Proportionality and the Development of Prealgebra Understandings
Thomas R. Post, Merlyn J. Behr, Richard Lesh *
Proportional reasoning is generally regarded as one of the components of formal thought acquired in adolescence. Relatively few junior high school students of average ability use proportional reasoning in a consistent fashion unaccompanied by physical actions (Goodstein 1983; Kurtz and Karplus 1979; Lawson and Wollman 1976; Lunzer 1965; Noelting 1980; Wollman and Karplus 1974). The issues involved in the teaching and learning of proportional reasoning are seemingly more complex than previously acknowledged. Repeatedly disappointing achievement results on the National Assessment of Educational Progress and international achievement comparisons further substantiate this perspective.
The majority of past attempts to define proportional reasoning (e.g., Karplus, Pulas, and Stage 1983; Noelting 1980) have been primarily concerned with individual responses to missing-value problems where three of four values in two rate pairs were given and the fourth was to be found. Those students who were able to answer successfully the numerically "awkward" situations containing noninteger multiples within and between the rate pairs were thought to be at the highest level and were considered proportional reasoners. 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 paper attempts to expand the previous view and suggests that proportional reasoning encompasses a wider and more complex spectrum of cognitive abilities. As more data are analyzed and additional research conducted, we and others will surely continue to modify and improve on our understanding of this important construct.
Proportional reasoning is one form of mathematical reasoning. It involves a sense of covariation, 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. The fact that many aspects of our world operate according to proportional rules makes proportional reasoning abilities extremely useful in the interpretation of real-world phenomena.
Proportional reasoning has aspects that are both mathematical and psychological. Mathematically, all proportional relationships can be represented by the function y = mx, the most fundamental type of linear equation. This equation represents a simple relationship between ordered pairs of numbers (x, y) that is multiplicative in nature. Traditionally, proportional situations have been embedded in missing-value problems. (a/b = c/x, with a, b, and c usually given explicit values. The task is to determine the value of x; the position of x can vary.)
Proportional reasoning is also required to compare two given rate pairs. In numerical comparison situations one is asked to compare a/b and c/d where a, b, c, and d are all given. The task is to deduce which rate pair is greater, faster, darker, more expensive, more dense, and so on.
Proportional reasoning involves qualitative thinking: "Does this answer make sense? Should it be larger or smaller?" Such thinking requires a comparison that is not dependent on specific values. For example, "If Nicki ran fewer laps in more time that she did yesterday, would her running speed be faster, slower, the same, or can't tell?" (How about fewer laps in less time?) In this kind of situation, qualitative reasoning requires the ability to interpret the meaning of two ratios, store that information, and then compare these interpretations according to some predetermined criteria. This process requires a mental capability that Piaget has equated with the formal operational level of cognitive development. He referred to this process as operating on operations. That is, the interpretation of each of the ratios is an operation in and of itself, and the comparison is yet another level of operations. Such processing requires multilevel comparative thinking quite different from an algorithmic approach, where a rule is used to solve predictable problems in predetermined ways.
In one sense qualitative reasoning is more general than quantitative reasoning, since one's conclusions relate to an entire class of values rather than specific entities. In another sense, qualitative reasoning is an important means to check the feasibility of responses and a way to establish broad parameters for problem conditions. It is well known that experts in a wide variety of areas use qualitative approaches to problems as a means to better understand the situation before proceeding to actual calculations and the generation of an answer. Novices, however, tend to proceed directly to a calculation or a formula without the benefit of prior qualitative analyses. It should also be pointed out that novices often answer problems incorrectly, suggesting that they could benefit from the use of qualitative procedures.
Another aspect of proportional reasoning involves a firm grasp of various rational number concepts such as order and equivalence, the relationship between the unit and its parts, the meaning and interpretation of ratio, and issues dealing with division, especially as this relates to dividing smaller numbers by larger ones. A proportional reasoner has the mental flexibility to approach problems from multiple perspectives and at the same time has understandings that are stable enough not to be radically affected by large or "awkward" numbers or by the context within which a problem is posed.
And lastly, proportional reasoners must be able to distinguish between proportional and nonproportional situations. This has direct implications for instruction.
THE STANDARD ALGORITHM
Although it can be effectively argued that students need to automatize certain commonly used mathematical processes (Gagné 1983), it can likewise be argued that the most efficient methods are often those that are the least meaningful and therefore are to be avoided during the initial phases of instruction. Unfortunately we sometimes confuse efficiency and meaning, and by default, even with the best of intentions, we introduce a concept in the most efficient but least meaningful manner. The standard algorithm for proportionality - a/b = c/x, a, b, and c given, find x - is one of these areas. The standard solution procedure is to cross multiply and solve for x. That is, ax = cb, or x = cb/a.
This algorithm in and of itself is a mechanical process devoid of meaning in a real-world context. It can, however, be approached in a rational manner, as we shall explain later.
First it must be stated that the standard algorithm is appropriate only after more understandable approaches have been developed. Some of these will now be discussed.
The Unit-Rate Method
The approach with the most intuitive appeal is undoubtedly the "how much (many) for one," or the unit-rate method. It has intuitive appeal because children have made purchases of one and many things and have had the opportunity to calculate unit prices and other unit rates. In fact, many standard one-step multiplication and division problems from third grade onward can be thought of as determining the unit rate (a division problem) or some multiple of the unit rate (a multiplication problem).
Let's look at two examples:
In the first example the unit rate is given and the student is to find some multiple of it. In the second example the student is asked to find the unit rate, given some multiple of it. Vergnaud (1983) suggests that many one-step multiplication and division problems can be thought of as missing-value (a/b = c/x) proportional situations where the unit rate is given or is to be found. Proportional situations where the unit rate is not given are, in fact, the standard missing-value problem types. In the problem that follows, note how a slight modification in the conditions of the problem changes the difficulty level. We now have a two-step problem whose solution encompasses precisely those elements discussed above.
Thus we see that the format for a standard missing-value problem is very much related to previously learned one-step multiplication and division problems. As the context (setting, rate type, etc.) and the numerical aspects of the problem are adjusted, the problem can be made more (or less) obscure to the student, but the format and the interpretation of proportional events remain unchanged.
This relationship to previously understood material is important pedagogically. Looking back, we see that "new" ideas (proportional situations in the form of missing-value problems) can now be rethought from familiar multiplication and division perspectives. Looking ahead, we see that the unit rate is also interpretable as the slope (m) of a linear function of the form y = mx. More will be said about that later.
It must be noted that
there are always two unit rates for a given rate pair, each being the
reciprocal of the other. One is usually more useful and more easily interpretable
than the other. In the previous example, 5 computer disks cost 450 cents.
Ratios can be expressed in two ways:
The first is interpreted as 90 cents for 1 disk, the result of the division 450/5 = 90. The second is interpreted as 0.0111 ... disks per cent, resulting from the division 5/450. This is a mathematically correct interpretation and useful if we are interested in how many disks could be purchased for $10. (Note that this latter problem is usually solved by dividing $ 10 by 90 cents.)
This choice causes students great difficulty. In many instances they are unable to interpret reciprocals of standard unit rates, probably because the idea is never formally considered and because they have been taught when dividing to ask how many times the bottom number "goes into" the top number. This leads to further problems when interpreting ratios because "there are no cents in computer disks."
Rates and their reciprocals (like functions and their reciprocals, graphs and their reciprocals, numbers and their reciprocals, etc.) are important mathematical concepts. Considering reciprocals in this relatively concrete context will facilitate later extensions of the concept to algebraic settings. For example, if Sally bought a disks for b dollars, how many disks could she buy for c dollars? Notice that here it is very important that both unit rates be interpretable by students. a/b has a very different meaning from b/a. Which unit rate is appropriate in this problem? Why?
A second, slightly less functional but very valid method for solving missing-value problems involves a "times as many" mentality. In proportional situations, if one variable is x times another within a given rate pair, this variable should likewise be x times the other in its equivalent rate pair. This is also true between rate pairs. Let's examine the idea using a modified version of our previous example:
An individual using the factor of change method would reason as follows: "Since I want three times the number of disks, the price should be three times as large, so the answer is 3 x $3.60." Similarly, one could reason that since 4 is 1/3 of 12, then $3.60 must be 1/3 of the required total cost, that is, 3 x $3.60. Notice that this argument has underpinnings related to rational numbers and is based on the idea that if I know 1/3 of something, then I can generate that something (the unit or whole) by multiplying by 3, since 3 x 1/3 = 1 (unit). (**)
You may be wondering why we suggested that the factor method was less functional than the unit-rate method. This is so because the ease of use with the factor method is very much related to the numerical aspects of the task. We chose example D because the numbers are very compatible with one another (integral multiples) and lend themselves to a "times as many" approach. What if the problem were like example C? Would we have been as likely to say, "Since 12 is 2 2/5 as large as 5 the price paid should be 2 2/5 as large as $4.50" (i.e., 12/5 x 4.50)? Probably not, although it is still conceivable. We have found that many teachers also have some difficulty with the factor interpretation when the numbers used are not multiples of one another. As would be expected, additional difficulties arise if the relationships are expressed with variables - that is, if Sally paid a cents for b disks, how much did she pay for four times as many (4b) disks?
To summarize, the factor method is a useful interpretation of proportionality and one that should be in every child's repertoire. It makes a certain subclass of word problems very easy to solve. Its use is, however, generally confined to those problems where corresponding numerical values in the rate pairs are integral multiples of one another.
APPLICATIONS TO NUMERICAL COMPARISON PROBLEMS
A second type of proportion-related problem (the first was the standard missing-value problem) involves a comparison of two rates and the determination of which one is less or greater. This problem type is known as a numerical comparison.
Notice that a "how much for one" mentality is a natural for this problem. Two simple calculations (divisions) will generate two unit rates, which then can easily be compared. Joanna had the better buy because she paid $0.85 a disk whereas Mary paid $0.90 a disk. A unit rate approach can be used effectively and is a natural way to deal with all numerical comparison problems. This is yet another reason why it should have a prominent place in the prealgebra curriculum. Our research with junior high school children (Heller, Post, and Behr 1985; Post et al. 1985) has determined that the unit rate method is far and away the most widely used by children who have not had formal instruction in the standard cross-multiplication algorithm. Ginsburg (1977) suggested that we as mathematics educators should attempt to exploit the mathematics that children already know and attempt to capitalize on and extend the thought processes that occur naturally. The unit rate procedure surely satisfies both these criteria.
Sets of equivalent
rate pairs, ratios, (***) and rational numbers
can be represented graphically with the exception that the two axes are
With one exception, (0,0), the vertical axis is uninterpretable, all entries having a zero in the numerator, and the horizontal axis is undefined, since entries on it would imply division by zero. The point (0,0) often has a viable interpretation, that is, zero cost for zero computer disks. Points representing equivalent fractions, ratios, or rate pairs will define a straight line through the origin. The reader will recognize this situation as a linear function of the form y = mx. Most proportional situations are typically restricted to the first quadrant because the values generated by "real life" are normally positive. As students become comfortable with interpreting physical situations through graphs in the first quadrant, extensions to more complex and nonproportional phenomena, such as the graphing of equations of the form y = mx + b and nonlinear forms (xy = k, y = x2), are appropriate.
Graphs can be used to generate equivalent ratios or rate pairs and to identify the unknown in the second rate pair of a missing-value problem. This is so because the linear function, of which the first rate pair is a part, completely defines the relationship between all equivalent rate pairs. A set of all equivalent rate pairs is called an equivalence class. In missing-value problems the second rate pair is always a member of this equivalence class and thus is represented on the same graph (line) as the first rate pair.
The process is illustrated below. Recall exercise C (Sally bought 5 disks for $4.50; how much for a dozen?). This can be represented in the form 5/450 = 12/x.
To find x graphically in this proportion, first plot the known ordered pair (5, 450). Then connect with the point (0,0). Remember (0,0) means that if there are no disks, then there is no cost, a reasonable proposition. Extend the line (suggested by these two points) through point (5, 450) as in figure 8.1 below. Next locate the desired number of disks (12) on the horizontal axis, proceed vertically until the function line is met, then proceed directly to the vertical axis. The point on the vertical axis (1080) represents the cost of twelve disks. A similar process could be followed for any number of disks. Conversely, the number of disks obtainable for a specific cost could be found by reversing the process.
Notice that the equation of this line is y = 90x. This is interpretable as cost = 90 x number of disks. But 90 is also the cost of a single disk - the unit rate. In addition 90 is also the slope of the line and often is referred to as the constant of variation. The graph can be used to identify the unit rate by locating 1 on the horizontal axis and following the procedure outlined above.
The slope of a line denotes the multiplicative nature of the relationship between the variables. Thus it is a very special number. The fact that 90 is also the unit rate (another special number) is no coincidence. The unit rate is always the slope of the line if that slope is expressed with a denominator of 1. You will see in the next section that any rate pair can be changed to the unit rate by simple division.
We already know that the slope (m) of a line of the form y = mx (straight line through the origin) is y/x (solving for m). But y/x is the same division used to define the rate pair and to determine the unit rate (recall example B). It follows, then, that the unit rate and the slope of any line expressed with a denominator of 1 (i.e., a slope of 4 is expressed as 4/1; a slope of 5/4 is expressed as [5/4]/1) are one and the same.
SOME EMPIRICAL RESULTS
In the spring of 1985, the Rational Number Project (Behr et al. 1983; Post et al. 1985) conducted a survey of over 900 seventh and eighth graders from an outlying suburban district near Minneapolis. Student achievement with missing-value, numerical-comparison, and qualitative-reasoning problems was very low (seventh-grade average was 50% correct; eighth grade, 66%) despite the fact that six of the eight sets of numerical values used were integral multiples of one another. Student responses were examined for strategies used. The unit rate and factor strategies were by far the most successful for the seventh graders and were used in about 30 percent of the problems by the eighth graders. This represented half of all the correct answers, even for students who had been taught the cross-multiplication algorithm.
In this section we relate the cross-multiplication algorithm to previous discussions and illustrate how it can be reinterpreted as a shortcut combination approach that employs the same type of thinking used in the unit-rate or factor method.
But what does Nicole's rate of 6 minutes/4 laps signify?
A bit of reflection will suggest that this is nothing more than the unit rate, denoting minutes per lap, which can be rewritten as
In fact, any rate pair can be rewritten as a unit rate by performing a simple division. Such a division will always reduce the denominator to 1, which is, by definition, the unit rate. Recall that division was the operation suggested in examples B and C for finding the unit rate. It follows that any rate pair or ratio is the unit rate expressed in a more complicated format, that is, with a denominator not equal to 1.
Now, if Nicole's unit rate is 6 minutes/4 laps (or 1 1/2 minutes per lap), then Colin's time can be found by multiplying this same unit rate (since they were running equally fast) by number of laps that he ran, in this instance ten.
Symbolically we write
We now see that the standard cross-multiplication algorithm is an application of the unit-rate approach as discussed earlier.
Solving, we again have
This could be rewritten as in (1), in which case the argument above applies, or it could be rewritten as
But the ratio 10 laps/4 laps can be rewritten as 10/4, or 2 1/2. (Recall that ratios, unlike rate pairs, are generally expressed numerically without labels.) Colin ran 2 1/2 times as far as Nicole, so his time should be 2 1/2 times as great. The solution to our problem now becomes 6 minutes x 2 1/2 = 15 minutes, obviously the same result.
The type of analysis discussed here can always be used to reinterpret the standard algorithm or cross-multiplication approach as either a unit-rate or factor strategy. This will make it more understandable to students. As a matter of principle, the more that students understand, the more they will view mathematics as an intricate and ever-expanding web of previously learned and interrelated ideas rather than a collection of arbitrary rules that have no apparent relationship or rationale.
A full understanding of the instances of simple (or direct) proportion discussed in this paper is a necessary prerequisite for later involvement with more "awkward" numerical entries; with more remote physical contexts such as density or solubility; with multiple and inverse proportions; and with other situations encountered in algebra, geometry, and the sciences that involve a series of magnitudes (variables) whose interrelationships cannot be represented by the simple linear equation y = mx.
Algebra is often defined as generalized arithmetic. Students must understand the connections between the abstract equations of algebra and the real world of arithmetic. The concept of using a variable as a placeholder will constitute an important conceptual underpinning for much of what is to follow. Introductory algebra must be based on the notion that variables can be manipulated in a manner that exactly parallels many aspects of the real world. This makes algebra both powerful and abstract. Proportional situations provide an ideal entry into the arena of algebraic representation, since its arithmetic precursors are justifiable through commonsense approaches.
The unit-rate method was suggested as the scaffolding on which other interpretations can be constructed because it is a powerful technique, yet one that capitalizes on children's natural thought patterns and inclinations to use this approach.
This paper has also suggested that approaches to proportionality be addressed from multiple perspectives. This position is firmly grounded in theory (Dienes 1967), and has been used extensively in the research of the Rational Number Project (Behr et al. 1983; Post et al. 1985). Generally, equipping students with a variety of perspectives and solution strategies fosters not only better understanding but also a more confident and flexible approach to problem solving.
** There are many situations in mathematics where a portion of the unit is given and the task is to generate the whole. These situations all have antecedents in rational number understandings. It is interesting to note that textbook-based experiences deal with finding parts given the whole but not the other way around. Is it any wonder, then, that so many children have such difficulty with these concepts and later, related ones occurring in an algebraic context, such as 2/3x = 8? (back to text)
*** It is generally agreed that ratios compare like quantities or measures, i.e., 4 dollars/6 dollars = 4/6, whereas rates compare unlike quantities or measures, i.e., 4 miles in 3 hours = 4 miles/3 hours. Ratios can be expressed numerically without labels; rates must retain their labels to be interpretable. Both are manipulated in the same manner. (back to text)
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* The development of this paper was supported in part by the National Science Foundation under Grant No. DPE-8470077 (The Rational Number Project). Any opinions, findings, and conclusions expressed are those of the authors and do not necessarily reflect the views of the National Science Foundation.