Re-visiting the competence/performance debate in the acquisition of the counting principles.

Abstract

Advocates of the "continuity hypothesis" have argued that innate non-verbal counting principles guide the acquisition of the verbal count list (Gelman & Galistel, 1978). Some studies have supported this hypothesis, but others have suggested that the counting principles must be constructed anew by each child. Defenders of the continuity hypothesis have argued that the studies that failed to support it obscured children's understanding of counting by making excessive demands on their fragile counting skills. We evaluated this claim by testing two-, three-, and four-year-olds both on "easy" tasks that have supported continuity and "hard" tasks that have argued against it. A few noteworthy exceptions notwithstanding, children who failed to show that they understood counting on the hard tasks also failed on the easy tasks. Therefore, our results are consistent with a growing body of evidence that shows that the count list as a representation of the positive integers transcends pre-verbal representations of number.

Full text

Page 1

Re-visiting the competence/performance debate
in the acquisition of the counting principles
Mathieu Le Correa,*, Gretchen Van de Walleb
Elizabeth M. Brannonc, Susan Careya
aHarvard University, USA
bRutgers University, USA
cDuke University, USA
Accepted 28 July 2005
Available online 20 December 2005
Abstract
Advocates of the ‘‘continuity hypothesis’’ have argued that innate non-verbal counting principles
guide the acquisition of the verbal count list (Gelman & Gallistel, 1978). Some studies have support-
ed this hypothesis, but others have suggested that the counting principles must be constructed anew
by each child. Defenders of the continuity hypothesis have argued that the studies that failed to sup-
port it obscured children?s understanding of counting by making excessive demands on their fragile
counting skills. We evaluated this claim by testing two-, three-, and four-year-olds both on ‘‘easy’’
tasks that have supported continuity and ‘‘hard’’ tasks that have argued against it. A few noteworthy
exceptions notwithstanding, children who failed to show that they understood counting on the hard
tasks also failed on the easy tasks. Therefore, our results are consistent with a growing body of evi-
dence that shows that the count list as a representation of the positive integers transcends pre-verbal
representations of number.
? 2005 Elsevier Inc. All rights reserved.
Keywords: Counting; Number words; Number; Cognitive development; Performance/competence
0010-0285/$ - see front matter ? 2005 Elsevier Inc. All rights reserved.
doi:10.1016/j.cogpsych.2005.07.002
*Corresponding author. Present address: Laboratory for Developmental Studies, Shannon Hall, 25 Francis
Av., Cambridge, MA 02138, USA.
E-mail address: lecorre@wjh.harvard.edu (M. Le Corre).
Cognitive Psychology 52 (2006) 130–169
www.elsevier.com/locate/cogpsych

Page 2

1. Introduction
When Leopold Kronecker said ‘‘The integers were created by God; all else is man-
made’’ (Weyl, 1949, p. 33) he was making a metaphysical claim. Yet, the remark inspires
a natural position concerning the cognitive foundations of arithmetical thought. If we
replace ‘‘God’’ with ‘‘evolution,’’ the position would be that evolution provided us with
the capacity to represent the positive integers, the natural numbers, and that the capacity
to represent the rest of the arithmetical concepts, including the rest of the number concepts
(rational, negative, 0, real, imaginary, etc.) was culturally constructed by human beings.
On this interpretation of Kronecker?s remark, he espoused the ‘‘continuity hypothesis’’
with respect to integer representations; i.e., he believed that these representations are avail-
able throughout human development, historically and ontogenetically.
In a series of important publications, Gelman and colleagues (Cordes & Gelman, 2005;
Gallistel & Gelman, 1992; Gelman, 1993; Gelman & Gallistel, 1978) have also put forward
the continuity hypothesis, arguing that representations of the positive integers are part of
our innate cognitive endowment. In their view, there is an innate system of non-verbal
symbols whose deployment conforms to three integer-constitutive counting principles:
the stable order principle, the one-to-one correspondence principle, and the cardinal prin-
ciple. The stable order principle captures the fact that the symbols are applied in a consis-
tent order across counting episodes. One-to-one correspondence means that every
individual is tagged with one and only one symbol and each symbol is applied to one
and only one individual. Finally, the cardinal principle entails that the last symbol of a
count represents the number of individuals enumerated during the count.
Gelman and Gallistel (1978) further suggest that these non-verbal counting principles
guide children?s acquisition of the verbal count list. ‘‘One could argue that skill in reciting
count-word sequences precedes and forms a basis for the induction of counting principles. We,
however, advance the opposite thesis: A knowledge of counting principles forms the basis for
the acquisition of counting skills’’ (p. 204). To support this proposal, they observed that
children are able to generate counting strategies that differ from the familiar left-to-right
sequence that parents typically employ in counting activities, suggesting that children are
not simply executing a meaningless routine. Further, some young children use an idiosyn-
cratic count list following a stable order (e.g., ‘‘one, three, four, six...’’ always in this
order), and others occasionally, if rarely, use the wrong type of ordered list, usually the
alphabet, as their count list. Gelman and Gallistel liken these behaviors to children?s over-
generalization of linguistic rules (e.g., ‘‘I goed to the store’’). That is, children are not
merely passively reproducing the patterns produced by their caregivers, but are actually
interpreting the input in terms of rich innate mental structures. When children establish
the numerical relevance of a memorized list that generally has a stable order, they do so
because they have interpreted their count list as the verbal instantiation of the non-verbal
counting principles.
A substantial body of research is at odds with the continuity hypothesis. Schaeffer,
Eggleston, and Scott (1974), for example, demonstrated that many children who could
successfully count arrays of between 5 and 7 objects nonetheless failed to provide a cardi-
nal response when asked how many objects there were. These children occasionally recog-
nized and constructed small arrays of 1–4 objects successfully, but did so without
counting. Similarly, Fuson and colleagues (Fuson, 1988; Fuson & Hall, 1983; Fuson,
Lyons, Pergament, & Hall, 1988) found that even when children provide the last number
M. Le Corre et al. / Cognitive Psychology 52 (2006) 130–169
131

Page 3

of a count in response to a cardinal query, they often seem to be following a ‘‘last word
rule’’ rather than demonstrating true understanding of the cardinal principle. For exam-
ple, children at times provide the last word following a count that has grossly violated sta-
ble order and/or one-to-one correspondence. They also at times relate the last word to the
last individual counted rather than to the entire set.
Wynn (1990, 1992) also found that for a period of about a year after they had first
memorized a short count list, children (1) only knew the exact meaning of a subset of
the number words in their count list; (2) rarely used counting to solve tasks that required
them to determine the cardinality of a set; and (3) often blatantly violated the counting
principles when they counted. Measures from two different tasks supported these general-
izations. In Wynn?s ‘‘Give a Number’’ task (GN), children were given a bowl full of objects
and were asked to give the experimenter various numbers of them. In the Point-to-X task,
children were shown two cards depicting objects (e.g., sheep), one with N objects and the
other with N + 1. Children were asked, ‘‘Can you show me the N sheep?’’ Both tasks
showed that children first learned the meaning of ‘‘one’’, then that of ‘‘two’’, then that
of ‘‘three’’. For example, ‘‘one’’-knowers1could only reliably give the correct number of
objects when asked for ‘‘one’’; they gave more than one when larger numbers were
requested, but the numbers they gave were not systematically related to those requested
(Wynn, 1990). These very children only succeeded on the Point-to-X task if one of the
two cards showed one object. If both showed more than one, they chose at chance. Sim-
ilarly, ‘‘three’’-knowers reliably gave only ‘‘one’’, ‘‘two’’, and ‘‘three’’ and failed on the
Point-to-X task if both cards showed more than three objects. However, when children
understood ‘‘four’’, their learning of the number words appeared to change radically:
Wynn found no children who knew ‘‘four’’ who did not also know the exact meaning
of the other number words in their count list, suggesting that children learned how their
count list represents number when they learned the meaning of ‘‘four’’. We will henceforth
refer to children who understand how counting represents number as ‘‘CP-knowers’’
(where ‘‘CP’’ stands for ‘‘cardinal principle’’) and will refer to ‘‘one’’-, ‘‘two’’-, and
‘‘three’’-knowers as ‘‘subset-knowers’’ because they know the exact cardinal meanings
of only a subset of the numerals in their count list.
Wynn identified this sequence in both cross-sectional (Wynn, 1990) and longitudinal
(Wynn, 1992) studies. Further, both data sets suggest that this developmental sequence
occurs over a highly extended timeframe. Wynn?s cross-sectional data showed that chil-
dren are subset-knowers until about age 31
showed that, on average, 4–5 months elapse between each ‘‘stage’’ such that about a year
elapses between the time at which children are ‘‘one’’-knowers and the time at which they
become CP-knowers.
Several measures suggested subset-knowers had a qualitatively different understanding
of counting from CP-knowers. First, subset-knowers were much less likely to use counting
in tasks requiring the exact determination of the cardinality of a set. Second, when asked,
‘‘How many X?s are there?’’ (the ‘‘How Many?’’ task) after counting a set of objects, sub-
set-knowers reported the last word of their count only about 20% of the time, whereas CP-
knowers did so about 70% of the time. CP-knowers were also three times more likely to
2(range 2;11–4;0) and her longitudinal data
1Henceforth, we will use the expression ‘‘N-knower’’ to refer to a child who only knows a subset of the number
words (e.g., a ‘‘two’’-knower is a child who only knows the adult meaning of ‘‘one’’ and ‘‘two’’), based on Wynn?s
GN task.
132
M. Le Corre et al. / Cognitive Psychology 52 (2006) 130–169

Page 4

repeat the last word of their count following a correct count than an incorrect count (84%
vs. 28%). Subset-knowers appeared insensitive to the accuracy of their counting, providing
a last word response equally often following correct and incorrect counts (Wynn, 1990).
Finally, subset-knowers sometimes violated the cardinal and stable order principles. When
these children did use counting to assemble a set, their count ended at a number other than
the requested number about 50% of the time (Wynn, 1992). In the GN task, these children
at times simply declared their response to be correct even when their own count had just
shown the number was incorrect. They also at times ‘‘fixed’’ the set by altering the count
sequence so that the last word was the number requested (e.g., ‘‘1, 2, 5’’ or ‘‘3, 3, 3, 3’’).
These behaviors were almost never observed in CP-knowers.
Insum,Wynn?sdatasuggestthatchildrenmemorizeapartoftheircommunity?scountlist
and establish the numerical relevance of the numerals in the list (e.g., learn the meaning of
‘‘one’’)almostayearbeforetheylearnhowtousethelisttodeterminethecardinalityofsets.
Thus, Wynn?s data suggest that the acquisition of the verbal count list may involve the con-
struction of a system of representation that is not innately available. Many have espoused
thisinterpretation,althoughtheexactnatureoftheprocesswherebythenewrepresentation-
alsystemisconstructedisundermuchdebate(e.g.,Bialystok&Codd,1997;Briars&Siegler,
1984; Carey, 2004; Cooper, 1984; Fuson, 1988; Karmiloff-Smith, 1992; Klahr & Wallace,
1976;Mix,Huttenlocher,&Levine,2002;Schaefferetal.,1974;Spelke&Tsivkin,2001;Star-
key & Cooper, 1995; Strauss & Curtis, 1984). Because these proposals hold that children?s
representational resources undergo a drastic, qualitative change when they acquire the
counting principles, we will refer to them (collectively) as the ‘‘discontinuity hypothesis.’’
It may be, however, that children?s failures on tasks assessing their understanding of
counting do not indicate a developmental discontinuity (Cordes & Gelman, 2005; Gelman,
1993; Gelman & Greeno, 1989; Greeno, Riley, & Gelman, 1984). Rather, children may
possess an innate understanding of the counting principles but nonetheless perform poorly
on tasks testing this understanding because of their limited abilities to understand what
they are being asked to do (what Greeno et al., 1984 call ‘‘utilization skills’’) and/or to
plan and execute counting procedures that will successfully meet the requirements of
the task (what Greeno et al., 1984 call ‘‘procedural skills’’).
Indeed, many of the tasks used to assess children?s understanding of counting arguably
make excessive performance demands. For instance, Gelman (1993) argues that whereas
the How Many? task presents a familiar means of eliciting the count list, it is a confusing
means of testing knowledge of the cardinal principle. To demonstrate cardinal knowledge
on this task, children must use the last word of their count to describe the set they just
counted. While some children do so spontaneously, many count without producing a car-
dinal answer (e.g., they say ‘‘1, 2, 3, 4’’ instead of ‘‘1, 2, 3, 4. Four duckies!’’). To elicit a
cardinal answer, the latter are asked again, ‘‘So how many Xs are there?’’ Conversational
pragmatics may lead children to infer that they had somehow erred in their first response
(or why would the experimenter ask again for the same information just provided?) and
should therefore repeat the counting procedure (see e.g., Freeman, Antonucci, & Lewis,
2000; Fuson, 1988; Gelman, 1993; Wynn, 1992). Children who have poorer utilization
and procedural skills may be more uncertain of their count. Thus, they may be more likely
to interpret this question as a challenge to the correctness of their count and so respond by
recounting. These same children may be less aware of whether their count was in fact accu-
rate, leading them to provide last word responses as often for correct counts as for incor-
rect ones (Gelman & Meck, 1986; Gelman, Meck, & Merkin, 1986).
M. Le Corre et al. / Cognitive Psychology 52 (2006) 130–169
133

Page 5

The other two tasks designed to test the cardinal principle—Point-to-X and GN—are
difficult because they require children to use counting to find or construct a set with some
previously specified cardinal value. For example, in GN, children are given a target (e.g.,
‘‘Can you give me five dinosaurs?’’) and then must remember this target while they use
counting to assemble an array (see Frye, Braisby, Lowe, Maroudas, and Nicholls, 1989
for an argument that this poses problems for young children). This may be more difficult
than deducing a cardinal value from a previous count. Also, each of these tasks requires
sophisticated counting strategies that may exceed the utilization and procedural skills of
younger children. To succeed in Point-to-X, children need to count each set, compare
the obtained cardinal values to that requested, and then select the card depicting the
requested cardinal value. In GN, children must coordinate counting with set construction,
a process that may be too taxing for poorer counters (Cordes & Gelman, 2005; Fuson,
1988). Moreover, younger children may be unable to fix incorrect sets not because they
are unaware of their error but rather because they are unable to implement some appro-
priate addition or subtraction strategy. Thus, these children may opt to extricate them-
selves from situations they cannot resolve by simply asserting that their response was
correct or by adopting a strategy they know is incorrect but that results in the correct last
word (e.g., having given three objects when five were requested, they may count the set as
‘‘5-5-5’’ or ‘‘1-2-5’’).
Finally, in their ‘‘videotape counting study’’, Gelman and Gallistel (1978) noted that
children with poor counting skills were less likely to engage in counting large sets than
children with better counting skills. Assuming that subset-knowers are really CP-knowers
with poor counting skills, this could explain why one of the main differences between sub-
set-knowers and CP-knowers is that the former are much less likely to count on the GN
and Point-to-X tasks. It could also explain why subset-knowers only seem to know the
meaning of ‘‘one,’’ ‘‘two,’’ and ‘‘three.’’ While counting is the only way of determining
the exact cardinality of large sets, that of small sets can be determined without counting
by subitizing. Therefore, given that even 2-year-olds can subitize (Starkey & Cooper,
1995), children who are reluctant to count could succeed on the small number trials of
the GN and Point-to-X tasks by relying on subitizing, but would fail on larger number
trials because they require counting.
If younger children have failed to show their understanding of counting on tasks such as
GN and Point-to-X because of performance limitations, it should be possible to reveal their
understandingofcountingbytestingthemontasksthatreducedemandsontheirprocedural
and utilization skills. Such evidence is in fact available. For example, when children are sim-
plyasked toassess the correctness of apuppet?s counting andthe resulting cardinal respons-
es,theyoftenperformquitewell(Gelman&Meck,1983;Gelmanetal.,1986;butseeBriars&
Siegler,1984;Fryeetal.,1989).Gelman,Meck,andMerkindemonstrated,forexample,that
children as young as 3 years of age appreciated that a puppet who simply repeated the last
word of a count in which it had skipped or double-counted an object had incorrectly
answered a ‘‘how many’’ question. Likewise, they appreciated that if a puppet had correctly
counted a set, its cardinal answer was only correct if it matched the last word of the count.
The same children further inferred that apuppet who counted the same set twice but arrived
at different values each time must have made a mistake in at least one of its counts. Finally,
children were better able to generate correct, non-canonical counting strategies (e.g., begin-
ning in the middle of a row) when they were allowed either to work first on small sets or to
work on the task for multiple trials (with no feedback).
134
M. Le Corre et al. / Cognitive Psychology 52 (2006) 130–169

Page 6

Further evidence in support of the performance account comes from Gelman?s (1993)
‘‘What?s on This Card?’’ (WOC) task. In this task, children were presented with sets of
cards depicting from one to seven stickers, and were simply asked, ‘‘What?s on this card?’’
On the first card of each set, the experimenter modeled the desired response: e.g., ‘‘That?s
right! It?s one bee.’’ On all subsequent trials, children were probed to elicit both counting
and production of a cardinal response. For example, if a child counted a set of four stick-
ers without producing a cardinal answer (e.g., ‘‘one, two, three, four!’’), the experimenter
asked, ‘‘So, what?s on this card?’’ or ‘‘How Many bees is that?’’ to elicit a cardinal answer.
If a child responded with a cardinal value alone (e.g., ‘‘That?s four bees!’’), the experiment-
er elicited counting with a prompt such as, ‘‘Can you show me?’’ Also, whenever it was
deemed necessary, the experimenter provided children with counting assistance by either
pointing to each object and/or by saying ‘‘one’’ and pointing to the first object to initiate
the count, and then pointing to the other objects one at a time.
According to Gelman, the WOC task minimizes performance factors in at least two
ways. First, the question, ‘‘What?s on this card?’’ capitalizes on children?s affinity for
and familiarity with kind labeling situations (as opposed to situations focused on cardinal
values). Second, counting assistance reduces the procedural demands of counting by help-
ing children keep track of counted and uncounted objects. This may also help children
overcome a lack of confidence in their counting ability, and it may increase their awareness
that counting is relevant to the task.
To determine whether WOC would reveal earlier numerical competence than Wynn
had shown, Gelman only tested children who were younger than the average age of
Wynn?s CP-knowers (i.e., 3;6). She determined the number of children who both produced
a correct count (allowing for one counting error) and a cardinal answer matching the last
word of the count on at least half of the trials. She found that between 70 and 90% of her
young 3-year-olds met this criterion for all numbers tested. Moreover, 60% of her 2.5-year-
olds met the criterion up to 5, but only 30% met it for larger numbers. These results led
Gelman to conclude that the WOC task provided a more sensitive measure of children?s
early counting competence in that it showed that even children as young as 2.5 years of
age were CP-knowers.
Taken together, Gelman et al.?s ‘‘Counting Puppet’’ studies and the WOC study sup-
port the view that young children failed Wynn?s tasks because of performance limitations,
supporting the continuity hypothesis. However, for several reasons, these studies do not
yet settle the continuity/discontinuity debate. First, none analyzed children?s performance
as a function of their Wynn stage. This is important because the age at which individual
children achieve different Wynn stages is quite variable; e.g., while the average age at
which children become CP-knowers is 3;6, some reach this stage at 2;11 (Wynn, 1990).
Thus, even though Gelman?s children were all younger than Wynn?s average CP-knower,
each age group may nonetheless have included some children who would have been CP-
knowers on Wynn?s tasks. Likewise, many of the 3-year-olds in the Counting Puppet stud-
ies could have been CP-knowers. Therefore, these studies may have overestimated the
extent to which they conflict with results that support discontinuity. The studies presented
here will address this problem directly. Experiment 1 will examine children?s performance
on WOC as a function of their GN stage, and Experiment 2 will examine children?s per-
formance on a Counting Puppet study as a function of GN stage. According to the per-
formance account, children should succeed on WOC and the Counting Puppet task
even if they are classified as subset-knowers on GN. On the other hand, according to
M. Le Corre et al. / Cognitive Psychology 52 (2006) 130–169
135

Page 7

the discontinuity hypothesis, only those children classified as CP-knowers on GN should
show that they understand counting on the WOC and Counting Puppet tasks.
Second, while showing some understanding of the counting principles in old 2-year-olds
and young 3-year-olds is impressive, it is not sufficient to show that children?s use of verbal
counting is guided by innate counting principles. Gelman and Gallistel?s continuity
hypothesis predicts that the counting principles will motivate children to actively seek
an ordered list of linguistic symbols and to interpret this list in terms of the counting prin-
ciples as soon as they find it and have established its relevance to number representations.
Previous studies have shown that some children have memorized a count list and have
established its numerical relevance (e.g., they have learned ‘‘one’’) by the time they are
24 months old (e.g., Le Corre & Van de Walle, 2001). Therefore, because their samples
did not include children that were young enough, the WOC and Counting Puppet studies
have not yet provided the critical piece of evidence: that children are CP-knowers as soon
as they have memorized a count list and have established its numerical relevance. We will
address this issue in Experiment 1 by including children who have just memorized their
count list in our sample.
The last two problems with the evidence for the continuity hypothesis concern the
WOC task itself. First, Gelman does not report the mistakes made by children who pro-
duced both cardinal responses and counts on at least 50% of trials. An analysis of mistakes
children make could reveal whether children who meet the 50% criterion really did under-
stand the counting principles. For example, one can imagine a child who counted on every
trial but whose cardinal answers matched her counts only 50% of the time, other times say-
ing things like ‘‘One, two, three. That?s two bears.’’ Such a child would have met Gelman?s
criterion, but clearly should not be considered a CP-knower. Experiment 1 will address
this issue by determining whether children ever produce such violations of the cardinal
principle, and whether their frequency is a function of their GN stage. Finding that chil-
dren who are subset-knowers on GN commit these errors would support the discontinuity
hypothesis.
Finally, Gelman (1993) argues that the fact that the large majority of her youngest chil-
dren met her criterion for small numbers (i.e., 2 and 3) is enough to show that these children
areCP-knowers.Whiletheseresultsareimpressive,theydonotnecessarilyimplyknowledge
of the counting principles. Indeed, Wynn?s subset-knowers could give or identify small sets
without counting (e.g., when ‘‘two’’-knowers correctly created or identified sets of two
objects, they did so without counting). This suggests that subset-knowers learn the meaning
of‘‘one,’’‘‘two,’’and‘‘three’’bymappingthesenumberwordsontotheoutputsofasubitiz-
ingprocessthatdeterminessmallcardinalvalueswithoutcounting(Carey,2001,2004;Klahr
&Wallace,1976;Siegler,1991,1998;Starkey &Cooper,1995).Moreover,wehaveseenthat
subset-knowers do sometimes count correctly, particularly if they are counting small sets.
Thus, ‘‘two’’- and‘‘three’’-knowers could correctlycount small sets andthen produce cardi-
nal answers matching their count (i.e., ‘‘two’’ or ‘‘three’’), not because they understand the
cardinalprinciple,butratherbecausethesearethenumberwordsthatmapontothecardinal
valuegivenbysubitizing.Toexaminethispossibility,alloftheanalysesofcountinginExper-
iment1willtreatsmall(1–3)andlarge(4–8)numbersseparately.Accordingtothecontinuity
hypothesis, children should respect the cardinal principle on both small and large numbers.
However, if young children?s respect of the cardinal principle on small numbers reflects an
accidental match of correct but numerically meaningless counting with subitizing, then they
should commit many errors on large numbers.
136
M. Le Corre et al. / Cognitive Psychology 52 (2006) 130–169

Page 8

In sum, developmental science has yet to determine whether the count-based represen-
tation of the natural numbers is the work of evolution or that of human culture. Each
position has its advocates, and each position is supported by rich but ambiguous evidence.
The present studies are the first to seek to advance this debate by analyzing within-child
consistency on tasks that have been taken to support each hypothesis. As both positions
predict that children who are classified as CP-knowers on GN will succeed on the easier
tasks, the critical question concerns the performance of children identified on GN as sub-
set-knowers. On the continuity account, even the least advanced subset-knowers (e.g.,
‘‘one’’-knowers) should show evidence of understanding counting on the easier tasks. In
contrast, if the subset-knower stages are real, ‘‘one’’-, ‘‘two’’-, and ‘‘three’’-knowers should
still prove to be subset-knowers, and not CP-knowers, on the easier tasks.
2. Experiment 1
2.1. Method
2.1.1. Participants
Fifty 2- to 4-year-old children (mean: 3;1, range: 2;0 to 4;0) participated. All were fluent
EnglishspeakersrecruitedintheNewYorkCityarea.Participantsweretestedeitheratauni-
versity child development laboratory or at local day care centers or nursery schools. Partici-
pants were initially recruited by letter and phone calls through commercially available lists
or by letters sent home by the day care centers. Parents who came to the laboratory received
reimbursementfortheirtravelexpensesandatokengiftfortheirchild.Themajorityofthechil-
drenwerefrommiddle-classbackgrounds,andmostwereCaucasianalthoughasmallnumber
ofAsian,AfricanAmerican,andHispanicchildrenparticipated.Anadditionalsixchildren?s
datawerediscarded,fivebecausetheydidnotknowthecountlistto‘‘six’’(meanage2;7,range:
2;3–3;8)andone because he could not be classified into aWynnstage on GN.
2.1.2. Stimuli
2.1.2.1. How Many?.
whales) presented in a single row. All the toys within each trial were identical. A Big Bird
puppet was used to introduce the task.
Stimuli consisted of small toy animals (e.g., frogs, puppies, and
2.1.2.2. What?s on this card? (WOC).
of 1–8 stickers placed on them in one or two rows. Each set had a distinct color and sticker
type (apples, bears, cows, and turkeys). For each set, the 1-card was always presented first,
followedbythe2-and3-cards,followedbythe4-and5-cards,andfinallybythe6-,7-,and8-
cards. The order within each subset varied across the four sets of cards. The apple set was
always presented first; the order of the remaining three sets was varied across children.
Stimuliconsistedoffoursetsofeightcardswithsets
2.1.2.3. Give-a-number (GN).
and dinosaurs. A Kermit puppet was used to request the toys.
Three sets of 15 small plastic toys were used: fish, horses
2.1.3. Procedure
The order of the WOC and How Many? tasks was counterbalanced. The GN task was
always conducted last. Testing for each child was completed in two sessions each of which
typically lasted 20–30 min. All sessions were videotaped and later transcribed.
M. Le Corre et al. / Cognitive Psychology 52 (2006) 130–169
137

Page 9

2.1.3.1. How Many?.
pet, saying, ‘‘Big Bird has a problem. He forgot how to count! Could you help Big Bird
count his toys?’’ The E then showed the child a tray of two, three, five, or six identical toys
on a plastic tray, and the child was asked to count the toys for Big Bird. The E presented
each set size twice in a pseudo-random order, and on one of the trials for each set size the
child was probed after the count for a cardinal response with the phrase, ‘‘OK, so can you
tell Big Bird how many toys he has here?’’ No feedback was given.
The experimenter (E) first introduced each child to a Big Bird pup-
2.1.3.2. What?s on This Card? (WOC).
For each set of 8 cards, the child was first shown the card with one object (e.g., one apple)
and asked, ‘‘What?s on this card?’’ The expected response was ‘‘an apple’’ or ‘‘apple.’’
Regardless of the child?s response the E responded, ‘‘That?s right, that?s one apple.’’ The
remainder of the procedure departed from Gelman?s in the following ways. First, our chil-
dren were only probed to produce both a count and a cardinal response three times per
card set rather than on every trial. On probe trials, if the child had given a cardinal
response (e.g., ‘‘two cows’’) the E asked, ‘‘Can you show me?’’ in an effort to elicit a count
response. If the child had spontaneously counted without providing a cardinal response,
the E asked, ‘‘So, what?s on this card?’’ which was meant to elicit a cardinal response. Sec-
ond, since we wanted to know whether children would independently infer that counting
was relevant to the task we never employed ‘‘How Many?’’ as a count probe. Children
often learn to produce a count sequence in response to ‘‘How Many?’’ as part of a social
routine without having any idea of what the sequence means or why they should produce
it (akin to children?s early production of the ABCs; Durkin, 1993). For the same reason, if
a child was reluctant to respond, we did not initiate counting by pointing to the first object
and saying, ‘‘one....’’ Finally, if a child gave non-numerical responses (e.g., ‘‘apples’’ or ‘‘I
do not know’’) to three consecutive cards, that set was terminated, and the E continued
with the next set of cards.
The method was modeled after Gelman (1993).
2.1.3.3. Give-a-Number (GN).
and said, ‘‘Kermit wants to play with a particular number of toys, and he?s going to ask
you for the number he wants. You see if you can help him!’’ The E placed a collection of
toys in front of the child and made Kermit ask, ‘‘Could you give me one dinosaur? Put it
right here, just one dinosaur!’’ The child was coaxed with phrases like, ‘‘he only wants X
dinosaur(s),’’ and ‘‘can you pick out X toy(s) for Kermit?’’ When the child provided a set
of any number, Kermit said things like, ‘‘Yay, I like to play with my dinosaurs!’’
After the initial demonstration, the E proceeded to ask for up to six toys. A titration
method modeled after Wynn (1992) was used whereby if the child succeeded at giving X
dinosaurs, Kermit requested X + 1 on the next trial. If the child then failed to give
X + 1 dinosaurs, X was requested on the subsequent trial. Children were tested only up
to the number that they could give correctly at least two out of three times.
When children did not count when producing a set, they were asked, ‘‘Can you count
and make sure you gave Kermit X toys?’’ regardless of the number they had given. If chil-
dren counted and the last number of their count did not match the number of objects
requested, the E then probed with, ‘‘But Kermit wanted X dinosaurs—can you fix it so that
there are X?’’ If children did not count in fixing the set, they were asked to verify the num-
erosity given by counting once more, and they were asked to fix it if their count failed to
match the number requested. Note that because children were allowed a single counting
To begin, the E introduced the child to Kermit the Frog
138
M. Le Corre et al. / Cognitive Psychology 52 (2006) 130–169

Page 10

error, they could be credited with having given the correct number even when they had
actually given X ± 1.
2.2. Results
2.2.1. How Many?
Evidence that a given child does not know what number words mean, or how the count
list represents number, would be of no interest if he or she does not know the words and
the counting routine. We thus analyzed children?s performance on How Many? to ensure
that they had learned a count list. To be included in the sample, each child had to have
memorized a stable count list containing at least six number word types. Because children
often counted during WOC and GN, and because both of these tasks involved sets greater
than six, the highest correct count each child produced often exceeded six. The averages
are reported in Table 1. All children used the standard English count list. As Gelman
and Gallistel (1978) documented, even our young 2-year-olds had learned a small count
list.
2.2.2. Give a Number (GN)
Before we report our results, a few comments on terminology. Children who can give all
numbers requested will be referred to as ‘‘CP-knowers’’, while children who only succeed
at giving a subset of the numbers requested will be referred to as ‘‘subset-knowers’’ or as
‘‘N-knowers’’ (where N is the highest number on which they succeeded). We are aware
that, insofar as the expressions ‘‘subset-knower’’ and ‘‘N-knower’’ imply that some chil-
dren do not understand all number words in their count lists and thus that they do not
understand counting, our terminology presupposes what we set out to determine. Despite
this shortcoming, we decided to use these expressions because we could not find others
that were theoretically neutral but nonetheless allowed for the easy integration of our
results with extant literature. The reader should bear in mind that, despite this terminol-
ogy, we will not come to a conclusion on the actual nature of subset-knowers? understand-
ing of counting until we have assessed their performance on the WOC task and on the new
Counting Puppet task presented in Experiment 2.
Table 1
Knower-level groups from Give a Number (GN)
Levels
N
Agea
Count list lengthb
Mean RangeMean Range
0-knowers
‘‘One’’-knowers
‘‘Two’’-knowers
‘‘Three’’-/‘‘four’’-knowersc
CP-knowers
7
7
2;7
2;6
3;4
3;1
3;8
2;1–3;2
2;1–3;11
2;10–4;1
2;7–3;11
2;9–4;0
8
8
6–10
6–12
8–11
6–11
8–12
10
11
15
10.1
8.8
8.7
aAges are in years and months (years;months).
bThe count list length is the number of distinct number words in each child?s count list. All children used lists
that followed the standard adult order.
cThis group comprised eight ‘‘three’’-knowers (mean age = 3;2) and three ‘‘four’’-knowers (mean age = 3;0).
M. Le Corre et al. / Cognitive Psychology 52 (2006) 130–169
139

Page 11

We first established that we could place our young participants in the same stages Wynn
found. Next, we asked whether the children who succeeded for all the numbers requested
(designated CP-knowers) were indeed the only children who provided unambiguous evi-
dence of understanding counting.
2.2.2.1. Highest number given.
the highest number of objects each child could give. To be considered to know the exact
meaning of a number word ‘‘N’’, children had to:
We adopted Wynn?s (1990, 1992) criteria for determining
(1) Give N objects at least 67% of the time when asked for that number.2
(2) Give N objects no more than half as often when asked for a different number.
(3) Satisfy conditions 1 and 2 for all numbers less than N.3
For example, if a child always gave two objects when asked for ‘‘two’’ but also gave two
objects on most trials on which she was asked for other numbers, she would not be con-
sidered to know ‘‘two’’. Also, because of criterion (3), if a child met our criterion for ‘‘one’’
and ‘‘three’’ but not for ‘‘two’’, she would be deemed to succeed only on ‘‘one’’. Children
who succeeded with all numbers requested (i.e., children who could give at least up to
‘‘six’’) were classified as CP-knowers.
Most of our children (47/50) fell into one of the five groups described by Wynn (0-,
‘‘one’’-, ‘‘two’’-, ‘‘three’’-, and CP-knowers; see Table 1). The three children who were clas-
sified as ‘‘four’’-knowers were the only ones who did not belong in any of the Wynn
groups. Because there were so few of them, ‘‘four’’-knowers will be combined with
‘‘three’’-knowers in most of the analyses below. Table 1 shows that all of the children
in Experiment 1, even the 0-knowers, could count at least to ‘‘six.’’ Since ‘‘six’’ was the
largest set ever requested, subset-knowers? failure to give the correct number of objects
asked for cannot be due to their lack of knowledge of the lexical items themselves or by
their counting range.
2.2.2.2. Spontaneous use of counting to produce sets.
als on which each child counted spontaneously, and then averaged these percentages for
each knower-level. Because most children will count when prompted, responses to E?s
request to count and check were excluded. Trials in which small numbers (2 or 3) were
requested and those in which large numbers (4 or more) were requested were analyzed sep-
arately. Fig. 1 illustrates means for small and large numbers for all knower-level groups
except 0-knowers; 0-knowers were not included because they did not have any large num-
ber trials. Consider first small numbers. Parametric analyses revealed a main effect of
We first computed the percent of tri-
2In some cases, children were considered to have given the correct number even if the actual number of objects
they had given was wrong. First, if they counted to give a set of objects or gave a set of objects and then counted
them, they were allowed one counting mistake. Thus, a child could give 5 when asked for ‘‘four’’ and be
considered correct if he had counted the set as 4. Second, on trials where they were given the opportunity to fix an
incorrect number, children?s fix was coded relative to their count. For example, if a child who had been asked for
‘‘five’’ gave 6 and counted them as 7, he was considered to have fixed the number correctly if he removed 2
objects.
3Only 4 of the 50 children were affected by this criterion. Two of them succeeded on ‘‘one’’ and ‘‘three’’ and
were classified as ‘‘one’’-knowers, one succeeded on ‘‘two’’ and ‘‘four’’ and was classified as a ‘‘two’’-knower, and
the last succeeded on ‘‘three’’ and ‘‘five’’ and was classified as a ‘‘three’’-knower.
140
M. Le Corre et al. / Cognitive Psychology 52 (2006) 130–169

Page 12

group, F(4,45) = 2.54, p = 0.05. This main effect stemmed from differences between the
CP-knowers and ‘‘three’’-/‘‘four’’-knowers who counted about 20% of the time when small
numbers were requested, and 0-, ‘‘one’’-, and ‘‘two’’-knowers who only counted about 1%
of the time. For large numbers,4we also found a main effect of group, F(3,45) = 10.54,
p < .001. Post hoc Tukey HSD tests showed that CP-knowers spontaneously counted
more frequently than all other groups (‘‘one’’-knowers, p < .01; ‘‘two’’-knowers,
p < .001; ‘‘three’’-/‘‘four’’-knowers, p < .005). No other differences were significant.
An additional analysis investigated whether children in each group used counting more
often to construct large sets than small ones. Zero- and ‘‘one’’-knowers? data were exclud-
ed from this analysis since they never counted to construct large sets. A difference
score (percentage of large sets constructed by counting minus percentage of small sets
constructed by counting) was computed for each of the children in the remaining groups.
CP-knowers were the only ones who were more likely to count on large number trials
(‘‘two’’-knowers, mean difference = 5%, t(9) = 1.47, p = .18; ‘‘three’’/-‘‘four’’-knowers,
meandifference = ?4%,
t(14) = 5.98, p < .001).
These data replicate Wynn?s findings that subset-knowers do not spontaneously use
counting to construct sets, not even for large ones. Of course, this analysis does not locate
the source of the failure—perhaps, as Gelman (1993) suggests, the difference between the
two groups is one of utilization or procedural skills. Perhaps the subset-knowers actually
understand counting but rarely spontaneously use it to construct sets in this task because
t(9) = .54,
p = .60;CP-knowers,meandifference = 40%,
4Zero-knowers were excluded from this analysis because only one was tested on large numbers. He never
counted.
0
20
40
60
80
100
small (2-3) large (4-6)
1-knowers
2-knowers
3- & 4-knowers
CP-knowers
Mean % spontaneous counting
Numerosity requested
Fig. 1. Mean percentage of trials in which children spontaneously used counting to construct sets in the GN task
as a function of set size requested and knower-level. Counting was deemed ‘‘spontaneous’’ if it occurred prior to
probing by the experimenter on a particular trial; e.g., counting that occurred in response to the experimenter?s
request to ‘‘check and make sure this is N’’ was not included.
M. Le Corre et al. / Cognitive Psychology 52 (2006) 130–169
141

Page 13

they are not aware that counting is a possible means of set construction or perhaps
because the co-ordination of counting and set construction places too many processing
demands on their fragile understanding of counting.
If the subset-knowers really do understand counting, their counts, however rare, should
at least end at the target number as frequently as those of the CP-knowers. To determine
whether this was the case, we computed the percentage of spontaneous counts ending at
the target for each child, and then computed the average percentage for each knower-level.
Because so few subset-knowers ever counted, we grouped all subset-knowers together. The
group was mostly composed of ‘‘three’’/‘‘four’’-knowers (5 out of 8). Almost all of the CP-
knowers were included (13/15). The remaining two CP-knowers were not included because
they never counted spontaneously; rather, they always put objects on the table without
counting and then correctly fixed the sets when they were asked to do so (see section on
fixing below).
CP-knowers were marginally more likely to end their spontaneous counts at the target
number than subset-knowers (94% vs. 58%; t(8) = 2.01, p = .08). Inspection of the perfor-
mance of individual subset-knowers suggests that the group?s positive performance was
exclusively due to the ‘‘three’’-/‘‘four’’-knowers. The few ‘‘one’’- and ‘‘two’’-knowers
who were included in this analysis never ended their counts at the target number, but
the ‘‘three’’-/‘‘four’’-knowers did so 93% of the time.
In sum, although the CP-knowers used counting to construct sets of all sizes, they were
much more likely to count when they were asked for large numbers; when asked for small
numbers, they often grabbed the correct number of toys without counting them. When
counting, CP-knowers almost always stopped at the target number. In contrast, all of
the subset-knowers predominantly used a non-count based strategy regardless of the size
of the requested sets; in fact most of them never counted. Of the subset-knowers who did
sometimes count, only those who were ‘‘three’’-/‘‘four’’-knowers ended their counts at the
target number; the few ‘‘one’’- and ‘‘two’’-knowers who sometimes counted never ended
their counts at the target.
2.2.2.3. Fixing sets with incorrect numbers of objects.
ing represents number, then when their counting reveals that the set they constructed is not
comprised of the requested number of objects, they should at least attempt to fix the set by
adjusting it in the right direction. To test this prediction, we selected trials for which chil-
dren initially gave the wrong number of objects (e.g., gave six objects when asked for four)
but then correctly counted the set they had given. Then we determined whether children
understood that their count revealed that they had given the wrong number by analyzing
what they did when they were asked to fix their answer. In particular, we determined the
probability that they would either fix the set in the wrong direction or leave it unchanged.
Children who understand how counting represents number should rarely leave the number
unchanged or change it in the wrong direction; in other words, they should at least change
their answer in the right direction. Trials on which children gave an incorrect number of
objects but did not count them correctly were not included because, under these condi-
tions, children could have left incorrect sets unchanged because they doubted the accuracy
of their counts.
Three ‘‘one’’-knowers, 8 ‘‘two’’-knowers, 8 ‘‘three’’-/‘‘four’’-knowers, and 10 CP-know-
ers were included in this analysis. The excluded CP-knowers did not have any fix trials
because they always gave the correct number on their own. In contrast, the excluded sub-
If children understand how count-
142
M. Le Corre et al. / Cognitive Psychology 52 (2006) 130–169

Page 14

set-knowers did not have fix trials either because they did not count in response to the E?s
request to check their answer or because they did not count correctly when they were
asked to check. The mean percentages of incorrect sets left unchanged or corrected in
the wrong direction are reported in Fig. 2. A one-way ANOVA showed a main effect of
group on percentage of incorrect sets left unchanged or changed in wrong direction,
F(3,25) = 5.45, p < .01. Tukey post hoc tests showed that CP-knowers were significantly
less likely to fail to fix correctly than ‘‘two’’-knowers (p < .01) and ‘‘three’’-/‘‘four’’-know-
ers (p < .05). That CP-knowers and ‘‘one’’-knowers did not differ is likely due to the small
number of ‘‘one’’-knowers who contributed to this analysis, because the ‘‘one’’-knowers
did not differ from the other subset-knowers. The difference between CP-knowers and
the subset-knowers reflects the fact that 70% of the CP-knowers (7/10) always fixed cor-
rectly, whereas about 60% of the subset-knowers (11/19) never fixed correctly. Subset-
knowers? most common response was to leave the sets unchanged and state that they
had given the right number.
2.2.3. Interim conclusions from GN analyses
Our initial sample of 56 children included younger participants than those of Wynn
(1990, 1992) or Gelman (1993). Of those, five were not analyzed because they could not
count up to ‘‘six’’, and seven were 0-knowers, suggesting that our sample indeed encom-
0
10
20
30
40
50
60
70
80
90
100
Mean % no change / incorrect fix
1-knowers CP-knowers3- & 4-knowers 2-knowers
Fig. 2. Mean percentage of trials in which children left their answer unchanged or changed it in the wrong
direction (e.g., adding objects when there already were too many) when they had given the wrong number of
objects but correctly counted them so that the experimenter asked ‘‘But I wanted N! Can you fix it and make it N?’’
M. Le Corre et al. / Cognitive Psychology 52 (2006) 130–169
143

Page 15

passed the earliest steps of acquiring an integer list representation of number. The substan-
tial number of 0-knowers among our sample is consistent with Fuson?s (1988) claim that
children learn the count list as part of a meaningless routine before they have assigned any
numerical meaning to any of the number words. Our results were consistent with those of
Wynn (1990, 1992). Like hers, the rest of our sample consisted of subset-knowers and CP-
knowers. Subset-knowers could all recite the count list at least up to ‘‘six’’ but could not
reliably construct sets of more than one, two, three, or four objects. CP-knowers could
construct sets of all sizes requested, suggesting that, unlike subset-knowers, they could
have constructed sets of any size represented in their count list. Subset-knowers almost
always used non-count based strategies to construct sets (e.g., randomly grabbing a set),
and their choice of strategy for producing sets was insensitive to the size of the requested
set. When they did count, they did not use the information provided by their count to fix
incorrect sets. In contrast, CP-knowers often used counting, particularly when they were
asked for numbers that lay outside of the subitizing range, and they consistently used
information provided by their counts to fix incorrect sets. One result is at odds with
Wynn?s. The five ‘‘three’’- and ‘‘four’’-knowers who used counting to construct sets were
as likely as CP-knowers to stop counting when they had reached the target set size. Thus,
GN may have underestimated understanding of the numerical meaning of counting in
some of the more advanced subset-knowers.
The difference in the fixing abilities of subset-knowers and CP-knowers is particularly
interesting because it demonstrates that the difference between the two groups was not just
a matter of accuracy. Rather, CP-knowers were the only ones who always fixed sets in the
right direction; subset-knowers not only left incorrect sets unchanged but also sometimes
fixedsetsinthewrongdirection(e.g.,addedmoreobjectstoasetthatalreadyhadtoomany).
This result supports the discontinuity hypothesis, for it suggests that subset-knowers truly
did not understand counting. Adherents of the continuity hypothesis might reply that these
children may have known that the number they had given was wrong but failed to fix the set
properly because they could not implement successful set transformation strategies.
If subset-knowers? knowledge of counting was indeed masked by procedural obstacles,
the WOC task, a ‘‘count to cardinality’’ task that makes fewer processing demands on chil-
dren, should reveal their knowledge. On the other hand, if the subset-knowers? failures
were caused by the fact that they do not yet understand how counting represents number,
then children should perform consistently on both tasks, despite the fact that WOC is an
easier task.
2.2.4. What?s on This Card? (WOC)
This task elicited a wide variety of response patterns. The three major response types
were (1) noun phrases without number words (e.g., ‘‘apple’’ or ‘‘apples’’); (2) cardinal
answers (e.g., ‘‘three’’; ‘‘three apples’’, or ‘‘one, two, three. That?s three apples!’’); and
(3) counts without cardinal answers (e.g., ‘‘one, two, three, four’’ or ‘‘one, two, three, four
apples’’). Most children produced all three response types, though the dominant type was
quite variable. Because this task is less familiar in the literature, Appendix A provides
excerpts of interviews of three children, each one illustrating a particular performance
type. Since our interest is in whether WOC will reveal that children?s failures on GN reflect
deficits in procedural or utilization skills, all of our analyses of WOC will be presented as a
function of GN knower level. As in our analyses of GN, ‘‘three’’- and ‘‘four’’-knowers will
be analyzed together.
144
M. Le Corre et al. / Cognitive Psychology 52 (2006) 130–169