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 years, 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 the first part of this extended essay 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 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 and on 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 essay 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 ideas in this essay are strongly influenced by it.

Nominal and Adjectival Quantities

    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 adjectives[2].  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 extended essay is to 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.

“If you have ten apples & want to divide them up among six people, what would you do?  Make apple sauce.”

     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]

Counting and Measuring

    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 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}

Discrete and Continuous Quantity

    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.                        {(5, ft), girl's height}

                  It took me an hour to drive home.        {(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.

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 precision[6].  Because of these considerations, it is clear that the result of a measuring act is necessarily a non-negative rational number[7] 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.

    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.