Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (93 page)

Although the closing lines of La Fontaine’s actual poem appear to lack any such incongruity, that impression is wrong, since it turns out that cicadas are not carnivorous, and so they have no use for flies or worms. The innocuous-seeming assumption that cicadas feed on small creatures of about their own size is simply erroneous. The famous French biologist and science writer Jean-Henri Fabre, in his autobiographical memoirs entitled “Entomological Souvenirs”, observed that cicadas have only a sucking tube with which to nourish themselves. In their larval stage, they get their nourishment from the sap of roots, and when they reach maturity, they suck sap from the branches of various trees and bushes. The plausible-seeming image of what cicadas eat is simply based on a naïve analogy with people or farm animals or other types of insects. La Fontaine’s poem would have been far more faithful to the true nature of cicadas if it had run this way:

If the great La Fontaine was so naïve, perhaps we should not judge Timothy, Janet, or Professor Alexander too harshly.

As people move into higher realms of abstraction, naïve analogies, with all their strengths and weaknesses, inevitably become trusted guides. The strengths of such analogies derive from their easy availability in long-term memory, in the form of efficient and ready-to-use mental structures. And their weaknesses stem from the fact that in certain contexts they are misleading. Naïve analogies are like skiers who sail with grace down well-groomed slopes but who are utterly lost in powder. In sum, naïve analogies work well in many situations, but in other situations they can lead to absurd conclusions or complete dead ends.

What the study of naïve analogies tells us about the human mind is of paramount importance for education, and this chapter is therefore oriented to some extent towards the educational payoffs connected with our ideas. A certain number of entrenched ideas about what it means to
learn
and to
know
will be called into question, and some new directions for education will be suggested. Below we list three key ideas that we will discuss in this chapter and the following one.

First of all, ideas that are presented in school classrooms are understood via naïve analogies; that is, children unconsciously make analogies to simple and familiar events and ideas, and these unconscious analogies will control how they will incorporate new concepts.

Secondly, naïve analogies are in general not eliminated by schooling. When teaching has an effect on a student, it usually just fine-tunes the set of contexts in which the student is inclined to apply a naïve analogy. The naïve notion does not displace the new concept being taught, but coexists with it. Both types of knowledge can then be exploited by a learner, but they will be useful in different contexts. And this is fortunate, since banishing naïve analogies from people’s minds would be extremely harmful. For example, looking at the world from the point of view of a professional physicist in everyday situations would often be hopelessly shackling. A physicist who sees a glass start to fall floorwards doesn’t need to wheel out Newton’s laws of motion and his universal law of gravitation in order to figure out what’s about to happen. Reaching out to catch the glass that’s about to be shattered is a straightforward consequence of pre-Newtonian, nontechnical world knowledge. And if two astronomy students are walking hand in hand on the beach admiring the pink-and-orange sunset, they most likely are not in the least thinking about the fact that it’s the earth that’s turning rather than the sun that is descending, and it’s most probable that they are enjoying the beautiful colors and the romantic feelings in much the same way as any other couple would.

Finally, a formal description of a given subject matter does not reflect the type of knowledge that allows one to feel comfortable in thinking about the domain. Humans do not generally feel comfortable manipulating formal structures; when faced with a new situation, they favor non-formal approaches. Learning is thus the building-up not of logical structures but of well-organized repertoires of categories that themselves are under continual refinement.

Familiarity is crucial in analogy-making for the simple reason that, in order to deal with an unknown situation, one intuitively feels more secure about extrapolating what one knows well than what one barely knows. We don’t mean that such choices are made consciously; they take place below one’s level of awareness. Thus unconscious analogical processes dominate the way we interact with our environment, forming the very basis of our understanding of the world and the situations we find ourselves in.

Quite obviously, we are not equally familiar with all the things that surround us. Certain notions seem totally natural to us, and others very little so. The notions of
addition, equality, adjective, verb, continent
, and
planet
strike us as familiar, while the notions of
partial differential equation, Fourier series, topological space, spinor, lepton, electrophoresis
, and
nucleotide synthesis
exude considerable strangeness for most of us. This is the case not only for abstract notions of that sort, but also for concrete objects. For most of us, rockets are less familiar than cars, and household robots are less familiar than computers. A category’s familiarity has to do with how much one has been exposed to it, with the amount of knowledge one has of it, and with the degree of confidence one has in one’s knowledge about it. One feels more comfortable with cars than rockets because one has seen many more of them, because one knows much more about them, and because one has greater clarity about how to get into and out of them, about what their control devices (steering wheel, brake, etc.) will do, and so forth. One is also, for similar reasons, far more familiar with gravity than with electromagnetism.

Familiarity has been studied in certain psychological experiments that explore the effect of the degree of entrenchment of a category. For instance, cognitive psychologist Lance Rips showed that a new piece of knowledge is more readily transferred from a
typical
member of a category to an
atypical
member of the same category than the reverse. For example, if participants in an experiment are told that robins are susceptible to a certain disease, they are likely to conclude that hawks, too, might catch this disease, whereas in the reverse situation, in which they are told that hawks are often afflicted with a certain disease, the chance is much smaller that they will infer that robins will suffer from it as well. The greater familiarity of the category
robin
than that of
hawk
is the source of this asymmetry. This means that the more entrenched a concept is (here,
robin
being more entrenched than
hawk
), the more likely it is to act as the source of an analogy.

A set of experiments conducted by Susan Carey, a developmental psychologist, on children of various ages and on adults as well, led to similar conclusions. Participants were told that all the members of a certain category —
dogs
, for instance — have a certain property, such as possessing an internal organ called an “omentum”. They were then asked if members of other categories, such as
humans
or
bees
, were also likely to have this internal organ. For very young children, but not for adults, the existence of such an organ was more easily transferred from
person
to
dog
than from
dog
to
person.
The explanation for this is that in the minds of young children, the concept of
person
is more entrenched than the concept of
dog.
Adults were perfectly happy extending
possession of such an organ to any species of animal whatsoever, as long as they had been told that two very different kinds of animals (such as dogs and bees) both possessed it. Young children’s thought processes were different; they would decide whether or not to extend a property to a new kind of animal depending on that animal’s perceived proximity to either of the two species (
i.e.
, depending on the strength of the analogy in question), and they did not use the superordinate category
animal
to make the analogy, because that abstract category was not sufficiently familiar to them.

A narrow vision of learning that we would be happy to see fully eliminated from educational programs presumes that knowledge acquired in school is independent of everyday knowledge. This philosophy would encourage the teaching of new ideas without any reference to everyday concepts, except for the most rudimentary notions, which are impossible to separate from everyday experience. Such a limited view of learning is based on a conception of the mind according to which we would keep facts and ideas that we pick up in school in a separate mental compartment from facts and ideas that we pick up in daily life, and year after year we would increase our school-based knowledge much as we would build a brick house: each brick added would be supported by previously installed bricks, and would in turn support new bricks.

Such a vision is appealing in some ways, since it would simplify the design of school curricula. It would make teaching easier, treating each discipline as an island disconnected from other disciplines, and trying, within any given discipline, to decompose its notions in a logical fashion and to arrange them in a strict, natural order. Teaching any complex idea would thus involve teaching a set of simpler ideas in the same domain, each of which would in turn be taught through yet simpler ideas in the same domain, and so forth.

It is a long-lived myth in the world of education that there is a watertight boundary between two types of knowledge: everyday knowledge, presumed to grow on its own with no need for formal teaching, and formal knowledge, conveyed in schools and presumed to be communicable independently of everyday knowledge. In this naïve view, school is seen as a magical shortcut that allows ideas arduously developed by humanity over thousands of years to be transmitted in just a few years to a random human being.

Another belief that exerts considerable influence on school curricula, particularly in science, is that scientific knowledge is best conveyed in precise, formal terms — especially through mathematical formulas — and that precise formalisms correspond to the way in which knowledge ought to be absorbed by beginners and in which it is manipulated by experts. It is presumed that the initially large gap between a printed formalism and one’s internal mental representation gradually approaches zero as one comes closer and closer to expertise.

While we’re on the subject of distances approaching zero, the following rather prickly formula expresses the idea that the function
f
(
x
) is continuous at the point
x
₀
:

(The upside-down “A” and the backwards “E” are shorthand notations for the words “for all” and “there exists”, and the arrow “⇒” can be read as “if… then…”.) Should one presume that all people who are comfortable with mathematics think of continuity in exactly these terms? Do people fluent in calculus really always imagine and mentally manipulate Greek letters, and is this what continuity means for them? Are all their prior intuitive notions of continuity totally dispensed with after they have absorbed this formula? If so, then in order to educate experts, or even to convey technical notions of this sort to ordinary students, the right way would be to transmit just such formulas, since they presumably embody the most distilled and precise essence of the notions. In particular, one would want the notion of
continuity
to be seen by students as synonymous with the above-displayed formula.

However, this view of education unfortunately conflates the actual way that experts think with the use of dense formalisms designed to capture subtle notions as rigorously and unambiguously as possible. It is the result of a long-standing philosophical assumption, reinforced by the broader culture, which is that logical thinking is superior to analogical thinking. More specifically, this view comes from misguided stereotypes of analogical thinking, which maintain that dependence on analogy, although possibly useful when one is just starting out in a field, is basically childish, and that analogies should rapidly be shed, like crutches or training wheels, when one gets down to brass tacks and starts seriously thinking in the domain. A related half-baked stereotype of analogical thinking is that it is like an untamable wild horse, so unpredictable and unreliable that it must be shunned, even if it might once in a while provide a spark of true insight; thus analogies belong not to the realm of reason but to that of “intuitions”, which, being irrational, cannot and should not be taught.