possible, the order relation between the quantities within the third pair. A solution of such a task includes two actions: finding out whether the third order relation is determinable, and, if so, then determining what that order relation is. Accordingly, the knowledge involved in solving these types of tasks relies on two categories of classes: the order determinability principles class and the order determination principles class. The order determinability principles specify the conditions under which order relations between quantities within two pairs can lead to the action of declaring whether the order relation between quantities within the third pair is determinate or indeterminate (e.g., if a and b are equal but c and d are unequal, then the order relation between the products a X c and b X d, or between the ratios a / c and b / d, is determinate). The order determination principles specify the conditions under which order relations between two quantities within two pairs can lead the action of declaring that the order relation between the two quantities within the third pair to be less than, greater than, or equals (e.g., if a and b are equal but c is greater than d, then a / c < b / d).
We have identified two pairs of classes of multiplicative principles: one pair concerns the order determinability principles, the other the order determination principles (Figure 2). Two classes, one from each pair, correspond to the invariance-of-product category - one is called the product order determinability principles class, the other the product order determination principles class. The other two classes correspond to the invariance-of-ratio category one is called the ratio order determinability principles class, the other the ratio order determination principles class. Each of the two product order (determinability or determination) principles classes is further divided into two subclasses product composition (PC) subclass and product decomposition (PD) subclass - depending on whether the principles solve tasks from the find-product-order subcategory or the find-factor-order subcategory, respectively. Similarly, each of the two ratio order (determinability or determination) principles classes is further divided into two subclasses - ratio composition (RC) subclass and ratio decomposition (RD) subclass - depending on whether the principles solve tasks from the find-rate-order subcategory or find-rate-quantity-order subcategory, respectively.
Figure 2 further presents the specific principles in each subclass. The product composition subclass consists of: PC1 the order relation between the products a X c and b X d is determinate if the order relations between a and b is the same as the order relation between c and d or if one of them is the equals relation; PC2 the order relation between the products a X c and b X d is indeterminate if the order relations between a and b conflicts with (i.e., is in the opposite direction) the order relationship between c and d; and PC3 the order relation between the products a X c and b X d is indeterminate if one of the order relations, either between a and b or between c and d, is unknown.
The second subclass of order determinability principles is the product decomposition (PD) subclass; it consists of: PD1 the order relation between the factors a and b in the products a X c and b X d is determinate if the order relation between the factors c and d conflicts with the order relation between the products a X c and b X d or if one is the equals relation; PD2 the order relation between the factors a and b in the products a X c and b X d is indeterminate if the order relations between the other two quantities, c and d, is the same as the order relation between the products a X c and b X d and neither is the equals relation; and PD3 the order relation between the factors a and b in the products a X c and b X d is indeterminate if one of the order relations, either between c and d or between a X c and b X d, is unknown.
The class of ratio determinability principles also consist of two subclasses. The first is the ratio composition (RC) subclass; it consists of the following principles: RC1 the order relation between the ratios a / c and b / d is determinate if the order relations between a and b conflicts with the order relation between c and d or one is the equals relation; RC2 the order relation between the ratios a / c and b / d is indeterminate if the order relations between a and b is the same as the order relation between c and d and neither one is the equals relation; and RC3 the order relation between the ratios a / c and b / d is indeterminate if one of the order relations, either between a and b or between c and d, is unknown.
The second subclass of the ratio order determinability class is the ratio decomposition (RD) subclass, which consists of the following principles: RD1 the order relation between the rate quantities a and b in the ratios a / c and b / d is determinate if the order relation between the rate quantities c and d conflicts with the order relation between the ratios a / c and b / d or if one is the equals relation; and RD3 the order relation between the rate quantities a and b in the ratios a / c and b / d is indeterminate if one of the order relations, either between c and d or between a / c and b / d, is unknown
Once a determinability principle is applied and the requested order relation is found to be determinable, then a determination principle can be applied to ascertain whether that relation is the less than, equals, or greater than relation. There is a determination principle which corresponds to each of the determinability principles which ascertains that the required order relation is the less than, equals, or greater than relation. There is a determination principle which corresponds to each of the determinability principles which ascertains that the required order relation is determinate (i.e., the first principle in each of the above subclasses). We denote them by [PC1], [PD1], [RC1], [RD1], respectively. For example, an instantiation of [PD1] is: if c > d and a x c < b x d, then a < b.
The MCF analysis, partly presented in this paper, can contribute to research on learning, curriculum development, and teaching, in several ways. First, an important objective for mathematics education is to find instructional ways that enable children to construct conceptual knowledge knowledge which is rich in relationships among concepts (Hiebert and Lefevre, 1986). Research on knowledge development has widely dealt with the characteristics of this knowledge, especially in mathematics, but has done little to identify the means by which such a knowledge is constructed. In our view, what constitutes a relationship among mathematical objects in a students mind is a set of mathematical principles which explicitly or implicitly employed by the student in the process of constructing that relationship. The analysis presented in this paper can guide researchers to design activities that help children construct mathematical principles which would facilitate the understanding of relationships among different concepts in the MCF. Second, these mathematical principles are believed to be the foundation for multiplicative and proportional reasoning; that is, these principles are the basic theorems in actions (ala Vergnaud, 1988), from which more complex theorem in actions can be constructed by children in solving advanced multiplicative and proportional reasoning problems. Finally, this analysis identifies several variables with subvariables which researchers could manipulate in multiplicative and proportion tasks and in experimental instruction to investigate the effect on children's performance. Finally, this analysis draws attention to the important issue of mathematical invariance, in investigating childrens reasoning and in designing instructional activities. Some related research questions are: (1) does an awareness of the invariance principles described here help children to solve multiplicative and proportion problems? (2) what instructional activities can help children construct these principles? (3) what constitutes children's ability to distinguish between additive invariance and multiplicative or proportional invariance? (4) how could additive and multiplicative concepts and relationships and distinction among them be introduced through the mathematical invariance idea?
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