Reprinted by permission of the publisher from Taylor, R., Ed., The Computer in School: Tutor, Tool, Tutee, (New York: Teachers College Press, © 1980 by Teachers College, Columbia University. All rights reserved.), pp. 161-176. To order copies, please contact www.teacherscollegepress.com. All rights reserved.
Teaching Children Thinking
Professor Emeritus, MIT
The phrase "technology and education" usually means inventing new
gadgets to teach the same old stuff in a thinly disguised version of the same
old way. Moreover, if the gadgets are computers, the same old teaching becomes
incredibly more expensive and biased towards its dullest parts, namely the kind
of rote learning in which measurable results can be obtained by treating the
children like pigeons in a Skinner box.
The purpose of this essay is to present a grander vision of an educational
system in which technology is used not in the form of machines for processing
children but as something the child himself will earn to manipulate, to extend,
to apply to projects, thereby gaining a greater and more articulate mastery
of the world, a sense of the power of applied knowledge and a self-confidently
realistic image of himself as an intellectual agent. Stated more simply, I believe
with Dewey, Montessori, and Piaget that children learn by doing and by thinking
about what they do. And so the fundamental ingredients of educational innovation
must be better things to do and better ways to think about oneself doing these
I claim that computation is by far the richest known source of these ingredients.
We can give children unprecedented power to invent and carry out exciting projects
by providing them with access to computers, with a suitably clear and intelligible
programming language and with peripheral devices capable of producing on-line
Examples are: spectacular displays on a color scope, battles between computer
controlled turtles, conversational programs, game-playing heuristic programs,
etc. Programmers can extend the list indefinitely. Others can get the flavor
of the excitement of these ideas from movies I shall show at the IFIPS meeting.
Thus in its embodiment as the physical computer, computation opens a vast
universe of things to do. But the real magic comes when this is combined with
the conceptual power of theoretical ideas associated with computation.
Computation has had a profound impact by concretizing and elucidating many
previously subtle concepts in psychology, linguistics, biology, and the foundations
of logic and mathematics. I shall try to show how this elucidation can be projected
back to the initial teaching of these concepts. By doing so, much of what has
been most perplexing to children is turned to transparent simplicity; much of
what seemed most abstract and distant from the real world turns into concrete
instruments familiarly employed to achieve personal goals.
Mathematics is the most extreme example. Most children never see the point
of the formal use of language. They certainly never had the experience of
making their own formalism adapted to a particular task. Yet anyone who works
with a computer does this all the time. We find that terminology and concepts
properly designed to articulate this process are avidly seized by the children
who really want to make the computer do things. And soon the children have
become highly sophisticated and articulate in the art of setting up models
and developing formal systems.
The most important (and surely controversial) component of this impact is
on the child's ability to articulate the working of his own mind and particularly
the interaction between himself and reality in the course of learning and thinking.
This is the central theme of this paper, and I shall step back at this point
to place it in the perspective of some general ideas about education. We shall
return later to the use of computers.
2. The Don't-Think-About-Thinking Paradox
It is usually considered good practice to give people instruction in their
occupational activities. Now, the occupational activities of children are learning,
thinking, playing and the like. Yet, we tell them nothing about those things.
Instead, we tell them about numbers, grammar, and the French revolution; somehow
hoping that from this disorder the really important things will emerge all by
themselves. And they sometimes do. But the alienation-dropout-drug complex is
certainly not less frequent.
In this respect it is not a relevant innovation to teach children also about
sets and linguistic productions and Eskimos. The paradox remains: why don't
we teach them to think, to learn, to play? The excuses people give are as paradoxical
as the fact itself. Basically there are two. Some people say: we know very little
about cognitive psychology; we surely do not want to teach such half-baked theories
in our schools! And some people say: making the children self-conscious about
learning will surely impede their learning. Asked for evidence they usually
tell stories like the one about a millipede who was asked which foot he moved
first when he walked. Apparently the attempt to verbalize the previously unconscious
action prevented the poor beast from ever walking again.
The paradox is not in the flimsiness of the evidence for these excuses. There
is nothing remarkable in that: all established doctrine about education has
similarly folksy foundations. The deep paradox resides in the curious assumption
that our choice is this: either teach the children half-baked cognitive
theory or leave them in their original state of cognitive innocence. Nonsense.
The child does not wait with a virginally empty mind until we are ready to stuff
it with a statistically validated curriculum. He is constantly engaged in inventing
theories about everything, including himself, schools and teachers. So the real
choice is: either give the child the best ideas we can muster about
cognitive processes or leave him at the mercy of the theories he invents or
picks up in the gutter. The question is: who can do better, the child or us'?
Let's begin by looking more closely at how well the child does.
3. The Pop-Ed Culture
One reads in Piaget's books about children re-inventing a kind of' Democritean
atomic theory to reconcile the disappearance of the dissolving sugar with their
belief in the conservation of matter. They believe that vision is made possible
by streams of particles sent out like machine gun bullets from the eyes and
even, at a younger age, that the trees make the wind by flapping their branches.
It is criminal to react (as some do) to Piaget's findings by proposing to teach
the children "the truth." For they surely gain more in their intellectual
growth by the act of inventing a theory than they can possibly lose by believing,
for a while, whatever theory they invent. Since they are not in the business
of making the weather, there is no reason for concern about their meteorological
unorthodoxy. But they are in the business of making minds—notably their
own—and we should consequently pay attention to their opinions about how
minds work and grow.
There exists amongst children, and in the culture at large, a set of popular
ideas about education and the mind. These seem to be sufficiently widespread,
uniform and dangerous to deserve a name, and I propose "The PopEd Culture.”
The following examples of Pop-Ed are taken from real children. My samples are
too small for me to guess at their prevalence. But I am sure very similar trends
must exist very widely and that identifying and finding methods to neutralize
the effects of Pop-Ed culture will become one of the central themes of research
EXAMPLES OF POP-ED THINKING
(a) Blank-Mind Theories. Asked how one sets about thinking a child
said: "Make your mind a blank and wait for an idea to come." This
is related to the common prescription for memorizing: "Keep your mind a
blank and say it over and over." There is a high correlation, in my small
sample, between expressing something of this sort and complaining of inability
to remember poetry!
(b) Getting-It Theories. Many children who have trouble understanding
mathematics also have a hopelessly deficient model of what mathematical understanding
is like. Particularly bad are models which expect understanding to come in a
flash, all at once, ready made. This binary model is expressed by the fact that
the child will admit the existence of only two states of knowledge often expressed
by "I get it" and "I don't get it." They lack—and
even resist—a model of understanding something through a process of additions,
refinements, debugging and so on. These children's way of thinking about learning
is clearly disastrously antithetical to learning any concept that cannot be
acquired in one bite.
(c) Faculty Theories. Most children seem to have, and extensively
use, an elaborate classification of mental abilities: "He's a brain",
"He's a retard," "He's dumb," "I'm not mathematical-minded."
The disastrous consequence is the habit of reacting to failure by classifying
the problem as too hard, or oneself as not having the required aptitude, rather
than by diagnosing the specific deficiency of knowledge or skill.
4. Computer Science as a Grade School Subject
Talking to children about all these bad theories is almost certainly inadequate
as an effective antidote. In common with all the greatest thinkers in the philosophy
of education I believe that the child's intellectual growth must be rooted in
his experience. So I propose creating an environment in which the child will
become highly involved in experiences of a kind to provide rich soil for the
growth of intuitions and concepts for dealing with thinking, learning, playing,
and so on. An example of such an experience is writing simple heuristic programs
that play games of strategy or try to outguess a child playing tag with a computer
Another, related example, which appeals enormously to some children with whom
we have worked is writing teaching programs. These are like traditional CAI
programs but conceived, written, developed and even tested (on other children)
by the children themselves.
(Incidentally, this is surely the proper use for the concept of drill-and-practice
programs. Writing such programs is an ideal project for the second term of
an elementary school course of the sort I shall describe in a moment. It is
said that the best way to learn something is to teach it. Perhaps writing
a teaching program is better still in its insistence on forcing one to consider
all possible misunderstandings and mistakes. I have seen children for whom
doing arithmetic would have been utterly boring and alienating become
passionately involved in writing programs to teach arithmetic and in the pros
and cons of criticisms of one another's programs like: "Don't just tell
him the right answer if he's wrong, give him useful advice.” And discussing
what kind of advice is "useful" leads deep into understanding both
the concept being taught and the processes of teaching and learning.)
Can children do all this? In a moment I shall show some elements of a programming
language called LOGO, which we have used to teach children of most ages and
levels of academic performance how to use the computer. The language is always
used "on-line," that is to say the user sits at a console, gives instructions
to the machine and immediately gets a reaction. People who know languages can
think of it as "baby LISP," though this is misleading in that LOGO
is a full-fledged universal language. Its babyish feature is the existence of
self-contained sub-sets that can be used to achieve some results after ten minutes
of instruction. Our most extensive teaching experiment was with a class of seventh
grade children (twelve year olds) chosen near the average in previous academic
record. Within three months these children would write programs to play games
like the simple form of NIM in which players take l, 2, or 3 matches from a
pile; soon after that they worked on programs to generate random sentences—like
what is sometimes called concrete poetry—and went on from there to make
conversational and teaching programs. So the empirical evidence is very strong
that we can do it, and next year we shall be conducting a more extensive experiment
with fifth grade children. The next sections will show some of the elementary
exercises we shall use in the first weeks of the course. They will also indicate
another important aspect of having children do their work with a computer: the
possibility of working on projects with enough duration for the child to become
personally—intellectually and emotionally—involved. The final section
will indicate a facet of how more advanced projects are handled and how we see
the effects of the kind of sophistication developed by the children.
5. You Can Take the Child to Euclid, but You Can't Make Him Think
Let's go back to Dewey for a moment. Intellectual growth, he often told us,
must be rooted in the child's experience. But surely one of the fundamental
problems of the school is how to extend or use the child's experience. It must
be understood that "experience" does not mean mere busy work: two
children who are made to measure the areas of two triangles do not necessarily
undergo the same experience. One might have been highly involved (e.g.,
anticipating the outcome, being surprised, guessing at a general law) while
the other was quite alienated (the opposite). What can be done to involve
the mathematically alienated child? It is absurd to think this can be done by
using the geometry to survey the school grounds instead of doing it on paper.
Most children will enjoy running about in the bright sun. But most alienated
children will remain alienated. One reason I want to emphasize here is that
surveying the school grounds is not a good research project on which one can
work for a long enough time to accumulate results and become involved in their
development. There is a simple trick, which the child sees or does not see.
If he sees it he succeeds in measuring the grounds and goes back to class the
next day to work on something quite different.
Contrast this situation with a different context in which a child might learn
geometry. The child uses a time-shared computer equipped with a CRT. He programs
on-line in a version of the programming language LOGO, which will be described
in more detail below.
On the tube is a cursor point with an arrow indicating a direction.
causes the point to move in the direction of the arrow through 100 units of
distance. The instruction
causes the arrow to rotate 90°.
The child knows enough from previous experience to write the following almost
The word "TO" indicates that a new procedure is to be defined, and
it will be called "CIRCLE." Typing
will now cause the steps in the procedure to be executed one at a time. Thus:
||The point creeps ahead 1 unit.
||The arrow rotates 1o.
||This is a recursive call; naturally it has the same effect as the command
CIRCLE typed by the child. That is to say, it initiates the same process:
||The point creeps on, but in the new,
slightly different direction. The arrow now makes an angle of 2° with
its initial direction.
||This initiates the same process all over again. And soon, forever.
It is left as a problem for the reader to discover why this point will describe
a circle rather than, say, a spiral. He will find that it involves some real
geometry of a sort he may not yet have encountered (See answer at end of paper.).
The more immediately relevant point is that the child's work has resulted in
certain happening, namely a circle has appeared. It occurs to the child to make
the circle roll? How can this be done? A plan is easy to make:
Let the point go around the circle once.
Then FORWARD 1
But there is a serious problem! The program as written causes the point to
go round and round forever. To make it go just once round we need to give the
procedure an input (in more usual jargon: a variable).
This input will be used by the procedure to remember how far round
it has gone. Let's call it "DEGREES" and let it represent the number
of degrees still to go, so it starts off being 360 and ends up 0. The way this
is written in LOGO is:
||: DEGREES means: the thing whose name is "DEGREES."
|IF:DEGREES= O STOP
|ROTATE LEFT 1
|CIRCLE:DEGREES - 1
||Each time round the number of degrees remaining is reduced by 1.
Now we can use this as a sub-procedure for ROLL:
Or, to make it roll a fixed distance:
TO ROLL :DISTANCE
IF :DISTANCE = 0 STOP
ROLL :DISTANCE 1
Or we can make the circle roll around a circle:
These examples will, if worked on with a good dose of imagination, indicate
the sense in which there are endless possibilities of creating even more, but
gradually more, complex and occasionally spectacularly beautiful effects. Even
an adult can get caught up in it! Not every child will. But if he does, the
result is very likely to be a true extension of his experience in Dewey's sense.
And evidence is accumulating for the thesis that there is scarcely any child
who cannot be involved in some computational project.
The next two sections will discuss two other peripheral devices suitable for
a computation laboratory in an elementary school: a programmable vehicle and
a music generator. There is, of course, no end to what one could invent. At
M.I.T. we are thinking in terms of soon adding mechanical manipulators, psychedelic
light shows in a reactive environment, apparatus for automated experiments in
animal psychology, etc., etc., etc.
6. The Love of the Turtle
At M.I.T. we use the name "Turtle" for small computer controlled
vehicles, equipped with various kinds of sense, voice and writing organs. Turtles
can be controlled by the same commands used in the previous section to describe
Graphics. They can be made to draw or to move about without leaving a visible
trace. Procedures to achieve this are exactly like the procedures for CRT Graphics.
However, sense organs allow another interesting dimension of work. An interesting,
simple one is a reflectivity sensor held close to the floor. A LOGO operation
called "LIGHT" has an integer value between 0 and 10, depending on
the reflectivity of the surface. Suppose we wish to program the turtle to follow
the left edge of a black line on a white floor. Using an important heuristic
we encourage the child to study himself in the situation, and try to simulate
his own behavior. The key idea, of course, is to use feed-back according to
the following plan:
Too far right
|Equal Black & White
This leads to the procedure:
IF LIGHT < 4 ROTATELEFT
IF LIGHT > 6 ROTATERIGHT
Notice that the child can think of the program as a very simple formal model
of himself, or, indeed, more justly, of a moth flying to a light. It is rare
that children in the traditional context of math-science get a chance to develop
a model so simple.
Turtles with touch, interactive behavior with several turtles, searching mazes,
and so on scarcely scratch the surface of what can be done with these beasts.
Just as the computer can be instructed to move a point on a TV display or make
a turtle move or print a word, it can be instructed to sing a note. The LOGO
instruction SING can be followed by an input to indicate a note (represented
by 1 ... 7) or a time. A program can be written thus:
will cause the computer to sing the tune.
will cause the tune to be repeated indefinitely. Programs can be written to
speedup, slowdown, raise, lower, transpose, sing in chorus, etc., etc. Children
can use the computer as a super musical instrument. They can compose at the
typewriter and hear their creations played perfectly. They can make and undo
small changes. They can cause the turtle or the CRT display to move to music
and so on endlessly.
8. Case Histories from the Muzzey Jr. High School Experiment
The following piece is extracted verbatim from a report on the seventh grade
teaching experiment performed at Muzzey Jr. High School at Lexington.
8.1. PROBLEM VS. PROJECT
The most exciting single aspect of the experiment was that most of the children
acquired the ability and motivation to work on projects that extend
in time over several days, or even weeks. This is in marked contrast with the
usual style of work in mathematics classes, where techniques are taught and
then applied to small repetitive exercise problems. It is closer, in ways that
are essential to the later argument here, to the work style of some art classes
where children work for several weeks on making an object; a soapcarving for
example. The similarity has several dimensions. The first is that the duration
of the process is long enough for the child to become involved, to
try several ideas, to have the experience of putting something of oneself in
the final result, to compare one's work with that of other children, to discuss,
to criticise and to be criticised on some other basis than "right or wrong."
The point about criticism is related to a sense of creativity that is important
in many ways which we shall talk about later—including, particularly,
its role in helping the child develop a healthy self-image as an active intellectual
Let's take an example. A continuing project over the last third of the year
was working on various kinds of "language generating" programs. The
children studied a program (given as a model) which generated two word sentences
The assignment was to study the model and go on to make more interesting programs.
The sample printout that follows brought great joy to its creator who had worked
hard on mastering the mathematical concepts needed for the program, on choosing
sets of words to create an interesting effect and on converting her exceedingly
vague (and unloved) knowledge about grammar into a useful, practical form.
INSANE RETARD MAKES BECAUSESWEET SNOOPY SCREAMS
SEXY WOLF LOVES THATS WHY THE SEXY LADY HATES
UGLY MAN LOVES BECAUSE UGLY DOG HATES
MAD WOLF HATES BECAUSE INSANE WOLF SKIPS
SEXY RETARD SCREAMS THATS WHY THE SEXY RETARD
HATES THIN SNOOPY RUNS BECAUSE FAT WOLF HOPS
SWEET FOGINY SKIPS A FAT LADY RUNS
The next class assignment was to generate mathematical sentences which were
later used in "teaching programs." For example:
8 * BOX + 6 = 48
WHAT IS BOX'?
Finally, in the last weeks, someone in the class said she wanted to make a
French sentence generator . . . for which she spurned advice and went to work.
In the course of time other children liked the idea and followed suit—evoking
from the first girl prideful complaints like "why do they all have to take
my idea?" The interesting feature was that although they took her idea,
they imprinted it strongly with their own personalities, as shown by the following
K.M. The girl who initiated the project. Thoughtful, serious about
matters that are important to her, often disruptive in class. Her approach to
the French project was to begin by writing procedures to conjugate all the regular
verbs and some irregular ones. The end of the school year fell before she had
made a whole sentence generator. But she did have a truly professional program,
completely debugged and working with great competence at conjugating—e.g.
given VOUS and FINIR as inputs it would reply: VOUS FINISSEZ.
MR A gay, exuberant girl who made the "SEXY COMPUTER" program
quoted above. Only half seriously she declared her intention of making the first
operational French sentence generator. In a sense she did—but with cavalier
disregard for the Academy's rules of spelling and grammar!
JC A clear mind with a balanced sense of proportion. Deliberately
decided to avoid the trap of getting so involved with conjugation that no sentence
would ever be generated. Too serious to allow his program to make mistakes.
Found a compromise: he would make a program that knew only the third person—but
was still non-trivial because it did know the difference between singular and
plural as well as the genders: thus it would say
LE BON CHIEN MANGE
LES BONNES FILLES MANGENT.
8.2. A DETAIL FROM A CHILD'S MATHEMATICAL RESEARCH PROJECT
The fine texture of the work on projects of this sort can only be shown by
case studies. The following vignette needs very little reference to LOGO—
thus illustrating how the projects are more than programming.
J is the author of the last French program mentioned. A little earlier he is
working on generating equations as part of a project to make "a program
to teach 8th grade algebra." He has perfected a program to generate equations
with coefficients in the range of 0-9 using a "random" number generator.
His present problem is to obtain larger coefficients.
First Solution: Almost everyone tries this: get bigger numbers by
adding smaller ones obtained from the old procedure. Amongst other
considerations, this looks like a good technique that has often paid well: use
old functions to define new ones.
Consequences: J chooses his equation generator but soon finds some
The new coefficients are in the range 0-18, which is unnatural and not very
There is a preference for some numbers e.g. 9 conics up ten times as often
Comment: The first problem can be alleviated by adding more numbers.
One can even add a random number of random numbers. But this aggravates the
second problem. J understands this qualitatively but does not see a way out.
It is interesting that children and adults often have a resistance to making
numbers by "non-numerical" operations. In this case the solution is
to concatenate the single digit random numbers instead of adding them. LOGO
has a simple way to express this and J is quite accustomed to making non-numerical
strings by concatenation. In fact this is how he makes the equation! Nevertheless
he resists. The problem is discussed in a class meeting and after some
prompting everyone suddenly "discovers" the solution.
New Solution: j changes his program, now making numbers up to 99 by
concatenation; he does some crude check of uniformity of distribution and tries
Disaster: For a while it seems to go well. But in the course of playing
with the "teaching program" a user types 5 and is surprised to get
a reply like:
YOU KNUCKLEHEAD; YOU TOOK 11 SECONDS AND YOUR ANSWER IS WRONG. THE ANSWER
IS 05. HERE IS SOME ADVICE... etc.
Comment: Poor J will get the sympathy of every mathematician who must
at some stage have tried to generalize a result by extending the domain of an
innocent looking function only to find that the extended function violates some
obscure but essential condition. He is also in the heart of the problem of representation.
Is `05" a good representation? Yes, no . . . have your choice but face
the consequences and be consistent. J's problem is that his procedures accept
"05" for arithmetic operations but not for the test of identity!
Solution: Change the identity test or peel off the leading zero, J
chose the latter. His program worked for a while and was used to great effect.
New Step: Later J was urged to allow negative numbers. He found a
good way: use the one digit random number generator to make a binary decision:
If less than 5, positive
That Problem Again: J had a program working perfectly with negatives.
Then one day decided to make it more symmetrical by using +5 and -5 for positive
and negative. This brought him back to the old problems raised by differences
between the machine’s representation and the human user’s. At this
point the year ended with J’s program not quite as effective as it had
been at its peak.
This work was supported by the National Science Foundation
under grant number GJ-1049 and conducted at the Artificial Intelligence Laboratory,
a Massachusetts Institute of Technology research program supported in part by
the Advanced Research Projects Agency of the Department of Defense and monitored
by the Office of Naval Research under Contract Number N00014-70-A-0362-0002.
This article originally appeared in: World Conference on Computer Education,
IFIPS, Amsterdam, 1970.