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Harel, G., Behr, M., Post, T. & Lesh, R. (1991). Variables Affecting Proportionality: Understanding of Physical Principles, Formation of Quantitative Relations, and Multiplicative Invariance. In F. Furinghetti (Ed.) Proceedings of PME XV Conference (pp. 125-133). Assisi, Italy: PME.


Guershon Harel, Purdue University

Merlyn Behr, Northern Illinois University

Thomas Post, University of Minnesota

Richard Lesh, Educational Testing Service


The analysis presented in this paper is part of several extensive analyses we have been conducting in the last few years dealing with multiplicative concepts in the global context of the multiplicative conceptual field. Here, we propose three global variables believed to affect multiplicative and proportional reasoning. The first variable deals with the physical principles underlying the problem situation. It intends to explain the conceptual basis for the variance in difficulty of proportion problems due to differences in context. The second variable ties theories on multiplication and division concepts (Fischbein, Deri, Nello, & Marino, 1985; Vergnaud, 1983, 1988; Nesher, 1988) with research on the concept of proportionality, by analyzing the type of multiplicative relations which exist between the problem quantities (e.g., whether this relation is of the multiplier-multiplicand type or product of measure type). The third variable deals with mathematical invariance. It is suggested that interpreting multiplicative and proportion situations in terms of variability and invariability and change and compensation can be powerful in problem solutions and should be a target for mathematics curricula.

In their literature review on proportional reasoning, Tourniaire and Pulos (1985) described the type of tasks used in research to investigate the concept of proportionality and the variables found in this research to affect children's performance on proportion problems. They classified these tasks into physical (e.g., the balance scale task), rate (where ratios of dissimilar objects are compared; e.g., speed problems), mixture, and probability tasks, and distinguished between missing-value proportion problems (e.g., "Find x In 3/x-2/7") and comparison proportion problems (e.g., "Which is greater, 3/7 or 4/9?). The variables affecting children's performance they divided Into task-centered variables - presence of integer ratio, order of the missing value, and numerical complexity - and context variables - presence of mixture, referential content (discrete versus continuous), familiarity, and availability of manipulatives.

Of the many traditional proportion tasks, three types of tasks have been widely used: rate and mixture tasks (e.g., Noelting, 1980a, 1980b), the balance scale task (Inhelder and Piaget, 1958; Siegler and Vago, 1978), and the fullness task (Bruner and Kenney, 1966; Siegler and Vago, 1980). In the discussion below we refer to these three types of tasks as representatives of traditional proportion tasks. Recently we have developed a class of proportion tasks we called "multilevel" tasks to distinguish them from the traditional tasks which we view as "single-level" tasks. They were so labeled because it can be shown (see Harel, Behr. Post, & Lesh, in press) that the "multilevel" tasks consist of three subtasks, two are isomorphic to two variables of a traditional task, and the third is an additive task which involves a coordination between two order relations. An example of a "multilevel" task is2:

2 blocks of kind ALFA weigh the same as three blocks of type BETA. If possible find whether 5 blocks of type ALFA weigh the same as, more than, or less than 7 blocks of type BETA?

In a previous work (Harel and Behr, 1988) we extended the task centered variables as defined by Tourniaire and Pulos (1985) and analyzed their impact on children's problem representations and solution strategies of missing value proportion problems. In this paper we present more global accounts for the reasoning and difficulties children encounter in dealing with multiplicative problems. These accounts will be discussed in the context of the two categories of tasks mentioned earlier: the three representatives from the traditional proportion tasks and the new class of "multilevel" tasks.



Tourniaire and Pulos reported that mixture problems have been found to be more difficult that other proportion problems. They gave the following explanation to account for this finding in the case of mixture and rate problems:

First the elements of the ratio in a mixture problem constitute a new object, e.g., red and yellow paint mixed makes orange, or a modified object, e.g., orange juice mixed with water makes a weaker orange juice. By contrast, no new object emerges in rate problems. Second, mixture problems require that the subject understands what happens when the two elements are mixed. Third, in most mixture problems, the quantities are expressed in the same unit, e.g., ounces, whereas in most rate problems the quantities involved are in different units, e.g., ounces and dollars. Dealing with quantities expressed in the same unit may be more confusing. (pp. 183-184).

These are peculiar explanations; it is not clear how they differentiate between mixture problems and rate problems or how they account for the relative difficulty between these two types of problems. First, the ratio elements in rate problems, as in mixture problems, do constitute a new object; the quantity of speed, for example, is a new quantity which emerges from the quantities of time and distance. Second, both rate problems and mixture problems require that the subject understands the physical situation when two quantities are combined. This is true whether this is a mixture of water and orange concentrate or a comparison between time and distance. Third, why should problems in which the quantities are expressed in the same unit be more difficult than those in which the quantities are expressed in different units?

In the discussion below we propose three variables which affect proportionality. The first two can account for the relative difficulty of proportion problems; these are the physical principles underlying the problem situation and the type of multiplicative relations which exist between the problem quantities. The third variable deals with advanced multiplicative reasoning in which ratios and products are compared in terms of changes and compensations. This type of reasoning is viewed as a culminating point in multiplicative reasoning and should be a goal for school mathematics. To reach this goal children must experience in early age situations that deal with multiplicative change and compensation and grasp an intuitive understanding of the mathematical principles (see Harel et al, in press; Harel, Behr, Post, & Lesh 1990) that constitute these situations.

Physical Principles Underlying the Problem Situation

In this section we hypothesize a conceptual basis for the variance in difficulty of proportion problems due to differences in context. A fundamental difference among proportion tasks lies in the principles underlying the physical interactions between or among the problem quantities. We believe that these principles are the basis for taking account of the proportionality constraints in solving the tasks. For example, in the problem above, it is assumed that the weight of each set of blocks is equally distributed among the individual blocks in the set; and if some of the numerical data in this problem were fractions, one must assume that the distribution of the weight within each block is homogeneous. The physical principle of homogeneous distribution of weight, or homogeneous density of matter, is intuitive and spontaneous, in the sense that it is acquired in everyday activities, such as lifting objects. Mixture and rate tasks require an intuitive understanding of other physical principles which are less spontaneous than the homogeneous density principle. In the mixture task the principle involved is about uniform diffusion between liquids; in the rate tasks the principle is about uniform rate, such as, speed or work. The fullness task involves the principle about liquid, which states that the pressure is the same at all points at the same level within a liquid at rest; this principle guarantees the uniform level of the liquid and the absolute separation of the water space from the empty space. The balance scale involves the principle of conservation of angular moment, which states the conditions of equilibrium that must be satisfied if a balance scale is to remain balanced or fall toward one of the two sides of the fulcrum. These conditions are non intuitive and less spontaneous when dealing with the summation of the products of weight and distance on each side of the fulcrum.

It should be emphasized that we are not claiming that in order for children to solve one of these tasks successfully they must explicitly know the physical principles underlying the task. Or argument is that an intuitive understanding of the physical principles underlying the situation of the task is necessary for solving the task correctly.

Multiplicative Relations Between Problem Quantities

The semantic relationship between the problem quantities in the "multilevel" problem presented above is conceptually different from those in the mixture task, the balance scale task, or the fullness task. In that problem the weight of each set of blocks can be thought of as a product of the number of blocks in the set and the weight of each block. Accordingly, the role played in the product by the first quantity (number of blocks) is conceived of as an (integral) multiplier and the role of the other (the weight of each blocks) as the multiplicand. The multiplier-multiplicand relationship — simple proportion relationship in Vergnaud’s (1983, 1988) terms, or mapping-rule relationship in Nesher’s (1988) terms — involves two measure spaces, number of building-blocks and weight. The use of this relationship is ordinary in everyday activities and in school word problems, and usually is expressed as repeated addition. It is based on a set-subset relationship, the operation unit of sets, and the concepts of cardinality and measure. All of these are acquired informally through everyday activities.

The balance scale task, in contrast, involves three measure spaces: the quantities multiplied are derived from two independent measure spaces — weight and distance — and their product creates the measure of space of moment. The semantic relationship between these three quantities — product of measure in Vergnaud’s (1983) terms — is formal, in the sense that it is acquired through instruction: moment is a vector quantity (not a scalar quantity) which is defined as the cross product of two vectors, weight (the net gravitational force acting on an object hung on one side of the fulcrum) and (directional) distance from a fixed point (the fulcrum) to the point on which an object is hung. Indeed for many balance scale task variations this definition is not necessary because they can be solved based on an intuitive knowledge acquired through inactive experience. These include many non-numeric variations, such as "If two boys, Tom and John, sit on opposite ends of a seesaw, in an equal distance from the center, and Tom is heavier than John, which side would go down, Tom’s side or John’s side? The definition of moment, however, is the foundation for the physical principle that
w1 • d1 = w2 • d2 (where w1 and w2 represent the weight of two objects each is hung on another side of the fulcrum and d1 and d2 their distances from the fulcrum, respectively) if and only if the two sides of the fulcrum are balanced, which children do not acquire spontaneously from everyday experience, and it must be used in solving numeric variations of the balance scale task.

The other two types of tasks, rate and mixture tasks and the fullness task, are of ratio types. The semantic relationships between the problem quantities within a ratio can have different meanings, among which the partitive division meaning, the quotitive division meaning, and the functional meaning. Consider, for example, the ratio 12 cups of water to 4 cups of orange concentrate. One can change 12:4 to the unit-rate 3:1 and think of 12:4 as the number of cups of water per one cup of orange concentrate, or change 12:4 to 1:1/3 and think of the number of cups of orange concentrate per one cup of water, both indicate a partitive division interpretation of the relationship between the ratio’s quantities. The ratio 12:4 can also be thought of as a rate, namely, that the amount of water (orange concentrate) is some number of times as much as the amount of orange concentrate (water); this relationship, in contrast to the former one, is a quotitive division interpretation. Finally, the ratio 12:4 can have a functional meaning; namely, one of the ratio quantities, 12 cups of water or 4 cups of orange concentrate, is dependent on the other. In the following section we will elaborate more on this viewpoint.

Multiplicative Invariance: Functional and Relational

Ratio and proportion problems can be interpreted in terms of functional dependencies, variability and invariance. Before we define this idea, we indicate that the term "invariance" is used in advanced mathematics in the context of whether a set of objects preserves its mathematical structure when it undergoes a certain change. For example, if U is a subspace of a vector-space W and T is a linear operator on W, it is important — for reasons related to transformation representations — to determine whether T(U) is still a subspace of W, or, using different words, whether U as a vector-space is invariant under T.

Questions of variability and invariability are important not only in advanced mathematics but also in the context of the mathematics taught in school, and in particular the context of the multiplicative conceptual field. Missing-value and comparison proportion problems and missing-value and comparison product problem (e.g., "Find x in 3•x = 2•7" and "If possible, determine which is greater, a•b or a'•b' if a>a' and b<b'?") can be analyzed by solvers in terms of changes and compensations. Consider, for example, the missing-value proportion problem, "Find x in 3/7 = 9/x." One can view this problem asking: if 3 undergoes a numerical change which results in the number nine, what change must 7 undergo so that the value of 3/7 is left unchanged? What constitutes this thinking is the understanding that the value of 3/7 is not invariant under the change 3 -> 9, and that a certain compensation, 7 -> x, for this change is needed to leave 3/7 unchanged (i.e., 3/7=9/x). We called this functional invariance because this change and its corresponding compensation each have only one outcome value, which is the "uniqueness to the right" property (i.e., for one input of a function there is exactly one output) that constitutes the idea of function.

Functional invariance reasoning can be applied to comparison proportion problems as well. Consider the problem, "Which is greater, 2/3 or 8/15?" In a manner similar to the above analysis one might view 8/15 as a result of a change made to 2/3. For example, 2 -> 8 and 3 -> 15 can be thought of as instantiation of the changes "times by 4" and "times by 5," respectively. Since the change 3 -> 15 does not compensate exactly for the change 2 -> 8, the value 2/3 is not viewed as the equal to the value of 8/15. One can go a step further and think that since the latter change (3 -> 15) EXCEEDS the compensational change which would leave 2/3 unchanged, (i.e., 3 -> 12), the result, 8/15, must be SMALLER than 2/3.

Relational invariance is when changes are interpreted in terms of the directionality of the order relation between corresponding quantities. Consider, for example, the problem: "Which is greater 6/7 or 3/8?" Similar to the above analysis, we can view 3/8 was a result of a change applied to 6/7, where 6 is changed to 3 (6 -> 3) and seven to 8 (7 -> 8). But, unlike the functional invariance in which 6 -> 3 and 7 -> 8 are specified as instantiations of functional changes, in relational invariance they are interpreted in terms of "increase" and "decrease" relations.

Functional invariance reasoning is conceptually very powerful because it is an abstraction of other interpretations such as those discussed earlier and can be applied not only in ratio problems but in a wide range of multiplicative problems. It emerges from the idea of variability and invariability which is central to mathematical reasoning and thus should be an important target for mathematics curricula.

In previous reports (Harel et al., In press; Harel, Behr, Post, and Lesh, 1990) we presented two classes of principles which are believed to constitute relational invariance reasoning: that is, they constitute the intuitive basis for deciding whether the order relation between two ratios or two products is determinable (the order determinability principles class). Or current experimental work examines teachers and children’s understanding of these principles and their ability to apply them in solving multiplicative and proportion problems.


Concluding Remark

To a large extent and until recently, research of the multiplicative domain has looked at the development of individual multiplicative concepts or subdomains, without much effort to deal with the interrelations and dependencies within, between, and among the mathematical/cognitive/instructional aspects of these concepts. For example, research on the learning of the rational number concept did not take into account children’s conceptions of multiplication and division (e.g., Fischbein intuitive models) or the role of unit in children’s development of different rational number interpretations; research on the learning of the decimal system is somehow separated from research on fractions and proportionality. It may well be that in the course of a scientific exploration, this path of research development was inevitable; namely, in the process of exploring models for the conceptual development of multiplicativity, research had to deal with separate multiplicative concepts before it could reflect on what was learned and raise questions about the interrelations between and among these concepts so that the accumulated knowledge can be described in general terms, in terms of a conceptual field. We believe, however, that we have reached a point where accumulated research knowledge in this domain is such that a reflection is necessary and possible. It is time to focus on questions concerning the acquisition of the multiplicative strictures by taking into account what is known from research on the learning of (seemingly) isolated multiplicative concepts.

In the last few years, we have made an effort in this direction. We have focused our research on theoretical analyses of the multiplicative conceptual field (MCF), based on our own and other’s previous work on the acquisition of multiplicative concepts — such as, multiplication, division, fraction, ratio, and proportion — and relationships among them. Our goal has been to better understand the mathematical, cognitive, and instructional aspects of multiplicative concepts in terms of a conceptual field. For example, in Harel and Behr (1989) we theorized models for problem representations and solution strategies of missing value proportion problems that take into account a wide range of multiplicative variables, such as the intuitive rules of multiplication and division and knowledge of the numerical component and the measure component of quantities; in Behr, Harel, Post, and Lesh (in press a, in press b) we introduced an analysis and notational system that focuses on the notion of "unit types" and "composite units" (Steffe, 1989) whereby we established links between research on rational numbers and research on other conceptual areas such as whole number arithmetic (e.g., Steffe, 1989), intuitive knowledge of multiplication and division (e.g., Fischbein, Deri, Nello, and Marino, 1985), exponentiation (e.g., Confrey, 1988), and beginning algebra (e.g., Thompson, 1989); and in Harel, Behr, Post, & Lesh (in press) we have presented an analysis, partially addressed in this paper, in which multiplicative situations were analyzed with respect to a wide range of task variables and basic principles upon which intuitive knowledge about fraction order and equivalence and multiplicative reasoning can be based were introduced.


1 This work was supported in part by the National Science foundation under Grant No. MDR-8955346. Any opinions, findings and conclusions expressed are those of the authors and do not necessarily reflect the views of the National Science Foundation.

2 These "multilevel" tasks were generate from another task called the Blocks task which was investigated with seventh graders (Harel et al., in press) and with experts (Harel & Behr, in press).
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