a.               referent-preserving operations - the case of averages

     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 =

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

    _________1__________

1 / x₁      +     1/x₂

           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 time-honored 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 {(4/3,hr),joint mowing time)}. See for example the

combining rates  challenge in the "modeling & formulating" challenges on this website. 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?

      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.

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.[1]

      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;

      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 four-function 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 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.[4]

     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.

     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.