Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (103 page)

As is probably quite clear, students who solve the ballet-lessons problem do not rely on its abstract, formal structure (as expressed in the above-stated “theorem”); they have no need at all to evoke the abstractions of set theory in order to solve it. When they carry out their reasoning, they don’t perceive the time intervals involved as
sets
(if they did, each set would contain infinitely many infinitesimal moments!), nor do they perceive the common age at which Laurel and Joan started their lessons as the
overlap
of two sets (their “intersection”, in set-theoretic language), nor do they see the lengths of time that the two girls took lessons as the
non-overlapping
parts of sets, nor do they see the girls’ ages when they stopped their lessons as the
sizes
of two sets. In fact, it would take careful intellectual work to recast this problem in set-theoretical language, because set theory is not the framework in which humans naturally perceive it, and that’s why efficient solving of the problem by a person should not be taken as showing that the formal rule (the “theorem”) was correctly applied. Rather, the act of perceiving the ballet-lesson situation in terms of familiar time-categories does the bulk of the work for the student, and there is no need whatsoever to code the situation into an arcane, abstract technical formalism such as set theory.

The diagrams on the facing page show three ways of conceiving these isomorphic word problems. The diagram at the top shows how the school-supplies problem is imagined by nearly everyone to whom it is given. The typical assumption is that in order to figure out the price of the T-square, you have to subtract the price of the binder from John’s total outlay (which itself is found through a subtraction), and the binder’s price is found by subtracting the price of the art kit from Lawrence’s total outlay — thus three subtractions
in toto
seem necessary. In this diagram, one doesn’t see that the difference between John’s and Lawrence’s total outlays is identical to the difference between the prices of the T-square and the art kit. That key idea, a prerequisite to solving the problem in a single step, is missing, and so three arithmetical operations seem to be needed.

The middle diagram shows the way that many people very naturally envision the problem of the ballet lessons. They begin with the idea that Joan and Laurel took up ballet at the same age, and so the difference between the
lengths of time
they took ballet has to equal the difference between their
ages
when they quit. This idea is screamingly obvious in the diagram, and that explains why the one-step solution is often found for the ballet-lessons problem. (Let’s not fail to note the frame blend on which this diagram is tacitly based, and which Gilles Fauconnier and Mark Turner would delightedly point out — namely, the fact that we are imagining aligning the two girls’ lives on a single horizontal time axis. Aligning their lives means placing their births at the same spot on the time axis, and as a result of this maneuver, their first lessons will also coincide.)

Each of the two upper diagrams (the top one, involving three operations, and the middle one, involving just one) was tailored to fit one specific problem, and it is not apparent that they have much in common with each other. Looking at the upper two diagrams alone, one might think that the problems they represent are of very different sorts; in fact, this is what most people claim who try to solve them both. But the bottom diagram not only shows how the two problems can be seen in a single unified manner; it also reveals how a one-step method works just as naturally for the school-supply problem as for the ballet-lesson problem. While the “theorem” enunciated above was abstruse and difficult to grasp, this visual encoding of the two problems in one single picture reveals their isomorphism (
i.e.
, the fact of having the same underlying structure), as well as how they both can be solved with just one arithmetical operation.

These three diagrams show how the way a person visualizes a word problem can either bring out or hide a pathway to the solution. The bottom diagram could be said to be more abstract than the upper two in that it unifies them in a single diagram, but on the other hand, the image of two boxes standing on the same pedestal is very concrete, and as soon as these problems are cast in a form that involves a shared “pedestal” (the binder, in the first problem, and the starting age, in the second one), the
abstruse idea expressed in the “theorem” suddenly becomes crystal-clear, as it has been fleshed out in a concrete manner, using simple, everyday images, such as boxes resting on a shared pedestal. Teaching a young student to see the “pedestal” in these two word problems imparts an elegant insight that is unavailable to most untrained adults.

A key challenge for educators is thus to take into account the way people manage to adroitly sidestep the formal encoding of situations by exploiting the way their familiar categories, built up over years of interactions with the world, work. Although most teachers are quite aware that the way in which a problem is “dressed” can profoundly affect its difficulty, educators have not yet figured out how to make the art of dressing mathematics problems into a powerful teaching tool. To achieve this would be a great advance, but of course doing so constitutes a great challenge as well.

Do you find it hard to see naïve categorizations such as
multiplication is repeated addition
and
division is sharing
as constituting
analogies
? Despite the multiple arguments we’ve mustered and the many situations we’ve dissected with a fine-tooth comb, perhaps a little voice inside you keeps insisting, “I’m sorry, but categorization and analogy-making are just not the same thing. Taking two notions that initially seem very far apart and then building a mental bridge between them because one sees that they have certain abstract qualities in common is a profoundly distinct type of act from seeing something and merely recognizing that it belongs to a familiar category, such as
table
.”

Why do so many people, perhaps even you, have an inner voice that so strongly resists the thesis that the building of analogies between things is just the same activity as the assigning of things to categories? How come your inner voice hasn’t gradually calmed down and grown silent over the course of reading this book? How come it hasn’t listened to all the reasons that we had hoped would convince you of our thesis?

The answer — a rather ironic one — is that our thesis itself explains why so many people have so much trouble accepting it: namely, the belief that analogy-making and categorization are separate processes springs from none other than a certain naïve analogy about the nature of categories. This naïve analogy, which has the dubious honor of having seriously held back progress in the field of analogy and categorization for a long time, has already been cited in this book. Here it is, once again:

Categories are boxes, and to categorize is to put items into boxes.

This is the everyday, down-home view of categorization. Let’s think about it a little bit.

If categories really
were
boxes and if there really
were
a reliable, precise mechanism for assigning things to their boxes, then it would make eminent sense to distinguish between two types of mental process. First would be
categorization
— a rigorous, exact algorithm reliably placing mental items in their proper boxes; second would be
analogy-making
— a subjective and fallible technique for dreaming up fanciful, unreliable bridges between mental items that do not enjoy a contents-to-container relationship.

However, as research in psychology has shown, and as we have stated throughout this book, the vision
categorization = placing things in their natural boxes
is highly misleading, for categorization is every bit as subjective, blurry, and uncertain as is analogy-making. A categorization can be outright wrong, can be partially correct, can be profoundly influenced by the knowledge, prior experiences, prejudices, or goals, conscious or unconscious, of the person who makes it, and can depend on the local context or the global culture in which it is made. In addition, categorizations can be just as abstract as analogies, can be nonverbal as well as explicitly verbal, can be shaded, and so forth.

Recent work, in fact, makes this point so obvious that the old view of categories as boxes is now usually called “the classical approach to categories”, because in cognitive science there is scarcely anyone around any longer who still puts stock in it; today the classical approach tends to be looked upon as simply a quaint historical stage in the development of a far more sophisticated theory of categories and categorization.

And yet, the “categories are boxes” naïve analogy is still seductive, and leads us all to fall victim to the nearly irrepressible belief that all objects and situations we encounter have a privileged category to which they belong, and which constitutes their intrinsic identity. Let’s recall the object Mr. Martin purchased; though it subsequently bounced merrily back and forth among such motley categories as
fragile object, dust-gatherer, spider carrier
, and
home for tadpoles
, it always seemed to remain
in truth
just one thing — namely, a
glass
— and this label constituted its genuine, intrinsic identity. The naïve unspoken analogy “categories are boxes” implies that each item in the world, just like Mr. Martin’s glass, has a proper box to which it belongs, and that this connection between a thing and its natural box is universally shared in all people’s heads, and finally that this is simply the nature of the world, having nothing to do with thinking or psychology. In this view, not only would category identity exist, but it would be precise and objective.

Over the years, the insidious motto “categories are boxes” has underwritten much scientific research reinforcing the belief that there is a clear-cut distinction between analogy-making and categorization. Indeed, the view of categorization that reigned in cognitive psychology for decades, though expressed in far more sophisticated terms, was essentially indistinguishable from this motto. That view was a restatement of the motto in technical terms borrowed from mathematical logic. It portrayed each mental category as possessing a set of “necessary and sufficient conditions” for membership. An entity belonged to the category
if and only if
it had all those properties. Thus each category-box was thought of as being precisely defined and having rigid, impermeable walls. In this theory of categories, there was no room for degrees of membership, nor for contextual effects on membership in categories. Only in the mid-1970s, when psychologist Eleanor Rosch published her seminal series of articles on categorization, was this erroneous but nearly universally accepted theory at last discredited.

And so, after finally being liberated from the motto, how do today’s researchers view analogy-making and categorization? Has a clear consensus emerged on what, if anything, makes the two different? Well, let’s listen to some of the top authorities. On the one hand, Thomas Spalding and Gregory Murphy write, “Categories let people treat new things as if they were familiar”; on the other, Mary Gick and Keith Holyoak
state, “Analogy is what allows us to see the novel as familiar”. If there is a distinction between these two characterizations, it eludes us! Or, to cite Catherine Clement and Dedre Gentner, “In an analogy, a familiar domain is used in order to understand a new domain, especially to predict new aspects of this new domain”, whereas for John Anderson, “If one establishes that a given object belongs to a certain category, then one can predict a great deal about the object.” Once more, these descriptions of supposedly different processes make them sound as alike as Tweedledum and Tweedledee.

The great overlap of these experts’ definitions confirms (if confirmation was needed) the tight relationship of analogy-making and categorization, and provides grist for the mill we are defending — namely, that the idea that analogy-making and categorization are separate processes is illusory. Still, someone who wished to play devil’s advocate might argue that by overhauling the definitions of analogy-making and categorization, one might be able to show that they are indeed
two
processes and not just
one.
In fact, after
Chapter 8
there is a dialogue that carefully explores this possibility, and we hope that that dialogue, in addition to closing our book, will also close the book on this issue.

However, if you, even after having read this far in our book, still feel reluctant to accept our thesis, rest assured that you are in excellent company, for even experts in the fields concerned fall victim all too often to this same naïve vision. Indeed, we — the authors of this book, the most fervent proponents of our thesis — find ourselves from time to time falling into the very trap we’ve tried so hard to warn our readers about! Yes, we too fall occasionally for the tempting illusion that categories are boxes. The following way of putting it would probably have warmed the cockles of Aristotle’s heart: