Before I was two years old I had developed an intense involvement
with automobiles. The names of car parts made up a very substantial
portion of my vocabulary: I was particularly proud of knowing about
the parts of the transmission system, the gearbox, and most especially
the differential. It was, of course, many years later before I understood
how gears work; but once I did, playing with gears became a favorite
pastime. I loved rotating circular objects against one another in
gearlike motions and, naturally, my first "erector set" project was
a crude gear system.
I became adept at turning wheels in my head and at making chains of
cause and effect: "This one turns this way so that must turn that
way so . . . " I found particular pleasure in such systems as the
differential gear, which does not follow a simple linear chain of
causality since the motion in the transmission shaft can be distributed
in many different ways to the two wheels depending on what resistance
they encounter. I remember quite vividly my excitement at discovering
that a system could be lawful and completely comprehensible without
being rigidly deterministic.
I believe that working with differentials did more for my mathematical
development than anything I was taught in elementary school. Gears,
serving as models, carried many otherwise abstract ideas into my
head. I clearly remember two examples from school math. I saw multiplication
tables as gears, and my first brush with equations in two variables
(e.g., 3x + 4y = 10) immediately evoked the differential. By the
time I had made a mental gear model of the relation between x and
y, figuring how many teeth each gear needed, the equation had become
a comfortable friend.
Many years later when I read Piaget this incident served me as a model
for his notion of assimilation, except I was immediately struck by
the fact that his discussion does not do full justice to his own
idea. He talks almost entirely about cognitive aspects of assimilation.
But there is also an affective component. Assimilating equations
to gears certainly is a powerful way to bring old knowledge to bear
on a new object. But it does more as well. I am sure that such assimilations
helped to endow mathematics, for me, with a positive affective tone
that can be traced back to my infantile experiences with cars. I
believe Piaget really agrees. As I came to know him personally I
understood that his neglect of the affective comes more from a modest
sense that little is known about it than from an arrogant sense of
its irrelevance. But let me return to my childhood.
One day I was surprised to discover that some adults---even most adults---did
not understand or even care about the magic of the gears. I no longer
think much about gears, but I have never turned away from the questions
that started with that discovery: How could what was so simple for
me be incomprehensible to other people? My proud father suggested
"being clever" as an explanation. But I was painfully aware that
some people who could not understand the differential could easily
do things I found much more difficult. Slowly I began to formulate
what I still consider the fundamental fact about learning: Anything
is easy if you can assimilate it to your collection of models. If
you can't, anything can be painfully difficult. Here too I was developing
a way of thinking that would be resonant with Piaget's. The understanding
of learning must be genetic. It must refer to the genesis of knowledge.
What an individual can learn, and how he learns it, depends on what
models he has available. This raises, recursively, the question of
how he learned these models. Thus the "laws of learning" must be
about how intellectual structures grow out of one another and about
how, in the process, they acquire both logical and emotional form.
This book is an exercise in an applied genetic epistemology expanded
beyond Piaget's cognitive emphasis to include a concern with the
affective. It develops a new perspective for education research focused
on creating the conditions under which intellectual models will take
root. For the last two decades this is what I have been trying to
do. And in doing so I find myself frequently reminded of several
aspects of my encounter with the differential gear. First, I remember
that no one told me to learn about differential gears. Second, I
remember that there was feeling, love, as well as understanding in
my relationship with gears. Third, I remember that my first encounter
with them was in my second year. If any "scientific" educational
psychologist had tried to "measure" the effects of this encounter,
he would probably have failed. It had profound consequences but,
I conjecture, only very many years later. A "pre- and post-" test
at age two would have missed them.
Piaget's work gave me a new framework for looking at the gears of
my childhood. The gear can be used to illustrate many powerful ädvanced"
mathematical ideas, such as groups or relative motion. But it does
more than this. As well as connecting with the formal knowledge of
mathematics, it also connects with the "body knowledge," the sensorimotor
schemata of a child. You can be the gear, you can understand how
it turns by projecting yourself into its place and turning with it.
It is this double relationship---both abstract and sensory---that
gives the gear the power to carry powerful mathematics into the mind.
In a terminology I shall develop in later chapters, the gear acts
here as a transitional object.
A modern-day Montessori might propose, if convinced by my story, to
create a gear set for children. Thus every child might have the experience
I had. But to hope for this would be to miss the essence of the story.
I fell in love with the gears. This is something that cannot be reduced
to purely "cognitive" terms. Something very personal happened, and
one cannot assume that it would be repeated for other children in
exactly the same form.
My thesis could be summarized as: What the gears cannot do the computer
might. The computer is the Proteus of machines. Its essence is its
universality, its power to simulate. Because it can take on a thousand
forms and can serve a thousand functions, it can appeal to a thousand
tastes. This book is the result of my own attempts over the past
decade to turn computers into instruments flexible enough so that
many children can each create for themselves something like what
the gears were for me.