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     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.

b.              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.

     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 multiplication as 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}                   corresponds to     {1, bag}

          {10, candies}                corresponds to     {2, bags}

          {15, candies}                corresponds to     {3, bags}

          {20, candies}                corresponds to     {4, bags}

          {25, candies}                corresponds to     {5, bags}

           {30, candies}                corresponds to     {6, bags}[4]

 The method clearly depends on our ability to translate the relationship expressed by the intensive quantity

                  {5.0, candies/bag, kind of party favor}

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.

      Finally, perhaps most lethal to the idea of multiplication as repeated addition, is the following example (and others of its ilk).

{0.5, mi/hr, average speed of moving walkway} x {1/50, hr, time spent on moving walkway}

= {0.01, mi. length of walkway}

The magnitude of the product is smaller than the magnitudes of each of the factors. How does one reconcile this result with the concept of repeated addition – the result of addition as an action normally means a sum that is larger than all of the addends.

      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}[5]

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}

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}[6]

     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 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.[7]

     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.

      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

{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.[8]

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.[9]

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

                  {(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.

      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}[10].

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}

Continuous x Continuous

                      {3.20, cm, width of rectangle}  x  {6.40, cm, length of rectangle}

                  =  {20.48, sq cm, 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 sq cm in this case.