Rational Number Project Home Page Lesh, R., Post, T., & Behr, M. (1988). Proportional Reasoning. In J. Hiebert & M. Behr (Eds.) Number Concepts and Operations in the Middle Grades (pp. 93-118). Reston, VA: Lawrence Erlbaum & National Council of Teachers of Mathematics.
Proportional Reasoning

Richard Lesh
WICAT Systems

Thomas Post
University of Minnesota

Merlyn Behr Northern
Illinois University

Proportional reasoning is a form of mathematical reasoning that involves a sense of co-variation and of multiple comparisons, and the ability to mentally store and process several pieces of information. Proportional reasoning is very much concerned with inference and prediction and involves both qualitative and quantitative methods of thought.

In our own research we have considered the essential characteristics of proportional reasoning to involve reasoning about the holistic relationship between two rational expressions such as rates, ratios, quotients, and fractions. This invariably involves the mental assimilation and synthesis of the various complements of these expressions and an ability to infer the equality or inequality of pairs or series of such expressions based on this analysis and synthesis. It also involves the ability to generate successfully missing components regardless of the numerical aspects of the problem situation. This perspective has not been universally employed by the research community.

All persons who solve a problem involving proportions do not necessarily use proportional reasoning. In fact, one could notice simple number relationships (since A is three times B, X must be three times D) or use a rote algorithm such as cross multiplication. To solve proportions of the type A/B = x/D, students are often taught the cross multiplication method A*D = x*B where x = A*D/B; yet research and experience have consistently shown that this method is (1) poorly understood by students (Post, Behr, & Lesh, 1988), (2) seldom a "naturally generated" solution method (Hart, 1984), and (3) often used by students to avoid proportional reasoning rather than to facilitate it. We suggest here that the use of this procedure precludes the use of proportional reasoning and does not involve proportional reasoning per se. We therefore prefer to speak of proportion related problems rather than proportional reasoning problems.

We view proportional reasoning as a pivotal concept. On the one hand, it is the capstone of children’s elementary school arithmetic; on the other hand, it is the cornerstone of all that is to follow. This chapter discusses this construct from both perspectives indicating what we believe to be transition mechanisms and student behaviors. We then explore requirements for a computer based model for the solution of proportion related problems and discuss the conditions under which this model is able to produce reasonable solutions. Finally, we raise questions as to the implications which this model might have for future research with children.

Previous attempts to assess proportional reasoning ability (Karplus, Pulos, & Stage, 1983a, 1983b; Noelting, 1980a, 1980b) have focused largely on individual responses to missing value problems. Those students who were able to answer successfully the numerically "awkward'' situations containing non-integer multiples within and between the rate pairs were thought to be at the highest level, and their answers were considered proportional responses. We believe that this is a limited perspective, a necessary but not a sufficient condition, especially since these problems lend themselves to purely algorithmic solutions. This chapter attempts to expand the previous view and suggest that proportional reasoning encompasses wider and more complex spectra of cognitive abilities which include both mathematical and psychological dimensions.

According to Piaget (Piaget & Inhelder, 1975), however, the essential characteristic of proportional reasoning is that it must involve a RELATIONSHIP BETWEEN TWO RELATIONSHIPS (i.e., a "second-order" relationship) rather than simply a relationship between two concrete objects (or two directly perceivable quantities). For example, Piagetians have considered a balance-beam task to be a prototype for proportional reasoning tasks, even though the reasoning involved does not fit the equation A/B = C/D but instead fits the equation AxB = CxD. In fact, Piagetians have argued that an early phase in children’s proportional reasoning capabilities often involves "additive reasoning" of the form A - B = C - D. We believe that it is desirable to restrict the term "proportional reasoning" to various aspects of multiplicative relationships between rational expressions.

In science education, Karplus et al. (1983a, 1983b) represent still a third perspective, which states that proportional reasoning must involve a LINEAR RELATIONSHIP BETWEEN TWO VARIABLES. So tasks characterized by the system of relationships Y = mX are considered to be proportion related tasks (which of course they are), even though the two sides of the equation are not symmetric. Tasks characterized by the system Y = (a/b)X + n must be differentiated from the proportional situations.

A major goal of this chapter will be to sort out and clarify aspects of proportional reasoning which preserve the intended mathematical meanings of this term and which research has shown to be educationally or psychologically significant. Another goal will be to identify neglected areas of research on proportional reasoning.

SOME IMPORTANT TYPES OF PROPORTIONAL REASONING TASKS

Critical insights underlying many of the most basic concepts in science, mathematics, and everyday problem solving often consist of recognizing similar patterns or structural similarity in two different situations. Because proportional reasoning deals with one of the most common forms of structural similarity, it is often linked to some of the most important elementary but deep concepts at the foundation level of many areas of science or mathematics. As stated earlier, we believe proportional reasoning is both the capstone of elementary arithmetic and the cornerstone of all that is to follow. It therefore occupies a pivotal position in school mathematics (and science) programs.

As each domain of knowledge uses this basic reasoning paradigm, it tends to be modified in subtle ways to fit the peculiar needs of the discipline. For example, in middle school arithmetic, slightly different forms of proportional reasoning are related to some of the foremost conceptual "trouble spots" in the curriculum:

 –(equivalent) fractions: 5/3 = n/m –long division: 805/23 = n/1 –place value and percents: n% = 75/100 –measurement conversion: n dollars = (2/3) m Canadian dollars –ratios and rates: 15 feet/2 seconds = n miles per hour

In the preceding kinds of topical areas, the following seven types of proportion related problems all arise naturally. Yet, types 3 through 7 have been neglected in textbook-centered instruction and research.

1. Missing value problems: A/B = C/D where three values (including one complete rate pair) are given, and the goal is to find the missing part of the second (and equivalent) rate pair.
2. Comparisons problems: A/B ‹= ? =› C/D where all four values are given, and the goal is to judge which is true:
A/B < C/D or A/B = C/D or A/B > C/D
3. Transformation problems:
(a) direction of change judgments:
An equivalence is given of the form A/B = C/D.
Then, one or two of the four values A, B, C, or D is increased or decreased by a certain amount, and the goal is to judge which relation ( <, >, or = ) is true for the transformed values.
(b) transformations to produce equality:
An inequality is given of the form A/B < C/D.
Then, for one of the four values A, B, C, or D, a value for x must be found so that, for example (A+x)/B = C/D.
4. Mean value problems: Two values are given, and the goal is to find the third.
(a)
geometric means: A/x = x/B
(b)
harmonic means: A/B = (A-x)/(x-B)
5. Proportions involving conversions from ratios, to rates, to fractions: The ratio of boys to girls in a class was 15 to 12. What fraction of the class was boys?
6. Proportions involving unit labels as well as numbers: (3 feet)/(2 seconds) = x miles per hour or 5 feet/second = x miles/hour
7. Between-Mode translation problems: A ratio (or fraction or rate or quotient) is given in one representation system, and the goal is to portray the same relationship using another representation system.

Realistic versions of proportional reasoning problems often involve between-representation comparisons. We have found that these tend to be surprisingly difficult for most students (Lesh, Behr, & Post, 1987).

Even when the two sides of a proportion involve the same representation system, students’ solutions to these problems often involve translations among various representation systems. For example, consider the word problem shown in Figure 1. We have found that success rates tend to range between 9.2% for fourth graders and 46.2% for eighth graders.

A student might think about the problem in Figure 1 by

 (a) paraphrasing (i.e., translating to simpler language): fifteen is to five as is to . (b) drawing a diagram (i.e., translating to a picture or diagram)
 HOUR 1 HOUR 2 HOUR 3 HOUR 4 HOUR 5 MMMMM MMMMM MMMMM MMMMM MMMMM
 (c) writing an equation (i.e., translating to written symbols): 15/5 = m/h

So even when a problem does not appear to involve more than a single representation system, its solution may involve several translations.

 Sue can walk 15 miles in 5 hours. Her ratio of miles per hour is: a. 5 to 15 b. 10 to 5 c. 3 to 1 d. not given Figure 1. A typical proportion related word problem

PROPORTIONAL REASONING IS A WATERSHED CONCEPT

To determine which aspects of proportional reasoning should be emphasized in the future, it is important to recognize the "conceptual watershed" role that proportional reasoning has played at the borderline which separates elementary from more advanced concepts. That is, it is both (1) one of the most elementary higher order understandings and (2) one of the highest level elementary understandings. For example, in the psychology of human learning, proportional reasoning is widely recognized as a capability which ushers in a significant conceptual shift from concrete operational levels of thought to formal operational levels of thought (Piaget & Beth, 1966).

The next two sections of this chapter examine proportional reasoning from two distinct perspectives:

 (1) Proportional reasoning as a cornerstone of algebra and other higher level areas of mathematics. (2) Proportional reasoning as a capstone of elementary arithmetic, number, and measurement concepts.

Proportional Reasoning as a Cornerstone of High School Mathematics

In the previous word problem (see Figure 1), suppose that we wanted to know how far Sue would walk in 3 hours. This problem can be solved in three steps:

 (a) Write an equation to describe the problem situation, for example, 15/5 = m/3. (b) Transform the "descriptive" form of the equation to an equivalent "computational" form, for example, m = 3x(15/5). (c) Compute by carrying out the indicated operations, m = 3x(3) = 9.

This describe-transform-compute procedure is one characteristic that differentiates algebra from arithmetic. Notice that it is not necessary for the equation which describes or models a problem to immediately specify a series of computations for producing an answer. In algebra, the description and solution phases of problem solving can be separated. In arithmetic, students usually proceed directly to a series of solution-producing computational procedures.

Proportional reasoning inherently involves some of the most important algebraic understandings having to do with equivalence, variables, and transformations. We will discuss each separately.

Levels of equality. The following kinds of equivalence classes occur frequently in proportion related situations:

 (a) equivalent numbers or ratios of numbers, for example, 1/3 = 2/6 = 3/9 (b) equivalent expressions having both measurement units and numbers, for example, kilometers / meters = 1000 / 1 <--> 1 kilometer = 1000 meters (c) equivalent expressions involving relations and/or operations as well as numbers and units of measure for example, 6 feet/2 seconds = 3 feet per second = 2.0455 miles per hour (d) equivalent equations, for example, 2/X = X/18 <--> X2 = 2x18 <--> X = 6 where the transformations preserve some important properties while changing others.

In arithmetic, equals signs (=) can usually be interpreted as meaning "results in" or "gives" (e.g., when the equation 5-3=2 is read as "five minus three results in two"). In algebra, however, the "=" sign usually stands for a more general type of equivalence. For example, two expressions may be treated as equivalent for any of the following reasons:

 (a) They are reducible to the same value, for example, 6/2 = 4 -1. (b) They are structurally similar; that is, they involve the same pattern of relations and operations, for example, (Of course this is true only for certain values of the variables.) (c) They have identical graphs on sets of non-zero measure, for example, f(x) = 2x2/3x and g(x) = 2x/3. (d) Their graphs cross the x axis at the same points. (e) One expression can be substituted for the other without gaining (or losing) interesting information.

An understanding of proportional reasoning must go beyond the simple notion that two sides of an equation are equal (in the sense of being reducible to the same value). For example, our intuitions tell us that an equation such as "6/2 = 4 - 1" should not be called a proportion (even though its two sides are equal) because the two sides of the equation are not structurally similar; that is, they do not involve the same pattern of relations or operations, and the components are not multiplicatively related.

The recognition of structural similarity appears to be an essential component for proportional reasoning to occur. However, if we demand a recognition of structural similarity as a necessary condition for a reasoning process to be called proportional reasoning, then, in general, tasks which are modeled by the equation Y = (a/b)X but which are not modeled by the equation Y/X = a/b should not be construed as involving proportional reasoning, even though Y = (a/b)X and Y/X = a/b are in some, but not all, ways equivalent.

Levels of variables. In addition to involving a variety of different levels of equivalence, simple proportional reasoning situations may also involve several levels of variables. It is not simply the ability to vary that makes something a variable. For example:

 (a) In many simple proportions, such as 3/x=x/27, the value of x cannot vary; yet this does not mean that x is not a variable. Here, the important thing to know about x is that it is an unknown which can be manipulated using rules similar to those that apply to known numbers. (b) Sometimes even fixed constants, such as # representing pi, can be assigned a range of values (e.g., 22/7 or 3.14 or 3.14159265356) depending on the level of precision that is chosen as appropriate in a given situation. Yet, this does not mean that # is a variable.

So the value of a symbol may be fixed, yet the symbol is a variable; or the value of a symbol may be able to vary, yet the symbol is a constant. Overly simplistic explanations, such as treating variables as though they were simply things that can vary, are bound to result in confusion for children.

Transformations and invariance. To generate a set of ratios equivalent to 3/9, a series of transformations can be used to "map" one expression to another: 1/3 --> 3/9 --> 9/27 -->.... Or, to find the unknown value for x in a proportion like 3/x= x/27, a series of transformations can be used to map one entire equation to another:
3/x = x/27 --> x = 27*3/x --> X2 = 81 --> x = 9. And, for any of the preceding kinds of transformations, issues related to levels of equivalence arise. Any time an object is transformed, some information is lost or gained; the issue is whether the altered information is of interest. Which properties remain invariant and which do not? For example, two equations may be considered equivalent for any of the following reasons. (Recall that in the previous section a similar list was generated for expressions.)

 (a) They have the same truth value (e.g., both true or both false). (b) They have the same solution set (e.g., same values satisfy both). (c) One is a simplified form of the other. (d) They can be transformed into one another using specified operations.

When educators treat transformations as though "everything is OK as long as you do the same thing to both sides of an equation," it should be no surprise that this simplistic notion often leads to errors in children's proportional reasoning. For example, in one of our recent clinical studies (Behr, Wachsmuth, Post, & Lesh, 19X4), students were shown equalities such as 3/4 = 6/X. Then a given number was subtracted from one of the four numbers, and the student was asked to change one of the remaining three numbers to restore equality, (e.g., (3 -1)/4 = (6- ?)/8). It was very common for students to simply increase or decrease the corresponding numbers by the same amount on both sides. In fact, one student even "proved" that (3-1)/4 = (6-1)/8; he erased the 1's from both sides, explaining "It's OK, 'cause I did the same thing on both sides!!"

The understanding that an equation (as a whole) represents an algebraic "object" which can he transformed in specified ways that leave certain interesting properties (such as the solution set) invariant is at the heart of algebraic reasoning. This is also essential in a simple form in the solution of simple proportions.

The equation A/B = C/D can be thought of as a static (=) relationship between two simple mathematical systems which are described independently by the ratio relationships A/B and C/D. However, it can also be thought of as a dynamic transformation which maps one simple mathematical system (described by the ratio relationship A/B) into an "equivalent" system (described by the ratio relationship C/D). It seems to us that the recognition of structural similarity is an essential component of proportional reasoning. It follows that issues related to structure transformation and invariance should be important themes in proportional reasoning, even at its most primitive levels.

In spite of the preceding facts, tasks dealing with transformations and invariance have been neglected in the research literature on proportional reasoning, even though Piaget (Piaget & Inhelder, 1956) and a few others have emphasized its importance in certain types of conservation tasks.

One reason that transformation tasks have been neglected is their tendency to involve dynamic actions which are difficult to portray in textbook or paper-and-pencil testing formats. A second factor is the artificial overemphasis on "x finding'' in the presecondary and secondary school mathematics curricula. The misconception that the essence of mathematics is "going around looking for lost x's" is an opinion shared by many. Yet, to understand the essence of proportional reasoning, it is important to understand that mathematics is essentially the study of structure and invariance, equivalence and non-equivalence under a variety of different transformations. As students progress to higher levels of mathematics, far fewer problem-solving activities fit the "x finding" stereotype. Far more fit into the category of studying structure-transformation-invariance.

Proportional Reasoning as a Capstone of Elementary School Mathematics

Many of the foremost conceptual stumbling blocks which occur in the elementary school curriculum are critical in the context of proportional reasoning. Examples include (1) the part-whole relationships described by Kieren (1976) and Behr, Lesh, Post & Silver (1983); (2) the composite units (i.e., units made up of other units) emphasized by Steffe (Steffe & von Glasersfeld, 1983; Steffe, this volume), Cobb (1987), and Post et al. (1988); (3) the representation-related abilities emphasized by Kaput (1987a; 1987b) and Lesh, Post, and Behr (1987); and (4) the measurement-related abilities emphasized by Vergnaud (1983), Streefland (1984, 1985), Post et al. (1988), and Kaput (1985). (These areas have been identified by researchers as the conceptual underpinnings of the previously mentioned trouble spots, all of which seem related to proportionality and ultimately to proportional reasoning.)

Pre-proportional reasoning. If the most critical characteristic of proportional reasoning is recognizing "the invariance of a simple mathematical system," then is it always necessary for this mathematical system to be described by the ratio relation A/B? Or would systems characterized by A - B (or A x B or A + B) also qualify? Mathematical systems characterized by A/B are among the simplest that can exist; they involve relationships between the two quantities A and B. So an equation such as A/B = C/D can be interpreted as meaning that the relationship "n more than" applies to both sides.

The relationship "n more than" can be interpreted in two distinct ways: (1) as the additive relationship A= n+B (i.e., n=A-B) or (2) as the multiplicative relationship A = n*B (i.e., n = A/B). In the latter situation it is generally referred to as "n times as many." In either case, reasoning about these situations is one of the simplest situations in which children go beyond comparisons between perceivable quantities to think about structural similarities between mathematical systems as a whole.

For reasons similar to those described above, tasks such as a balance-beam task, which are characterized by the equation A*B=C*D (rather than by the equation A/B = C/D), are sometimes referred to as "multiplicative proportional reasoning" tasks; and tasks characterized by the equation A + B = C + D are usually considered to involve additive reasoning.

Tasks characterized by A/B=C/D or A*B=C*D tend to involve a multiplicative relationship, but do they involve proportional reasoning? Our rule for answering this question is that if there is no evidence that the child recognizes the structural similarity represented by the two sides of the equation, then there is no evidence of proportional reasoning. For example if A*B corresponds to a directly perceivable quantity (rather than to a relationship between two quantities), then a task which might otherwise have been characterized by A*B=C*x may actually be reduced (in the mind of the child) to a task characterized by P = C*x, where P is a "new" element in the system. In this case, structural similarity is not recognized, and proportional reasoning is not required. Therefore, the simple ability to give correct answers to problems of the form A*B = C*x does not guarantee that proportional reasoning is being used. The same is true for problems of the form A/B= x/D (where the position of x can vary).

Similar concerns arise with regard to additive tasks characterized by the equations A + B = C + x or A - B = x - D. That is, the tasks that these equations describe tend to be naturally reducible to non-proportions of the forms P = C + x or P = x - D, where P is a directly perceivable quantity.

If children's inclinations to use addition to solve problems characterized by A/B=x/D are not very reliable indicators of proportional reasoning, then why have researchers (largely developmental psychologists) referred to "additive proportional reasoning" in the research literature? The answer is that during early stages in children's understanding of proportional reasoning, additive reasoning strategies are often used to respond to tasks which should have involved multiplicative relationships (Hart, 1984; Noelting, 1980a, 1980b; Vergnaud, 1983). For example, the following interview shows how additive reasoning arose naturally (but incorrectly) during one of our investigations of multiplicative tasks.

A seventh grader was given a 2x3 rectangle (see Figure 2) and asked to "enlarge it." The student responded (correctly) by doubling the lengths of each of the sides to produce a 4x6 rectangle. So the request was made to "enlarge it again so that the base will be 9." This time the student drew a7x9 rectangle, explaining, "If I doubled, it would have been 12; so I added3 on, so the other side is 9." There are a number of interesting things to notice about this. First, additive reasoning does in fact often appear "naturally" as an early stage in the development of proportional reasoning. Second, the reasoning paradigm that a child uses often varies within a given task (as in the preceding example) or from one task to another, depending on particular task characteristics such as the following: the complexity of the number relationships (Karplus et al., 1983a, 1983b), perceptual distractors (Behr et al., 1983), and the placement of the "unknown" quantity (Bezuk, 1986), for example, x/B=C/D versus A/x=C/D versus A/B=x/D versus A/B = C/x.

In general, tasks characterized by additive equations (i.e., A-B=C-D or A + B = C + D) should not be referred to as proportional reasoning tasks; and even multiplicative tasks (such as balance-beam tasks or the preceding kinds of area tasks) characterized by the equation A*B = C*D may be poor indicators of true proportional reasoning, especially when these tasks are relegated to an algorithmic solution. So, in general, "proportional reasoning" is a term reserved for the solution of tasks characterized by a relationship between two rational expressions: that is, A/B=C/D. We believe, however, that this is a necessary but not a sufficient condition. Other types of situations also reflect true proportional reasoning abilities.

 Figure 2. Enlarging a rectangle

Transitions from pre-proportional reasoning to proportional reasoning. Even though Piaget and other developmental psychologists often speak of proportional reasoning as though it were a global ability, or a manifestation of a general cognitive structure, it appears that the evolution of proportional reasoning is characterized by a gradual increase in local competence (e.g., Lesh, Post, and Behr, 1987; Tourniaire & Pulos, 1985; Karplus et al.. 1983a 1983b). Proportionality is initially mastered in small and restricted classes of problem settings. Competence is then gradually extended to larger classes of problems.

In this section we suggest that the preceding "gradually increasing local competence view" of cognitive development has important implications for research and instruction that involve proportional reasoning. This view also provides guidance for research and instruction related to problem solving.

The example that we discuss illustrates some of the more important mechanisms that enable students to develop from pre-proportional (additive) reasoning to general forms of multiplicative proportional reasoning. The problem shown in Figure 3 was used in Lesh's research on using mathematics in everyday situations. The problem is especially interesting because

 (a) Solutions to it validated developmental sequences in proportional reasoning, which have been hypothesized by Piaget (Inhelder & Piaget, 1958; Piaget, Grize, Szeminska, & Bang, 1968), Noelting (1980a, 1980b), Karplus et al. (1983a, 1983b), Karplus and Peterson (1970), Hart ( 1981), and many others. (b) Solutions showed how mechanisms which have been shown to facilitate general conceptual evolution may also play important roles in problem solving. Before we consider a "typical'' solution to this problem, it is useful to first outline some of the most important stages which developmental psychologists have observed in the evolution of children's general proportional reasoning capabilities. (a) In their most primitive responses, students tend to ignore part of the data, perhaps, for example, by comparing numerators only in the equation A/B = C/D. (b) At a slightly more sophisticated level, students may notice relationships among the four factors in the proportion A/B=C/D but may relate them only in a qualitative manner. (c) Early attempts at quantifying often involve constant additive differences (i.e., A - B = C - D) rather than multiplicative relationships. (d) The earliest use of multiplicative reasoning is often based on a sort of "pattern recognition and replication" strategy, which some have called a "build-up" strategy (Hart, 1984; Karplus & Peterson, 1970; Piaget et al., 1968). For example: A candy store sells 2 pieces of candy for 8 cents. How much do 6 pieces of candy cost? The solution might be recorded as: –2 pieces for 8 cents –4 pieces for 16 cents –6 pieces for 24 cents Given a table of values, children may notice a pattern which is then applied to discover an unknown value. However, as we pointed out earlier in this chapter, success in the use of this strategy is a relatively weak indicator of proportional reasoning. Piaget et al. (1968) referred to this stage as "pre-proportionality" because children have the intuition that the differences change with the size of the numbers and that the change may be multiplicative in nature, but they do not necessarily realize that they need to consider the constantly increasing difference between the related terms of each rate pair, that is, of each ratio. According to Piaget, pre-proportionality is brought about by coordinating functions, whereas proportionality is based on reversible operations. The main difference between function related thinking and operation-related thinking is that the first type is essentially non-reversible; that is, given a change in one of the four variables in a proportion, the child cannot compensate by changing one of the remaining variables. (e) Piaget's "logical proportions" indicates a level of thought in which a multiplicative relationship is noticed between two terms; this relationship is then applied to the other two terms.

 Materials: calculator, Sears catalogue from 10 years ago, current Sears catalogue, newspaper from 10 years ago, current newspaper. Students: This problem has been undertaken by students working either individually or in groups of three, raging from middle school through adults. The group described in this section were average ability seventh graders. Problem: Fred Finley began teaching 10 years ago in Centerville. He and his new bride rented and apartment at 3188 Main Street for \$250 per month. He also bought a new VW rabbit for \$4500. His starting salary was \$14,000. This year, Fred’s brother, Tom, also began teaching in Centerville. He and his new bride rented the very same apartment that Fred had rented 10 years earlier, only now the rent was \$430 per month. He too bought a new VW rabbit; the price was now \$8900. Other price differences can be found in the catalogs and newspapers that you have been given. What should Tom’s current starting salary be, so that it will be equivalent to Fred’s salary from 10 years ago? Figure 3. The inflation problem.
(return to text reference)

In general, according to Piaget, adolescents' proportional reasoning develops from (1) a global compensatory strategy (often additive in nature) to (2) a multiplicative strategy without generalization to all cases to (3) a final formulation of a law of proportions. However, in attempts to verify Piaget's theory, it has been noted that the level of reasoning that a child uses is often not consistent across tasks or even within a given task (e.g., when the number relationships or perceptual distractors or contextual variables are changed slightly). Even though the stages that Piaget describes have proven to be quite robust for describing children's behaviors on a given task, variability across tasks is often quite large. This has been referred to as "horizontal decalage." Conceptual development in the area of proportional reasoning seems to be characterized by gradually increasing local competence more than by the acquisition of some general all-purpose reasoning strategy. Environmental interactions obviously play a large role in this development.

Solutions like the one described below usually took our students between 20 and 40 minutes to produce. The solution described here is typical of those produced by our students, who were average ability middle schoolers through adults. Students generally went through five distinct reconceptualization cycles during their 40 minute session. Notice the similarity between the stages in their solution and the stages noted by Piaget and others.

Conceptualization 1: The students' first conceptualization of the problem used additive reasoning based on only a biased subset of the given information. The group subtracted to find price differences for pairs of presumably comparable old and new items. But only a few items were considered; these were simply the ones that were noticed first, such as the car and one or two items in the catalog. Furthermore, nothing was said explicitly to indicate how these subtracted differences would allow a new salary to be determined.

Conceptualization 2: Why did the students shift to a second conceptualization of the problem? Two possible explanations are suggested. (1) As they worked out details related to their first conceptualization, they began to recognize difficulties that had at first been ignored (e.g., "Which items should we consider?" and "What are we going to do with this information anyway?"). (2) Because it was tedious to carry out the processes associated with the first conceptualization, other ways to think about the problem were considered.

The students' second conceptualization of the problem was based on an extremely primitive multiplicative relationship involving an even more biased subset of the given information. Whereas the first conceptualization had lost sight of the overall goal when attention was focused on details (individual subtractive differences), the second conceptualization ignored details when attention was focused on the relationship between the two salaries. Because 10 years had passed, the students guessed that perhaps Tom's salary should be 10 times as large as Fred's!

Although this "brainstorm" was quickly recognized as foolish, it served a very positive function; it introduced a multiplicative way to think about relationships between old prices and salaries and current prices and salaries.

Conceptualization 3: The students' third conceptualization of the problem used a pattern recognition and replication type of pre-proportional reasoning. The students noticed a pattern of items whose price increases were (approximately) a simple integer ratio (i.e., approximately a factor of 2). So they guessed that Tom's salary should be approximately two times Fred's old salary. This insight showed real promise; however, the new clarity of thought that it introduced enabled the students to notice that, for example, some items had actually decreased in price even though most of them had increased.

Conceptualization 4: The students' fourth conceptualization used a true multiplicative proportion, but it was still based on only a biased subset of information; that is, the students began speaking about "percent increases" (but actually calculated ratios based on simple integer factors). Percent increases were calculated for several items, and these values were then averaged.

Conceptualization 5: The students' fifth conceptualization used a multiplicative proportion which was based on a clear and explicit sampling procedure. Here, for the first time, the students actually wrote a (crude) mathematical sentence of the form "A is to B as C is to D." The values for A and B were based on sums of prices for a number of "typical" items, and the students noticed that after a sufficient number of items had been included in the sum, the ratio was not affected much by the addition of more items.

The preceding solution was not atypical. Our students usually progressed through 2 to 10 distinct reconceptualization cycles during their 40-minute problem solving sessions; these cycles nearly always resembled compact versions of Piaget's developmental sequences. Recall that these sequences describe the gradual evolution of children's general proportional reasoning capabilities over several years. Therefore, because of these striking similarities, we began to refer to our problem solving sessions as "local conceptual development sessions."

From a theoretical standpoint, the idea of interpreting problem solving as local conceptual development has enormous implications. Mechanisms which appear as important in local conceptual development sessions can be used to help explain general conceptual development from early additive forms of reasoning to proportional reasoning and other higher order understandings.

Thinking of proportional reasoning as a gradually increasing local competence rather than as a global manifestation of some general cognitive structure should result in more, not less, importance for developmental theories and for proportional reasoning as a very rich research site. The ultimate importance of proportional reasoning results from its power to facilitate problem-solving capabilities. One of the most important reasons to make problem solving a central part of the school mathematics curriculum is the contributions that such experiences make to children's understandings of other central concepts, many of which in turn are related to and involve proportional reasoning.

PROPORTIONAL, REASONING AND RATIONAL NUMBER CONCEPTS

In earlier sections, we suggested that research on proportional reasoning should pay greater attention to tasks involving (1) dynamic transformations, (2) more than a single representation system, (3) measurement units as well as numbers, and (4) more than a single type of rational expression (rate, ratio, quotient, fraction). On the other hand, we believe that the term "proportional reasoning" should be restricted to situations characterized by the equivalence of two rational expressions (A/B=C/D). Even with this restriction, however, a number of ambiguities remain:

 (a) Even if a mathematician (or educator or psychologist) characterizes a task using the proportion A/B=C/D, this does not necessarily mean that such an equation describes the processes and relationships that a child uses to interpret and solve the task. (b) Among leading researchers in the area of rational numbers (or rational expressions), there is disagreement about the essential characteristics that distinguish, for example, rates from ratios (Freudenthal, 1983; Kaput, 1985; Karplus et al., 1983a, 1983b; Streefland, 1984; Noelting, 198Oa, 1980b; Tourniaire & Pulos, 1985; Vergnaud, 1983). In fact, it is common to find a given author changing terminology from one publication to another, perhaps to conform to common usage, which is itself inconsistent.

A Computer Based Model

This section describes the defining characteristics that we have found to be most useful for distinguishing among the various types of rational expressions. The power and internal consistency of our definitions were tested using a computer-based model; this model was used to investigate the range of situations in which the rules and definitions that drive our model will produce appropriate results and, more importantly, to help identify appropriate variables and perspectives for use in research with children.

Some alternative views about the nature of "rates". Among the participants at this conference. Vergnaud and Schwartz are two who have taken considerably different points of view about the most essential characteristics that distinguish rates and ratios.

Vergnaud (1983, this volume) follows a tradition established by the early Greeks: rates are defined as having to do with quantities in two different measure spaces (e.g., 30 miles/5 hours), whereas ratios are defined as having to do with quantities within a single measure space (e.g., 15 cookies/10 cookies). The Greeks favored this perspective because they were particularly interested in ways that earlier whole-number-based measurement concepts flowed into the domain of rational numbers and rational expressions. Vergnaud, being a developmental psychologist, no doubt favors this perspective for similar reasons.

Kaput, Luke, Poholsky, and Sayer (1986) and Schwartz (1983, this volume) follow a tradition similar to Gauss. This position is reflected today by Freudenthal (1973), Lebesque (1966), Whitney (1968a; 1968b) and other mathematicians who have been less concerned about reconciling rational number concepts with lower order whole-number-based concepts and more concerned about finding links to higher order topics related to various types of functions and more complex measure spaces. Theirs is based on a mathematics of quantity as contrasted with the more commonly used mathematics of number. Kaput and Schwartz begin with a distinction between two basic types of quantities:

 (a) Extensive quantities include examples such as 5 miles, 5 degrees (temperature), 5 degrees (angle), or 5 servings (food). Extensive quantities tell "how much" (i.e., the "extent") of a quantity is associated with a given object. (b) Intensive quantities (or "per" quantities) include examples such as 30 miles-per-hour or 30 dollars-per-item. Intensive quantities do not tell "how much" of a given quantity in absolute terms; instead, they express relationships between one quantity and one unit of another quantity.

Note that scalar quantities are treated as special types of intensive quantities in which the two quantities being related involve the same kind of units: for example, 30 dollars-per-dollars (earned-money/saved-money). According to the view of Schwartz and Kaput, a rate is a single (intensive) quantity, whereas a ratio is a relationship between two quantities.

Both Vergnaud's and Schwartz's points of view adequately clarify issues in their own primary areas of interest but each leads to ambiguities when extended to the other's area of interest. For example, according to Schwartz and Kaput, should "30 miles/5 hours" be considered a rate (a single intensive quantity) or a ratio (a relationship between two quantities) or a quotient (an operation involving two quantities)? According to Vergnaud, does "3 Canadian-Dollars/2 U.S. Dollars" involve two measures within a single measure space (a ratio) or two different measure spaces (a rate)? What about "4 quarters/1 dollar"?

One difficulty with the Greek perspective has to do with determining when two measure spaces are the same. What if a single quantity is measured using several different units as in the examples above? Or what if a single unit is used to measure two different types of quantities? For example, in sewing, a length measure (e.g., 3 yards of cloth) describes an area; similar situations occur in cooking, carpentry, farming, and so on. In fact, these kinds of situations are especially common in the sciences, where indirect measures of basic quantities must often be used, so a unit of one type of quantity is used to measure a second type of quantity.

A fundamental difficulty for psychologists and educators is that mathematicians have seldom taken the trouble to provide rigorous definitions which highlight many task characteristics that are educationally or psychologically significant; this is because a mathematician's objective is usually to focus on structural similarities between tasks rather than on the psychological distinctions between them. So for many task characteristics which are psychologically significant, there do not exist corresponding "correct" definitions. A need for greater agreement and consistency is clear in mathematics education research and instruction. The goal of the next section is to describe similarities and differences among rates, ratios, fractions, and quotients using language that is sufficiently powerful and consistent so that:

 (a) Our computer-based model, PAT (which stands for "Problem Transformer''), can produce appropriate results for a reasonably large class of proportional reasoning problems. (b) The major distinctions which have been noted by researchers such as Freudenthal (1983), Kaput (1985), Karplus et al. (1983a, 1983b), Noelting (1980a, 198()h), Streefland (1985), Tourniaire and Pulos (1985), and Vergnaud (1983) will be taken into account.

To accomplish these goals, we will (1) extend Vergnaud's perspective by clarifying when two measure spaces are the same or different and (2) extend Schwartz's perspective by defining fractions, rates, ratios, and quotients in such a way as to remove ambiguities within the class of proportional reasoning problems in school mathematics and science textbooks.

A brief description of the PAT (Problem Transformer) utility. For the purposes of this chapter, the main point to understand about PAT is that it was designed to enable students to "write" (i.e., assemble from an on-line dictionary, which can be extended gradually) word problems like those given in their textbooks which PAT then transforms when the student gives commands such as (1) UNDERLINE by highlighting the key words and data, (2) PARAPHRASE using simpler sentences and no irrelevant information, (3) OUTLINE using a list of "givens" and "goals," (4) SIMPLIFY by presenting the same problem with simpler numbers, (5) GIVE AN ANALOGY with the same structure in a different context, (6) SUB-STEP by identifying an "intermediate question" which can be answered using the available data, (7) BACK-STEP by identifying an intermediate question based on working backwards from the goals to the givens, (8) MODEL using electronic versions of well-known concrete manipulatives, and (9) ABSTRACT by writing an algebraic equation, function, or expression to describe the problem. It is our view that these transformation-related skills required by PAT are also important indicators of children's proportional reasoning abilities.

The critical understandings needed to create a utility such as PAT are those which enable researchers to describe similarities in the underlying structure of problems (within some rich but sufficiently simple conceptual domain).

A second relevant PAT characteristic is that its computational abilities are similar to those of Schwartz's (1983) SEMCALC; that is, PAT refuses to allow students to enter expressions that have no unit labels. For example, if a student types in "3" (or 3x, or 3ax), PAT responds by asking, "3 whats?!!" whereupon the student is prompted to enter (1) how much, (2) the unit of measure, and (3) the kind of underlying quantity, for example, `'3 miles (distance)" or "3 miles-per-hour (speed)."

If a student enters "3 apples + 2 oranges" then (1) if the knowledge that both apples and oranges are fruits is already contained in PAT's on-line library, PAT will simplify this statement to "5 fruits," or (2) if the library does not contain the preceding information, PAT responds by asking "What is the relationship between apples and oranges?" (multiple choice): ٱ All apples are also oranges. ٱ All oranges are also apples. ٱ All apples and oranges are {fill in the blank}. PAT's library is then modified to include the new information gained from the student's response.

When PAT is in "computation mode," it manipulates mathematical expressions, which include (1) unit labels (e.g., feet, miles per hour), (2) quantity types (e.g., length, speed), (3) variables (e.g., x and y), (4) literal constants (e.g., a, b, m), and (5) "pure" numbers.

Fractions, rates, ratios and quotients. For our purposes, two measure spaces will be considered different whenever they involve (1) a different set of objects, (2) a different type of underlying quantity (e.g., length, weight, time, etc.), (3) a different unit of measure, or (4) a different scheme for assigning numbers (i.e., measures) to objects in the space. For example, if students heights are measured in inches and also in feet then these measurements will be considered to be in two distinct spaces, and the conversion from one type of unit to the other will be treated as a mapping (or transformation or translation) from one measure space to another.

With the preceding understandings in mind, fractions, rates, ratios, and quotients can be distinguished according to whether they are (1) single (extensive or intensive) quantities, (2) relationships between pairs of quantities, or (3) operations performed on pairs of quantities. These distinctions are critical to a simple-minded student such as PAT, because the rules for combining (or linking or adding) are often quite different depending on whether the "objects" being operated on are quantities, relations, or operations It is important to emphasize that the following definitions were required in order that PAT not produce gibberish. Our purpose is to raise the question as to whether PAT's requirements have implications for future research with children. We think the answer may be yes. The definitions which follow have the consistency necessary for PAT but are not necessarily ones which we are advocating for general usage.

 (a) Fractions are special kinds of extensive quantities; they tell the size of a single object, for example, 3/4 pizza.

Note: (1) PAT really interprets "3/4 pizza" as "3 fourths-pizza," where "fourths-pizza" is the unit of measure, and 3 tells the size of the object being measured. So "3 fourths-pizza" is treated in the same way as "3 centimeters." (2) Mathematically, a quantity like "3 fourths-pizza" can be represented as a point on an integer number line, which is labeled using units of "fourths-pizza" (see Figure 4).

 (b) Rates are intensive quantities; they can be recognized by the "per" in their unit labels, for example, 3 miles-per-hour. Fractional rates such as "3/4 miles-per-hour" can be interpreted in a straightforward way as "3 fourth-miles-per-hour."

Note: (1) Some authors refer to "unit rates" and to "non-unit rates." However, according to the terminology used here (and PAT's requirements), only unit-rates will be called rates. Recall that any division of two extensive quantities will produce a unit rate, for example, 4 hamburgers/2 persons = 2 hamburgers per person, or 30 miles/10 hours = 3 miles per hour. Thus any indicated comparison (A/B) of two quantities can be converted to a unit rate and expressed in the more usual form "so many A's per 1 B" by simply carrying through with the indicated division. PAT, however, is more comfortable with the interpretation of rates being restricted to unit rates. (2) Mathematically, a quantity such as "3 miles-per-hour" can again be represented as a point on an integer number line, which is labeled using units of "miles-per-hour" (see Figure 5).

 (c) Ratios are binary relations which involve ordered pairs of quantities (of either the extensive, intensive, or scalar types).

Note: Mathematically, a relation between two measure spaces, P and B, is usually represented using ordered pairs (or equivalence classes of order pairs) in the measure space formed by the cross-product PxB (see Figure 6).

 (d) Quotients are binary operations which combine two quantities (extensive, intensive, or scalar) by mapping them to a quantity in a third measure space.

Note: (1) If two extensive quantities are mapped to an intensive quantity, then the operation is often called a "partitive" division (e.g., 3 pizzas/4 boys →3/4 pizzas-per-boy). (2) If an extensive and an intensive quantity are mapped to an extensive quantity, the operation is often called a "quotitive" division (e.g., 3 pizzas/ (3/4 pizzas-per-boy) →4 boys). (3) Mathematically, a binary operation from two measure spaces P and B into a third measure space S is usually represented as a mapping from PxB into S.

All four types of rational expressions can be seen in different regions of diagrams such as Figure 7. For example: (1) because fraction and rates are both quantities, they will occur as axes related to P, B, or S; (2) because ratios are relations, they will occur as points in PxB; and (3) because quotients are operations, they will occur as mappings from P and B into S.

 Figure 4. Integer number line labeled using units of fourths-pizza.

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 Figure 5. Integer number line labeled using units of miles-per-hour.

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 Figure 6. Measure space formed by the cross-product PxB.

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 Figure 7. Mapping from two measure spaces P and B in to a third measure space S.

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At an even higher level of abstraction, all four basic types of rational expressions (fractions, rates, ratios, and quotients) can be represented using a single mathematical model, a homogeneous space consisting of 3x3 matrices. Objects, relations, operations, and transformations can all be represented as matrices within this single vector space. From an intuitive point of view, this is not surprising because, for example, even though fractions and rates both refer to single quantities, both also implicitly involve comparisons between two quantities. For fractions (e.g., 3/4 pizza), the comparison is concealed within the "number half" of the expression (e.g., 3/4), whereas for rates, the relationship is concealed in the "units half" of the expression (e.g., pizzas-per-boy). The relationship that is concealed within 3 fourths-pizza is between the size of the unit (i.e., a fourth-pizza) and the size of the object being measured (i.e., 3).

From a psychological perspective, there are dangers in the preceding kinds of mathematical generality. It should not deceive us into believing that just because rates, ratios, quotients, and fractions can all be interpreted as equivalent types of objects (in a 3x3 vector space), they are viewed as equivalent by young students. For example, it is only at a rather sophisticated level of understanding that a rate such as "three-fourths miles-per-hour" is considered to be equivalent to the ratio "three miles to four hours."

Operations with fractions, rates, ratios, and quotients. According to the definitions of fractions, rates, ratios, and quotients that have been given here, somewhat different rules apply to the four basic types of rational expressions when we add them, multiply them, or equate them. So, to end this chapter, examples will be given to show how PAT deals with some of these differences at the computational phase.

Because fractions and rates are quantities, they can be added and multiplied using the standard rules that apply to simpler types of quantities. For example:

 (1) Only quantities which are in the same measure space can be added. 3 APPLES + 2 APPLES = 5 APPLES 3 MILES-PER-HOUR + 2 MILES-PER-HOUR = 5 MILES-PER HOUR (2) If two quantities are in different but related measure spaces, then they must be mapped into the same measure space before they can be added. 3 FOURTHS-PIZZA 2 THIRDS-PIZZA 9 TWELFTHS-PIZZA + 8 TWELFTHS-PIZZA Note: The ability to transform between "related" measure spaces is a critical skill in the preceding operations. (3) The standard rules for addition usually do not make sense for ratios (i.e., ordered pairs of quantities). For example, what would it mean to "add" a boy-to-pizza ratio of two-to-three to a boy-to-pizza ratio of three-to-four? Seventeen-to-twelve would not be a sensible answer. (4) Quantities from different measure spaces can be multiplied using the usual rules of multiplication (although the results may or may not have "sensible" interpretations). 5 MEN x 3 HOURS-OF-WORK = 15 MAN*HOURS-OF-WORK 5 MILES-PER-HOUR x 3 HOURS = 15 MILES 5 FEET x 3 FEET = 15 FEET x FEET = 15 FEET2 = 15 SQUARE-FEET

Note: An extensive quantity times another extensive quantity corresponds to the "cross-product" interpretation of multiplication and an extensive quantity times an intensive quantity corresponds to "repeated addition."

Part of the goal in using a computer-based model such as PAT is to make explicit certain thought processes which may be important in children's thinking, even though these processes may be used only implicitly or without much reflection.

Are unit labels and transformations among measure spaces and conversions among the various types of rational expressions really as important as PAT's solutions seem to suggest? In general, our research suggests that the answer is "yes"; translations among various quantity types and conversions among various "rational expression" types (rates, ratios, quotients and fractions) appear to be real psychological factors. Students' abilities to do "unit label arithmetic" and to switch flexibly from one type of rational expression to another, were, in our work, particularly reliable indicators of everyday problem solving ability. We suspect that the number of conversion-related steps that PAT takes to solve a problem is an excellent predictor of item difficulty.

The most important task characteristics required by PAT to generate "similar problems" are those directly related to unit types and to rational expression types

SUMMARY

In this chapter, we have attempted to develop a coherent framework which focuses attention on the major insights that have evolved out of past research on proportional reasoning. We have also attempted to focus attention on some neglected categories of tasks and to suggest a few new ones for investigation in future research and development. We gave attention to clarifying language difficulties because, in the domain of proportional reasoning, confusing and inconsistent terminology are serious impediments to future progress.

Whether we looked at proportional reasoning as a capstone of elementary mathematics or as a cornerstone of advanced mathematics, similar themes appeared. These invariably had to do with transformations and equivalences and with translations within and among various measurement systems and representation systems.

Although developmental psychologists have tended to speak of proportional reasoning as though it were a global ability related to some general cognitive structure, mathematics education research clearly shows that the evolution of proportional reasoning is characterized by a gradual increase in local competence. It is our perspective that research characterized by local conceptual development will provide critical insights about the mechanisms by which children evolve to higher order forms of reasoning. Proportional reasoning seems to be an especially productive research area for studying this phenomenon.

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ACKNOWLEDGEMENTS

This research was supported in part by the National Science Foundation under grant No. DPE-847tX177. Any opinions, findings and conclusions expressed are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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