Semantic Aspects of Quantity - CiteSeerX

Semantic Aspects of Quantity Judah L. Schwartz

Professor of Engineering Science & Education, Emeritus Massachusetts Institute of Technology and Professor of Education Harvard Graduate School of Education

January 1996

Semantic Aspects of Quantity

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© 1996 Judah L. Schwartz

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TABLE OF CONTENTS Preface PART I - The Arithmetic of Quantity I. II. III. IV.

V.

VI.

VII.

Representing how many and how much Adjectival Quantities Equality and Ordering - Comparing Adjectival Quantities Computing with Adjectival Quantities - Referent-Preserving Operations a. addition of discrete adjectival quantities b. addition of continuous adjectival quantities c. subtraction of discrete adjectival quantities d. subtraction of continuous adjectival quantities Computing with Adjectival Quantities - Referent-Transforming Operations. a. intensive and extensive quantity b. what's wrong with repeated addition and subtraction c. multiplication and division of adjectival quantities d. the conversion of units e. do we need scalars? f. more about multiplication and division Revisiting Some Earlier Notions a. referent-preserving operations b. negative numbers and vectors Some Thoughts on School Arithmetic PART II - Quantity in Secondary Mathematics

VIII.

IX.

X. XI.

XII. XIII.

Toward an Algebra of Adjectival Quantity a. representing functions symbolically b. the power of symbolic manipulation c. representing functions graphically d. the power of graphical manipulation e. linking symbolic and graphical representations of functions f. two special functions g. fitting functions to counted and measured data Operations with Functions a. referent-preserving operations with functions b. referent-transforming operations with functions Comparing Functions of Adjectival Quantity Composing Functions of Adjectival Quantity a. polynomials and rational functions b. periodic and transcendental functions c. acquiring a taste for dimensionless adjectival quantity Toward a Differential & Integral Calculus of Adjectival Quantity Conclusion(s)


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Preface This paper is the result of a kind of personal intellectual odyssey that started almost three decades ago. I wrote the first version of the paper in May of 1976 and circulated it to some friends but never published it. Since then, people have been kind enough from time to time to quote it and appreciate the importance of some of the ideas. Some people have even urged me to publish the paper. Having been recently prodded in this direction, I reread the paper and decided that while it contained many good ideas, it was flawed in ways that I now think I know how to fix. This is a revised and substantially expanded version of that paper. Like the blind men palpitating the elephant, this paper will probably be read in very different ways by different readers. The mathematician is likely to say that it belongs in the domain of psychology; the psychologist will say that it deals with linguistics; the linguist will say that it deals with applied mathematics. They are each correct. There is a body of phenomena that this paper seeks to explore, i.e. the ability of people to make quantitative statements about their surroundings and to communicate those statements and their implications to others. These phenomena are real and worthy of attention. My interest in them grows out of many years of close observation of the performance of mathematical acts by people, mainly students, in both formal and informal settings. In this paper I attempt to analyze some of the ways in which people use quantities that are meaningful to them. This analysis is then used to shape the definition of a formal mathematics of quantities with referents in the world around us and to deriving the consequences and entailments of that formal mathematical structure. Implicit in this effort is an agenda for education and schools. Mathematics holds the position it does in the curriculum of our culture because we believe it to be a source of analytic tools that can help people negotiate with the social and physical world in which they find themselves. Of necessity this means that the mathematical abstractions of quantity, shape, space, pattern, function, arrangement, etc. that they deal with all have their counterparts in reality. I believe it behooves us to brings the abstractions and the realities that provoke their generation closer together. It is toward this end that this paper is devoted. Lincoln, Massachusetts January 1996


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PART I - The Arithmetic of Quantity I.

REPRESENTING HOW MANY AND HOW MUCH

We perceive, recognize and interpret complex situations of all sorts. In so doing, we choose to attend to some elements of situations, and ignore, suppress and aggregate others. We choose to attend to some of the relationships among elements and to neglect or dismiss others. This view of the intertwined processes of perception and recognition has been addressed by psychologists for some years now, sometimes with deep insight. I have found this formulation to be a suggestive and indeed a productive way of thinking about the problem of constructing mathematical models of real situations. Real situations are characterized by a wealth of elements and relationships among the elements. Certainly, an essential step in the process of formulating a mathematical model of a situation is deciding on what elements of the situation and what relationships among them are worth attending to. There are subtle questions here. For example, when are two objects identical? The answer is neither obvious nor straightforward. For example, the common-sense notion of identicalness is one that the scientific community has had to struggle long and hard to overcome, in order to develop a consistent quantum mechanical theory of atomic particles. What is implied about the attribute lists of two objects judged to be similar? Is there a formal way of taking into account the fact that the same two objects may be judged to be similar or different depending on context? What sorts of relationships exist among objects? Spatial relationships that have metric or topological characters shape the ways in which we analyze structure and define structural hierarchy. Temporal relationships can be causal, correlative or apparently random. They play an important role in the way we analyze function. Logical relationships such as conjunction, disjunction and negation help us quantify and define measures. In this paper I will consider how people manipulate those models of real situations in which the elements that are attended to and represented by symbols are such quantities as numbers of apples volumes of milk weights of cars lengths of string and in which the relationships represented by symbols are the arithmetic operations and ordering and equality relations. It is obvious that at least some of the quantities we use as elements of the models we build of the world around us have to refer to entities in our surround which is to say that they are adjectival in nature. I believe a stronger statement can be made, i.e. that all of the


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quantities we employ in our models of the surround are adjectival in nature. If indeed this claim is true and further if it is true that the fundamental reason for the presence of mathematics in the school curriculum is to equip people with a set of analytic tools for modeling their world, then it is at least plausible to explore the question of why it is that we do not teach the mathematics of such quantity in schools but rather the mathematics of number devoid of referents. The teaching of the mathematics of pure quantity begins early in the grades. For example, children are taught the number facts for the most part as if the numbers were disembodied entities with no referents. Seven is deemed to be more than three even if one is talking about seven bacteria and three elephants. While it is true in this example that the cardinality of the set of bacteria is greater than the cardinality of the set of elephants (if you don't count the bacteria in the elephants), it is hard to imagine many attributes of the set of bacteria that are greater in magnitude than the corresponding attributes of the set of elephants. Later in the curriculum one encounters the task of using the arithmetic operations with more than one known quantity to construct new quantities with new referents. For example, 5 apples and 4 oranges can be “added” to produce 9 pieces of fruit, and 4 blouses and 3 skirts can be “multiplied” to yield 12 outfits. Still later the algebra curriculum requires students to extend these notions to the use of arithmetic operations with quantities whose values may not be known. For example, if the 1000 lb. of water in a 50 lb. drum leaks out of the drum at a constant rate and takes a total of 10 hr. to do so, what is the weight of the drum and the water in it 3 hours after it has begun to leak?1 Is the mathematics of quantities with referents different from the mathematics of pure number? Should it be? Should it be taught? These are some of the questions this paper attempts to answer. There seems to be little prior work in this area. What work there is by mathematics educators (see for example Bell etal 1984) does not build on a formal theory of quantity with referent. There is a minor tradition in the mathematics research community (Lebesgue 1966, Whitney 1968a, 1968b) that considers seriously the importance of quantity with referents and the work described here is strongly influenced by it.

1It

is common for algebra curricular materials to pose problems in this sort of sterile and disembodied fashion. The reader may wish to think of the situation more generally as the leaking of a fluid from a reservoir such as oil from a tanker, blood from a wounded human, a species from an ecological system, etc.


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ADJECTIVAL QUANTITY

In the English language number words such as two, five and twenty-three, can be used either as nouns or as adjectives. All the numbers in the sentences The sum of two and three is five. The square root of eighty-one is nine. are nouns. In contrast, the numbers that occur in the sentences Four books and three books are seven books. Ten women and two men are twelve people. are adjectives2. I shall refer to these two forms of number words as nominal numbers and adjectival numbers respectively. The reader will recognize that those settings in which people are asked to model real situations using mathematical statements are settings that will require the use of adjectival quantity. If this observation were to be carried no further there would be little point in drawing the distinction between nominal and adjectival quantity. There is however, a further observation to be made, i.e. in schools students are introduced to both nominal and adjectival quantity. For the most part, however, students are taught arithmetic manipulation of nominal quantity only. The implicit assumption is made that all the manipulations and operations that are appropriate to nominal quantity are also appropriate to adjectival quantity. This assumption will be seen to be both logically and psychologically false. The central purpose of this section of the paper is explore comparison and computation of adjectival quantity, i.e. the arithmetic manipulations appropriate to the kind of quantity that people use in making mathematical statements about the world around them. Real objects are characterized by many attributes. Apples, for example, have color, mass, position, cost, etc. Milk also has color, mass, position, cost., etc. In this respect they seem to be similar, and indeed they are. Nonetheless, they differ in a way that was recognized by the Marx brothers long ago: “If you have ten apples and want to divide them up among six people, what would you do?” “Make apple sauce.”

2I

am aware of the fact that there are those who question whether this is an appropriate use of the notion of adjective. To be sure these numbers are quantifiers, and as such resemble words such as all, some, none, etc.. However, they are certainly adjectival in character in that they are descriptive of their referents.


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Partitioning an apple results in pieces that are no longer called apples. In contrast, partitioning water results in smaller volumes of water. A sharper example of this distinction can be seen in the biblical story of Solomon, who when confronted by two women, each claiming to be the mother of a new-born infant, suggests that the infant be cut in half. The unacceptable nature of this solution to this dispute derives from the fact that the action would result in two entities, neither of which is an infant. This distinction between count nouns and mass nouns will play a prominent role in the analysis of the arithmetic of adjectival quantity. It is the distinction between discrete and continuous quantity. When we talk of discrete quantity, we use count nouns, and we may have many of them or few of them. On the other hand, when we talk of continuous quantity, we use mass nouns, and we may have much or little of the referent entity. In comparing discrete quantities, we talk about more and fewer while in comparing continuous quantity we use more and less.3 These distinctions are not limited to the English language, but can be found in many other languages as well.4 In order to sharpen the distinction, note that it is possible to use the word one with count nouns but not with mass nouns. Thus, one may say, one apple or one person but not one clay or one water. Indeed, we may use any integer as an adjective with count nouns. In attaching a magnitude to a count noun, we have to make a judgment that takes all the attributes of the referent object into account and which results in a single yes-or-no decision as to whether the object we are considering does or does not belong to the class of things to which we are assigning number. Thus, before being able to say one apple, we must take into account the size, color, shape, weight, etc. of the object before us, and conclude, yea or nay, does this object, with all the values of all its attributes, satisfy our requirements for being an apple. Similarly, the problem of counting all the chairs in a house requires that we make decisions about objects that are as disparate as easy chairs, stools, chaise lounges and probably even step ladders. For each object considered, color, form, size and materials need to be taken into account and a single dichotomous decision arrived at, i.e. is the object in question a chair? It goes without saying that the only sort of number that can result from the act of counting is an integer. On the other hand, in the case of mass nouns, we can introduce adjectival quantity only after we have singled out the attribute of the referent object that we wish to quantify. We say, for example, three hundred grams of clay. In so doing we ignore the shape of the clay and its color and focus solely on its weight (more properly its mass, in this case). Because of these considerations, it is clear that the result of assigning size to some attribute of a 3It

is interesting to note that many of Piaget's conservation experiments ask subjects to indicate which entity has more of the quantity in question, e.g. number, volume, linear extent. The discrete-continuous distinction disappears in exactly this situation. Might this degeneracy be a contributory factor to the “lack of conservation” exhibited by young children? 4The linguistic distinction between discrete and continuous quantity extends to many computer languages that find it useful to introduce distinct data types such as reals and integers for quantities invoked in the programs. (Needless to say, the reals aren't real; they are rationals as long as one has computers with finite word length.)


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mass noun, an act we normally call measuring, necessarily results in a non-negative rational number.5 Formally, adjectival quantity may be thought of as having the following structure: {measure, attribute} For discrete quantities, labeled by count nouns, this structure takes on the form {cardinality of set, definition of set} as in the case of {4, apples} for example. For continuous quantities, labeled by mass nouns, the measure component of this structure has internal structure {(magnitude, unit), attribute} The process of assigning size to an adjectival quantity described by a count noun is an act we normally call counting, while the process of assigning size to some attribute of an entity described by a mass noun is the process we normally call measurement. Here are some examples: The girl is five feet tall. It took me an hour to drive home.

{(5, ft), girl's height} {(1, hr), driving time}

It is clear that this formulation of the act of measuring implies a number of sub-acts, each of which is dictated by the structure of continuous adjectival quantities. These sub-acts include

1.

the singling out of the attribute to be quantified Probably the most explicit encounter that youngsters have with this issue is the problem of sorting out perimeter and area. The confounding of these two attributes of shape is a serious obstacle to the learning of area measure.

5It

would seem that occasionally one encounters a count noun modified by a non-integer, as in the statement The average family in that area has 1.9 children. As we shall see later the semantic structure of the referent situation in such cases is very different from those situations in which a counting act leads to the standard {cardinality, definition} structure.


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The issue of identifying the attribute to be measured is not simply a problem for young people at school. For example, unless one is willing to settle for the tautological “IQ is what the IQ test measures”, this problem plagues a good part of the adult world as well. 2.

the choosing of a unit appropriate to that attribute It is clear that it is possible to measure the distance to the moon in millimeters, or the thickness of a piece of paper in light-years. Somehow, it is inappropriate to do so. The competent measurer ought to have some degree of control over the size of the unit measure. This question of the choice of appropriate unit size is intimately bound up with the question of the appropriate level of precision.

3.

determining the magnitude of the measure in the chosen unit In an analog world, a scale of some sort must be read. This means that a clear mental model of the number line must be understood. Specifically, matters of 'betweenness' must be understood, as well as the appearance of a zero on the scale.

Not apparent from the structure of continuous adjectival quantity, but nonetheless central to the measurement act is the need for a judgment to be made about the adequacy of the precision for the context at hand. Calendars and clocks both measure time but they are not interchangeable instruments. Similarly, the measurement of the heights of people will not be to millimeter precision nor their weights (strictly speaking, their masses) to milligram precision - indeed, one might reasonably argue that it is not sensible to talk about humans having heights that can be ascertained to millimeter precision or weights that can be ascertained to milligram precision6. Because of these considerations, it is clear that the result of a measuring act is necessarily a non-negative rational number7 and never a real number in the mathematical sense. It is not always easy to make a clear distinction between discrete and continuous quantity. Sand is certainly continuous if you need to buy a cubic yard of it. On the other hand, it is quite clearly discrete if you have grain of it stuck between two teeth. Are air and water continuous? Reasonably so on the scale of human beings, but not at all so on an atomic scale. In the end, of course, it would seem that the particulate nature of nature would dictate that all quantity should be discrete.

6Most

adults are about one centimeter taller in the morning when they first get out of bed because of the overnight expansion of the cartilage between the vertebrae. One's body weight can easily change by many milligrams after a sneeze or the cutting of fingernails. Drinking a glass of water may increase the mass of a person by several hundred thousand milligrams. 7Strictly speaking, the result of a measurement is a range of non-negative rational numbers. The size of the range is determined by the precision of the measuring instrument.


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This vacillation between the discrete and the continuous may even affect our decision to count rather than to measure or vice versa. For example, consider the problem of removing 10 six-penny nails from a large barrel of such nails. One simply removes a handful and counts out 10 nails. Suppose on the other hand, one wishes to remove 10,000 such nails from the barrel. Even quite young primary school children quickly reach the conclusion that counting is not a very promising way to address the problem. Redefining the task so that it becomes a measuring rather than a counting task is seen as a more reasonable way to proceed. Thus, one might choose to remove an amount of nails whose weight is about 100 times the weight of 100 nails or about 200 times the weight of 50 nails (Why not 10,000 times the weight of one nail or twice the weight of 5,000 nails?) It might seem that a reasonable criterion for judging discreteness versus continuity of quantity might be the size of the entity in question relative to the size of humans. This is often, but not always, a reasonable criterion. For example, the models that traffic engineers make of traffic flow often depend on the idea of a fluid of automobiles. Similarly, the analysis of population is often done with a model that treats discrete human beings as an aging fluid in a container with sources and sinks.

III.

EQUALITY & ORDERING - COMPARING ADJECTIVAL QUANTITY

We now turn our attention to the words less than or fewer than, not equal, or not the same number as, equal or as many as, more than and the mathematical symbols that correspond8 to them . How do we use these words in talking about adjectival quantity. Logically we must consider three sorts of comparisons. These are •

comparing a discrete quantity to a discrete quantity

•

comparing a continuous quantity to a continuous quantity

•

comparing a discrete quantity to a continuous quantity

These cases need to be investigated separately.

8Note

the differences between what the words indicate and what the symbols indicate. The words in some cases distinguish between discrete and continuous quantity. On the other hand, the symbols allow us to distinguish the construct less than or equal to from the construct less than and the construct greater than or equal to from the construct greater than. Express check-out lanes in supermarkets are frequently insensitive to these matters. It is not uncommon to encounter a sign saying Ten items or less !


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a. Comparing discrete quantities If we wish to compare two discrete quantities to one another we must consider several possibilities. These are •

the referents of the quantities are identical

•

the referents of the quantities are distinct (and disjoint)

•

the referents of the quantities are neither identical nor disjoint i.

identical discrete referents - apples and apples For example, Six apples are fewer than eight apples. {6, apples} < {8, apples}. The quantities we are comparing have the structures

{x1 , y0 }and {x2 , y0 } where x1 and x2 stand for the numbers (or cardinalities) of elements in the two sets (6 and 8 in this case) and y0 stands for the definition of the set (apples in this case). These sets have different cardinalities but the same set definition, and the equality and inequality symbols are used in the same way they are used for nominal quantity. ii.

distinct discrete referents - apples and oranges For examples, suppose we wish to compare Six apples and Eight oranges. The quantities we are comparing have the structure

{x1 , y1 } and {x2 , y2 } . where x1 and x2 stand for the numbers (or cardinalities) of elements in the two sets (6 and 8 in this case) and y1 and y2 stand for the definitions of the sets (apples and oranges in this case).


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People are willing to consider such questions as “Are there more apples than oranges in the fruit bowl?” They treat this question as a question about cardinalities only and interpret it as “Is the number of apples in the fruit bowl greater than the number of oranges in the fruit bowl?” They ordinarily answer this question by establishing a one-to-one correspondence. iii.

discrete referents that are neither identical nor distinct - apples and fruit For example, suppose we wish to compare Six apples and Eight pieces of fruit. The quantities we are comparing have the structure

{x1 , y0 } and {x 2 , Y0 },Y0 ⊃ y0 , where x1 and x2 stand for the numbers (or cardinalities) of elements in the two sets (6 and 8 in this case), y0 and Y0 stand for the definitions of the sets (apples and pieces of fruit in this case), and the set y0 is a subset of the set Y0 (apples are pieces of fruit). Consider the question “Are there more apples than fruits in the fruit bowl?” The confusion surrounding the answering of this question is traceable directly to the sub-acts involved in the act of counting. Specifically, one has to consider all the attributes of the entity to be counted and to decide whether it can be included in the set one is attaching magnitude to. Suppose, for example there are six apples and two oranges in the fruit bowl. The word fruit in the question may be interpreted by some as meaning non-apple fruit. In that case one can expect a quite different answer that if the respondent considers the word fruit to refer to both apples and oranges. The difficulty probably lies with the way we use language. In normal discourse the word or is used as if it were part of the implicit phrase either...or, that is to say, to separate possibilities that are disjoint. We are embedded in a linguistic surround that abounds in the use of such phrases as hot or cold, tall or short, sedan or station wagon etc.9

b. Comparing continuous quantities 9Piaget,

in an oft-criticized experiment would ask youngsters to look at a collection of flowers and consider the question, “Are there moretulips of flowers?” #####


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If we wish to compare two continuous quantities to one another we must consider several possibilities. These are • •

the comparison of identical attributes (e.g. weight) of the two quantities - within this case we must consider two cases, i.e. identical and distinct units of measurement the comparison of distinct attributes of the two quantities i.

identical attributes of referents with identical units For example, suppose we wish to compare six lbs. of peas and eight lbs. of paper. The quantities we are comparing have the structures

{( x1 , u 0 ), y0 } and {( x2 , u 0 ), y0 } where x1 and x2 stand for the magnitudes of the measures of the two quantities (6 and 8 in this case), u0 stands for the units of the measure of the two quantities (lbs. in this case) and y0 stands for the attribute that is being measured (weight in this case). Here we are dealing with distinct magnitudes but the same attribute measured in the same units, and the equality and inequality symbols are used in the same way they are used for nominal quantity, e.g. The 1.4 kg. of water in the brown pitcher is heavier than the .7 kg piece of meat. ii.

identical attributes of referents with distinct units Suppose we wish to compare two quantities such as A 2 meter long stick A 200 centimeter tall person These quantities have the structures

{(x1 , u1 ), y0 } and {(x2 , u 2 ), y0 } where x1 and x2 stand for the magnitudes of the measures of the two quantities (2 and 200 in this case), u1 and u2 stand for the units of the © 1996 Judah L. Schwartz

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measures of the two quantities (meters and centimeters in this case) and y0 stands for the attribute that is being measured (weight in this case). Here we are dealing with distinct magnitudes and distinct units of the same attribute. This case raises a set of issues that forces us to delay resolving this question until after we have introduced the notions of extensive and intensive quantity and have considered some of the mathematical operations that transform the referents of adjectival quantity. iii.

distinct attributes of referents Ordinarily we are not willing to compare the magnitude of adjectival quantity if the quantities themselves refer to distinct attributes. Thus we do not regard as sensible such statements as The maximum speed of the car is greater than its weight The girl is younger than her height. Occasionally, we permit ourselves a somewhat poetic violation of this apparent rule with the our use of such statements as He has more luck than brains.10

A remark about the comparison of continuous adjectival quantity is in order. We have a set of specialized words that we use for the ordering of measures of specific attributes. These include taller, shorter

height

heavier, lighter,

weight

longer, shorter

duration

wider, narrower

horizontal extent

It is quite likely that in specialized situations people develop sets of comparative adjectival forms that permit them to make ordering statements succinctly.11 10It

is sad to note that all too often our youngsters are asked to solve problems in middle school in which an area of a rectangle is said to be equal (sic!) to its perimeter. 11 It is logically possible to consider comparing discrete to continuous quantities - One can imagine trying to compare the weight of 20 apples to the weight of a gallon of milk. Note that such comparisons are only superficially about comparing discrete to continuous quantity in that it is the measure of continuous attribute of the count noun apples that is being compared to the corresponding attribute of the mass noun milk.


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COMPUTING WITH ADJECTIVAL QUANTITIES REFERENT-PRESERVING OPERATIONS

In this section we begin the consideration of how the arithmetic operations that we know how to carry out with nominal numbers may or may not be extended to adjectival quantities. The operations we refer to are addition, subtraction, multiplication and division. We will discover that there are systematic differences between the way a given operation works with nominal quantity and the way it works with adjectival quantity. For example, a problem that has been studied extensively in the mathematics education research community is the following: How many school buses are needed to take all the students in the school on a field trip if there are 500 students in the school and each bus carries 30 students?12 The expected answer is 17 buses even though a straight forward computation leads to the result 16 and 2/3 buses. Students are expected to recognize that buses are count nouns and that the quantity {2/3 bus} is not acceptable. How do arithmetic operations with adjectival quantity work? We observe that in the case of nominal numbers, the arithmetic operations provide ways for us to take two numbers and combine them so as to produce a new number. If we try to extend these operations to adjectival quantities we must face the question of what happens to the referents of the quantities being combined and how is the referent of the resulting quantity determined. As in the case of comparing adjectival quantities that we considered in the previous section we must take into account the nature of the quantities being combined, i.e. are they discrete, continuous or both. Broadly speaking, we consider two quite different kinds of operations. The first kind of operation, which I call referent-preserving, allows us to combine two quantities with the same (or equivalent) referents and produce a new quantity with the same referent. A simple example might be {3, apples} + {6, apples}. In this case the result {9, apples} has the same referent as each of the quantities that were composed in order to construct the result. Addition and subtraction are the most common examples of referent-preserving operations. The second kind of operation, which I call referent-transforming, allows us to combine two quantities, with the same or differing referents, to produce a new quantity whose 12Actually the

problem is posed in a less useful way. The numbers of students and the pupil-carrying capacity of the buses are not “simple” numbers and thus it is not easy to ascertain whether students who have difficulty with the problem do not understand the discrete/continuous quantity issue or simply have difficulty with the mechanics of division.


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referent differs from either one or the other or both of the referents of the original two quantities. A simple example might be {30, apples/bushel} x {10, bushels}. In this case the referent of the result, i.e., bushels, is not the same as either as the referent of either of the quantities that were composed in order to construct the result. Multiplication and division are the most common examples of referent-transforming operations.

a.

addition of discrete adjectival quantities

Addition would seem to be a referent-preserving operation. We are willing to say Two apples and three apples are five apples. and define an addition for adjectival quantity so that formally

{2, apples} + {3, apples} = {5, apples}. However, people are also willing to say Two apples and three oranges are five pieces of fruit. Clearly, the referents of the two and the three are not identical. In fact, they are disjoint, i.e. no apple is an orange and no orange is an apple. On the other hand, both apples and oranges are pieces of fruit. In this case, when the referents of the two quantities are disjoint and there exists a superordinate class to which both referents belong, we can extend the operation of addition with nominal number to adjectival quantity in a straightforward fashion. Formally we have

{n1 , A0 } + {n2 , B0 } = {n1 + n2 , M }, A0 I B0 = ∅, and A0 , B0 ⊂ M . 13 If the superordinate class is too remote from the referents being composed then it is not likely that we will find people “adding” them. For example, people do not say Two elephants and three computers are five things. One must be cautious about judgments in this area because statements of the following sort might clearly be made in the appropriate context. 13In

this case n1 stand for two, A0 stands for apples, n2 stands for three, B0 stands for oranges, M stands for pieces of fruit. A0 ∩ B0 = ∅ means that no object is both an apple and an orange. The expression A0, B0 ⊂ M means that both apples and oranges are pieces of fruit.


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Two toasters and three overcoats are five inventory items. Finally, people do not often say Two apples and three pieces of fruit are five pieces of fruit. This statement is clearly ambiguous. The referents of the two and the three may or may not be disjoint, i.e. some, all or none of the apples may be among the pieces of fruit and we have no way of telling from the utterance alone. It seems to be the case that the operation of addition that we know for nominal numbers may be extended in a reasonably straightforward way to the addition of discrete adjectival quantities, provided we are careful about disjointness and superordinate classes. Does it all go as easily for continuous adjectival quantity? b.

addition of continuous adjectival quantities

If we combine two volumes of water, 100 cm3 and 50 cm3, we obtain a new quantity of water whose volume is the sum of the two original volumes computed as if the quantities were nominal numbers. In this case we have identical referents, and we are considering common attributes measured in the same units. Formally

{(x1 , u 0 ), y0 } + {(x2 , u 0 ), y0 } = {(x1 + x2 , u 0 ), y0 } where x1 and x2 stand for the magnitudes of the measures of the two quantities (100 and 50 in this case), u0 stands for the units of the measures of the two quantities (cm3 in this case) and y0 stands for the attribute that is being measured (volume in this case). The first problem we encounter when we seek to move beyond this case is one that we encountered before when we analyzed the comparison of adjectival quantities. Suppose we combine two volumes of water, 0.1 liters and 50 cm3. If we physically do so, we obtain a new quantity of water, 0.15 liters or 150 cm3, whose volume is the combined volume of the two original volumes. But how do we compute the magnitude of the combined volume? As before, we must beg off at this point and delay resolving this question until after we have introduced the notions of extensive and intensive quantity and have considered some of the mathematical operations that transform the referents of adjectival quantity. The second problem that we encounter when we try to extend our notion of addition to continuous adjectival quantity arises when we seek to combine a common attribute of distinct referents, for example, 100 cm3 of water and 50 cm3 of alcohol. The volume of the resulting fluid is in fact smaller than 150 cm3! If this example seems esoteric, consider combining 100 cm3 of water and 50 cm3 of air.


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It is not the case that the common attribute of distinct referents can never be added algorithmically in the same way that nominal numbers can. One can add one's shoulder height and the length of one's arm and the height above the ground of the ladder rung that one is standing on in order to determine just how high up the wall one can reach with a paintbrush. Is there a clear, procedural way of deciding when and under what circumstances one can extend the operation of addition with nominal numbers to the combining of common attributes of distinct referents for continuous quantity? It would seem not. The detailed nature of the referent situation must be taken into consideration and a judgment made in each instance. I suspect the ability to exercise reasonable judgments about such matters is part of what we regard as common sense. c.

subtraction of discrete adjectival quantities

The situations that people try to describe and analyze using the operation of subtraction are for the most part situations in which there is a prior state, and then something happens to change it resulting in a final state. For example, Jimmy has five apples. He then gives two apples to Joey. Jimmy now has three apples. When cast in the form of a school mathematics problem, e.g. Jimmy has five apples. He then gives two apples to Joey. How many apples does Jimmy now have? such situations are referred to as cause-change problems.14 Consider the statement Removing two apples from five apples leaves three apples. This suggest that we can define a subtraction operation for discrete adjectival quantity if the quantities in question have common referents. Formally.

{5, apples} − {2, apples} = {3, apples}. On the other hand, how shall we treat the statement Removing two apples from five pieces of fruit leaves....? People are willing to end the sentence with the phrase three pieces of fruit. This suggests an interesting distinction between the operations of addition and subtraction applied to nominal numbers and the same operations applied to adjectival quantity. In the case of nominal numbers, subtraction and addition are inverse operations. If subtraction with 14Pearla

Nesher once pointed out to me that cause-change problems can be the source of some consternation. Consider the following two examples; Five people come into a room and three people leave the room. How many people are in the room? and Three people leave a room and five people come into the room. How many people are in the room? The second formulation makes clear the necessity for clarifying what is the original situation in a cause-change problem.


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adjectival quantity were the inverse of addition with adjectival quantity, then people would be willing to say that Two apples and three oranges are five pieces of fruit. and Removing two apples from five pieces of fruit leaves three oranges. are equivalent statements. Formally it would seem that the allowable inverse operations to Two apples and three oranges are five pieces of fruit.

{n , A }+ {n , B } = {n 1

0

2

0

1

+ n2 , M }, A0 I B0 = 0, A0 , B0 ⊂ M

are Removing two apples from five pieces of fruit leaves three pieces of fruit.

{n

1

+ n2 , M }-{n1 , A0 } = {n2 , M

}

and Removing three apples from five pieces of fruit leaves two pieces of fruit.

{n

1

+ n2 , M }-{n 2 , B0 } = {n1 , M }.

For subtraction of discrete adjectival quantity with disjoint referents to make any sense, one referent must be superordinate to the other referent, e.g. apples are pieces of fruit as are oranges. If the referents are distinct and neither is superordinate to the other, there does not seem to be any reasonable definition of subtraction. For example, Removing three apples from five oranges leaves...?

d.

subtraction of continuous adjectival quantity

If we remove one volume of water, say 50 cm3, from a larger volume of water, say 150 cm3, we obtain a new quantity of water whose volume is the difference of the two original volumes computed as if the quantities were nominal numbers. In this case we have


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identical referents, and we are considering common attributes measured in the same units. Formally

{( x1 , u 0 ), y0 } - {( x2 , u 0 ), y0 } = {( x1 − x2 , u 0 ), y0 } As in the case of addition with continuous adjectival quantity, we encounter several problems when we seek to move beyond this straightforward case. The first is one that we encountered before when we analyzed the comparison of adjectival quantities. Suppose we remove a volume of water, 50 cm3 say, from a larger volume of water, 0.15 liters say. If we physically do so, we obtain a new quantity of water, 0.10 liters or 100 cm3, whose volume is the difference in the volumes of the two original volumes. But how do we compute the magnitude of the remaining volume? As before, we must beg off at this point and delay resolving this question until after we have introduced the notions of extensive and intensive quantity and have considered some of the mathematical operations that transform the referents of adjectival quantity. The second problem that we encounter when we try to extend our notion of subtraction to continuous adjectival quantity arises when we seek to find the difference of two quantities that have a common attribute of distinct referents, for example, 150 cm3 of fluid and 50 cm3 of alcohol. As was the case with the subtraction of quantities, if one of the referents is superordinate to the other, it is possible to make sense of the operation. On the other hand, the fact that it is possible to make sense of the operation does not guarantee that the operation of subtraction on nominal numbers can be generalized to continuous adjectival quantity. Consider, for example, the problem of removing 15 cm3 of salt from 150 cm3 of salt water. In the case of 150 cm3 of fluid and 50 cm3 of alcohol, removing 50 cm3 of alcohol from a mixture of alcohol and water will leave one with more than 100 cm3 of fluid. It is not the case that the common attribute of distinct referents can never be subtracted algorithmically in the same way that nominal numbers can. One can subtract the weight of a person's clothes from the weight of the clothed person to obtain the weight of the unclothed person. Is there a clear, procedural way of deciding when and under what circumstances one can extend the operation of subtraction with nominal numbers to the problem of finding differences between continuous adjectival quantities with common attributes of distinct referents? It would seem not. The detailed nature of the referent situation must be taken into consideration and a judgment made in each instance. V.

COMPUTING WITH ADJECTIVAL QUANTITY REFERENT-TRANSFORMING OPERATIONS

a.

intensive quantity


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Before we examine referent-transforming operations with adjectival quantity, we need to make a kind of digression. The most common referent-transforming operations are multiplication and division. By far the majority of the situations in which we are called upon to do either multiplication or division will involve a different sort of “quantity” that is neither directly counted nor measured, i.e. intensive quantity. To understand what kind of quantity this is and why it appears in almost all multiplication and division problems, let us consider the following situation with, say, coffee beans (assumed for our purpose to be a continuous quantity). Suppose we have a pile of coffee beans characterized by the three adjectival quantities; {(5.0, lb), weight of coffee} {(40.00, $), cost of coffee} {(8.00, $/lb), price/lb of coffee} If we have two such piles of coffee beans and coalesce them, it is clear that the appropriate mode of composing the quantities describing the weight and the cost of the coffee differs from the appropriate mode of composing the quantity that describes the price/lb. The weight of the coalesced pile of coffee beans is {(10.0, lb), weight of coffee} and the cost of the coalesced pile of coffee beans is {(80.00, $), cost of coffee} while the unit price of the coalesced pile of coffee beans is {(8.00, $/lb), price/lb of coffee}. The fact that the mode of composition of the price/lb adjectival quantity is different from the others is a clue. The price/lb quantity is a different sort of descriptor of the coffee. Whereas the first two quantities describe the entire pile of coffee beans, the price/lb quantity describes not only the entire pile of coffee beans but it can equally well describe a single coffee bean or a freight car full of them. It is a quantitative descriptor of a quality of the coffee and does not describe any aspect of the coffee that depends on the amount of coffee. It is called an intensive quantity. For the most part mathematical intensive quantities can be recognized by the fact that their unit measures almost always contain the word per.15

15

It should be pointed out that an intensive quantity need not have the word per appear explicitly in its unit measure. Often it is implicit as in the case of the knot or nautical mile/hour for example. Another interesting case of an intensive measure in which the word per does not appear explicitly in the unit measure is temperature. A degree on the temperature scale is a measure of the average kinetic energy per


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In general, one can think of an intensive quantity as a generalization of the notion of density. Think of a solid block of pure copper. Copper at room temperature has a density of 8.92 grams per cm3. If the block has a volume of 1000 cm3 it will have a mass of 8920 grams. A smaller block, with a volume of 10 cm3 will have a mass of about 89.2 grams. A still smaller block with a volume of .01 cm3 will have a mass of 0.0892 grams. In each case the ratio of the mass of the copper block to its volume give the same intensive quantity, i.e., 8.92 grams per cm3. This notion of density is a mass density in that it is a measure of the mass per unit volume. In the case of the copper block we considered a pure material whose mass is uniform throughout its volume. The more general notion of density does not necessarily refer to mass nor does it really depend on uniformity. For example, suppose that following a collision, all of the bags of different varieties of instant coffees in a freight car broke open and mixed to some degree. There would be a huge pile of unevenly mixed coffee powder on the floor of the freight car. The price per pound of the coffee in the pile is a costdensity that would vary from place to place in the freight car. If you chose a cupful of coffee from one part of the freight car it would have a value that would not likely be the same as the value of a cupful of coffee chosen from a different part of the freight car. Despite the fact that the pile as a whole has a price per pound that varies from place to place in the freight car, the total price of the coffee in the freight car can be calculated. Another common example of an attribute density that varies from place to place is the population-density of a country. This density is the ratio of a number of people to an area rather than a volume. Taken as a whole, Canada has a population density of 7 persons per square mile. Clearly this does not mean that in every square mile of Canadian territory one will find 7 people. The province of Ontario has a population density of about 22 persons per square mile and the city of Toronto has a population density of about 20,000 persons per square mile. However, the population density of Toronto cannot be said to imply that in every .001 square miles one will find 20 people living. The speed of a vehicle during the course of a trip can be thought of as a kind of generalized density. In this case we consider the ratio of a distance to a time. If I drive a total of 120 miles in 3 hours my average speed for the entire trip is 40 miles per hour. Suppose that at the beginning of the trip I spent one minute accelerating from rest to a speed of 60 miles per hour. During that minute my average speed was 30 miles per hour. During a later part of the trip I might have spent 20 minutes driving a distance of 25 miles. During that period my average speed was 75 miles per hour or 1.25 miles per minute. If during that twenty minute period I drove at a constant speed then I would have driven 5 miles in 4 minutes, .25 miles in .2 minutes, .125 miles (660 feet) in .1 minutes (6 seconds), 110 feet (about 7 car lengths!) in 1 second. The ratio of the distance traveled to the time it took to travel that distance remaining constant is entirely analogous to our

particle in a material. The non-obvious intensivity may contribute to the substantial difficulty that many adults as well as students have in distinguishing between heat and temperature.


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considering smaller and smaller blocks of copper in which the ratio of the mass of the copper to the volume of that mass remained the same.16 It should be pointed out that the assertion of an intensive quantity such as {(34.5, mi/hr), speed}, {(13.6, gm/cm3), density of mercury} or {(5, candies/bag), party favor} gives no information whatsoever about the number or amount of the relevant extensive quantities, in these cases miles, hours, grams, cubic-centimeters, candies or bags. The statement of an intensive quantity is a statement of a relationship between quantities. In general, the related quantities are extensive quantities, but this need not be the case. Consider, for example, acceleration which has the units [(distance / time) / time]. If the referent is homogeneous, then the relationship holds for all parts of the referent entity. If not, then the intensive quantity is a local property of the referent entity. In our example of the instant coffee in the freight car, the mass/volume density varied from place to place in the freight car. In our example of the car trip the distance/time density was different at different times. Although it may seem odd to refer to a relationship as a quantity, an important reason for doing so is that these relationships can be quantified and ordering and arithmetic operations can be carried out with these quantities. The fact that the core of the idea of an intensive quantity is a relationship makes the representing of it difficult. Let us consider how we might represent different sorts of quantities graphically. An extensive quantity such as {(5, meters), distance} can be represented as a suitably placed point on a suitable (in this case, distance) number line, as can the extensive quantity {(2, seconds), time}. The intensive quantity {(5/2, meters/seconds), speed} cannot be represented on either of these number lines17 but rather must be represented as an infinite number of points that lie on a straight line in the meters-seconds plane. (Figure 1.)

16

It should not be assumed from these examples that an intensive quantity is necessarily a scalar quantity. There can be vector, second rank tensor, etc. intensive quantities. 17To be sure this quantity can be represented as a point on a single meter/second number line. Doing so, however, hides its relationship to both distance and time as quantities.


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meters

1 1

seconds

In the case of discrete adjectival quantities, such as {5, fish} and {2, bicycles} and the related intensive quantity {5/2, fish/bicycle} we have a similar situation. We might indicate {5, fish} as a suitable point on a “fish number line” and {2, bicycles} as a suitable point on a “bicycle number line”. The intensive quantity {5/2, fish/bicycle} cannot be represented on either of these number lines but rather must be represented as an infinite number of points that lie on a straight line in the fish-bicycle plane. Because of the discrete nature of both fish and bicycles, all of the points must have integer coordinates in the fish-bicycle plane. fish

10

10

bicycles

It is probably desirable to devise some sort of pictorial representation of this kind of quantity in addition to the formal graphical representation. We know how to display pictorial representations of {5, fish} and {2, bicycles} by drawing pictures of five fish and two bicycles. But how can we make a pictorial representation of {5/2, fish/bicycle} that has the property that no specific number of fish or specific number of bicycles is being referred to but only the relationship between these two quantities? One possibility is to call upon the entire fish-bicycle plane (Kaput etal, 1986). Suppose we tessellate the plane with icons of small fish and bicycles18 and define a suitable set of sample sizes with which to sample the icons in the plane. If the tessellation is made properly, then it will turn out that a rectangular sample of size 1 x 7 or 7 x 1 will enclose 5 fish and 2 bicycles no matter where in the plane it is placed.

18In

the figure we use the words FISH and BIKE rather than icons.


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FISH

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[Imagine this plane continuing both vertically and horizontally.] Similarly, no matter where it is placed in the plane a rectangular sample of 2 x 7 or 7 x 2 will enclose 10 fish and 4 bicycles, and a rectangular array of 10 x 14 will enclose 100 fish and 40 bicycles, etc. Since an intensive quantity is formed (in general) from two extensive quantities, each of which might be either discrete (D) or continuous (C), we have the following possibilities; •

Intensive quantities of the form D/D are relationships between two sets of discrete extensive quantities, e.g. children/family or candies/bag

•

Intensive quantities of the form C/D and D/C are relationships between a set of discrete and a set of continuous extensive quantities, e.g. gallons/bucket or persons/year

•

Intensive quantities of the form C/C are relationships between two sets of continuous extensive quantities, e.g. miles/hour or grams/cubic centimeter

One final remark about intensive quantity. Suppose we think about a penny that is dropped down a well. The speed of the penny when you first release it from your hand is {(0, meters/second), speed}. The penny falls faster and faster until it hits the bottom. At different times after its release from your hand the penny has different speeds. Acceleration is a measure of how fast the speed of the penny is changing. In our terms we might think of it as a speed-time density. This seems to mean that speed is an extensive quantity even though we treated it earlier as an intensive quantity. Speed is indeed an intensive quantity if we are interested in its relation to the quantities from which it is composed. If we are not, then there is no reason not to treat it as an extensive quantity. It may seem that the notion of intensive quantity, so suggestive of the concept of derivative in the calculus, is an esoteric one and cannot seriously be expected to play an important role in the mathematics of the early grades. It will turn out, however, that intensive quantity is essential to the understanding of the vast majority of situations that call for the arithmetic acts of multiplication and division.


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what's wrong with repeated addition and subtraction

We now turn to the major business of this section, i.e., referent-transforming operations with adjectival quantity. By far the most common such operations are multiplication and division. Normally, multiplication is introduced to youngsters as an efficient way of doing repeated addition, and division as an efficient way of solving the problem of equitably sharing some set of desirable objects among children. There are serious flaws with each of these approaches. Some of these flaws are procedural in nature and give rise to students having mechanical difficulties and others are conceptual in nature and stand in the way of a proper understanding of multiplication and division. We turn first to the procedural flaws. Suppose we seek to count the number of dots in an array that contains 3 rows of dots with each row containing 4 dots. Clearly the array looks like:

A multiplicative procedure that does not yield {12, dots} is {4, dots} x {3, dots} = {12, dots2}

and not

{12, dots}

It is possible to construct a product of nominal and adjectival quantities such as 3 x {4, dots} or 4 x {3, dots} but this suffers from two difficulties. The first is that we have not defined operations between nominal and adjectival quantities, and second, how are we supposed to know to what the nominal quantity refers? So far we have not properly mathematized the quantities referred to in the original verbal statement of the problem. They are in fact {3, rows} and {4, dots/row}. Note that the {3, rows} x {4, dots/row} = {12, dots} does lead to the correct number of dots. Note that this multiplication involves the product of one factor that is intensive in nature and one that is extensive. We will return to this observation in the next section.


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The reader will note that the equivalent formulation of a product {3, dots/column} x {4, columns) = {12, dots} yields two factors with entirely different referents from those in the first product. Thus we see that the factoring of the magnitude of an adjectival quantity does not in any way constrain the “factoring” of its referent. Here, for example, is the same array “factored” in yet a different way

namely {3, dots/L-block} x {4, L-blocks} = {12, dots}. Let us continue our consideration of repeated addition. Here are two different multiplication problems; {5.0, candies/bag, kind of party favor} x {6, bags} and {14.5, mi/hr, average speed on trip} x {3.2, hr, time trip takes}. The repeated addition model, which works easily in the first instance, suggests that we build sequentially the following table of associated extensive quantities; {5, candies} {10, candies} {15, candies} {20, candies} {25, candies} {30, candies}

corresponds to corresponds to corresponds to corresponds to corresponds to corresponds to

{1, bag} {2, bags} {3, bags} {4, bags} {5, bags} {6, bags}19

The method clearly depends on our ability to translate the relationship expressed by the intensive quantity {5.0, candies/bag, kind of party favor}

19Note

that the product of a quantity with the referent candies/bag and a quantity with the referent bags gives rise to a quantity whose referent is neither candies/bag nor bags.


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into an association between two discrete extensive quantities, i.e. candies and bags which can then be iterated the requisite integer number of times, in this instance 6. It also depends quite explicitly on the presumption of homogeneity, i.e. that the intensive quantity {5.0, candies/bag, kind of party favor} characterizes each of the bags in the ensemble. If we attempt to apply this same sort of analysis to the second of the problems cited above. {14.5, mi/hr, average speed on trip} x {3.2, hr, time trip takes}. we immediately discover that the strategy of iterating the association of miles and hours an integer number of times simply can't be carried out in this instance. In a similar fashion we can consider two division problems that will serve to underline the difficulties with the way division is normally presented to youngsters in schools. Suppose one has a total of {30, candies} distributed uniformly at the rate of {5, candies/bag} among {6, bags}. The quantity {30, candies} / {6, bags} can be assumed to characterize each of the bags and be a local property of the referent situation. The equivalent quantity {5, candies/bag}20 can be computed by repeatedly subtracting 6 from 30 and recognizing that this can be done 5 times. What is happening when one does this sharing of candies among bags is an acting out of {1, candy/bag}+{1, candy/bag}+{1, candy/bag}+{1, candy/bag}+{1, candy/bag} This division by repeated subtraction is the mirror image of multiplication by repeated addition. Suppose, on the other hand, one wishes to divide the quantity {33, candies}

20We

note that the result of executing the division of one discrete quantity by another need not be a new discrete quantity. In fact, it will almost always be the case that such a division will result in a quotient that is 'continuous' (strictly speaking rational rather than real) and that is not restricted to integer values.


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by the quantity {6, bags} It is clear that the mental image of sharing or partitioning one quantity evenly does not work easily here. When first introduced to division students are taught that 33/6 is 5 R 3 (read as 5 remainder 3) When dealing with adjectival quantity, how do we answer the question of the referents of the symbols 5 and 3? They are clearly different from one another. Specifically, they are {5, candies/bag} and {3, candies}21 There is yet another procedural way in which the repeated addition model of multiplication, or its mirror image, the sharing model of division, fails. These models, which, as we have seen, can sometime be used with discrete adjectival quantity, lead students to expect that multiplication always results in a new quantity which is larger in magnitude than either of the factors in the product and that division always leads to a quotient which is smaller in magnitude than the dividend. Indeed, the common parlance usage of the English words multiply and divide reinforce this expectation. In addition to these procedural flaws of “repeated addition” and “sharing” models of multiplication and division, there is, in my view, a deep conceptual flaw that inheres in these formulations. They lead the student to believe that the resulting computed quantity is of the same sort and has the same referent as one of the quantities that entered into the computation. As we have seen in the examples above, this is in general not true. Multiplication and division are referent-transforming compositions of adjectival quantity and addition and subtraction are referent-preserving. A pedagogic strategy for teaching referent-transforming compositions that depends on extending referent-preserving compositions misses the essential feature of that which is being introduced, i.e. that referent-transforming composition gives rise to a quantity of a new kind!

c.

multiplication and division of adjectival quantities

The (I E E') Semantic Triad Fortunately, we are not obliged to rely on “repeated addition” and “sharing” to provide our students with mental models of multiplication and division. It is possible to approach the problem of providing students with a mental model of the multiplication and division 21Later,

after having been introduced to the decimal system students are taught that the result of this division is {5.5, candies/bag}. If the candies are discrete, then no single bag can have this number of candies and the quantities can only be statistically descriptive of the situation. To make the point more strongly, think of children/family rather than candies/bag.


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of adjectival quantity that is not limiting in the ways we discussed above. In the interests of specificity let us consider the previous problem involving the three related quantities E' {(46.4, mi), distance traveled} E {(3.2, hr), time trip takes} I {(14.5, mi/hr), average speed on trip} These three quantities can be related to one another via multiplication operations or via division operations. However, because of the commutativity of multiplication, there is only one distinct multiplication operation relating the three quantities, i.e. I x E = E' while there are two division operations relating the three quantities, i.e. E'/ E = I and E'/ I = E The intensive quantity can be thought of explicitly as a relationship between distance and time for an infinite ensemble of trips, each of which is traversed at an average speed of 14.5 mi/hr. The correspondence between distance and time for this ensemble of trips can be illustrated in a graph such as the one we showed earlier. In the distance-time plane, a line of slope 14.5 mi/hr is drawn through the origin. Every point on the line describes a trip in the ensemble of trips described by the intensive quantity 14.5 mi/hr. The triad of quantities are linked to one another semantically and students would be well served if they were taught to think of a referent situation involving distance, time and speed. The relationship among these quantities can be expressed in three different ways.22 There is another graphical encoding of the relationship among these quantities that can easily be constructed. In the velocity-time plane a horizontal line whose ordinate is 14.5 mi/hr is drawn. Every point on this line corresponds to a trip of a given duration. Each such point also defines an area bounded by the average velocity axis, the average velocity, the time axis and the value of the time of the trip in question. The measure of this area is the distance traveled in the course of that trip. Here then are the two graphical representations of this (I E E') triad presented side by side.

22I

became acutely aware of the price of not doing this when a student one day described to me the three Ohm's Laws he learned in physics. He quoted them as voltage = current x resistance, current = voltage / resistance and resistance = voltage / current.


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distance (mi)

speed (mi/hr)

slope 14.5 mi/hr 14.5 46.4

area 46.4 mi

3.2

3.2

time (hr)

time (hr)

It is interesting to note that these two graphical representations of the relationship among the three quantities of interest capture the essential idea of the differential calculus that the rate of change of a function is given by its slope, and the essential idea of the integral calculus that the cumulative area under the graph of a function is a measure of cumulative effect of the changes in the function. These ideas can be introduced quite early in the curriculum to help think about and represent the semantically related triad of adjectival quantities involved in multiplication and division. When in later years, students in learning calculus are asked to think about slopes and areas under curves, they will be extending the power of some familiar ideas. There do not seem to be important differences between the behavior of discrete and continuous adjectival quantity under the referent-transforming operations of multiplication and division. All of the following cases are to be found D/D x D

e.g.

children / family

x

families

D/C x C

e.g.

people / hour

x

hours

C/D x D

e.g.

pounds / person

x

number of people

C/C x C

e.g.

miles / gallon

x

gallons

Before leaving our discussion of the (I E E') triad, it is worth addressing an issue that seems to cause a great deal of difficulty for many people, i.e. the division operation E' / E = I when both extensive quantities are discrete adjectival quantities. An example is {100, children} / {40, families} = {(2.5, children/family), average number of children per household} Note that although both children and families are count nouns and that


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{100, children} and {40, families} are discrete adjectival quantities the quantity {(2.5, children/family), average number of children per household} is a continuous adjectival quantity. Many people find this quantity moderately amusing, seemingly because they think it refers in some way to a portion of a child.23

d.

the conversion of units

The (I E E') semantic triad allows us to return to an issue that we left unresolved earlier, i.e., the conversion of units. Consider the quantity {(13.5, ft), length of bookshelf}. Another way to describe the same attribute of the same bookshelf is to use a different measure. Suppose we wish to measure the length of the bookshelf in yards rather than feet. The quantity {(3.0, ft/yd), length conversion factor} has the property that when used in a multiplication or a division, it does not change the nature of the referent but only the numerical description of its measure. Unit measure conversion factors are thus special sorts of intensive quantities that can be used freely in order to achieve consistency of measures.24 One can now return to the question of how to compare or find the sum and/or difference of the two volumes {(2, ft3), volume} and {(144, in3), volume} Using suitable unit measure conversion factors allows us to say that {(2, ft3), volume} and {(3456, in3), volume} are equivalent volumes as are

23Even

more disturbing to people is the reciprocal of this quantity {(.4, families/child), ...}! should not conclude that any quantity whose unit is the ratio of common attributes such as length/length or time/time is a unit conversion factor. For example one can characterize a clock that runs slowly as losing time at the rate of {(1.5, seconds/hour), clock's rate of losing time} or {(1/2400, seconds/second), clock's rate of losing time}. 24One


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{(144, in3), volume} and {(0.0833, ft3), volume}. While it is straightforward enough to make comparisons of magnitude of the two volumes expressed in common units, the question of what sense is there in forming their sum and/or difference remains exactly as it was before - the arithmetic operations of addition and subtraction may not be reasonable models of the referent situation.

e.

do we need scalars?

There is another intensive quantity that merits special attention. This kind of intensive quantity is like the unit measure conversion factor in that it does not change the nature of the referent of the quantity on which it operates. However, it does change the magnitude of its measure. It is normally called a scalar and is assumed to have no referent (Vergnaud 1983). When one says that the weight of a second grader is {52.5, lb, body weight} and that the weight of an eighth grader is twice as much, is not the weight of the eighth grader obtained from that of the second grader by multiplying by the pure nominal number 2? It is possible to adopt the position that the scalar quantity is better represented by {2., lb/lb, ratio [8th grader/2nd grader] body weight} It may seem awkward to “complicate” the problem of doubling the weight by forcing the 2 to carry the unit lb/lb. Another example will make the need for doing so clear. Suppose one wants to enlarge a photograph and one asks the photographer, “Please make me a copy of this picture that is twice as big.” The request seems to be ambiguous. Is it intended that the enlarged picture have linear dimensions that are twice as big as the original, or is it intended that the area of the enlarged picture be twice as big as the original? This issue is immediately resolved if the word “twice” is understood to mean either {(2, cm/cm), ratio of lengths} or {2, cm2/cm2), ratio of areas} It should be pointed out that this sort of intensive quantity, the scale conversion factor, has the property that its numerical magnitude is independent of the particular (dimensionless) intensive units in which it is measured.


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Finally, let us observe that quantities of this form will allow us later to extend our generalization of the arithmetic of adjectival quantity to algebra and calculus in such a way as to make those subjects broadly accessible tools for modeling the world around us.

f.

more about multiplication and division

The (I I' I") and (E E' E") Semantic Triads Although the (I E E') triad accounts for the vast majority of the multiplication and division operations we ask students to undertake, there are other kinds of triads of quantities related by multiplication and division operations. These other structures are I x I' = I" and E x E' = E". as well as their associated division structures. An example of a multiplication of the first sort, I x I' = I", is {(33.3, mi/gal), fuel efficiency of car on trip} x {(1.5, gal/hr), car's rate of burning fuel on trip} = {(49.95, mi/hr), average speed on trip}25. The second sort of multiplication, E x E' = E", is an example of the formation of a Cartesian product. Here are some specific examples; Discrete x Discrete {3, blouses} x {5, skirts} = {15, outfits} Discrete x Continuous {400, passengers} x {(1000, miles), distance} = {400000, passenger-miles), volume of traffic} 25Strictly speaking,

the product should be {(50.0, mi/hr), average speed} because the measured quantities in the first factor of the product is known to be larger than {(33.25, mi/gal), fuel efficiency}and smaller than {(33.35, mi/gal), fuel efficiency} and in the second factor, larger than {(1.45, gal/hr), rate of burning fuel} and smaller than {(1.55, gal/hr), rate of burning fuel}.


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Continuous x Continuous {3.20, cm, width of rectangle} x {6.40, cm, length of rectangle} = {20.48, cm2, area of rectangle} Both the (I I' I") and the (E E' E") semantic triads share with the (I E E') triad the property that the multiplication operation maps a quantity in one space onto a quantity in another space. On the other hand, they differ from the (I E E') triad and from one another in significant ways. The I" intensive quantity, {(49.95, mi/hr), average speed on trip} in the case above, is itself a relationship, i.e. one that describes any of, or indeed all of, an infinite ensemble of trips, each of which involves a time and a distance that are related through the intensive quantity. On the other hand, the E" extensive quantities, {15, outfits} {(400000, passenger-miles), volume of traffic} {20.48, cm2, area of rectangle} are not relationships between two extensive quantities. Each of these quantities is an extensive quantity in its own right. In contrast, however, to the extensive quantity E' which is produced in the (I E E') triad, the referent E" in the (E E' E") triad is an entirely new kind of referent. This new extensive quantity represents an expansion of the semantic space of the problem and it remains to be defined along with its measure. In our examples, we note the fact that a blouse-skirt is precisely what we mean by an outfit, that passenger-miles is a possible measure of volume of traffic and that the product of two lengths is an area measured in cm2 in this case.

VI.

REVISITING SOME EARLIER NOTIONS

a.

referent-preserving operations

Addition and subtraction of adjectival quantity, as generalized from addition and subtraction of nominal quantity, are paradigmatic referent-preserving operations. However, they are not the only possible referent-preserving operations. Vector addition, for example, of lengths, velocities, forces, etc. also preserves the referent of the original quantities being added. Stated this way, it may seem a bit arcane and not germane to most students. Consider, however, that the hypotenuse of a right triangle with legs of {(3, meters), length of side} and {(4, meters), length of side} is, as Pythagoras tells us,


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{(3, meters ), length of side}2 + {(4, meters ), length of side}2 = {(5, meters), length of hypotenuse} and is a special case of vector addition. Pythagorean addition is not the only common case of a referent-preserving operation that is not simple addition. Normally we think of the result of an addition of two adjectival quantities as yielding a quantity which is larger that either of the two original quantities. Nonetheless, if we compute the average value of a given attribute of a collection of things, or of a continuously distributed thing, we obtain a quantity with the same referent whose magnitude is certainly smaller than the largest of its “ingredients” and larger than the smallest of them. Thus, consider for example, the average of the three quantities {(4, meters), length} {(5, meters), length} {(8, meters), length} The usual recipe for computing averages would tell us to add the three quantities and divide by three. Three what? Here is a different way to think about the average. We wish to construct a new quantity which is a length. We want each of the “ingredient” quantities to determine 1/3 of the resulting quantity. What we mean by this is that 1/3 of each meter of the average should be determined by each constituent. We could form the resulting quantity in the following fashion: {(1/3, meter/meter), ratio of lengths} x {(4, meters), length} + {(1/3, meter/meter), ratio of lengths} x {(5, meters), length} + {(1/3, meter/meter), ratio of lengths} x {(8, meters), length} = {(17/3, meters), average length} It is now reasonably clear why an average formed in this manner has the property that its value is always larger than the smallest of its constituents and smaller than the largest of its constituents. There is yet another sort of addition that one encounters from time to time. Symbolically it can be written as


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1 . 1 1 + x1 x 2 This form of combining quantities is familiar to engineers and physicists who are interested in the value of the resistance of two resistors in parallel or two capacitors in series. It has the interesting property that this quantity is smaller than both of the quantities x1 and x2, that are used in its evaluation. If this sort of addition seems remote and specialized, consider the following familiar timehonored algebra problem. Two people A and B set out to mow a lawn. If A, working alone, can mow the lawn in {(2, hr), time} and B, working alone, can mow the lawn in {(4, hr), time} then working together they can mow the lawn in

{1, lawn } {(1 / 2, lawn/hr), A' s mowing speed} + {(1 / 4, lawn/hr), B' s mowing speed} = {(4/3, hr), joint mowing time}. This is precisely the sort of combining of mowing times that is needed to obtain a joint mowing time that is smaller than either A's time or B's time when working alone. Each of these instances of referent-preserving operations has the property that the computation of the final composite quantity requires more than one elementary computation, e.g., Pythagorean addition requires multiplication and addition and extracting a square root. In each of the instances described above intermediate resultant quantities are constructed that do not have the same referent as the original quantities being composed. It is only the final composed quantity that necessarily has the same referent as the original two quantities. From the point of view of the mathematics of adjectival quantity, what is important is that each of these operations, whether simple or composite, is referent-preserving.

b.

negative numbers and vectors

A special case of vector addition and subtraction is of particular importance in the elementary curriculum. Suppose instead of permitting vectors to have any direction in the plane (or in space) we consider only vectors that lie along a line. If, for the sake of specificity, we consider the line to be horizontal, then we may have vectors that point to the right and vectors that point to the left. How shall we add two right-pointing vectors? two left-pointing vectors? two vectors pointing in opposite directions?


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It is soon clear that the rule we know from the arithmetic of nominal quantity for the addition and subtraction of signed numbers is precisely what is called for here. But what does this have to do with the addition and subtraction of adjectival quantity? In order to answer this question, we must consider how signed adjectival quantity arises. When we count or measure entities in our surround, it often happens that we wish to assign not only size but also a sense. For example, in speaking about turning an automobile steering wheel through some angle, we might like to distinguish a clock-wise rotation of the wheel from a counter-clock-wise one. In describing a temperature change we might like to distinguish a ten degree rise in temperature from a ten degree fall in temperature. We talk about body weight changes, both up and down. All of these are examples of adjectival quantity that have in addition to a size a sense, e.g., clockwise, up, left, etc. If we are to capture the essence of these quantities in our mathematics, then we must encode not only their sizes but their senses, as well. These quantities are vectors on “quantity lines” (as opposed to “number lines”). Now that we seek to mathematize both the sense and the size of the quantity, it is convenient to use quantity lines that extend in both directions from zero. All of our previous discussion about referent-preserving and referent transforming operations can be readily extended to quantities that have both magnitude and sense.

VII.

SOME THOUGHTS ON SCHOOL ARITHMETIC

To a large extent the arithmetic curriculum of the elementary school as well as the algebra curriculum of the middle and high school focus on the manipulation of symbols representing mathematical objects rather than on using mathematical objects in the building and analyzing of arithmetic or algebraic models. Thus, in the primary levels, most of the mathematical time and attention of both teachers and students is devoted to the teaching and learning of the computational algorithms for the addition, subtraction, multiplication and division of integers and decimal and non-decimal fractions. Later, the teaching and learning of algebra, becomes, in large measure, the teaching and learning of the algebraic notational system and its formal, symbolic manipulation. All too often, the problem of using the mathematical objects and actions as the basis for modeling one's surround is a minor and neglected piece of the mathematics education enterprise. The arithmetic curriculum, for the most part begins with the place value system and the teaching and learning of what are called the “number facts”. In the early grades, the “facts” are the facts of addition and subtraction of nominal numbers. As we saw above, these facts are useful in modeling situations that call for the description of cause-change referent situations with the assumption that the referent sets in the model are disjoint but in subtler situations they may mislead.26 26

As a consequence, one finds the following problem causing a great deal of puzzlement.


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Even if one is content with an arithmetic curriculum that is concerned primarily with formal, manipulative skills rather than with modeling, there are still a number of vexing questions to consider in the design of an arithmetic curriculum. These include; How far should the “facts” extend? The usual answer is up to 10 + 10 for addition and up to 12 x 12 for multiplication.27 Why or why not? Would it not be valuable to learn the squares of integers up to about 30 x 30 for example? What about fact tables for non-integer quantities? etc. What algorithms for the operations should be taught? The most commonly taught computational algorithms are peculiarly opaque with respect to both the place value structure of the number as well as to the issue of making reasonable estimates of the size of the number that results from the computation.28 What other operations, if any, should be taught? Why or why not? There was a time when a square root algorithm was taught widely in the schools. There are other unary operations that might be taught. There are also other binary operations that may have some utility. Perhaps the hardest question of all derives from the fact that the pedagogic time and attention that is invested in the teaching of the arithmetic curriculum is largely directed toward trying to convey an understanding of how the various computational algorithms work. Is this a useful use of the time and energy of both students and teachers? The traditional mathematics curriculum at the elementary levels concentrates on the acquisition of computational skills, specifically getting students to master with some degree of automaticity the algorithms for adding, subtracting, multiplying and dividing whole numbers, fractions and decimals. However, we live in an age when a simple fourfunction calculator can be bought for less than the cost of a weekly newsmagazine. With the remarkable exception of the elementary grades of the schools of our country, almost all the calculation done in the country is done electronically. Thus, in preparing students to calculate “by hand” the schools are not preparing our students for the world they will encounter. The counter-argument is often made that students need to understand the conceptual underpinnings of the computations that are done in the world around them. Indeed they do! However, such conceptual understanding does not flow from mindless repetition of un-understood mathematical ceremonies, but rather from a direct addressing of the There are three people in our family who like soft-boiled eggs and 4 people in our family who like hardboiled eggs. How many people are there in our family? 27The special role of 12 in arithmetic instruction probably has its origin in the fact that the United States was an English colony at a time when that country's currency had 12 pence to the shilling. 28The most commonly taught procedures for doing addition, subtraction and multiplication all begin with actions on the least significant digit of the numbers rather than the most significant digit.


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conceptual issues involved in computation with whole numbers, fractions and decimals. Thus, even at the youngest levels, we should stress the importance of the order properties of numbers and estimation much more than we normally do.29 Given that the tiresome repetition of computational exercises, often without understanding, (how many educated adults understand why the procedures for long division or division of fractions work?) •

does not prepare students for the kinds of applications of mathematics that they are likely to encounter, and,

•

uses up time that might better be spent in helping students develop a conceptual understanding of, and appreciation for, the subject of mathematics, and,

•

that filling school and homework time with this sort of activity deadens the students' interest and curiosity about mathematics,

we would be well advised to reconsider what we think is important mathematics in the elementary grades. We see that there is much to change in the school arithmetic curriculum even if we adhere to its traditional content. I believe, however, that sticking with the traditional content, in the face of the extraordinary failure of mathematics education to engage generations of students, and in the fact of a rapidly changing technological world, is irresponsible. I believe that the focus of an arithmetic curriculum, and indeed all required school mathematics, should be on its use as a set of tools for modeling the world around us, for analyzing these models, for making inferences and drawing conclusions from them and for communicating with others. If this is deemed to be a reasonable set of goals for a school arithmetic curriculum, perhaps it is time to think of replacing present school arithmetic, which is largely the arithmetic manipulation of nominal quantity, with the arithmetic of modeling and problem posing and solving with adjectival quantity.

29For

example, while a great deal of time is spent in the middle grades teaching students to find the least common denominator of two fractions in order to add them, most students have difficulty producing a fraction whose value lies between the values of two given fractions. Similarly, while a great deal of effort is expended teaching students to compute such products as 315 x 876, few of them are taught to think of the product as reasonably approximated by 300 x 900 or 270,000 which differs from the exact result by about 2%.


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PART II - Quantity in Secondary Mathematics VIII.

TOWARD AN ALGEBRA OF ADJECTIVAL QUANTITY

It would seem that mastering the arithmetic of adjectival quantity consists of being able to extract quantity from the surround by counting and measuring, and to concatenate such quantities to generate new quantities that describe different elements of the surround. Is there more mathematics that all students should be expected to learn? The usual answer that our society, and indeed all others, seems to give to this question is yes. We teach, and expect all children to learn, a subject called algebra. The subject, as organized in the texts we use, as perceived by teachers and as presented in the classrooms of the country, is remarkably incoherent and fragmented (Schwartz etal, 1993). If the teachers of the subject believe it to be incoherent and fragmented there should be little surprise that it is badly taught and badly learned. If, as is universally agreed, the subject is badly taught and badly learned, and if the society expects people to learn the subject, it behooves us to search out ways to repair the situation. One of the reasons that algebra is badly taught and badly learned is that there seems to be no consensus as to what are the fundamental objects of the subject in the way that all seem to agree that the fundamental object of arithmetic is number, or as argued in this paper, quantity. It is my thesis that one can build a coherent approach to the subject by taking as the fundamental object of the subject the mathematical object called the function.30 The function can be thought of as a procedure for obtaining the value of a quantity given the values of the quantity (or quantities) on which it depends. In this sense the function can be thought of as a “recipe” for producing a quantity given the necessary ingredient quantity or quantities. Functions, to the extent to which they are taught in the algebra curriculum, are almost always taught as relationships among nominal quantities. Clearly, in the spirit of the present paper, it is interesting to ask what building an algebra of functions of adjectival quantity might be like. This leads us to consider the idea of a “quantity recipe” that produces an adjectival quantity given the one or more adjectival quantities on which it depends. In arithmetic we obtained the mathematical objects we work with, i.e. quantities, by modeling sizes of entities in our surround such as lengths and weights and times. In algebra we obtain the mathematical objects we work with, i.e. functions, by modeling relationships among quantities.31

30There

are those who will argue that the concept of variable is central to the subject of algebra. I would simply point out that the concept of variable implies the concept of function, and vice versa. When I say that I regard function as fundamental, I mean the linked concepts of function and variable. 31To be sure there are functions on domains other than the domain of numbers and one wants to be sure that these are not excluded. On the other hand, one can't do everything at once.


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How shall we represent these objects called functions? There are several ways to do so. The two most powerful representations, in my view, are symbolic and graphical. The representations are complementary in the sense that they each highlight and make salient different aspects of the concept of function, in general, and of the particular functions represented, in particular. The following sections of the paper introduce the concept of function of adjectival quantity in each of these representations.

a

representing functions symbolically

As we have seen, composing two adjectival quantities whose magnitudes are known results in a third adjectival quantity of known magnitude. Successive composition of this sort leads to a succession of adjectival quantities of known magnitude. If the entire procedure is repeated with different starting values, a different set of resulting values may well ensue. Tables which list which resulting values are associated with which starting values can then be formulated and a picture of the behavior of the system being modeled may begin to emerge. As an example, suppose we wish to build a storage shed whose width is constrained to be 10 ft but whose depth we are free to vary somewhat. If we make it deeper, it will be able to hold more. On the other hand, the deeper it is, the more expensive it will be. Suppose in particular, that it will cost {100., $/ft, cost/depth of shed} over and above {500. $, fixed cost of constructing shed}. A shed that is {15., ft, depth of shed} deep can store a board diagonally that is {18.03, ft, length of board} and will cost {2000., $, cost of shed}. A shed that is {20., ft, depth of shed} deep can store a board diagonally that is {22.36, ft, length of board} and will cost {2500., $, cost of shed}. A shed that is {25., ft, depth of shed} deep can store a board diagonally that is {26.93, ft, length of board} and will cost {3000., $, cost of shed}. A shed that is {30., ft, depth of shed} deep can store a board diagonally that is {31.62, ft, length of board} and will cost {3500., $, cost of shed}. Here is a summary table; depth of shed (in feet) 15 20 25 30


maximum length of storable board (in feet) 18.03 22.36 26.93 31.62

cost of shed (in $) 2000. 2500 3000 3500

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Representing the relationships among these three quantities in this sort of tabular form can clearly get to be awkward. Further, its utility may be limited by the fact that it may not be possible to anticipate all the possible depth-of-shed values that may turn out to be of interest. One would like to be able to have both a more parsimonious as well as a more generative way of encoding this information. Suppose one adopts the convention of recording the procedure for relating say, the maximum length of storable board to the depth of the shed. Such a procedure can be described in the following way; {(L, ft), maximum length of storable board} ← SQUARE ROOT OF [ {(10.00, ft), width of shed} x {(10.00, ft), width of shed} + {(D, ft), depth of shed} x {(D, ft), depth of shed} ] This representation of the relationship is to be understood as a generative one, i.e. given a value of the magnitude of the quantity {(D, ft), depth of shed} the procedure allows one to calculate a value for the quantity {(L, ft), maximum length of storable board}. This procedure can be written as {(L, ft), maximum length of storable board} ← {( 100 + D 2 ), ft), maximum length of storable board}. In a similar fashion, one can write the relationship between {(C, $), cost of shed} and {(D, ft), depth of shed} as

{(C, $), cost of shed} ← {(500., $), fixed cost of constructing shed} + {(100., $/ft), cost/depth of shed} x {(D, ft), depth of shed} = {(500. + D 100.), $, cost of shed}.

We have constructed two quantities


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{(L, ft), maximum length of storable board} and {(C, $), cost of shed} whose magnitudes are not specified, but are known to depend on the quantity {(D, ft), depth of shed}. Each of these entities, L and C, is said to be a function of the variable quantity which is the depth of the shed. Since, in each instance, the way in which the quantity in question depends on the depth of the shed is well-defined and known, each of these functions is said to be a known function of the depth of the shed. It should be noted that the variable quantity in each of these functions is a length, but that the function itself is not necessarily a length. In our case, one of our functions is a length measured in feet and the other is a cost measured in dollars.32 Although there is a great deal of economy in describing relationships in this symbolic form, this representation does not make transparent several important sorts of features of relationships of this kind. Specifically, in this representation it is not clear how the rate of change of cost of the shed or the maximum length of storable board varies with the depth of the shed. Nor is it clear whether or not each of these quantities has an extreme value, i.e. is there any value of depth of shed that makes the cost of the shed or the maximum length of storable board either a maximum or a minimum? For the purpose of making these properties of the relationship salient, a graphical representation of the function is much to be preferred. In a later section we will consider how the concept of the function and its behavior might be approached graphically. Generating functions of both fixed and variable adjectival quantity is at the heart of algebraic modeling. Doing this, however, is something that is seemingly difficult for many youngsters. It is often the case that students can carry out the necessary successive concatenations of known and fixed starting quantities, but find themselves at a loss for what to do when the value of the starting quantity is not given explicitly, but rather, is indicated by a letter rather than a number. In our illustrative example above, such students would be perfectly capable of building the table we constructed, but would have difficulty in writing down in symbolic form the function that they themselves used to produce the table.

b.

the power of symbolic manipulation33

It has become fashionable in mathematics education reform circles to call for a substantial reduction in the amount of time devoted to symbolic manipulation in algebra (NCTM 32

Because the referents of the domain and range of functions of adjectival quantity may differ two functions of adjectival quantity can be composed only if the range of one function matches the domain of the other. 33I am indebted to my colleague Dan Chazan for the insight that led to the argument in this section of the paper.


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1989). If the algebraic entities being manipulated are expressions or functions of nominal quantities, then there is little reason to argue with this position. However, if one is dealing with functions of adjectival quantities, then symbolic manipulation offers an unusual opportunity for the development of insight and understanding of the real situation that provides the referents of the functions and their variables. Here are two examples. Suppose a water tank initially holds 1000 gallons of water. The owner of the tank begins to drain the water at a constant rate of 5 gallons/hour. We are interested in a function whose value is the volume of water in the tank at any time t after the draining of the tank has begun.34 The quantities known by name and magnitude are {(1000, gallons), initial volume of water} {(5, gallons/hr), rate of draining tank} A quantity referred to by name but not by magnitude in the statement describing the situation is {(t, hours), elapsed time variable} These quantities may be combined in the following way to yield a symbolic expression for volume as a function of elapsed time: {(V(t), gallons), volume of water in tank}

←

{((1000 - 5t), gallons), volume of water in tank}35.

It is common for texts to ask students to factor expressions such as (1000 - 5t) expecting them to write an expression such as 5(200 - t). If we take our commitment to the algebra of adjectival quantity seriously then we are led to observe that while we recognize in this new form of the function the quantity {(5, gallons/hr), rate of draining tank}, the quantity {((200 - t), hr), ???} seems to be new. Clearly it must be a time, but what time does it refer to in the referent situation? In fact, it is the quantity

34Stated

in this form, the problem seems sterile and remote from the concerns of both high school students and adults. However, the reader is invited to restate the problem in terms of an oil tanker spilling oil, a wounded person losing blood, an ecosystem losing a species, etc. 35Strictly speaking the function has this form only for values of t such that V(t) is not negative. The statement describing the situation does not state explicitly what was happening to the level in the tank preceding the beginning of draining (was the tank being filled?) nor does it state what happens after it is emptied.


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{((200 - t), hr), time remaining before tank is empty}. This quantity, however, may be thought of as the difference between the quantities {(200, hr), ???} and {(t, hr), elapsed time variable}. Where did the 200 come from and what does it describe in the referent situation? It describes the total time it takes to empty the tank. In contrast to the corresponding situation with functions of nominal quantity, we note that the seemingly mechanical act of factoring a simple expression leads to an insight into the nature of the real situation that the mathematics describes. Here is a second example similar to one that might be found in any algebra text: Suppose one throws a ball straight upward with a speed of 4 ft/sec from a height of 6 ft. above the ground. Write an expression for the height of the ball above the ground as a function of elapsed time. You may use the value of 32 ft/sec2 for the acceleration of gravity. Students will no doubt make use of the formula printed in the book to write the function {(Height(t), ft), height above the ground after time t}

←{(6 + 4t - 16t2, ft), height above the ground after time t} As an expression of a function of nominal quantity this may be factored to yield (3 - 4t)(2 + 4t). If one seeks to interpret each of these factors in terms of the referent situation one discovers that each of them describes the square root of a length. It is difficult to see what elements of the referent situation that are important or interesting are described by quantities whose measure is given in [feet]1/2. On the other hand, it is also possible to factor this expression so as to obtain the form {( 6 + (4 - 16t)t, ft), height above the ground after time t}. This quantity may be profitably decomposed into its components as follows:


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{(6, ft), initial height above the ground} {(4 - 16t, ft/sec), net vertical speed after time t} {(t, seconds), elapsed time variable}. The quantity {(4 - 16t, ft/sec), net vertical speed} can be further decomposed into {(4, ft/sec), upward speed} and {(16t, ft/sec), average downward speed after time t}. It seems reasonably clear that there is insight to be gained from the thoughtful application of symbol manipulation to functions of adjectival quantity. Both of these examples are instances of revisiting something we earlier found to be true when we were factoring adjectival quantity, i.e., that decomposing a magnitude into a product of factors does not dictate a necessary decomposition of the referent into two referent factors. In the first case described above the quantity {(1000 - 5t), gallons), volume of water in tank} could have been written as {(t2, min2), ???} x {(1000/t2 - 5/t, gallon/min2), ???}. Because there is not a unique way to do so, the factoring of referents does not guarantee insight into the referent situation. However, if done reasonably and in terms of what one already knows about the referent situation, it heightens the likelihood that one can discern further, as yet not explicit, properties of the situation.


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representing functions graphically

Earlier in this paper it was suggested that graphical representations of functions may be complementary to symbolic ones in a number of important ways. We introduced the symbolic representation of functions by the generalization of relatively simple computational procedures. If functions are mathematical encodings of relationships among quantities in our surround, what shall we do when the relationship is too complex to be described in simple computational terms? or if we only know some qualitative features of the relationship36? It is in these cases that the graphical representation of the relationship may be of greatest help. Many of the relationships among quantities that we might wish to mathematize are temporal in nature, i.e., we seek a description of how the value of some quantity varies in time. When one seeks to mathematize this sort of relationship among quantities in the surround it is often the case that the relationship is best thought of in terms of events, occurring at “instants” of time, and processes that take place in the intervals of time linking those instants. Let us consider a specific example. Here is a description of a relationship between the height of a tree and elapsed time. Our story opens. A tree is planted in the ground. At first it grows very slowly as it takes root and adjusts to its new surroundings. At some point in time it starts to grow rapidly. After a while the rate of growth slows down and the tree reaches its mature height. The story may separated into a sequence of events. This parsing of the story into a sequence of events suggests that a useful graphical representation of the situation might be had by plotting these events and the processes joining them qualitatively in a Cartesian speed-time plane. Here is a possible representation of that variation.

height

time

36The

essential qualitative features of a relationship that we may know might include the number of zero crossings, the number and nature of the extreme values of the function as well as the behavior of the function for very large values of its argument(s).


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There is no simple symbolic way of describing this relationship between the height of the tree and elapsed time. Symbolic representations of this relationship can be generated but they are cumbersome and complicated. Moreover, they are likely to be more precise statements about the referent situation than we are justified in making. On the other hand, many important properties of the relationship between the height of the tree and elapsed time can be comfortably represented and made immediately evident, if not salient, in a graphical representation. In a symbolic representation such features of the relationship as the initially constant rate of growth or the final constant height are not likely to be readily apparent. It is important to point out not all the relationships we represent graphically are temporal in nature. However, such relationships do form an important special class of relationships in that by far most of the graphs that the public is asked to look at and understand are graphs of functions of a single variable, i.e., time.

d.

the power of graphical manipulation

Earlier we saw that manipulating functions depicted symbolically could lead to new insights about the referent situations that the functions described. A corresponding possibility exists with functions that are depicted graphically. Consider our earlier example of the height of a tree as a function of time. We can imagine rigidly displacing the graph vertically like this for example; height

height

goes to time

time

This change in the graph describes a different but related situation, namely one in which a taller sapling was initially planted. Subsequently, this taller sapling grew at the same rate as the shorter one. The taller sapling reaches a larger mature height, the difference in mature heights being the same as the difference in initial heights. Whatever the symbolic representation of the function may be, H(t) say, our manipulation of the graphical representation of the function corresponds to

H (t ) → H (t ) + constant height = H (t ) + h0 , say.


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Rigidly displacing the original graph horizontally, to the left for example, also describes a different situation, albeit not as interesting a one. The horizontally displaced graph depicts the same tree sapling being planted earlier. Symbolically this means

H (t ) → H (t + constant time interval) = H (t + t0 ), say. Our original graph may be modified not only by displacing it rigidly but also by stretching or squeezing it. Here, for example, is a vertical squeeze. height

height

time

goes to

time

The situation this modified graph describes is one in which a shorter sapling is initially planted and which then grows more slowly but takes the same time as the original tree to reach its mature height which is smaller than the mature height of the original. Symbolically, H ( t ) → αH ( t )

where α is a scale conversion factor whose units are length/length. Stretching (or squeezing) our original graph horizontally gives rise to a description of yet another different but related situation. height

height

goes to time

time

The initial height of the sapling is the same as in our initial graph. The tree grows more slowly. However, this tree reaches all of the same intermediate heights as the original tree and also reaches the same mature height as the original tree. It just does so more slowly. Note that this slower rate of growth is different that the slower rate of growth described by the vertical squeezing of the original graph. Symbolically this modification of the function can be written as


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H (t ) → H (αt ) where α is a scale conversion factor whose units are time/time.

e.

linking symbolic and graphical representations of functions

It is interesting to consider the links between the two forms of representing functions that we have discussed. In order to do this, let us return to the situation we discussed in an earlier section where we considered the building of a shed and how the maximum length of boards that could be stored in the shed as well as the cost of the shed depended on its depth. The two functions we derived were: {(L, ft), maximum length of storable board} ← {( 100 + D 2 ), ft), maximum length of storable board}. {(C, $), cost of shed} ← = {(500. + D 100.), $, cost of shed}. Is there a way to represent these functions graphically? Here we display the graphs of these two functions. Values of the depth variable in feet are read on the horizontal axis which is common to both the length function and the cost function. Values for the length function can be read on the left vertical axis in feet and values for the cost function can be read on the right vertical axis in dollars.


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This figure exhibits both some of the power and some of the perils associated with the graphical representation of functions. The power inheres, at least in part, from allowing the viewer to grasp instantly something of the overall behavior of the functions in question. Specifically the cost function behave linearly - the graphical equivalent of saying that each additional foot of depth of shed adds $100 to the cost of the shed. Somewhat less apparent is the fact that the length function is curved for very small depths but rapidly seems to approach a straight line as we consider deeper and deeper sheds. How are we to understand this? Also, how are we to understand what the graph seems to be telling us about the maximum length of storable board if the depth of the shed is zero feet? One of the perils of the graphical representation, deliberately presented here as a way of raising the issue, is the specious question, “What is the significance of the depth for which the length function and the cost function intersect?” These are two entirely distinct functions that happen to depend on the same variable. Since they are distinct, they may be separately manipulated. Consider then what might happen if vertical scale of one of the functions is changed. Clearly, the function does not change, but its appearance on the paper does. Since the appearance on the paper of the other function has not changed in any way, the position of the intersection of the two functions has changed!

f.

two special functions

In the algebra of nominal quantity there are two functions that have special conceptual importance. These are the constant function f ( x ) = 1, −∞ < x < ∞ and the identity function f ( x ) = x , −∞ < x < ∞ . It is often difficult for students to accept the constant function as a function since as a recipe for accepting input numbers and generating output numbers, it doesn't seem to do anything. Moreover, it doesn't “look like” a function but rather seems to be a number. This last point is particularly puzzling. Does the symbol 1 signify a number or a function of x ? In fact, might not the symbol 1 signify a function of two variables x and y as in f ( x , y ) = 1, −∞ < x < ∞, −∞ < y < ∞ ? For that matter might not the functions


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f ( x ) = x 2 or g( x ) = sin x be regarded as functions of more than one variable? A similar but related issue arises when we consider expressions with parameters. e.g., linear expressions of the form mx + b. Might these not be considered as follows: f ( m, x , b ) = mx + b, −∞ < m < ∞, −∞ < x < ∞, −∞ < b < ∞ Is there any principled way of deciding how many variables a function depends on? At this point it will suffice to say that this difficulty does not vanish when one moves to the algebra of adjectival quantity.37 While fewer students have difficulty with the identity function, it too has the uncomfortable feature of seemingly not being a very interesting number recipe - it simply produces a number whose value is that of the number that was inputted. This seeming symmetry between input and output is clearly an artifact of our considering a function of one variable. Since a function is an object that produces a single output value when supplied with appropriate input values a function of more than one variable cannot output that which was inputted. In the algebra of adjectival function there is an identity function for every referent. For example a time identity function might look like Itime(t)

← {(t, hr), time}

while a distance identity function might be Idistance(d)

← {(d, meters), distance}.

37Knowing

that a function of adjectival quantity should be constant is often of great value in modeling situations. For example, The total energy of an oscillating mass on a spring is a function of the speed of the mass as well as its position. If the system can be regarded as lossless, then this energy is constant in time, despite the fact that both the speed of the mass and its position separately vary in time.


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f.

unfinished DRAFT

fitting functions to counted and measured data

Suppose we have a metal bar which at room temperature (20° Celsius) is 100 cm long. We are interested in ascertaining how the length of this bar increases as the temperature rises. We set up an arrangement that allows us to heat the bar in a controlled way and measure its length at a variety of different temperatures. If we plot our measured data we might get something like this: length [in cm]

100

20

temperature

30

[in degrees Celsius]

We believe several things about the relationship between the length of the bar and its temperature. These beliefs include • • •

the length of the bar is a “simple” function of temperature the unmeasured length of the bar at temperatures intermediate to the measured ones can be found by simple interpolation the “irregularities” in the data are due to errors of measurement

Knowing what we know about linear functions we can write the length of the bar as a functions of temperature {(L(T), cm), length of bar}

← {(α, cm/degree Celsius), expansivity of metal} x {(T - 20, degrees Celsius), difference from room temperature} + {(100, cm), length at room temperature} = α(T - 20) + 100


cm.

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where α is a small positive quantity that is a measure of how much the bar expands for each degree Celsius rise in temperature.38 Now suppose that we had performed our experiment using a Fahrenheit thermometer rather than a Celsius thermometer. In that case our plot of the data would have looked like this: length [in cm]

100

68

86

temperature [in degrees Fahrenheit]

If we were to fit these data with a simple function we would obtain {(L(T), cm), length of bar}

← {(β, cm/degree Fahrenheit), expansivity of metal} x {(T - 68, degrees Fahrenheit), difference from room temperature} + {(100, cm), length at room temperature} = β(T - 68) + 100

cm

where β is a small positive quantity that is a measure of how much the bar expands for each degree Fahrenheit rise in temperature. Note that in the first instance the variable is measured in degrees Celsius and that in the second case the variable is measured in degrees Fahrenheit. There is something disturbing about this situation. Nature doesn't really care whether we use a Celsius or a Fahrenheit thermometer. Why should the functions we fit in the two different cases

38The

reader will note that this expression implies that at a temperature of 20 - 100/α the bar has no length at all! Moreover, at temperatures lower than this, the bar acquires a negative length. Clearly one must be careful with models.


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α(T - 20) + 100

cm

β(T - 68) + 100

cm

and

look so different from one another? Looking back to our discussion of the graphical manipulation of functions, it would seem that these two functions might be thought of as two instances of a linear function of a variable T that is manipulated graphically. Specifically, the function

α(T - 20) + 100

cm

can be translated horizontally so as to become the function

α(T - 68) + 100

cm.

This function can then be stretched or squeezed horizontally39 by ratio of β/α to become

β(T - 68) + 100

cm.

Note that in the course of this transformation the referent of the symbol T changed from degrees Celsius to degrees Fahrenheit. Now suppose that we had performed our experiment using a Fahrenheit thermometer and ruler calibrated in inches. In that case our plot of the data would have looked like this: length [in inches]

39.37

68

86

temperature [in degrees Fahrenheit]

39...depending

on the relative sizes of α and β.


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If we were to fit these data with a simple function we would obtain {(L(T), in.), length of bar}

← {(γ, in/degree Fahrenheit), expansivity of metal} x {(T - 68, degrees Fahrenheit), difference from room temperature} + {(39.37, in.), length at room temperature} = γ(T - 68) + 39.37

in

where γ is a small positive quantity that is a measure of how much the bar expands for each degree Fahrenheit rise in temperature. This too is a linear function and it would seem that our graphical manipulations would enable us to transform it into either of the temperature functions we obtained using a meter stick instead of an English ruler. Note that in the course of such a transformation the referent of the function L(T) will change from centimeters to inches. We shall return to this point in a later section when we discuss the composing of functions of adjectival quantity.

IX.

OPERATIONS WITH FUNCTIONS

a.

referent-preserving operations with functions

Suppose we have two functions each depending on a single variable. There are four possibilities. Clearly, if we wish to consider referent-preserving operations, then only the first two cases below are germane. • • • •

the referents of the variables and the referents of the functions are the same. the referents of the variables differ but the referents of the functions are the same. the referents of the variables are the same but the referents of the functions differ. the referents of the variables differ and the referents of the functions differ.

We have already considered a case of the first type in discussing a projectile thrown vertically upward. In that instance the quantity {(4 - 16t, ft/sec), net vertical speed} was formed by summing {(4, ft/sec), upward speed} and {(-16t, ft/sec), average downward speed after time t}.


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It is important to note that we must regard the object {(4, ft/sec), upward speed} as a constant function of the variable t rather than simply the adjectival quantity {(4, ft/sec), upward speed} which we write in an indistinguishable form. Here is an example of a case of the second type. The cost of renting a car is dependent on the number of days D the car is rented and also on the number of miles M driven. For the sake of specificity let us say that the part of the cost derived from the number of days rented can be written as {(C1(D), dollars), cost of renting car for D days} and that the part of the cost derived from the number of miles driven can be written as {(C2(M), dollars), cost of driving rental car for M miles}. Each of these functions is a function of a single variable, the first of time and the second of distance. It is entirely plausible to say that the total cost of renting a car for D days and driving it M miles is {(C(D,M), dollars), cost of renting car for D days & driving it for M miles}

{(C1(D)+C2(M), dollars), cost of renting car for D days & driving it for M miles} How are we to interpret the fact that C(D,M) is a function of two variables that seems to be the sum of two functions each of which is a function of a different single variable? I believe the answer lies in exactly the same sort of semantic reasoning that underlies referent-preserving operations with quantities that have different referents. In those cases, we expand the referent set of each of the addends so that we can then add directly with the expanded set. For example; Two apples and three oranges are five pieces of fruit. In a similar fashion we must now reinterpret the functions C1(D) and C2(M). We must no longer think of C1(D) as a function on the single variable D and C2(M) as a function on the single variable M, but rather each of these functions is now to be thought of as a function of two variables, i.e., C1(D,M) and C2(D,M). The function C1(D,M) does not depend explicitly on M and the function C2(D,M) does not depend explicitly on D. b.

referent-transforming operations with functions

Thinking about referent transforming operations on functions is also informed by the analogies to similar operations on adjectival quantity. Consider for example the (I E E') triad. One would like to generalize each of the elements of the triad and allow it to be a


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function of some variable. Suppose for example we wish to generalize a triad we considered earlier: distance (mi)

speed (mi/hr)

area as measure of distance

slope as measure of speed time (hr)

time (hr)

to a case in which the speed varies as a function of time. distance (mi)

speed (mi/hr)

slope as a measure of speed ?

area as a measure of distance ?

time (hr)

time (hr)

One could imagine constructing the quantity speed(time) x time which has the dimensions of a distance but which value of speed shall we use? If we wish to use the situation with adjectival quantity as a guide, one might be tempted to replace speed(time) by some sort of “average speed” and then use the area under this “average speed” from some fixed time to some variable time t as the magnitude of a function that describes the distance traveled from that fixed time to t. Similarly one could imagine constructing the quantity distance(time) / time which has the dimensions of a speed but here we have the problem of deciding which distance to use. In this case, if we wish to use the situation with adjectival quantity as a guide, then we must recognize the fact that the slope of the distance vs. time graph varies from point to point.


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Let us now consider the (I I' I'') triad. Here is an interesting situation. Suppose we are interested in the cost of filling a glass cylinder of a given radius, say R, with a liquid with a given cost per cubic centimeter, say C. If the cylinder is of height h then the cost of the additional liquid per additional centimeter increase in the height of the cylinder is given by  $  cm 3 3 2 C , , cost/ cm of liquid } × ( π R ⋅ 1 , ), increase in volume/cm increase in height    3 cm  cm 

{

}

which is independent of h. Here is what this situation might look like graphically. cost of liquid $

volume of cylinder cm3

slope

slope 2

π R cm 3/cm

C $/cm3

height cm

volume cm3

Our symbolic expression above is essentially the product of the slopes of these two graphs. Suppose now that we alter the situation somewhat. Let us assume that the cost per cubic centimeter of the liquid goes down as the volume of liquid purchased goes up and that we are interested in changing not the height of the cylinder but rather its radius. cost of liquid $

volume of cylinder cm 3

slope slope ? $/cm3

volume cm 3

? cm 3/cm

radius cm

How do we compute the cost of the additional liquid when we increase the radius of the cylinder? There are, to be sure, other sorts of referent-transforming operations with functions other than those that are generalizations of the (I E E') and (I I' I") triads. The ones I discussed here are of special interest because of the role they play in the differential and integral


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calculus of adjectival quantity. These matters will be discussed in greater detail in a later section.

X.

COMPARING FUNCTIONS OF ADJECTIVAL QUANTITY

We have discussed what is meant by the comparison of adjectival quantity. We now turn to the question of what is meant by the comparison of functions of adjectival quantity. In doing so, it is appropriate to review what is meant by the comparison of functions of nominal quantity. Consider a simple linear equation such as one that might be encountered in an algebra book in common use. x−3= 5− x An equation like this is an interesting mathematical object. It seems to have a function on either side of an equal sign. What could it possibly mean to say that the function x - 3 is equal to the function 5 - x ? These functions are clearly not equal as can be seen immediately if one plots them, or evaluates them for some value of x such as 0 or 1, or almost any other value, for that matter. Nonetheless, the equation is asking us to compare the functions x - 3 and 5 - x in some fashion. It is possible to make sense of the equation by reinterpreting what is meant by the = sign that it contains. We can think of the = sign as meaning “for what value(s), if any, of the variable(s) does the function on the left of the = and the function on the right of the = have the same value?” Iconically it might be better if the character we used were written this way:

? With these notions in mind let us ask once again what is meant by the comparison of two functions. Our previous discussion suggests that we consider two functions separated by an equal40 sign to be a comparison of the functions on either side of that symbol. Finding a value of x for which the two functions have the same value means that for slightly smaller values of x one of the functions will be larger and for slightly larger values of x the other function will be larger.41 This corresponds rather closely to what we meant by comparing two quantities. In that case we were interested in which of the two quantities was larger. 40For

the sake of simplicity I have limited the discussion to comparisons of functions that contain the = sign. A similar argument can be made for comparisons involving >, ≥,

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