Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (58 page)

Our chart of quadrilateral types, given above, doesn’t allow us to predict how Tom’s teacher would react if Tom, on his first assignment, were to write four different labels in each square (indicating that it is simultaneously a square, a rectangle, a rhombus, and a parallelogram), and if he were to write two labels in each non-square rectangle (indicating that it is both a rectangle and a parallelogram), and if he were also to write two labels in each non-square rhombus (indicating it is both a rhombus and a parallelogram). Such labels would come directly from a careful application by Tom of his own definitions (in his second answer) — definitions that were so warmly applauded by his teacher and that were given top grades. Would she have given him full credit for answering exercise #1 in such an unorthodox fashion, when we know that in fact she gave him full credit for a totally different, much more conventional set of answers? One would hope that she would be delighted to find in her class a student who is so insightful, and that such thoughtful answers would be rewarded with full credit, and without reservation — but that would depend on whether she appreciated or disdained unconventional viewpoints.

Indeed, categories similar to
non-square rectangle
— for example,
non-square rhombus
and
parallelogram that is neither a rectangle nor a diamond
(and thus not a square either) — are needed to bridge the gap between rigorous mathematical definitions and informal human concepts. To be more precise, we tend unconsciously to presume “non-square” when we think “rhombus” — and yet square rhombi exist, of course. However, they are such a special case that we have to distinguish them from “normal” rhombi, which are not square. Once again, we find ourselves square in the territory of marking. For example, we could posit two new categories,
rhombus
₁
and
rhombus
₂
, analogous to
car
₁
and
car
₂
. Just as the marked category
car
₁
excludes trucks while the unmarked category
car
₂
includes them,
rhombus
₁
would exclude squares while
rhombus
₂
would include them.

Every square is thus a member of
rectangle
₂
but not necessarily of
rectangle
₁
, and every rectangle (whether of type 1 or 2) is a member of
parallelogram
₂
but not necessarily of
parallelogram
₁
. And so we see that the various types of quadrilaterals fall into a more complex diagram than the one we saw earlier. It looks as follows:

There are now eight categories, organized on four levels of abstraction; all eight are needed to capture what we might call “expert knowledge”, which comes from two sources: first, an understanding of rigorous and formal mathematical definitions, and second, a sense for how these words are actually used in various contexts by someone who has fully mastered this small but slippery mini-domain of math. It’s not hard to see why this can give schoolchildren headaches.

We aren’t saying that this kind of hierarchical diagram is an explicit structure (like the sequence of letters of the alphabet) that gets consciously committed to memory by all people who understand quadrilaterals well. To the contrary, no memorization is involved at all. If one has a mathematical bent, each link in the diagram comes directly from a clear understanding of how two particular concepts are related, and the consequences flow out almost trivially. And yet ironically, this kind of crystal-clear understanding rests tacitly on an ambiguous way of using words: sometimes a square is seen
in contrast to
a rectangle, while other times it is seen as
a kind of
rectangle.

All this goes to show that even in mathematics, where utter precision is expected, it’s commonplace to have terms that stand for two or more concepts, hence are ambiguous. In this particular case, to be an expert in the domain, one needs to have various categories such as
non-square rectangles
(that is to say,
rectangle
₁
), and
rectangles whether square or not
(that is,
rectangle
₂
). Altogether, the number of such terms comes to eight, which exceeds the number of lexical items by three. And it is by no means easy for people to acquire the proper organization of the categories associated with the words “square”, “rectangle”, “rhombus”, “parallelogram”, and “quadrilateral”.

Indeed, a study we undertook in France showed that the majority of university and middle-school students are unaware of how some of these categories include others, and many simply refuse to accept the idea that squares are rectangles or rhombi. They may go so far as to invent new properties, if one asks them for definitions. For example, they may insist, “A rectangle has to be wider than it is high” or “A rectangle has two pairs of equal-length sides, but not all four sides have the same length”, or then again, “A rhombus has four equal-length sides, but it has no right angles”.

This reveals that the most general interpretation of these words — that is, the unmarked sense — is usually not recognized, and that students’ improvised definitions mostly describe
marked
categories (such as
non-square rectangle, non-square rhombus
, etc.). The most frequently observed idea in the minds of these students, all of whom had supposedly mastered all of these notions (indeed, these matters were very fresh in the minds of the middle-school participants, and were assumed to be part of the background knowledge of the undergraduate participants), is just a two-tier hierarchy with
quadrilateral
on top, and
parallelogram, rectangle, rhombus
, and
square
below.

Students who gave slightly more sophisticated answers tended to add just one middle level —
parallelogram
— which they interpolated between the top level (where
quadrilateral
was found) and the bottom (on which resided
rectangle, rhombus
, and
square
):

Our investigations showed that an expert-like hierarchy having
four
levels (as in the figure two pages back) was present in only a very small percentage of the participants.

As we have just seen, part and parcel of becoming an expert in a domain involves, over and above possessing many categories, organizing them efficiently. Thus, to understand quadrilaterals well, one has to know a good deal more than just the fact that squares, rectangles, rhombi, and parallelograms are all special types of quadrilaterals; one also has to know the relationships that interconnect them. This theme — that an efficient organization of categories is part and parcel of expertise — is important, because expertise is far from being limited to narrow, technical domains. Expertise is what anyone acquires who has deep knowledge of
any
domain; everyone is an expert in their everyday environment, as well as in their profession and in their various hobbies. To be sure, expertise does not always require high degrees of creativity or insight, although of course it doesn’t exclude those.

You might aspire to become a past master in dog breeding, or perhaps in the etymology of English words, or in the varieties of tea, or in the world of banking, or in Spanish literature, or opera or history or anatomy… If you succeed, some will call you a “walking encyclopedia” of your specialty. Mr. Martin takes great pleasure in spending afternoons reading about dogs. Aside from learning the names of many breeds, including “poodle”, “bulldog”, “German shepherd”, “basset”, “chihuahua”, “golden retriever”, “grayhound”, “pit bull”, “borzoi”, “Saint Bernard”, “fox terrier”, “Dalmatian”, “doberman”, “labrador”, “dachshund”, “spaniel”, and so on, he has
learned to recognize them all in photos. He has also learned various little factoids about each breed, such as that golden retrievers have a very gentle character, that German shepherds have almond-shaped eyes and ears that stand up, that greyhounds are very attached to their masters, that bulldogs may have originated in Tibet, that Dalmatians are the mascots of firehouses, that chihuahuas are the smallest breed, that fox terriers owe their existence to fox-hunting, that Saint Bernards are named after a hospice in the Swiss Alps, that labradors are excellent guide dogs, that poodles are probably descended from water spaniels, and so forth.

This is typical of anyone who is tackling a new domain. One familiarizes oneself with the most salient entities belonging to the domain and one learns to tell them apart, since one surely can’t claim know a domain well with recognizing its principal denizens. Thus categorization involves being able to make
distinctions.
But since the categories constituting any domain are also linked by many types of relationships, acquiring a new category means connecting it mentally with one or more prior categories; thus categorization also involves making
associations
.

The distinction between
genus
and
species
, borrowed from biology, is another way of describing this process. Any new entity belongs to a certain
species
(a narrow category), which in turn belongs to a certain
genus
(a wider category). Thus, the first time you run into a certain animal you might simply call it a
dog
, and later you might learn that it is a
dachshund.
Possessing or acquiring such knowledge gives one the sense of having reached a decent level of mastery, even if one is not yet an expert. The simple act of categorizing an entity as a
dog
allows one to tap into prior stocks of knowledge one has, such as
barks
;
has a stomach
;
may bite
;
may have and transmit rabies; is prone to drooling
;
has a life expectancy of between ten and fifteen years; may be dangerous to children and even adults;
and so on. To these general pieces of knowledge one can add specific facts about golden retrievers, such as
has golden fur
;
has a gentle disposition
;
is rather large
;
weighs between 60 and 90 pounds
;
is playful
; and so on. Just as Mr. Martin proceeds in this manner as he learns about dogs, so do we all as we learn about any domain.

Is this what becoming knowledgeable in a specific area is all about? Does the acquisition of knowledge consist in learning about more and more species that belong to a given domain? Is a beginning enthusiast someone who has acquired ten or fifteen categories, all subsumed under one umbrella category, and who knows a little bit about each particular one? Does an expert differ from this in having hundreds of examples all of which are richly mentally annotated?

Yes, in part; but expertise goes considerably beyond that. Aside from certain very technical and esoteric domains, being an expert does not mean having memorized a list, even if the list is enriched with specific tidbits about each of its items. If Mr. Martin wished to become an expert on his town’s telephone directory, then memorization would suffice; he would be an expert simply by virtue of having learned many names, addresses, and phone numbers by heart. But in more conceptual domains, expertise doesn’t come from amassing knowledge of ever more species belonging to a single genus; to be sure, experts will generally have more species under their belt than will novices, but that is by no means the full story.