Unpacking the Cardinal Principle of Counting:
A Last-Word Rule + the Successor Function
Barbara Sarnecka
University of California - Irvine
Alexandra Cerutti
Columbia University
Susan Carey
Harvard University
The cardinal principle of counting (Gelman & Gallistel, 1978) states that the last number word uttered in a counting sequence expresses the cardinal number
of items in the set that was counted. One way to test children’s knowledge of the cardinal principle is to have them count and then ask “how many.” However,
some children might learn a ‘last-word rule’ that allows them to answer the “how many” question without really understanding the cardinal principle (Fuson,
1988). If the cardinal principle is more than just a ‘last-word rule,’ then what else is it? We suggest that the missing piece may be understanding of the
successor function (the function describing how numbers are formed: N, N+1, [N+1]+1 . . . , etc.) The present study used a series of tasks to assess
children’s knowledge of how to count, and two tasks exploring children’s awareness of the successor function. These principles are (1) Direction (words
appearing later in the count list indicate greater set sizes) and (2) Unit (moving one word forward in the list means adding one item to the set). Children were
also given a standard Give-N task (Give N bananas to the monkey) to determine their level of number knowledge. Children who could give five or more items
successfully (which requires the use of counting) are called Cardinal-Principle Knowers, or CP-knowers. Children who could give only 1, 2, 3, or 4 items and
did not use counting to generate sets or check answers are called Subset Knowers, because they know a subset on number-word meanings (e.g., ‘one’ and
‘two’ only) but do not yet use counting to generate sets or check answsers. Results indicate that most of the children knew the counting-word list and knew
how to coordinate counting with pointing. Furthermore, even subset-knowers followed a ‘last-word rule’ nearly 70% of the time. However, subset-knowers
answered randomly on tasks probing understanding of the successor function, whereas CP-knowers performed significantly above chance. We conclude that
researchers should think of children’s cardinal-principle knowledge as a last-word rule plus understanding of the successor function.
Understanding how to count
Participants
Reciting the Count List
Counting and Pointing
“First, let’s count. Can you count to ten?”
(Score indicates how high children
counted before making an error.)
“Now show me how you count these.”
Score indicates how many objects
(on an array of 10) children counted
before skipping or double-counting.
RESULTS: Most subset-knowers, and
every CP-knower, recited the
sequence perfectly up to “ten.”
Last-Word Rule
RESULTS: Most children in both groups
counted 10 or nearly 10 objects.
10.00
“I have a picture of some things here,
and I am going to count them.
(Child cannot see picture.)
And then I’m going to ask you
how many there are. You try to
guess the number of things
in the picture, by listening to my
counting. Ready? 1, 2, 3,
4, 5! OK, how many things?”
9.64
ten
nine nine
8.76
8.36
eight eight
seven seven
six six
RESULTS: CP-knowers always
answered correctly. But
subset-knowers also answered
correctly 68% of the time,
indicating that many of them
had learned a last-word rule
without understanding
the cardinal
principle.
five five
four four
three three
two two
one one
Subset-knowers CP-knowers
Subset-knowers CP-knowers
19 subset-knowers, aged 32 to 48 mos. (mean 40 mos).
14 CP-knowers, aged 38 to 47 mos (mean 43 mos).
All participants were monolingual and
native speakers of English.
100%
68%
Subset-knowers CP-knowers
Understanding the successor function
Direction Task
Unit Task
Probes children’s understanding that forward in the count list means increasing set size,
and backward in the count list means decreasing set size.
“OK, I’m putting five Xs in here, and five Xs in here.
And now I’ll move one.” (E transfers 1 object from one bowl to the other.)
“OK, now there’s a bowl with four, and a bowl with six. And I’m going to ask you about
the bowl with six. Are you ready? Which bowl has six?”
Probes children’s understanding that a move of one step in the count list
corresponds to a change of one item in the set size.
“OK, I’m putting five things in here. How many things?
. . . Right! And now I’ll put in one more.
So we started with five, and I put in one more.
Now is it six, or seven?”
RESULTS: Subset knowers responded at chance (50%).
CP-knowers performed significantly above chance, indicating a correlation
between knowledge of the cardinal principle (as measured by the Give-N task)
and understanding of the direction aspect of the successor function.
RESULTS: Subset knowers responded at chance (50%).
CP-knowers performed significantly above chance, indicating a correlation
between knowledge of the cardinal principle (as measured by the Give-N task)
and understanding of the unit aspect of the successor function.
71%
69%
chance
44%
39%
Subset-knowers
CP-knowers
Subset-knowers
K. C. (1988). Children's counting and concepts of number. New York: Springer-Verlag.
REFS: Fuson,
Gelman, R. & Gallistel, C. R. (1978). The child's understanding of number. Cambridge, MA: Harvard University Press.
CP-knowers
QUESTIONS to: sarnecka@uci.edu