The Math Gene pp. 27-31 "The Rise and Fall of Piaget", by Keith Devlin:
Much of our current popular wisdom about small children's mental
abilities originates in the work of the cognitive psychologist Jean
Piaget fifty years ago. Piaget's influence can be found not only in
many of our current beliefs about the way children learn, but also in
our educational systems. Unfortunately, as often happens with
ground-braking research, subsequent investigations have show that many
of Piaget's conclusions were almost certainly wrong. (I say "almost
certainly" because some psychologists still maintain that Piaget was
right, and that the experimental results I shall describe admit
alternative conclusions.)
In the 1940s and I950s, Piaget developed a "constructivist" view of
child development. According to this view, a newborn baby enters the
world with a cognitive clean slate and, by observing the world around
it, gradually pieces together a coherent and steadily increasing
understanding of that world. In other words, the child constructs a
mental model or conceptualization of the world.
Piaget did not arrive at his conclusions by armchair speculation. He
was an experimentalist, and his experiments are one reason why his work
was so influential.. It took great ingenuity and equipment not
available in Piaget's time for subsequent generations to devise more
reliable experiments. Whcn they did so, they reached very different
conclusions.
For example, according to Piaget, children younger than ten months old
have no proper sense of physical objects as things that endure in the
world. Piaget based this conclusion on his observation that, when an
object such as a toy is hidden under a cloth, a baby ten months old or
younger will fail to reach for it. According to Piaget, "object
permanency," as he called it, is not innate but is acquired sometime
after ten months of age.
Similarly, Piaget believed that children do not have a number sense
until they acquire it at around four or five years of age. In one of
Piaget's experiments, repeated many times by different groups, a
psychologist would show a four-year-old child two equally spaced rows
of six glasses and six bottles and ask whether there were more glasses
or more bottles. The child invariably answered that there were the same
number. Presumably the child observed a one-to-one correspondence
between the l rows. The experimenter then spread out the glasses to
form a longer row l and asked the child again whether there were more
glasses or more bot-l ties. Now the child would answer that there were
more glasses, apparently l misled by the longer length of that row.
"Obviously," Piaget concluded, l "this shows that the child does not
have a properly developed number sense." In particular, Piaget claimed,
four- and five-year-old children have not yet grasped the idea of
number conservation: the notion that rearranging the objects in a
collection does not change their number.
At the time, Piaget's experiments were held up as triumphs of
experimental science in psychology. As a pioneer, Piaget was blazing a
trail for future generations. And that is good science. Unfortunately,
his methods had serious flaws. He relied on the motor actions of the
babies in the object permanency test and on a dialogue between the
experimenter and the subject for the various number tests performed on
older children.
In the case of object permanency, a baby's failure to reach for an
object hidden under a blanket does not support the rather dramatic
conclusion that the baby thinks the object has ceased to exist. Perhaps
he simply does not yet have sufficient hand-arm coordination to reach
for a hidden object. In fact, we now know that this explanation is
correct. Recent experiments, more sophisticated than Piaget's, indicate
that even very young babies have a well-developed sense of object
permanency.
Likewise, dialogue with a small child is highly unreliable.
Communication via language is never loo percent objective and free of
the influences of context, emotion, social factors, and possibly
several other things. Just how unreliable dialogue can be was
demonstrated by Jacques Mehler and Tom Bever at MIT during the late
1960s.
In one experiment, Mehler and Bever carried out the original Piaget
experiment to test for number conservation, but with two- and
three-year old children instead of Piaget's four- and five-year-olds.
The children succeeded perfectly. Consequently, unless we believe that
children temporarily lose their sense of number conservation between
the ages of four and six, we clearly need some alternative explanation
for Piaget's results. One is readily available.
Around five years of age, children begin to develop the ability to
reason about another person's thought process ("What Daddy means by
this is . . . "). This provides the most likely explanation of Piaget's
observations. Remember the way the experiment was performed. First the
experimenter arranges the glasses and bottles in two equally spaced
rows and asks the child which row has more objects. Then the
experimenter rearranges one of the rows, making it longer, and again
asks the child, "Which row has more objects?"
Now, by four or five years of age, a young child knows that adults are
powerful and are knowledgeable. Moreover, she has probably observed the
respect her parent showed the experimenter when they arrived at the
laboratory. How is this child likely to react when she sees the
experimenter rearrange the objects in one of the two rows and then ask
the very same question as a moment earlier, "Which row has more
objects?" She may well reason, "Hmm. That's the same qucstion she just
asked me. Adults are not dumb, and this is a special kind of adult who
knows a lot. We can both see that the number of objects hasn't changed.
So I must have misunderstood the question the last time. I thought she
was asking me about the number of objects in the row, but obviously she
was really asking me about the length, since that's what she just
changed." And so the child gives the answer she thinks is expected of
her.
Of course, we can't know for sure. Attempts to find out by
interrogating the child are unlikely to yield conclusive evidence, for
the same reason that the original Piaget experiment is suspect! This is
where the Mehler and Bever experiment came into its own. The kind of
"what-does-she-really-want?" reasoning just described is beyond two- or
three-year olds. Mehler and Bever's younger subjects took the
experimenters' questions literally, and counted correctly.
What Piaget's original experiment really showed is that four- and five
year-old children can reason rationally about the motivations and
expectations of another person. That's an important and useful
discovery. But it's not the one Piaget thought he had made!
To confirm that children from age two upward have a good sense of
number, Mehler and Bever redesigned the Piaget test to avoid the
reliance on language. Their idea was breathtakingly simple. Instead of
glasses and bottles they presented the child with two rows of M&Ms. One
row contained six M&Ms, the other had four. Sometimes the rows were the
same length; sometimes the row of six M&Ms was longer; other times the
row of four M&Ms was longer. Instead of being asked to indicate which
row had more candies, the child was simply told he could pick one row
and eat them. The outcome was precisely what any parent would predict.
The child invariably plumped for the row of six candies, regardless of
its length. He knew full well which row had more members, and moreover
realized that the number was not dependent on the arrangement. The
result was just as conclusive with two-year-old children as with four
year-olds.
Another ingenious variation of the original Piaget experiment reached
the same conclusion. This time, James McGarrigle and Margaret Donaldson
of the University of Edinburgh carried out their experiment in a small
puppet theater. Like Piaget, they started by aligning two rows of the
same number of objects and asking the child which row had more objects.
After the child responded correctly, the experimenter pretended to look
away while a teddy bear puppet lengthened one of the rows. Turning
back, the experimenter exclaimed, "Oh dear, that silly teddy has mixed
up the rows. can you tell me which row has more objects again?"
Children from two to five invariably gave the correct answer. Since the
teddy bear had rearranged one of the rows, unseen by the experimenter,
the child presumably found it reasonable for the adult to ask the same
question again. Yet when the experimenter repeated the process with the
same children but rearranged the objects him- or herself, the four- and
five-year-old children responded exactly as they had for Piaget, basing
their answer on length. -Keith Devlin, The Math Gene (2000)
----------------------------------------------------
Sorry for the re-post, but I was very annoyed that I miscalculated
line lengths in my original post, and the text was almost unreadable.
-Lee Corbin
This archive was generated by hypermail 2b30 : Fri Oct 12 2001 - 14:40:11 MDT