Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (97 page)

Unfortunately, no tool at that level of sophistication exists to help us locate lost items in the physical world, and for that reason the on-line world’s concept of
search
, which by now is second nature to nearly everyone, has become the source of analogies in the physical world, rather than the other way around. It is very tempting to think that objects that we’ve misplaced over and over again, such as our keys, our wallet, our checkbook, or our glasses (or their case), should be just as easily coaxed to chime out their presence to us as is our misplaced cell phone. In short, we’d like to be able to summon anything that we can’t find and have it instantly chirp back at us, telling us just where to find it.

By now, the Web and cell phones have given us the sense that pretty much anything should be within easy reach at any time. Wherever we are located, just about anything that matters to us is within clicking range, emailing range, cell-phoning range, or text-messaging range. This gives us the illusory feeling that anything of any sort should be available to us instantaneously.

The great ease of obtaining things from afar these days has had the effect of reinforcing the sense of presence of people who are in fact absent. Some people, for example, have started to feel ill at ease about making the slightest critical comment about anyone, any time their cell phone is nearby — as if the person being criticized might somehow overhear the comments even without any call having been initiated. This scenario is a bit reminiscent of living in an apartment with very thin walls, where one never knows for sure whether one might be overheard by others in neighboring apartments. Along much the same lines, certain people, when in a chat room on the Web, often start whispering to physical people who are physically near them, as if the virtual occupants of the virtual room could overhear every word they say aloud. In short, some computer addicts develop the jittery feeling that the physical world all around them is populated by virtual people who are able to overhear conversations about themselves.

One very useful feature of computers is that they offer us all sorts of chances to undo mistakes that we have made. Thus when we are using a word processor or a photo editor, it seems only reasonable that we should instantly be able to reverse any action that we’ve done — including massive deletions carried out by accident — and we get very used to such luxuries, taking them for granted. The habit then spills over into other domains of life and becomes an expectation, and when we realize there is no “undo” button in nearly any of them, we become frustrated. Here are two anecdotes illustrating this tendency:

Not only the act of “undoing” but many other types of frequent computer actions can become such strongly ingrained habits that they wind up shaping the way people see and behave in the material world, as the following set of anecdotes demonstrates:

These days, amusing mental contaminations due to these types of crossed wires involving computer concepts and pre-technological concepts are a dime a dozen, and there are Web sites aplenty “where” (if we may analogically borrow this concept from the physical world) people gleefully report and laugh at their own gaffes of this sort.

Now that we have taken a careful look at the ways that naïve analogies originating in recent technology have insidiously invaded our lives, we can turn back to the field of education, focusing specifically on the role that naïve analogies play in how children pick up basic mathematical concepts in school.

Is the equation “3 + 2 = 5” completely clear? Is there just one way to understand it? Do all educated adults understand the equals sign in the same way? Theoretically, an equation symbolizes a perfect equivalence or interchangeability; that is, an equals sign tells us that the two expressions flanking it stand for one and the same thing. The notion of equality, when described this way, seems so simple and straightforward that it would seem hard to imagine any other way of interpreting it. And yet there is another side to the notion of equality, and it comes out of a naïve analogy that we will call “the operation–result analogy”.

In this alternate interpretation, the left side of an equation represents an
operation
, while the right side is the
result
of the operation. This is a naïve analogy in which equations are tacitly likened to processes that take time, and it crops up in situations that have nothing to do with school or mathematics, and which influence everyone, including young children. For instance:

Here, the equals sign is a symbol that links some sort of action in the world to its outcome, and it can be read as “gives” or “yields” or “results in”. When seen that way, “3 + 2 = 5” is not the statement of an equivalence at all; rather, it expresses the idea that the process of adding 3 and 2 results in 5.

The ideas of
interchangeability
and
operation–result
are different. The second point of view clearly embodies an asymmetrical conception of equations, in which the two sides play different roles, one side always standing for a process and the other always representing its outcome. To write “5 = 5” would be incompatible with this viewpoint, since no process is indicated. Likewise, writing “7 – 2 = 8 – 3” is also troublesome, since now there is no result. And lastly, writing “5 = 3 + 2” would be disorienting, because the operation and its result occupy the wrong sites. Indeed, many first-and second-graders understand equality in just this fashion, insisting that “5 = 3 + 2” is “backwards”, and that “7 – 2 = 8 – 3” makes no sense because “after a
problem
there has to be an
answer
, not just another problem”. Some even balk at “5 = 5”, replacing it with something such as “7 – 2 = 5”.

The operation–result naïve analogy guides children before they encounter the concept of
equivalence
, because the notions of a
process
and its
result
are familiar even to toddlers. These notions are close cousins to the notions of
cause
and
effect
, as well as to the idea that certain means have to be used to reach certain ends.

Although today’s children may acquire a fairly decent understanding of equality in elementary school, coming up with the symbol “=” took a long time for humanity as a whole. A symbol for equality in mathematics first appeared only in the year 1557, in a book by the Welsh mathematician Robert Recorde. He wrote:

The word “gemowe” means “twins”, and the “twinnedness” of the upper and lower horizontal lines was intended to symbolize the general idea of equality. The fact that a symbol for equality took such a long time to occur to anyone, even though mathematics had existed for at least two millennia, reveals that it is far from a self-evident notion.

Although for many adults today the idea that “equality equals equivalence” may seem obvious in a mathematical context, it doesn’t follow that the operation–result view of
equality
has disappeared from their minds. In fact, people often write down, and read aloud, equations in a way that reflects their unconscious understanding. For instance,
in reading “4 + 3 = 7”, many people will say “four plus three makes seven”, whereas for “7 = 4 + 3” they might say “seven is the sum of four and three”. If education always resulted in equations being seen as statements of interchangeability, then by the end of high school, the operation–result view of the equals sign would surely have disappeared for once and for all. The order of the two sides in an equation would be completely irrelevant, and both ways of writing an equation down would elicit exactly the same commentary. However, this turns out not to be the case. Let’s take a look at some specific cases, starting out with some that are very remote from mathematics.

It’s standard practice for advertisements to appeal to the child inside each of us rather than to the budding mathematician. Here are a few examples of “equations” culled from real ads:

Just to convince ourselves that interchangeability is not the idea behind these equations, let’s flip them around. As you will see, the resulting “equations” sound utterly silly, even nonsensical.

As these examples reveal, people’s first glimmerings of understanding of the equals sign come from the naïve analogy of an operation followed by a result, and even if the concept of interchangeability gains some ground in the course of twelve years of education, the
operation–result
viewpoint is never fully eradicated. It can always be coaxed out of dormancy when the right cues are presented. We thus see that education does not eliminate the first naïve ideas about a mathematical notion — even one that we tend to think of as completely trivial because it is taught in elementary school, when children are only six or seven years old. In childhood and even when one is fully grown, the naïve view coexists with a different view, which is instilled at school but which is also dependent on a pre-existent and familiar notion: that of
same thing
(that is,
identity).
The transition from the earliest viewpoint (operation–result) to the more sophisticated one (identity) does not obliterate the earlier viewpoint, which remains
triggerable, and on which one frequently relies on a day-by-day basis, sometimes even in scientific contexts, such as mathematics or physics, as we shall now see.

For physicists, the most fundamental formula of classical mechanics is doubtless Newton’s second law, which describes how a force affects the motion of an object. The basic idea of this celebrated law is compatible with the naïve analogy that says that one side of an equation should represent a
process
, with its other side representing the
result
of that process. In this case, the process (ideally occupying the left side of the equation) would be the action of a force of size
F
on a mass of size m, and the result (ideally on the right) would be an acceleration of size
a
imparted to the mass. Rendered symbolically, this yields the equation “F/m
= a
”. Unfortunately, though, Newton’s law is virtually never written this way. Instead, it is almost always cast as follows: “F
= ma
”. This famous formula is quite confusing to many students, since neither side of it cleanly symbolizes either the process or the result. The alternative notation
“F/m = a
” encodes the naïve analogy much more clearly, and would therefore be easier for students to relate to, but it is seldom if ever found in textbooks. From a purely
logical
standpoint, these two versions of Newton’s law are completely equivalent and interchangeable, but from a
psychological
and
pedagogical
standpoint, they certainly are not.

Luckily, physicists are often sensitive to such psychological pressures, and most of the time they try hard to cast their equations in the form of clean and clear cause-and-effect relationships, with one side giving rise to the other side. Take, for instance, the first of Maxwell’s four equations for electromagnetism: