[Its.an.education.project] From Piaget
echerlin at gmail.com
Fri May 16 09:55:18 CEST 2008
I am reading in the literature of Constructivism, and will post my
observations and interesting quotations from the pioneers from time to
time to stimulate discussion. It is a curious fact, and one that seems
quite important to me, that I come to understand my own thinking about
a subject much better when I have to explain it to someone else.
The following is the best example I have so far seen of Piaget's
approach to Genetic Epistemology and to Constructivism. Genetic
Epistemology suggests that to understand the nature of knowledge, we
must understand the process by which knowledge has come about. Since
we cannot study prehistoric humans, our best source is to study the
origins of ideas in children. This study requires an understanding of
human psychology, and also the current state and historical
development of logic, mathematics, and the progress of science. Such
interdisciplinary study has not been popular, in part because so few
people have enough knowledge, and because people in different
disciplines have great difficulty communicating with others.
Jean Piaget (1968)
Source: Genetic Epistemology, a series of lectures delivered by Piaget
at Columbia University, Published by Columbia Univesity Press,
translated by Eleanor Duckworth. First lecture reproduced here.
"This example, one we have studied quite thoroughly with many
children, was first suggested to me by a mathematician friend who
quoted it as the point of departure of his interest in mathematics.
When he was a small child, he was counting pebbles one day; he lined
them up in a row, counted them from left to right, and got ten. Then,
just for fun, he counted them from right to left to see what number he
would get, and was astonished that he got ten again. He put the
pebbles in a circle and counted them, and once again there were ten.
He went around the circle in the other way and got ten again. And no
matter how he put the pebbles down, when he counted them, the number
came to ten. He discovered here what is known in mathematics as
commutativity, that is, the sum is independent of the order. But how
did he discover this? Is this commutativity a property of the pebbles?
It is true that the pebbles, as it were, let him arrange them in
various ways; he could not have done the same thing with drops of
water. So in this sense there was a physical aspect to his knowledge.
But the order was not in the pebbles; it was he, the subject, who put
the pebbles in a line and then in a circle. Moreover, the sum was not
in the pebbles themselves; it was he who united them. The knowledge
that this future mathematician discovered that day was drawn, then,
not from the physical properties of the pebbles, but from the actions
that he carried out on the pebbles. This knowledge is what I call
logical mathematical knowledge and not physical knowledge."
This states the importance of such ideas, and calls attention to the
need for research. The research itself is a long, arduous, and
contentious process, with a vast literature, as is all scientific
observation, experiment, and theory-building, with many false leads
and backtracks. It has not simply given us a body of definite
knowledge in the way that we can say elementary physics does. It is
more like the parts of physics out near the frontiers, where the
variety of models of reality under consideration changes from year to
year. But some lessons can be, and have been drawn from all of this,
of which more another time.
For the moment is it enough to have a powerful example of integrated
thought, action, and discovery. This child did not only discover what
we may call the Law of Conservation of Pebble Number. He discovered
the existence of laws amenable to experiment and rational thought, but
not dependent on the particular objects used in the discovery. In
short, he discovered mathematics itself. Most children don't do this
in this way, and of course, most don't become mathematicians as a
What do children discover? How? What do they become as a result? These
questions amply repay any amount of attention and effort aimed at the
continued creation of ever-more-accurate understanding, because they
allow us to explain ourselves better to ourselves and to others, a
conversation that has always been rewarding, but never more so than
Please let me know if this makes sufficient sense to you, and what
else might need explaining. Also if you have been taught some quite
different view of the nature of knowledge, or simply find this one
obviously false, as some do.
End Poverty at a Profit by teaching children business
"The best way to predict the future is to invent it."--Alan Kay
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