Please read this chapter and write down one word, from every sentence ( I have uploaded for you)Please then make a word puzzle out of these words, intersecting them and crossing them, using letters as intersections.This word-puzzle won’t be a game to play so it doesn’t have to be perfect just try making it into a puzzle, but it will be a design and art piece to visually express the words that may have stayed with you from the reading.Please choose words that are interesting – or that you don’t know – or words that vibrate with meaning and richness for you.Please then scan or take a picture of this word-puzzle (please keep it all on one page) and it should be in hand written.THE PROCESS OF EDUCATION
inferred is the only known way of reducing the quick
rate of loss of human memory.
Designing curricula in a way that reflects the basic
structure of a field of knowledge requires the most
fundamental understanding of that field. It is a task that
cannot be carried out without the active participation
of the ablest scholars and scientists. The experience of
the past several years has shown that such scholars and
scientists, working in conjunction with experienced
teachers and students of child development, can prepare
curricula of the sort we have been considering. Much
more effort in the actual preparation of curriculum
materials, in teacher training, and in supporting research
will be necessary if improvements in our educational
practices are to be of an order that will meet the challenges of the scientific and social revolution through
which we are now living.
There are many problems of how to teach general
principles in a way that will be both effective and interesting, and several of the key issues have been passed
in review. What is abundantly clear is that much work
remains to be done by way of examining currently effective practices, fashioning curricula that may be tried out
on an experimental basis, and carrying out the kinds of
research that can give support and guidance to the general effort at improving teaching.
How may the kind of curriculum we have been discussing be brought within the intellectual reach of
children of different ages? To this problem we turn next.
W E begin with the hypothesis that any subj ect
can be taught effectively in some intellectually honest
form to any child at any stage of development. It is a
bold hypothesis and an essential one in thinking about
the nature of a curriculum. No evidence exists to contradict it; considerable evidence is being amassed that
supports it.
To make clear what is implied, let us examine three
general ideas. The first has to do with the process of
intellectual development in children, the second with
the aCt of learning, and the third with the notion of the
“spiral curriculum” introduced earlier.
Intellectual development. Research on the intellectual
development of the child highlights the fact that at each
stage of development the child has a characteristic way
of viewing the world and explaining it to himself. The
task of teaching a subject to a child at any particular
age is one of representing the structure of that subject in
terms of the child’s way of viewing things. The task can
be thought of as one of translation. The general hypothesis that has just been stated is premised on the considered
judgment that any idea can be represented honestly and
usefully in the thought forms of children of school age,
and that these first representations can later be made
more powerful and precise the more easily by virtue of
this early learning. To illustrate and support this view,
we present here a somewhat detailed picmre of the
course of intellectual development, along with some suggestions about teaching at different stages of it.
The work of Piaget and others suggests that, roughly
speaking, one may distinguish three stages in the intellecmal development of the child. The first stage need
not concern us in detail, for it is characteristic principally
of the pre-school child. In this stage, which ends (at
least for Swiss school children) around the fifth or sixth
year, the child’s mental work consists principally in
establishing relationships between experience and action;
his concern is with manipulating the world through
action. This stage corresponds roughly to the period
from the first development of language to the point at
which the child learns to manipulate symbols. In this
so-called preoperational stage, the principal symbolic
achievement is that the child learns how to represent the
external world through symbols established by simple
generalization; things are represented as equivalent in
terms of sharing some common property. But the child’s
symbolic world does not make a clear separation between
internal motives and feelings on the one hand and external reality on the other. The sun moves because God
pushes it, and the stars, like himself, have to go to bed.
The child is little able to separate his own goals from the
means for achieving them, and when he has to make
corrections in his activity after unsuccessful attempts at
manipulating reality, he does so by what are called intuitive regulations rather than by symbolic operations,
the former being of a crude trial-and-error nature rather
than the result of taking thought.
What is principally lacking at this stage of develop34
ment is what the Geneva school has called the concept
of reversibility. When the shape of an object is changed,
as when one changes the shape of a ball of plasticene,
the preoperational child cannot grasp the idea that it
can be brought back readily to its original state. Because
of this fundamental lack the child cannot understand
cenain fundamental ideas that lie at the basis of mathematics and physics-the mathematical idea that one conserves quantity even when one partitions a set of things
into subgroups, or the physical idea that one conserves
mass and weight even though one transforms the shape
of an object. It goes without saying that teachers are
severely limited in transmitting concepts to a child at
this stage, even in a highly intuitive manner.
The second stage of development-and now the child
is in school-is called the stage of concrete operations.
This stage is operational in contrast to the preceding
stage, which is merely active. An operation is a type of
action: it can be carried out rather directly by the
manipulation of objects, or internally, as when one
manipulates the symbols that represent things and relations in one’s mind. Roughly, an operation is a means
of getting data about the real world into the mind and
there transforming them so that they can be organized
and used selectively in the solution of problems. Assume
a child is presented with a pinball machine which
bounces a ball off a wall at an angle. Let us find out what
he appreciates about the relation between the angle of
incidence and the angle of reflection. The young child
sees no problem: for him, the ball travels in an are,
touching the wall on the way. The somewhat older
child, say age ten, sees the two angles as roughly related
-as one changes so does the other. The still older child
begins to grasp that there is a fixed relation between the
two, and usually says it is a right angle. Finally, the
thirteen- or fourteen-year-old, often by pointing the
ejector directly at the wall and seeing the ball come back
at the ejector, gets the idea that the two angles are equal.
Each way of looking at the phenomenon represents the
result of an operation in this sense, and the child’s thinking is constrained by his way of pulling his observations
An operation differs from simple action or goaldirected behavior in that it is internalized and reversible.
“Internalized” means that the child does not have to go
about his problem-solving any longer by overt trial and
error, but can actually carry out trial and error in his
head. Reversibiliry is present because operations are seen
as characterized where appropriate by what is called
“complete compensation”; that is to say, an operation
can be compensated for by an inverse operation. If
marbles, for example, are divided into subgroups, the
child can grasp intuitively that the original collection of
marbles can be restored by being added back together
again. The child tips a balance scale too far with a
weight and then searches systematically for a lighter
weight or for something with which to get the scale
rebalanced. He may carry reversibiliry too far by assuming that a piece of paper, once burned, can also be restored.
With the advent of concrete operations, the child
develops an internalized structure with which to operate. In the example of the balance scale, the structure is
a serial order of weights that the child has in his mind.
Such internal structures are of the essence. They are the
internalized symbolic systems by which the child represents the world, as in the example of the pinball machine
and the angles of incidence and reflection. It is into the
language of these internal structures that one must translate ideas if the child is to grasp them.
But concrete operations, though they are guided by
the logic of classes and the logic of relations, are means
for structuring only immediately present reality. The
child is able to give structure to the things he encounters,
but he is not yet readily able to deal with possibilities
not directly before him or not already experienced.
This is not to say that children operating concretely
are not able to anticipate things that are not present.
Rather, it is that they do not command the operations
for conjuring up systematically the full range of alternative possibilities that could exist at any given time. They
cannot go systematically beyond the infortnation given
them to a description of what else might occur. Somewhere between ten and fourteen years of age the child
passes into a third stage, which is called the stage of
“formal operations” by the Geneva school.
Now the child’s intellectual activity seems to be
based upon an ability to operate on hypothetical propositions rather than being constrained to what he has
experienced or what is before him. The child can now
think of possible variables and even deduce potential
relationships that can later be verified by experiment or
observation. Intellectual operations now appear to be
predicated upon the same kinds of logical operations
that are the stock in trade of the logician, the scientist,
or the abstract thinker. It is at this point that the child
is able to give formal or axiomatic expression to the concrete ideas that before guided his problem-solving but
could not be described or formally understood.
Earlier, while the child is in the stage of concrete
operations, he is capable of grasping intuitively and
concretely a great many of the basic ideas of mathematics,
the sciences, the humanities, and the social sciences.
But he can do so only in terms of concrete operations.
It can be demonstrated that fifth-grade children can
play mathematical games with rnIes modeled on highly
advanced mathematics; indeed, they can arrive at these
rules inductively and learn how to work with them.
They will flounder, however, if one attempts to force
upon them a formal mathematical description of what
they have been doing, though they are perfectly capable
of guiding their behavior by these rnIes. At the Woods
Hole Conference we were privileged to see a demonstration of teaching in which fifth-grade children very
rapidly grasped central ideas from the theory of functions, although had the teacher attempted to explain to
them what the theory of functions was, he would have
drawn a blank. Later, at the appropriate stage of development and given a certain amount of practice in concrete operations, the time would be ripe for introducing
them to the necessary formalism.
What is most important for teaching basic concepts
is that the child be helped to pass progressively from
concrete thinking to the utilization of more conceptually
adequate modes of thought. But it is futile to attem?t
this by resentin formal explanations based on a 10 IC
tfiat IS !Stant from t e c
s manner of t”
e in its implicanons for hlffi. Much teaching
mathematics is 0 this sort. The child learns not to undermathematical order but rat er to apply certain
devices or reci es without understanding their significance and connecte ness.
ey are 0 trans ate mto
IUs way of thinking. Given this inappropriate start, he
is easily led to believe that the important thing is for him
to be “accurate”-though accuracy has less to do with
mathematics than with computation. Perhaps the most
striking example of this type of thing is to be found in
the manner in which the high school student meets
Euclidian geometry for the first time, as a set of axioms
and theorems, without having had some experience with
simple geometric configuratiQllS- and the intuitive means
whereby one deals with them. If the child were earlier
given the concepts and strategies in the form of intuitive
geometry at a level that he could easily follow, he might
be far better able to grasp deeply the meaning of the
theorems and axioms to which he is exposed later.
But the intellectual development of the child is no
clockwork sequence of events; it also responds to influences from the environment, notably the school environment. Thus instruction in scientific ideas, even at the
elementary level, need not follow slavishly the natural
course of cognitive development in the child. It can also
lead intellectual development by providing challenging
but usable opportunities for the child to forge ahead in
his development. Experience has shown that it is worth
the effort to provide the growing child with problems
that tempt him into next stages of development. As
David Page, one of the most experienced teachers of
elementary mathematics, has commented: “In teaching
I have been
from kindergarten to graduate
amazed at the intellectual similarity of human beings at
all ages, although children are perhaps more spontaneous,
creative, and energetic than adults. As far as I am concerned young children learn almost anything faster than
adults do if it can be given to them in terms they understand. Giving the material to them in terms they understand, interestingly enough, turns out to involve knowing the mathematics oneself, and the better one knows
it, the better it can be taught. It is appropriate that we
warn ourselves to be careful ·of assigning an absolute
level of difficulty to any particular topic. When I tell
mathematicians that fourth-grade students can go a
long way into ‘set theory’ a few of them reply: ‘Of
course.’ Most of them are startled. The latter ones are
completely wrong in assuming that ‘set theory’ is intrinsically difficult. Of course it may be that nothing is
intrinsically difficult. We JUSt have to Watt until the
proper point of view and corresponding TanguageTor
, ‘n (;jven particular subject matter
or a pamcular concept, It IS easy to ask trivial questions
or to lead the child to ask trivial questions. It is also easy
to ask impossibly difficult questions. The trick is to find
the medium questions that can be answered and that
rake you somewhere. This is the big job of teachers and
textbooks.” One leads the child by the well-wrought
“medium questions” to move more rapidly through the
stages of intellectual development, to a deeper understanding of mathematical, physical, and historical principles. We must know far more about the ways in which
this can be done.
Professor Inhelder of Geneva was asked to suggest
ways in which the child could be moved along faster
through the various stages of intellectual development
in mathematics and physics. What follows is part of a
memorandum she prepared for the Conference.
“The most elementary forms of reasoning-whether
logical, arithmetical, geometrical, or physical-rest on
the principle of the invariance of quantities: that the
whole remains, whatever may be the arrangement of its
parts, the change of its form, or its displacement in space
or time. The principle of invariance is no a priori datum
of the mind, nor is it the product of purely empirical
observation. The child discovers invariance in a manner
comparable to scientific discoveries generally. Grasping
the idea of invariance is beset with difficulties for the
child, often unsuspected by teachers. To the young
child, numerical wholes, spatial dimensions, and physical
quantities do not seem to remain constant but to dilate
or contraCt as they are operated upon. The total number
of beads in a box remains the same whether subdivided
into two, three, or ten piles. It is this that is so hard for
the child to understand. The young child perceives
changes as operating in one direction without being able
to grasp the idea that certain fundamental features of
things remain constant over change, or that if they
change the change is reversible.
“A few examples among many used in studying the
child’s concept of invariance will illustrate the kinds of
materials one could use to help him to learn the concept
more easily. The child transfers beads of a known quantity or liquids of a known volume from one receptacle
to another, one receptacle being tall and narrow, the
other flat and wide. The young child believes there is
more in the tall receptacle than the flat one. Now the
.” -If…
child can be confronted concretely with the nature of
one-to-one correspondence between two versions of the
same quantity. For there is an easy technique of checking: the beads can be counted or the liquid measured in
some standard way. The same operations work for the
conservation of spatial quantity if one uses a set of sticks
for length or a set of tiles for surface, or by having the
child transform the shape of volumes made up of the
same number of blocks. In physics dissolving sugar or
transforming the shapes of balls of plasticene while conserving volume provides comparable instruction. If teaching fails to bring the child properly from his perceptual,
primitive notions to a proper intuition of the idea of
invariance, the result is that he will count without having
acquired the idea of the invariance of numerical quantities. Or he will use geometrical measures while remaining
ignorant of the operation of transitivity-that if A includes B, and B includes C, then A also includes C. In
physics he will apply calculations to imperfectly understood physical notions such as weight, volume, speed,
and time. A teaching method that takes into account the
natural thought processes will allow the child to discover
such principles of invariance by giving him an opportunity to progress beyond his own primitive mode of
thinking through confrontation by concrete data-as
when he notes that liquid that looks greater in volume
in a tall, thin receptacle is in fact the same as that quantity
in a flat, low vessel. Concrete activity that becomes
increasingly formal is what leads the child to the kind
of mental mobility that approaches the naturally reversible operations of mathematics and logic. The child
gradually comes to sense that any change may be men42
tally cancelled out by the reverse operation-addition by
subtraction-or that a change may be counterbalanced
by a reciprocal change.
“A child often focuses on only one aspect of a phenomenon at a time, and this interferes with his understanding. We can set up little teaching experiments in
such a way that he is forced to pay attention to other
aspects. Thus, children up to about age seven estimate
the speed of two automobiles by assuming that the one
that gets there first is the faster, or that if one passes the
other it is faster. To overcome such errors, one can, by
using toy automobiles, show that two objects starting at
different distances from a finish line cannot be judged
by which one arrives first, or show that one car can pass
another by circling it and still not finish first. These are
simple exercises, but they speed the child toward attending to several features of a situation at once.
“In view of all this it seems highly arbitrary and very
likely incorrect to delay the teaching, for example, of
Euclidian or metric geometry until the end of the primary
grades, particularly when projective geometry has not
been given earlier. So too with the teaching of physics, ;
which has much in it that can be profitably taught at an
inductive or intuitive level much earlier. Basic notions in
these fields are perfectly accessible to children of seven
to ten years of age, provided that they are divorced from
their mathematical expression and studied through materials that the child can handle himself.
“Another matter relates particularly to the ordering
of a mathematics curriculum. Often the sequence of
psychological development follows more closely the
axiomatic order of a subject matter than it does the his-

torical order of development of concepts within the field.
One observes, for instance, that certain topological notions, such as connection, separation, being interior to,
and so forth, precede the formation of Euclidian and
projective notions in geometry, though the former ideas
are newer in their formalism in the history of mathematics than the latter. If any special justification were
needed for teaching the structure of a subject in its
proper logical or axiomatic order rather than its order
of historical development, this should provide it. This
is not to say that there may not be situations where the
historical order is important from the point of view of
its cultural or pedagogical relevance.
“As for teaching geometrical notions of perspective
and projection, again there is much that can be done by
the use of experiments and demonstrations that rest on
the child’s operational capaciry to analyze concrete experience. We have watched children work with an apparatus in which rings of different diameter are placed at
different positions between a candle and a screen with
a fixed distance between them so that the rings cast shadows of varying sizes on the screen. The child learns how
the cast shadow changes size as a function of the distance
of the ring from the light source. By bringing to the
child such concrete experience of light in revealing situations, we teach him maneuvers that in the end permit
him to understand the general ideas underlying projective geometry.
“These examples lead us to think that it is possible to
draw up methods of teaching the basic ideas in science
and mathematics to children considerably younger than
the traditional age. It is at this earlier age that systematic
instruction can lay a groundwork in the fundamentals
that can be used later and with great profit at the secondary level.
“The teaching of probabilistic reasoning, so very common and important a feature of modem science, is hardly
developed in our educational system before college. The
omission is probably due to the fact that school syllabi
in nearly all countries follow scientific progress with a
near-disastrous time lag. But it may also be due to the
widespread belief that the understanding of random
phenomena depends on the leamer’s grasp of the meaning of the rarity or commonness of events. And admittedly, such ideas are hard to get across to the young. Our
research indicates that the understanding of random
phenomena requires, rather, the use of cenain concrete
logical operations well within the grasp of the young
child-provided these operations are free of awkward
mathematical expression. Principal among these logical
operations are disjunction (‘either A or B is true’) and
combination. Games in which lots are drawn, games of
roulette, and games involving a gaussian distribution of
outcomes are all ideal for giving the child a basic grasp
of the logical operation needed for thinking about probability. In such games, children first discover an entirely
qualitative notion of chance defined as an uncertain
event, contrasted with deductive certainty. The notion
of probability as a fraction of cenainty is discovered
only later. Each of these discoveries can be made before
the child ever learns the techniques of the calculus of
probabilities or the formal expressions that normally go
with probability theory. Interest in problems of a
probabilistic nature could easily be awakened and de-
veloped before the introduction of any statistical
processes or computation. Statistical manipulation and
computation are only tools to be used after intnitive
understanding has been established. If the array of computational paraphernalia is introduced first, then more
likely than not it will inhibit or kill the development of
probabilistic reasoning.
“One wonders in the light of all this whether it might
not be interesting to devote the first two years of school
to a series of exercises in manipulating, classifying, and
ordering objects in ways that highlight basic operations
of logical addition, multiplication, inclusion, serial ordering, and the like. For surely these logical operations are
the basis of more specific operations and concepts of all
mathematics and science. It may indeed be the case that
such an early science and mathematics ‘pre-curriculum’
might go a long way toward building up in the child
the kind of intnitive and more inductive understanding
that could be given embodiment later in formal courses
in mathematics and science. The effect of such an approach would be, we think, to put more continuity into
science and mathematics and also to give the child a
much better and firmer comprehension of the concepts
which, unless he has this early foundation, he will mouth
later without being able to use them in any effective
A comparable approach can surely be taken to the
teaching of social studies and literature. There has been
little research done on the kinds of concepts that a child
brings to these subjects, although there is a wealth of
observation and anecdote. Can one teach the struCture
of literary forms by presenting the child with the first
part of a Story and then having him complete it in the
form of a comedy, a tragedy, or a farce-without ever
using such words? When, for example, does the idea of
“historical trend” develop, and what are its precursors
in the child? How does one make a child aware of literary style? Perhaps the child can discover the idea of style
through the presentation of the same content written in
drastically different styles, in the manner of Beerbohm’s
Christmas Garland. Again, there is no reason to believe
that any subject cannot be taught to any child at virtually any age in some form.
Here one is immediately faced with the question of
the economy of teaching. One can argue that it might
be better to wait until the child is thirteen or fourteen
before beginning geometry so that the projective and
intnitive first steps can immediately be followed up by
a full formal presentation of the subject. Is it worth
while to train the young inductively so that they may
discover the basic order of knowledge before they can
appreciate its formalism? In Professor Inhelder’s memorandum, it was suggested that the first two grades might
be given over to training the child in the basic logical
operations that underlie instruction in mathematics and
science. There is evidence to indicate that such rigorous
and relevant early training has the effect of making later
learning easier. Indeed the experiments on “learning set”
seem to indicate JUSt that-that one not only learns specifics but in so doing learns how to learn. So important
is training per se that monkeys who have been given
extensive training in problem solving suffer considerably
less loss and recover more qnickly after induced brain
damage than animals who had not been previously thus
educated. But the danger of such early training may be
that it has the effect of training out original but deviant
ideas. There is no evidence available on the subject, and
much is needed.
The act of learning. Learning a subject seems to involve three almost simultaneous processes. First there is
acquisition of new information-often information that
runs counter to or is a replacement for what the person
has previously known implicitly or explicitly. At the
very least it is a refinement of previous knowledge. Thus
one teaches a student Newton’s laws of motion, which
violate the testimony of the senses. Or in teaching a
student about wave mechanics, one violates the student’s
belief in mechanical impact as the sole source of real
energy transfer. Or one bucks the language and its builtin way of thinking in terms of “wasting energy” by introdncing the student to the conservation theorem in
physics which asserts that no energy is lost. More often
the situation is less drastic, as when one teaches the details of the circulatory system to a student who already
knows vaguely or intuitively that blood circulates.
A second aspect of learning may be called transformation-the process of manipulating knowledge to make it
fit new tasks. We learn to “unmask” or analyze information, to order it in a way that permits extrapolation or
interpolation or conversion into another form. Transformation comprises the ways we deal with information
in order to go beyond it.
A third aspect of learning is evaluation: checking
whether the way we have manipulated information is
adequate to the task. Is the generalization fitting, have
we extrapolated appropriately, are we operating proper48
ly? Often a teacher is crucial in helping with evaluation,
but much of it takes place by judgments of plausibility
without our actually being able to check rigorously
whether we are correct in our efforts.
In the learning of any subject matter, there is usually
a series of episodes, each episode involving the three
processes. Photosynthesis might reasonably comprise
material for a learning episode in biology, fitted into a
more comprehensive learning experience such as learning
about the conversion of energy generally. At its best a
learning episode reflects what has gone before it and
permits one to generalize beyond it.
A learning episode can be brief or long, contain many
ideas or a few. How sustained an episode a learner is
willing to undergo depends upon what the person expects to get from his efforts, in the sense of such external
things as grades but also in the sense of a gain in understanding.
We usually tailor material to the capacities and needs
of students by manipulating learning episodes in several
ways: by shortening or lengthening the episode, by
piling on extrinsic rewards in the form of praise and
gold stars, or by dramatizing the shock of recognition
of what the material means when fully understood. The
unit in a curriculum is meant to be a recognition of the
importance of learning episodes, though many units drag
on with no climax in understanding. There is a surprising lack of research on how one most wisely devises
adequate learning episodes for children at different ages
and in different subject matters. There are many questions that need answers based on careful research, and
to some of these we turn now.
There is, to begin with, the question of the balance
between extrinsic rewards and intrinsic ones. There has
been much written on the role of reward and punishment in learning, but very little indeed on the role of
interest and curiosity and the lure of discovery. If it is
our intention as teachers to inure the child to longer and
longer episodes of learning, it may well be that intrinsic
rewards in the form of quickened awareness and understanding will have to be emphasized far more in the
detailed design of curricula. One of the least discussed
ways of carrying a student through a hard unit of
material is to challenge him with a chance to exercise
his full powers, so that he may discover the pleasure of
full and effective functioning. Good teachers know the
power of this lure. Students should know what it feels
like to be completely absorbed in a problem. They
seldom experience this feeling in school. Given enough
absorption in class, some students may be able to carry
over the feeling to work done on their own.
There is a range of problems that have to do with
how much emphasis should be placed on acquisition,
transformation, and evaluation in a learning episodegetting facts, manipulating them, and checking one’s
ideas. Is it the case, for example, that it is best to give
the young child a minimum set of facts first and then
encourage him to draw the fullest set of implications
possible from this knowledge? In short, should an episode for a young child contain little new information
but emphasize what can be done to go beyond that bit
on one’s own? One teacher of social studies has had
great success with fourth-graders through this approach:
he begins, for example, with the fact that civilizations
have most often begnn in fertile river valleys-the only
“fact.” The students are encouraged in class discussion
to fignre out why this is the case and why it wonld be
less likely for civilizations to start in mountainous country. The effect of this approach, essentially the technique of discovery, is that the child generates information on his own, which he can then check or evaluate
against the sources, getting more new information in
the process. This obviously is one kind of learning episode, and doubtless it has limited applicability. What
other kinds are there, and are some more appropriate
to certain topics and ages than others? It is not the case
that “to learn is to learn is to learn,” yet in the research
literature there appears to be little recognition of differences in learning episodes.
With respect to the optimum length of a learning
episode, there are a few commonsense things one can
say about it, and these are perhaps interesting enough
to suggest frnieful research possibilities. It seems fairly
obvious, for example, that the longer and more packed
the episode, the greater the pay-off must be in terms of
increased power and understanding if the person is to
be encouraged to move to a next episode with zest.
Where grades are used as a substitute for the reward
of understanding, it may well be that learning will cease
as soon as grades are no longer given-at graduation.
It also seems reasonable that the more one has a sense
of the structure of a subject, the more densely packed
and longer a learning episode one can get through without fatigne. Indeed, the amount of new’ information in
any learning episode is really the amount that we cannot
qnite fit into place at once. And there is a severe limit,
as we have already noted, on how much of such unassimilated information we can keep in mind. The estimate is that adults can handle about seven independent
items of information at a time. No norms are available
for children-a deplorable lack.
There are many details one can discuss concerning
the shaping of learning episodes for children, but the
problems that have been mentioned will suffice to give
their flavor. Inasmuch as the topic is central to an understanding of how one arranges a curriculum, it seems
obvious that here is an area of research that is of the
first importance.
The “spiral curriculum.” If one respects the ways of
thought of the growing child, if one is courteous
enough to translate material into his logical forms and
challenging enough to tempt him to advance, then it
is possible to introduce him at an early age to the ideas
and styles that in later life make an educated man. We
might ask, as a criterion for any subject taught in primary school, whether, when fully developed, it is worth
an adult’s knowing, and whether having known it as a
child makes a person a better adult. If the answer to
both questions is negative or ambiguous, then the material is cluttering the curriculum.
If the hypothesis with which this section was introducedis true-that any subject can be taught to any
child in some honest form-then it should follow that
a curriculum ought to be built around the great issues,
principles, and values that a society deems worthy of
the continual concern of its members. Consider two
examples-the teaching of literarure and of science. If
it is granted, for example, that it is desirable to give
children an awareness of the meaning of human tragedy
and a sense of compassion for it, is it not possible at the
earliest appropriate age to teach the literarure of tragedy
in a manner that illuminates but does not threaten?
There are many possible ways to begin: through a retelling of the great myths, through the use of children’s
classics, through presentation of and commentary on
selected films that have proved themselves. Precisely
what kinds of materials should be used at what age with
what effect is a subject for research-research of several
kinds. We may ask first about the child’s conception of 41.,. “••.1
tragic. and here one might proceed in mucn tne same 1;’, …. Or
“”ay that Piaget and his colleagues have proceeded In
srudying the child’s conception of physical causality, . tcH”1
of morality, of number, and the rest. It is only when ‘of
we are e ui ed with such knowledge that we wdl be eJ.Jc..
ever we present to him into his own su Jectlve terms.
Nor need we wait for all the research findings to be in
before proceeding, for a skillful teacher can also experiment by attempting to teach what seems to be inruitively
right for children of different ages, correcting as he
goes. In time, one goes beyond to more complex versions
of the same kind of literarure or simply revisits some of
the same books used earlier. What matters is that later
teaching build upon earlier reactions to literarure, that
it seek to create an ever more explicit and marure understanding of the literarure of tragedy. Any of the great
literary forms can be handled in the same way, or any
of the great themes-be it the form of comedy or the
theme of identity, personal loyalty, or what not.
So too in science. If the understanding of number,
measure, and probability is judged crucial in the pursuit
of science, then instruction in these subjects should begin
as intellectually honestly and as early as possible in a
manner consistent with the child’s forms of thought. Let
the topics be developed and redeveloped in later grades.
Thus, if most children are to take a tenth-grade unit in
biology, need they approach the subject cold? Is it not
possible, with a minimum of formal laboratory work if
necessary, to introduce them to some of the major biological ideas earlier, in a spirit perhaps less exact and more
Many curricula are originally planned with a guiding
idea much like the one set forth here. But as curricula
are actually executed, as they grow and change, they
often lose their original form and suffer a relapse into
a certain shapelessness. It is not amiss to urge that acmal
curricula be reexamined with an eye to the issues of
continuity and development referred to in the preceding
pages. One cannot predict the exact forms that revision
might take; indeed, it is plain that there is now available
too little research to provide adequate answers. One can
only propose that appropriate research be undertaken
with the greatest vigor and as soon as possible.
MUCH has been said in the preceding chapters
about the importance of a student’s intuitive, in contrast
to his formal, understanding of the subjects he encounters. The emphasis in much of school learning and student examining is upon explicit formulations, upon the
ability of the student to reproduce verbal or numerical
formulae. It is not clear, in the absence of research,
whether this emphasis is inimical to the later development of good intuitive understanding-indeed, it is even
unclear what constitutes intuitive understanding. Yet we
can distinguish between inarticulate genius and articulate
idiocy-the first represented by the student who, by his
operations and conclusions, reveals a deep grasp of a subject but not much ability to “say how it goes,” in contrast
to the student who is full of seemingly appropriate words
but has no matching ability to use the ideas for which
the words presumably stand. A careful examination of
the nature of intuitive thinking might be of great aid to
charged with curriculum construction and teachmg.
Mathematicians, physicists, biologists, and others stress
the value of intuitive thinking in their respective areas.
In mathematics, for example, intuition is used with two
rather different meanings. On the one hand, an individual is said to think intuitively when, having worked for

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