Gelman’s Counting Principle
How do we know whether children can count ?
If knowing how to count just means reciting
the numbers (i.e., ‘‘one, two, three...”) up to “five” or
“ten,” perhaps pointing to one object with each numeral,
then many two-year-olds count very well.
Of course, counting only tells you number of things if you do it correctly,
following the three ‘how-to-count’ principles identified by Gelman and
Gallistel (1978). These are :
(1) The one-to-one principle, which says that ‘‘in
enumerating a set, one and only one [numeral] must
be assigned to each item in the set.”
counting must be used in the same order in any one
count as in any other count.” and
(3) The cardinal principle, which says that ‘‘the [numeral] applied to the
final item in the set represents the number of items in the set.”
As Gelman and Gallistel pointed out, so long as the
child’s counting obeys these three principles, the numeral
list (‘‘one,” ‘‘two,” ‘‘three,”... etc.) represents the cardinalities
1, 2, 3,... etc.
However, observations have shown that three-year-old
children often violate the one-to-one principle by skipping
or double-counting items, or by using the same numeral
twice in a count.
Children also violate the stable-order principle, by producing different
numeral lists at different times.
The cardinal principle can also be viewed as something more profound
– a principle stating that a numeral’s cardinal meaning is determined by
its ordinal position
in the list. This means, for example, that the fifth numeral in
any count list – spoken or written, in any language – must
mean five. And the third numeral must mean three..
If so, then knowing the cardinal principle means having
some implicit knowledge of the successor function – some
understanding that the cardinality for each numeral is generated by
adding one to the cardinality for the previous numeral.
Thus Gelman feels that preschoolers learn to count as a result of the
perfecting of counting procedures.
Do you agree ?