Developmental Science 5:2 (2002), pp 186–212
ARTICLE WITH PEER COMMENTARIES AND RESPONSE
Blackwell Publishers Ltd
How infants process addition and subtraction events
Leslie B. Cohen and Kathryn S. Marks
Department of Psychology, The University of Texas at Austin, USA
Abstract
Three experiments are described that assess 5-month-old infants’ processing of addition and subtraction events similar to those
reported by Wynn (1992a). In Experiment 1, prior to each test trial, one group of infants was shown an addition event (1 +
1) while another group was shown a subtraction event (2 − 1). On test trials, all infants were shown outcomes of 0, 1, 2 and
3. The results seemed to require one of two dual-process models. One such model assumed that the infants could add and subtract
but also had a tendency to look longer when more items were on the stage. The other model assumed that infants had a preference
for familiarity along with the tendency to look longer when more items were on the stage. Experiments 2 and 3 examined the
assumptions made by these two models. In Experiment 2, infants were given only the test trials they had received in Experiment
1. Thus, no addition and subtraction or familiarity was involved. In Experiment 3 infants were familiarized to either one or two
items prior to each test trial, but experienced no actual addition or subtraction. The results of these two experiments support
the familiarity plus more items to look at model more than the addition and subtraction plus more items to look at model.
Taken together, these three experiments shed doubt on Wynn’s (1992a) assertion that 5-month-old infants can add and subtract.
Instead they indicate the importance of familiarity preferences and the fact that one should be cautious before assuming that
young infants have sophisticated numerical abilities.
Learning the number system and how to manipulate it
is one of the most difficult tasks a young child encounters; it is a slow and laborious process taking years to
complete (for example, Fuson, 1988). Children study
mathematics from their earliest school days to high
school graduation and beyond. However, like most areas
of psychology, there are multiple perspectives on this
topic. Three major views on the development of numerical competence can be distinguished. The empiricist
view argues that children learn about numbers by
observing numerical transformations and noting the
consistencies between events (Kitcher, 1984). Piaget’s
constructionist view argues that the number concept is
built from previously existing sensorimotor intelligence
(Piaget, 1941/1952). In contrast, a more recent nativist
view argues that sensitivity to number is innate and even
young infants possess strikingly mature reasoning abilities in the numerical domain (Wynn, 1992b, 1992c).
Over the course of the last 20 years, researchers have
explored questions about the roots of numerical knowledge using looking time techniques with infants. The
first area to be investigated was called subitization. Subitization is the rapid, perceptual enumeration of small
sets, usually from one to four items. It is thought that
adults subitize unless a display contains more than four
or five items, in which case they revert to counting
(Balakrishnan and Ashby, 1992). Some researchers have
suggested that infants may also have the ability to subitize small arrays of items. Starkey and Cooper (1980),
the first to propose infant subitization, found that
infants at 5.5 months of age were able to discriminate
two from three dots, but not larger numbers of dots.
Further research has since replicated Starkey and
Cooper’s (1980) findings both with neonates (Antell &
Keating, 1983) and with 10- to 12-month-olds, the latter
using common objects instead of dots (Starkey & Cooper,
1980). Together this research may provide evidence for
the presence of numerical knowledge during early infancy.
However, more recent research is telling a different
story. In contrast to previous studies, Clearfield and Mix
(1999a, 1999b) systematically manipulated contour
length and area in the standard subitization paradigm
with 6- to 8-month-old infants. They reported that
infants dishabituated to a change in either contour
length or area, but not to a change in number. As a
result, they concluded that infants may actually be using
Address for correspondence: Leslie B. Cohen, Department of Psychology, Mezes Hall 330, University of Texas, Austin, TX 78712, USA; e-mail:
cohen@psy.utexas.edu
© Blackwell Publishers Ltd. 2002, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA.
Infant addition and subtraction
continuous quantity rather than number to discriminate
between displays, and thus may not be subitizing. Holding mass constant, Feigenson and Spelke (1998) reached
a similar conclusion with 7-month-old infants. Thus, the
conclusion that infants are subitizing remains controversial. Further investigation is still necessary to determine
infants’ actual subitizing ability as well as the age at
which subitizing first occurs.
A second body of evidence indicates that infants may
be able to process numerical information in one modality and then transfer it to another. Starkey, Spelke and
Gelman (1983, 1990) were the first to show that 6- to 9month-old infants might be able to enumerate sounds
and match them correctly with a visual display depicting
that number. These results are even more remarkable than
those for subitizing because they suggest some primitive
counting ability by infants (Starkey et al., 1990). However, this research is also controversial. Other laboratories,
using infants of the same age, have been unable to replicate the original findings (Moore, Benenson, Reznick,
Peterson & Kagan, 1987; Mix, Levine & Huttenlocher,
1997). In addition, Mix, Huttenlocher and Levine (1996),
using a procedure adapted for preschoolers, found that
3-year-olds are unable to correctly match auditory to
visual numerosity. Thus, as with the subitizing results,
there is no uniform agreement that infants under 6 months
of age, or even young children, are able to enumerate
sounds and then match them with a visual display.
Given the evidence, albeit tentative, that young infants
have some understanding of number, Wynn (1992a) took
the next step and asked to what extent infants are able
to actively manipulate the number system. In what has
become a frequently cited paper, Wynn (1992a) argued
that infants as young as 5 months of age ‘are able to
calculate the precise results of simple arithmetical operations’ ( p. 750, emphasis added). In her first set of
experiments, Wynn showed infants a large stage on
which various objects were inserted and removed. In the
1 + 1 condition, infants saw a doll placed on the stage.
A screen then rotated up to occlude the middle of the
stage. Infants then saw a second doll placed behind the
screen. When the screen rotated back down, infants saw
one of two outcomes. In the arithmetically possible
event, there were two dolls standing on the stage. In the
arithmetically impossible event, there was only one doll
standing on the stage. A similar course of events took
place in the 2 − 1 condition. Initially two dolls were
placed on the stage one at a time. After the screen rose
to occlude the dolls, a hand entered and removed one of
the dolls. At the end of the trial, either one (arithmetically possible event) or two dolls (arithmetically impossible event) were present on the stage. Wynn found that
infants looked significantly longer at the impossible
© Blackwell Publishers Ltd. 2002
187
outcome than at the possible outcome. In a separate experiment (Wynn, 1992a, Experiment 3), she showed
infants 1 + 1 = 2 or 3. As with the original experiment,
she found that infants looked significantly longer at the
impossible outcome of 3. She argued that this was evidence that infants were actually predicting the precise
outcome of the event, rather than relying on simpler
mechanisms such as directionality.
Based on the results of her experiments, Wynn (1995a,
1995b) has argued that infants are not only sensitive to
number, they are able to manipulate small numerosities.
In the course of her work, Wynn has made three major
claims about infants’ abilities. The first is that infants understand the numerical value of small collections of objects.
Number is abstracted over varying perceptual details
(Wynn, 1992a, 1995b, 1996). Second, and related to the first,
is that infants’ knowledge is general and can be applied to
varying items and different modalities (for example,
Starkey et al., 1990). Third, she claimed that infants are
able to reason at the ordinal level and compute the result
of simple arithmetic problems (i.e. add and subtract).
In contrast to Wynn’s innate, domain-specific, numerical approach, Simon (1997, 1998) has argued that a
non-numerical, domain-general set of competencies can
account for the data Wynn (1992a) presented. One is
memory and discrimination. Another is the ability to individuate a small set of items. The third is object permanence, and the fourth is the ability to represent objects in
terms of spatio-temporal characteristics (Simon, 1998).
Both Simon and Wynn have made predictions regarding what infants should do in the Wynn task. In fact, both
approaches make the assumption that infants will compare
sets of items based upon a one to one correspondence
between the sets and will respond more (i.e. look longer)
at arithmetically impossible events than at arithmetically
possible events. However, there are other possible reasons why infants may respond more to the impossible
events than to the possible events in the Wynn task.
One possibility is that infants understand that when
material is added, the outcome should be more, or less
in the case of subtraction, but they don’t know how
much more or less. This ‘directional’ explanation would
be consistent with a rudimentary understanding of the
ordinal property of numbers, an assumption made by
Simon as well. Reasoning at this level would involve
comparing the final outcome to the initial display based
upon the relative amount of material. As long as the
outcome is consistent with the direction of transformation, i.e. greater than or less than, the infants should
look less than when the direction of transformation is
violated. As we noted earlier, Wynn (1992a) presented
some preliminary evidence (Exp. 3) that infants are
doing more than making an ordinal transformation, but
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Leslie B. Cohen and Kathryn S. Marks
as we shall see from our Experiment 2, there may be
another explanation for her results. In any case, it seems
reasonable at least to consider this directional hypothesis.
Another possible explanation, one that has not been
addressed before in the context of infant addition and
subtraction, is that the infants are simply responding
more to familiar than to novel displays. That familiarity may be based upon either the number of objects in
the display, or as Clearfied and Mix (1999a, 1999b) and
Feigenson and Spelke (1998) have proposed, the overall
quantity created by the objects. Since the 1960s theorists
have proposed that organisms are most interested in an
intermediate, optimal level of stimulation (e.g. Berlyne,
1963; Hunt, 1965; McCall & Kagan, 1967). According
to Berlyne, for example, that level of stimulation is based
upon the overall novelty, complexity and incongruity of
the display. Several experiments in the 1960s and 1970s
demonstrated that indeed young infants often responded
more to a familiar stimulus than to a novel stimulus. In fact,
recently Roder, Bushnell and Sasseville (2000) reported
another example of 41/2-month-old infants having a
familiarity preference, prior to a novelty preference.
These results regarding infants’ preferences for familiarity and novelty have been summarized in the threedimensional model presented by Hunter and Ames (1988).
According to this model, with repeated presentations of
a stimulus, infants should display a familiarity preference
prior to a novelty preference. Furthermore, the extent of
this familiarity preference should depend upon the age of
the infant, with younger infants showing a greater familiarity preference than older infants, and the complexity
of the stimulus or event, with more complex events producing greater familiarity preferences than simple events.
Based upon this type of model, one might well predict
that the Wynn task could be producing familiarity preferences. The infants are relatively young, approximately
5 months of age. The task is quite complex, with noises,
multiple objects and a hand moving in and out, and
both a screen and a Venetian blind going up and down.
Furthermore, the infants have not been habituated to
any of the events so they would be expected to be in an
early stage of processing the objects, a stage when a
familiarity preference is likely to occur. It is also the case
that in the addition event they receive many more exposures to the single object (the incorrect result) than to
the two objects (the correct result) whereas in the subtraction event they receive many more exposures to the
two objects (the incorrect result) than to the single
object (the correct result). Therefore, based upon the
model provided by Hunter and Ames, the conditions
would seem optimal for infants to look longer at the
impossible event, not because it is impossible, but
because it is more familiar.
© Blackwell Publishers Ltd. 2002
Table 1 Predicted looking times based upon three different,
but possible explanations
Addition Task
1
1
1
1
+
+
+
+
1
1
1
1
=
=
=
=
0
1
2
3
Familiarity
Directional
Computational
Short
Long
Short
Short
Long
Long
Short
Short
Long
Long
Short
Long
Subtraction Task
2
2
2
2
−
−
−
−
1
1
1
1
=
=
=
=
0
1
2
3
Familiarity
Directional
Computational
Short
Short
Long
Short
Short
Short
Long
Long
Long
Short
Long
Long
One way of testing among these three explanations is
to present infants with 0 and 3 objects as well as the 1
and 2 objects typically used in this task. The three different explanations and the unique predictions generated
from them are presented schematically in Table 1. Note
that all of them predict the same pattern of results
reported by Wynn (1992a) and Simon et al. (1995) in the
1 and 2 object tests. However, they make different predictions regarding 0 and 3 objects. In the familiarity
preference case infants would be responding most to the
outcome they had seen the most. That is, they would
find the outcome that matched the initial state of the
display most interesting because they were in an early
stage of encoding the display. In essence, this explanation assumes only a same–different comparison between
the initial and final displays.
In contrast to this explanation, according to the directional predictions, the infants may have some ordinal
understanding of the changes to the display. It is possible that infants understand that when material is added,
the outcome should be more (or less, in the case of subtraction). This explanation would be consistent with a
rudimentary understanding of the ordinal property of
numbers. It would also be consistent with comparing the
final outcome to the initial display based upon the
amount of material rather than the number. As long as
the outcome is consistent with the direction of transformation, i.e. the ‘directional’ hypothesis shown in Table 1,
the outcomes that result in more (for addition) and less
(for subtraction) will result in lower levels of looking.
Finally, the most sophisticated possibility shown in
Table 1 would be the actual computation of the outcome
based not only upon direction, but also on the actual
number. In essence, this hypothesis assumes that infants
Infant addition and subtraction
are able to add and subtract small numbers, consistent
with Wynn’s (1992a) view.
The main purpose of our first experiment was to replicate previous findings within the context of a larger set
of test events that includes outcomes with 0 and 3
objects as well as 1 and 2 objects. In so doing we hoped
to be able to evaluate the plausibility of the three explanations just discussed.
Experiment 1
Method
189
in a separate control room. One experimenter monitored
the image of the infant’s face, but remained blind to the
stage. When the infant fixated on the stage, the experimenter pressed a button connected to an Apple Power
Macintosh 8500 computer programmed to record the
duration of fixation. The computer signaled the beginning
and ending of the trials. A Panasonic color camera,
positioned behind the infant, recorded the stage. In
addition, the signal from each camera was routed into a
Videonics digital video mixer where the images were
combined on to one tape. This tape was used for reliability purposes. Looking time data from 20 participants
were recoded. The mean correlation between the two
observation sessions was 0.93 (SD = 0.06).
Participants
Eighty healthy, normally developing infants, 5 months
of age (M = 21.90 weeks, SD = 1.22 weeks) participated
in this study. Of the 80 infants, 85% were Caucasian. The
majority of parents had at least a four-year college
degree. An additional 31 infants participated but were
excluded from data analysis due to fussiness (N = 28) or
persistent inattention (N = 3).
Stimuli
The toys used in this study were brightly colored stuffed
toy monkeys. The main body of the monkey was light
blue. There were colored dots on the abdomen and
stripes on the paws. The ears of the monkey were bright
red. A squeaker inside the monkey was pressed repeatedly allowing the experimenters to draw attention to the
object as it was moved across the stage.
Apparatus
The apparatus was modeled as closely as possible on the
Wynn (1992a) experiment. Each infant sat in a car seat
attached to a low table, facing a large puppet stage,
60 cm high by 80 cm wide by 30 cm deep. The infant
was separated from the stage by 100 cm. The parent sat
behind and to the right of the infant. The stage was
constructed from bright yellow foam board. A door, cut
in the right side of the stage allowed the visible addition
and subtraction of the objects (toy monkeys). A hidden
trap door in the back of the stage allowed an experimenter to add or remove monkeys surreptitiously.
A screen, 35 cm wide by 18 cm high, rotated on a
horizontal rod connected to the front of the stage. An
attached mechanism allowed a second, hidden experimenter to rotate the screen up and down. When in the
upright position, the side of the screen exposed to the
infant was white. A reading light with a 40-watt bulb
was aimed at the center of the stage from above. Dim
recessed lights in the ceiling of the room provided
additional indirect light. Finally, a large dark green
mini-blind hung over the stage and could be dropped
down in front of the stage between trials. Dark
green curtains surrounding the stage concealed both
experimenters.
A Panasonic, low-light, black and white camera was
positioned under the stage and aimed at the infant’s face.
This camera was connected to a monitor and Sony VCR
© Blackwell Publishers Ltd. 2002
Procedure
Infants were randomly assigned to either the addition
or subtraction condition. Equal numbers of male and
female infants were assigned to each condition. Order of
presentation of the test trials was counterbalanced using
a Latin Square.
Three experimenters worked together to run the
experiment. The first two experimenters were in the testing room, behind the puppet stage. One experimenter
controlled the screen and visibly placed the monkeys on
the stage. A second experimenter operated the miniblind between trials and secretly inserted and removed
the monkeys through a trapdoor in the back of the stage
to create the correct outcome for each test trial. The
third experimenter sat in a control room and recorded
looking times on-line. Digi-Tech, hands-free walkietalkies allowed the experimenter in the control room
to communicate the beginning and end of trials to the
second experimenter in the testing room.
Pre-test. Infants were presented with two trials to
familiarize them with the rotation of the screen, the
movement of the hand, and the sight and sound of the
monkey. In the first pre-test trial, no items were placed
on the stage. The screen began flat against the front of
the stage and rotated up to vertical. An empty hand
entered the stage through the side door, ‘tiptoed’ across
the stage from the side, went behind the screen, paused,
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Leslie B. Cohen and Kathryn S. Marks
and left the display empty. The screen then rotated back
in the opposite direction, until it returned to its starting
point. Infants were allowed to continue looking at the
end of the event for an additional two seconds. The
mini-blind was lowered to end the trial.
In the second pre-test trial, infants saw an event identical to the test event except that the outcome was not
shown. In the addition condition, the stage began empty.
A hand holding one monkey entered the stage through
the door on the side, placed the monkey on the stage,
and exited. The screen was rotated up to vertical. A second monkey was then added to the stage through the
side door and placed behind the screen. The empty hand
left the stage through the side door. For the infants in
the subtraction condition, the stage also began empty.
Two monkeys were placed on the stage by the hand, one
at a time, and the screen then was rotated up. An empty
hand entered the stage and removed one monkey from
behind the screen and carried it off the stage through the
door on the side of the stage. Infants in both the addition and subtraction condition were allowed to continue
looking at the end of their second pre-test event for an
additional two seconds. Once again the mini-blind was
lowered to end the trial.
Test. Infants were shown eight test trials with each of
the four outcomes presented twice. The screen began flat
against the front of the stage. The experimenter placed
one monkey (in the addition condition) or two monkeys
(in the subtraction condition) on the stage, and the
screen was rotated up to vertical. The first experimenter
then either added or subtracted one monkey from the
display through the door on the side of the stage. Hidden from sight, the second experimenter added or
removed monkeys as needed through the trap door to
produce the outcome for that trial. The screen was
rotated down to its starting position to reveal the outcome of the event. See Figures 1 and 2 for a schematic
diagram of the experimental events.
Looking times to each display were recorded by the
third experimenter in the control room. A look was considered valid if it was longer than one continuous second. A trial was terminated when the infant looked away
from the display for longer than one continuous second
or when the infant looked at the display for a maximum
of 60 continuous seconds. At the end of each trial, the
mini-blind was lowered across the opening in the stage
to allow the first and second experimenters to reset the
display.
Results
A 2 Condition (addition vs subtraction) × 2 Trial Block
(first vs second) × 4 Outcome (0, 1, 2 and 3) × 2 Gender
© Blackwell Publishers Ltd. 2002
Figure 1 Schematic drawing of the sequence of events in the
addition condition. First one object is placed on the stage. The
occluding screen is raised and a second object is placed on
the stage. Then the screen is dropped to reveal either 0, 1, 2
or 3 objects.
(male vs female) ANOVA yielded a number of significant results. The main effect of outcome was significant,
F(3, 228) = 5.38, p < 0.01, as was the main effect of trial
block, F(1, 76) = 32.24, p < 0.01. The main effect of trial
block indicated significantly longer looking times during
the first block (M = 8.29 s, SD = 6.15 s) than during the
second block (M = 6.26 s, SD = 4.49 s) of test trials.
There was also a significant main effect of gender, F (1,
76) = 4.91, p < 0.05. Overall, males (M = 8.03 s, SD =
6.12 s) looked significantly longer than did females (M
= 6.53 s, SD = 4.64 s). However, gender did not interact
with any other main effects or interactions, so it was
excluded from subsequent analyses.
The main question addressed by this study was
whether looking times to zero, one, two and three items
Infant addition and subtraction
191
Figure 3 Infant looking times in Experiment 1 to 0, 1, 2 and 3
items on block 1 and block 2 test trials when in either addition
or subtraction conditions.
Figure 2 Schematic drawing of the sequence of events in
the subtraction condition. Two objects are placed on the stage.
The occluding screen is raised and one of the objects is
removed. Then the screen is dropped to reveal either 0, 1,
2 or 3 objects.
varied as a function of the addition or subtraction
manipulation. The interaction of interest was the Outcome
× Condition interaction, which was significant, F(3, 228)
= 3.12, p < 0.05. The three-way Trial Block × Outcome
× Condition interaction was also marginally significant,
F(3, 228) = 2.61, p = 0.052 and is shown graphically in
Figure 3.
In order to understand this interaction we first ran
separate analyses for each trial block just on outcomes
one and two, the same outcomes tested by Wynn. As
shown in Table 1, all three explanations predicted a
Condition × Outcome interaction (i.e. they predicted
that infants should look longer at an outcome of one
than an outcome of two in the addition condition, but
infants should look longer at an outcome of two than at
an outcome of one in the subtraction condition).
© Blackwell Publishers Ltd. 2002
A 2 Condition × 2 Outcome ANOVA on the first block
of test trials produced only one significant result, the
predicted interaction, F (1, 78) = 11.38, p = 0.001. In the
addition condition infants looked significantly longer
at an outcome of one (M = 10.94 s, SD = 7.36 s) than at
an outcome of two (M = 7.82 s, SD = 4.97 s), F (1, 39) =
6.40, p = 0.02. In the subtraction condition, on the other
hand, infants looked significantly longer at an outcome
of two (M = 9.95 s, SD = 7.67 s) than at an outcome of
one (M = 7.09 s, SD = 6.96 s), F(1, 39) = 5.04, p = 0.03.
The only significant effect in the second block of trials
was the main effect of outcome, F(1, 78) = 7.92, p < 0.01.
Infants looked significantly longer at an outcome of
two (M = 7.24 s, SD = 5.56 s) than an outcome of one
(M = 5.53 s, SD = 3.24 s).
One final set of analyses examined just those outcomes that were novel, that is, zero, two and three in the
addition condition and zero, one and three in the subtraction condition. Each explanation shown in Table 1
predicts a different pattern of results. The familiarity
explanation predicts short looks to all outcomes in both
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Leslie B. Cohen and Kathryn S. Marks
addition and subtraction conditions. The directional
explanation predicts an interaction with long looks to
an outcome of zero in the addition condition and long
looks to an outcome of three in the subtraction condition. Finally the computational explanation predicts the
quadratic looking pattern of long, short, long in both
addition and subtraction conditions.
Once again separate analyses were run for each trial
block. In these analyses Outcome had three levels: zero,
middle and three, where the middle outcome was two for
the addition condition and one for the subtraction condition. The 2 Condition × 3 Outcome ANOVA for trial block
one yielded only a significant main effect for Outcome,
F(2, 156) = 4.60, p < 0.02. Subsequent linear and quadratic
trend tests produced a significant linear trend, F(1, 156)
= 9.124.60, p < 0.005. Infant looking times increased
regularly from the zero outcome (M = 6.43 s, SD = 4.43
s), to the middle outcome (M = 7.46 s, SD = 6.02 s), to
the three outcome (M = 8.85 s, SD = 5.63 s). The quadratic trend did not approach significance, F(1, 156) < 1.0.
No significant effects were found for trial block two.
Discussion
The results of this study, at least those in the first block
of trials, replicated the major original findings of Wynn
(1992a). In the subtraction condition, infants looked
significantly longer at an outcome of two than one and
in the addition condition, infants looked significantly
longer at one than two. The main question, though, is
what is the best way to explain these findings? One possibility is that the infants were actually adding and subtracting. However, other explanations are also possible
and the present experiment was designed to enable us to
decide among them.
By adding outcomes of zero and three to the original
outcomes of one and two, we were able to assess three
distinct processing explanations. The familiarity explanation states that infants should look longest at the outcome that is most familiar to them. The most familiar
outcome will correspond to the first number of items on
the stage (prior to the addition or subtraction manipulation). In the case of addition, this would be an outcome of one. In the subtraction condition, it would be
an outcome of two. The directional explanation assumes
only a directional understanding of addition and subtraction. In this case, infants would look longer at outcomes in the opposite direction than expected. In the
case of addition, outcomes of zero and one are directionally incorrect and should both be looked at longest.
In subtraction, outcomes of two and three are directionally incorrect and should be looked at longest. Finally,
a pure computational explanation would predict that
© Blackwell Publishers Ltd. 2002
infants should look longest at all of the arithmetically
incorrect outcomes.
The present data, particularly the obtained linear trend,
do not unequivocally support any of these explanations.
The first one outlined was a simple familiarity preference. Although infants do show a preference for one in
the addition condition and two in the subtraction condition, as predicted by a familiarity preference, their
looking times also should be equally low at the other
novel outcomes, zero, middle and three in both conditions. Instead, their looking times displayed an increasing linear trend. Thus, a simple familiarity preference, by
itself, cannot account for the data.
The second alternative was a qualitative understanding of the direction of the operation. Infants should look
longer at events that violate the directionality of the
operation. In the addition condition, infants should look
longer at the zero outcome than at the middle or three
outcome. In the subtraction condition, they should look
longer at the three than at the middle or zero outcome.
Whereas the data from the subtraction condition are
consistent with this strategy, the data from the addition
condition are not. Infants showed the opposite pattern
of looking times.
The final possibility outlined was true computational
reasoning. This explanation predicts that infants should
show increased looking times to all of the impossible
events. In the case of the addition condition, two is the
only possible outcome. Looking times to the outcomes of
zero, middle (i.e. two in addition) and three should follow a high (impossible), low (possible), high (impossible)
pattern. Similarly in the subtraction condition, one is the
only possible outcome. Thus, looking times to the outcomes of zero, middle (i.e. one in subtraction) and three
should follow the same high, low, high pattern. The data
clearly do not support this model either. Infants’ looking
times to the novel outcomes of zero, middle and three
showed a strong linear increasing trend and no hint of a
quadratic trend as predicted by the computational
model. Thus, when both zero and three are included as
alternatives, the results from this study are not consistent
with knowledge of addition and subtraction.
From the point of view of a purely arithmetic reasoning interpretation, the most troubling finding was that in
both the addition and subtraction conditions infants did
not look very long at the zero outcome even though in
both conditions it was an impossible event. Wynn
(1995a) has attempted to account for the zero problem
by assuming that infants look longer at impossible
events, except when the outcome is zero. However, her
argument is not entirely clear. On the one hand, she
argues that zero is a privileged entity that cannot be
represented using Meck and Church’s (1983) accumu-
Infant addition and subtraction
lator mechanism, the mechanism she uses to explain the
addition and subtraction. Because the accumulator’s
neutral position is the same as the result of an operation
ending in zero, the mechanism cannot distinguish
between the two conditions. Thus, infants cannot form
numerical expectations when the outcome of an event is
zero. However, the correct outcome in the present experiment is not zero, so that argument does not seem to
apply. On the other hand, Wynn and Chiang (1998)
report that infants can distinguish between outcomes of
zero in a magical versus expected disappearance situation, with infants tending to look longer at zero in magical as opposed to expected disappearances. Because
both our addition and subtraction conditions could be
considered cases of ‘magical disappearances’ (zero is
never the correct result of the operation), we assume
Wynn would predict infants will look longer at zero in
those situations as well. However, since we found that
infants in both the addition and subtraction conditions
tended to look less at zero, this prediction fails to
account for the results found in Experiment 1.
On the other hand, one could make the common
sense assumption that infants should look more when
there is more to look at, i.e. when there are more objects
on the stage. That assumption clearly fits with the linear
trend found in looking times to novel outcomes. A twoprocess explanation combining this ‘more to look at’
prediction with Wynn’s arithmetical reasoning hypothesis
would probably fit the present data.
However, combining the ‘more to look at’ assumption
with a familiarity preference would also fit the data.
Considering familiarity first, as we noted earlier, Hunter
and Ames (1988) outlined a theoretical model in which
familiarity and novelty preferences are based on the age
of the infants, the complexity of the task and the amount
of time infants have to process the events. Recently Bogartz,
Shinskey and Schilling (2000), Schilling (2000) and
Cashon and Cohen (2000) have all reported familiarity
preferences with the violation of expectation method
(Baillargeon, 1987; Baillargeon, Spelke & Wasserman,
1985) used for studying object permanence.
In the present experiment, which can also be considered an example of the violation of expectation method,
on each trial the infants saw either one object on the
stage and then another one added or they saw two
objects on the stage and then one subtracted. In the
addition condition, over the course of the 8 test trials
they saw one object 10 times and zero, two and three
objects only 2 times each. Infants looked longer at the
outcome that they saw most frequently, namely one
object. Infants in the subtraction condition saw two
objects 10 times, and zero, one and three objects only
twice each. They showed this same pattern of looking
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193
longer at the outcome that was most frequent, in this case
two objects. More support for the familiarity preference
comes from considering each block of four trials separately. In both the addition and subtraction conditions,
the results from the first block of trials alone mirrored
the results of the data as a whole. In contrast, in the second
block of trials, some evidence of the ‘more to look at’
assumption was present. Infants looked significantly
longer at an outcome of two than at an outcome of one
regardless of condition. This disappearance of the familiarity effect would be expected given the large number of
repetitions of the basic addition or subtraction event.
On the other hand there is no reason to expect such a
disappearance based upon the computational explanation.
Wynn’s (1992a) claim is that infants are able to precisely calculate the result of simple arithmetic problems.
To examine the hypothesis that infants were using an
imprecise, directional strategy, she also showed infants 1
+ 1 = 2 or 3. Presumably this design would rule out any
explanation based upon familiarity as well since both 2
and 3 would be novel. She reported that infants looked
longer at the impossible event, 1 + 1 = 3. However, this
evidence should only be considered suggestive given that
the difference in looking times to two versus three did
not reach statistical significance using a traditional twotailed test. We also found no significant difference
between 2 and 3 in our addition condition. Thus, at this
point, evidence that infants look longer at 3 than at 2
items after an addition manipulation should still be considered tentative. Even if one found that 3 items were
looked at more than 2 items, it is imperative that controls for looking longer when there are more items to
look at be included.
The possibility that infants look longer the more there
is to look at was really an ad hoc assumption based
upon an inspection of the test data in Experiment 1. In
order to provide an independent test of this assumption
we conducted Experiment 2. The procedure of Experiment 1 was simplified to its most basic elements, just a
presentation of the 8 test trials. No warm-up was given,
no addition or subtraction was presented and infants
were not familiarized to any of the 4 outcomes.
Experiment 2
Experiment 2 was designed to examine the possibility of
a simple preference for more items over fewer items.
Infants were given the same test trials infants had
received in Experiment 1. That is, they received two
blocks of 0, 1, 2 and 3 items presented in a Latin Square
order. However, unlike Experiment 1, prior to each test
they did not see a hand adding or subtracting items.
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They also did not receive any familiarization with 1 or 2
items prior to each test trial.
Method
Participants
Sixteen healthy, normally developing infants, 5 months
of age (M = 21.79 weeks, SD = 1.23 weeks) participated
in this study. Of the 16 infants, 69% were Caucasian. The
majority of parents had at least a four-year college
degree. One additional infant participated but was
excluded from data analysis due to fussiness.
Apparatus and stimuli
The setup of the room and stage were identical to that
used in Experiment 1. Looking time data from four randomly chosen participants were recoded as a test for
reliability. The mean correlation between the two observation sessions was 0.96 (SD = 0.02).
Procedure
Infants were randomly assigned to one of four presentation orders, counterbalanced using a Latin Square.
Equal numbers of male and female infants participated
in the experiment.
Two experimenters worked together to run the experiment. The first experimenter was in the testing room, behind
the puppet stage. She had control of the mini-blind and
presentation of the objects. The other experimenter sat in
a control room and recorded looking times on-line. DigiTech hands-free walkie-talkies allowed the experimenter
in the control room to communicate the beginning and
end of each trial to the experimenter in the testing room.
Infants were shown four different test trials in a counterbalanced order, with each of the four outcomes presented twice. When the mini-blind in front of the stage was
raised, 0, 1, 2 or 3 objects were sitting on the stage. No
manipulation of the display took place in this experiment.
Looking times to each display were recorded by one
of the experimenters. A look was considered valid if it
was longer than one continuous second. A trial was terminated when the infant looked away from the display
for longer than one continuous second. At the end of
each trial, the mini-blind was dropped in front of the
stage to allow the experimenter to reset the display.
Results
Figure 4 provides looking times to each number of
objects for each block of trials. A 2 Test Block (first or
© Blackwell Publishers Ltd. 2002
Figure 4 Infant looking times in Experiment 2 to 0, 1, 2
and 3 items on block 1 and block 2 test trials.
second) × 4 Outcome (0, 1, 2 or 3) ANOVA revealed a
main effect of outcome, F(3, 45) = 4.48, p < 0.01 and a
main effect of trial block, F(1, 15) = 13.96, p < 0.01.
There was a significant linear increase in looking time as
the number of items presented on the stage increased,
F(1, 45) = 10.82, p < 0.01; but no significant quadratic
trend, F(1, 45) = 2.56, n.s. Infants also looked significantly longer at the first block of trials (M = 9.85 s,
SD = 6.97 s) than at the second block of trials (M = 6.05 s,
SD = 4.46 s). Although the interaction between outcome
and trial block was not significant, we were interested
in comparing the looking times across the two blocks
of trials. Results of the first trial block revealed no
significant difference among outcomes. In contrast, in
the second block, there was a significant main effect
of outcome F(3, 45) = 4.02, p < 0.05, and once again, a
significant linear trend, F(1, 45) = 11.61, p < 0.005.
Infant addition and subtraction
Discussion
The primary result from this experiment was that infants
showed increased looking times as the number of items
to look at increased. That result was significant in the
overall analysis, and it was significant in the second trial
block, but not the first trial block. Apparently, infants
must be given sufficient time to process the overall testing situation before this preference is evident. These
results are consistent with the results of Wynn’s (1992a)
Experiment 3. In that experiment, she found that during
the pre-test, there were no significant differences in looking at 2 and 3 items. However, when presented with 1 +
1 = 2 or 3 in the test trials, infants appeared to look
longer (albeit not significantly) at the impossible event
with 3 items. The two blocks of our experiment can be
compared to the pre-test and test trials of Wynn (1992a).
In our first four trials, there was no preference for more
objects. However, there was a linear increase in looking
time, as more items were placed on the stage in the second block. Based upon the present results, one could
argue that Wynn (1992a) did not find pre-test differences
because the infants in her experiments were not sufficiently familiar with the testing situation to show such a
preference. Thus her apparent, albeit not significant,
demonstration that infants in an addition condition
looked longer at 3 items than at 2 items, could simply
have reflected an emerging tendency for longer looking,
the more items there were to look at.
Experiment 3
Experiment 3 was designed to be an independent test of
the possibility that a familiarity preference would
develop in this type of complex event. The experiment
examined what would happen if infants were familiarized with either 1 or 2 objects prior to receiving the test
items used in Experiment 2. Unlike Experiment 2, half
of the infants were shown 1 item prior to each test trial.
The other half were shown 2 items prior to each test
trial. Thus, in this experiment the infants had an opportunity to develop a familiarity preference, but no
opportunity to respond on the basis of addition or subtraction. Unlike Experiment 1 the infants did not receive
warm-up trials, the sight of a moving hand or other
features of that experiment’s procedure. The goal of
Experiment 3 was not to replicate directly all aspects of
Experiment 1 except addition and subtraction. The goal
was simply to add familiarization experience with either
1 or 2 objects to the test trials of Experiment 2. The
reason was to determine whether a familiarity preference
would be superimposed on the previously found linear
© Blackwell Publishers Ltd. 2002
195
trend of looking longer as the number of items on the
stage increased. As in previous experiments, infants were
given two blocks of test trials with 0, 1, 2 or 3 items.
Method
Participants
Sixteen healthy, normally developing infants, 5 months
of age (M = 21.22 weeks, SD = 0.85 weeks) participated
in this study. Of the 16 infants, 56% were Caucasian. The
majority of parents had at least a four-year college
degree. An additional 3 infants participated but were
excluded from data analysis due to fussiness.
Apparatus and stimuli
The setup of the room and stage were identical to that
used in Experiment 1. Looking time data from five randomly chosen participants were recoded for reliability
purposes. The mean correlation between the two observation sessions was 0.99 (SD = 0.001).
Procedure
Infants were randomly assigned to either the 1-item
familiarity condition or the 2-item familiarity condition.
Equal numbers of male and female infants were assigned
to each condition. Two experimenters worked together
to run the experiment. The first experimenter was in the
testing room, behind the puppet stage. She was responsible for controlling the mini-blind as well as the events
taking place on the stage. The other experimenter sat in
a control room and recorded looking times on-line.
Digi-Tech hands-free walkie-talkies allowed the experimenter in the control room to communicate the beginning and end of each trial to the experimenter in the
testing room.
Test trials
As in Experiments 1 and 2, infants were shown two sets
of four test trials in a counterbalanced, Latin Square
order, with each of the four outcomes presented twice.
The screen began flat against the front of the stage.
Either 1 or 2 objects were on the stage at the beginning
of each trial. Infants saw this configuration for approximately 2 seconds. The screen was then rotated up to
vertical to hide the stage. The experimenter added or
removed objects as needed through the trap door to produce the outcome for that trial. The screen was then
rotated down to its starting position to reveal the outcome of the event.
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The experimenter in the control room recorded
looking times to each display. A look was considered
valid if it was longer than one continuous second. A trial
was terminated when the infant looked away from the
display for longer than one continuous second. At the
end of each trial, the mini-blind was dropped across
the front of the stage to allow the experimenter to reset
the display.
Results
The results are shown separately for each block of trials
in Figure 5. A 2 Familiarization Condition (familiarization to 1 or 2) × 2 Trial block (first vs second) × 4 Outcome
(0, 1, 2 or 3) ANOVA revealed a significant three-way
interaction F(3, 42) = 6.17, p < 0.01. The ANOVA also
revealed significant outcome, F(3, 42) = 4.93, p < 0.01,
and test block, F(1, 14) = 29.55, p < 0.01 main effects.
Overall, infants looked longer at 1, 2 and 3, than they
did at 0, F(1, 42) = 14.01, p < 0.01. Infants also looked
longer during the first block of trials, M = 15.57 s, SD =
11.17 s than during the second block of trials, M = 7.32 s,
SD = 3.84 s.
To investigate the three-way interaction, the same
analyses conducted in Experiment 1 were run. We first
performed a separate analysis for each trial block on
outcomes one and two. On the first block of trials the 2
Familiarization Condition × 2 Outcome ANOVA produced only a significant interaction, F(1, 14) = 10.59,
p < 0.01. In the one-object familiarization condition
infants looked significantly longer at one object (M =
22.47 s, SD = 8.87 s) than at two objects (M = 11.82 s,
SD = 5.57 s), F(1, 7) = 10.77, p = 0.01. In the two-objects
familiarization condition looking times were in the
opposite direction with longer looking at two objects
(M = 20.60 s, SD = 16.84 s) than at one object (M =
14.69 s, SD = 10.07 s). However, the difference between
these two means did not reach statistical significance,
F(1, 7) = 2.27, < 0.20, perhaps because the N was so small.
As in Experiment 1, a final set of analyses examined
only those outcomes that were novel, that is, 0, 2 and 3
in the one-object familiarization condition, and 0, 1 and
3 in the two-object familiarization condition. Once
again, for the purpose of the analyses, the outcomes
were treated as zero, middle and three, and separate
analyses were run for each block of trials. Outcome was
significant for both the first block, F(2, 28) = 3.54, p <
0.05 and the second block, F(2, 28) = 3.97, p < 0.05. The
first block revealed a significant linear trend, F(1, 28) =
7.04, p = 0.01, with looking times increasing regularly
from the zero-object outcome, (M = 9.76 s, SD = 4.79 s),
to the middle-object outcome, (M = 13.26 s, SD = 8.00 s)
to the three-object outcome, (M = 17.73 s, SD = 13.50 s).
© Blackwell Publishers Ltd. 2002
Figure 5 Infant looking times in Experiment 3 to 0, 1, 2
and 3 items on block 1 and block 2 test trials when
familiarized with either 1 or 2 items prior to each trial.
The second block of trials produced both a marginally
significant linear trend, F(1,28) = 3.20, p = 0.08 and a
significant quadratic trend, F(1, 28) = 4.73, p < 0.05.
These trends occurred because infants looked less at the
zero outcome (M = 5.56 s, SD = 2.76 s) than at either
the middle outcome (M = 8.88 s, SD = 4.68 s) or the
three outcome (M = 7.70 s, SD = 3.9 s).
Discussion
The results of Experiment 3 are consistent with the twoprocess view that incorporates a preference for familiarity (e.g. Hunter & Ames, 1988) with longer looking when
there are more items on the stage. Infants who repeatedly saw one item at the beginning of the event had a
significant preference for one item over two items. In
contrast, infants who repeatedly saw two items tended to
have a preference (albeit not significant) for two items
over one item. Also, this tendency to look longer in the
test at the number of items presented prior to the test
occurred in trial block one but not trial block two. The
disappearance of the tendency with repeated exposure
Infant addition and subtraction
(i.e. trial block two) is consistent with a familiarity
effect as described by Hunter and Ames (1988).
In the first block of trials infants demonstrated a clear
increase in looking time as the number of test items
increased. An increase also occurred in the second block
although the tendency was for infants to look less at 0
items than at more than 0 items. Thus, in Experiment 3,
an experiment that included no addition or subtraction
manipulation, we found evidence for both a familiarity
effect and a tendency to look longer when more items
were on the stage.
Direct Comparison between Experiment 1
and Experiment 31
In Experiment 2 we asked whether infants would look
longer when more items were on the stage. The infants
did. In Experiment 3 we added familiarization experience
with either 1 item or 2 items to the test trials in Experiment 2 and asked whether infants would show a familiarity preference as well as a tendency to look longer when
more items were on the stage. They did. Strictly speaking, Experiment 3 was not designed to be a control for
Experiment 1. One can identify a number of differences
between Experiments 1 and 3 in addition to the fact that
Experiment 1 included addition and subtraction whereas
Experiment 3 did not. For example, Experiment 3 did
not contain the warm-up trials found in Experiment 1.
Experiment 1 also had the repeated appearance and disappearance of a hand, which was not present in Experiment 3. Nevertheless, in Experiment 1 we argued that one
possible reason for the results was that the infants were
displaying a familiarity preference on top of a preference
for looking more when there were more items on the stage.
Since those same two effects were found in Experiment
3, it might be instructive to directly compare the results
from Experiment 1 with the results from Experiment 3.
Sixteen infants were tested in Experiment 3. In order
to make the Ns comparable in the two studies, we
selected 16 infants from Experiment 1 that comprised
the last complete, counterbalanced group of infants run
in the study. That is, the group included 8 males and 8
females. Four infants of each sex were in the addition
condition and 4 were in the subtraction condition. Also,
each subgroup of 4 infants was assigned test trials
according to a counterbalanced Latin Square design.
We duplicated the types of analyses we had run previously in Experiment 1 and Experiment 3 except that we
added Experiment 1 versus 3 as an additional factor. As
1
We wish to thank one of the outside reviewers for suggesting this
comparison.
© Blackwell Publishers Ltd. 2002
197
in those experiments, separate ANOVAs were computed
for each block of trials.
Our first set of analyses compared infants’ looking times
to outcomes of 1 versus 2 items. On the first block of trials
a 2 Experiment (Experiment 1 vs Experiment 3) × 2
Familiarization Condition (familiarization to 1 vs 2 items,
which is also the same as addition versus subtraction in
Experiment 1) × 2 Outcome (1 vs 2 items) ANOVA yielded
a significant main effect of Experiment; F(1, 28) = 471,
p < 0.05. On the first block of trials infants looked longer
overall during Experiment 3, (M = 17.40 s, SD = 11.44 s)
than during Experiment 1, (M = 10.87 s, SD = 9.98 s).
The only other significant result was the Familiarization
Condition × Outcome interaction, F(1, 28) = 15.27,
p < 0.001. In both experiments infants looked longer at
an outcome of 1 item if they had been familiarized to 1
item, and they looked longer at an outcome of 2 items if
they had been familiarized to 2 items. The three-way interaction of Experiment × Familiarization Condition × Outcome
did not approach significance. It produced an F < 1. The
same ANOVA was run on the block two data, but no
significant differences were found. In summary, it appears
that although infants looked longer in general during
Experiment 3 than Experiment 1, they produced the same
pattern of looking in the two experiments. They looked
longer at the familiar outcome than at the novel outcome.
Our final set of analyses compared Experiments 1 and
3 on infants’ tendency to look longer when there were
more items on the stage. Once again, separate analyses
were performed for each block of trials. On trial block
one the 2 Experiment × 2 Familiarization Condition × 3
Outcome (zero, middle and three) ANOVA yielded two
significant main effects. As in the previous block one
analysis, infants looked longer in general during Experiment 3 (M = 13.58 s, SD = 9.94 s) than during Experiment 1 (M = 8.48 s, SD = 7.55 s), F(1, 28) = 6.75, p <
0.05. Infants also looked longer overall when more items
were on the stage as indicated both by a main effect of
Outcome F(2, 56) = 5.17, p < 0.01, and by the significant increasing linear trend, F(1, 56) = 10.189, p < 0.005.
The quadratic trend did not approach significance. No
significant differences were found for the block two data.
So once again, these analyses indicate that although
infants looked longer in general during Experiment 3
than Experiment 1, they produced the same pattern of
looking in both experiments. In this case the pattern was
to look longer when more items were on the stage.
General discussion
Three experiments were conducted to evaluate Wynn’s
(1992a) claim that 5-month-old infants can add and
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Leslie B. Cohen and Kathryn S. Marks
subtract. Experiment 1 was designed to test three competing hypotheses concerning why infants would look
longer at the incorrect number (1 test item) in the addition
problem and (2 test items) in the subtraction problem.
One hypothesis was that infants were actually adding
and subtracting. A second hypothesis was that they were
responding at an ordinal level to more versus fewer
items. A third hypothesis was that the infants were simply
demonstrating a greater response to the familiar test display. It should be noted that either of the last two alternatives could be accomplished by attending to the overall
quantity of objects rather than the exact number of
objects as suggested by Clearfield and Mix (1999a, 1999b).
The results of Experiment 1 did not support any of
the three hypotheses independently. However, the results
were consistent with two possible dual-process explanations. One explanation posited that infants could, in
fact, add and subtract, but that their tendency to look
longer at the incorrect number was superimposed on a
tendency to look longer when there were more items on
the stage. The other hypothesis was that infants were
responding more to a familiar outcome, but that this
preference for familiarity also was superimposed on a
tendency to look longer when there were more items on
the stage.
Experiment 2 tested whether, in fact, infants would
look longer when more items were on the stage. In
Experiment 2, infants were given only the test items
from Experiment 1 without any prior warm-up, familiarization or addition and subtraction experience. Evidence was found (overall and particularly on the second
block of test trials) for a linear increase in looking as the
number of items in the stage increased.
In Experiment 3 infants were familiarized with either
1 item or 2 items before encountering each test event.
Thus, their experience was similar to that of Experiment
1, except that there was no warm-up period and no hand
added or subtracted any items. Nevertheless, in most
respects their behavior mirrored that of infants in Experiment 1. As both the analyses of individual experiments
and the direct comparison of Experiment 1 with Experiment 3 indicated, in both experiments infants familiarized with 1 item looked longer at 1 item than at 2 items
in the test, whereas infants familiarized with 2 items
looked longer at 2 items than at 1 item in the test. There
was also a tendency in both experiments for infants to
look longer the more test items there were to look at.
Thus, Experiment 3 provided support for the familiarity
plus more items hypothesis over the addition-subtraction plus more items hypothesis.
One consistent difference between Experiment 1 and
Experiment 3 was also found. Infants looked considerably
longer overall in Experiment 3 than in Experiment 1.
© Blackwell Publishers Ltd. 2002
Although the reason for this difference in looking time
is unclear, the nature of the events themselves may help
to explain it. In Experiment 1, infants saw items placed
on a stage, and a hand enter and leave the stage. These
actions took approximately 20 seconds in the addition
condition and 23 seconds in the subtraction condition.
During the majority of this time, infants were looking at
the stage. In contrast, in the third experiment none of
these actions took place. Infants saw an item on a stage
for approximately 2 seconds, the screen rotate up, and
the screen rotate down. The entire sequence of events
took approximately 10 seconds. Assuming that there is a
maximum amount of time infants will look at any event,
the shorter procedure in Experiment 3 gave infants more
time to process the end of the event, possibly resulting
in longer looking times. In any case, despite the overall
difference in looking times and the physical differences
between Experiments 1 and 3, since type of experiment
did not interact with the main findings of a familiarity
preference and a longer looking with more items preference, these two preferences should be considered viable
explanations for the results in Experiment 1. The present
results also raise the distinct possibility that other studies using the Wynn procedure, including Wynn’s original
experiment, that have found apparent evidence for addition and subtraction, may merely have found evidence
for a familiarity preference.
These experiments are not the only ones that have
contradicted Wynn’s (1992a) assertion that young infants can add and subtract. In another recent report,
Wakeley, Rivera and Langer (2000) attempted to replicate Wynn’s studies with a more controlled procedure.
They found that infants did not look longer at the
impossible events in the addition or the subtraction conditions. Based on their findings, they argued that infants’
ability to compute the outcome of arithmetic problems
is fragile and inconsistent at best.
In response to this counter-argument, Wynn (2000)
reported a number of studies that have replicated the
original results using that procedure as well as modified
procedures. In addition, Wynn discussed three potential
methodological differences that may have affected
Wakeley et al.’s results. The first two relate to infant
attentiveness to the events. The final one relates to subject exclusion due to fussiness. The controls used in our
procedure (i.e. presenters being blind to the participant
during trials) more closely matched those of Wakeley
et al., yet we did find the same differences (i.e. looking
longer at 1 item in the addition condition and longer at
2 items in the subtraction condition) reported by Wynn
(2000). Thus, it seems that these methodological differences cannot account for the null results found by
Wakeley et al.
Infant addition and subtraction
Why, then, did we find differences when Wakely et al.
did not? According to our predictions, infants should
have shown a familiarity preference, just as they did in
previously published studies. We are not certain. One
potential difference between our Experiment 1 and the
Wakeley et al. study is the length of the intertrial interval. In Wynn (2000) and in our procedure, as soon as the
stage was reset, a new trial began. On average, the intertrial interval was less than 6 seconds with a standard
deviation of 1 second. In contrast, Wakeley, Rivera and
Langer used a consistent 10 s intertrial interval. Allowing more time to elapse between trials may have made it
more difficult for infants to become sufficiently familiar
with the 1 object. The lack of a comparable subtraction
condition also makes comparison between the two studies
difficult.
Perhaps Wakeley et al. are correct that the evidence
for infant addition and subtraction is fragile and inconsistent. However, no matter how carefully a study is
done, it is difficult to mount a convincing challenge
against previously reported evidence when one fails to
find a significant difference. In essence it amounts to
trying to prove the null hypothesis. That difficulty is
compounded when, as Wynn (2000) correctly points out,
several other studies have replicated her results. In fact,
we did so in Experiment 1. The problem with Wynn’s
explanation is that we also replicated her results in our
Experiment 3, an experiment in which no addition or
subtraction was involved. It is much more difficult to
counter a challenge when a set of experiments first replicate the results in question and then show that those
results can be accounted for by a different, and in this
case simpler, set of reasons.
The other studies reported by Wynn (2000) that have
replicated her results all tested infants on 1 and 2 items
after a 1 + 1 event or a 2 − 1 event. To our knowledge,
no previous study has included controls for a possible
familiarity preference. The one that may come closest
was reported recently by Uller, Carey, Huntley-Fenner
and Klatt (1999). They argued they were testing an
‘Object-file’ model versus an ‘Integer-symbol’ model.
But from our point of view they may also have been
varying the familiarity of the objects during their test
trials. In their experiments they showed infants 1 + 1 =
1 or 2 when the items were either placed on the stage
first (object first condition) or the screen was placed on
the stage first and the objects were dropped behind the
screen (screen first condition). In the object first condition infants had more of an opportunity to build up a
familiarity preference for one, the incorrect number. It is
not surprising, then, that in their first two experiments
Uller et al. (1999) found 8-month-old infants responding
more to the impossible event (or from our point of view
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199
the familiar event) only in the object first condition.
In contrast, in Experiment 3, 10-month-old infants
responded to the impossible event even in the screen first
condition. Perhaps, as suggested by Hunter and Ames
(1988), older infants need less familiarization time with
the objects before showing a familiarity preference.
Uller et al.’s final experiment is more difficult to interpret from a familiarity preference point of view. In this
experiment two separate small screens were used instead
of a single large screen. In contrast to the first experiments, 8-month-old infants in the screen first condition
now looked longer at 1 item than at 2 items in the test.
One could make the argument that with two small
screens and one object dropped behind each screen during familiarization, the infants may have treated the
familiarization period as two examples with 1 object
rather than as a single example with 2 objects. Perhaps
that produced enough familiarization with 1 object for
8-month-old infants to respond more during the test to
1 object than to 2 objects. Admittedly, this interpretation
of Uller et al.’s Experiment 4 is highly speculative. But
the interpretation could easily be tested by running a
subtraction condition as well as an addition condition.
When two screens are used, we would expect 8-montholds to have more ‘trouble’ with subtraction than with
addition. If the infants are becoming more familiarized
with one object in the two screen condition, they should
tend to prefer one object in the test, which would be the
‘impossible’ result in an addition problem, but the ‘possible’ result in a subtraction problem.
It is clear that future research should follow Uller
et al.’s example by testing older infants and considering
possible developmental changes in the processes underlying how infants treat these events. An important question is whether infants progress from a simple preference
for familiarity to more sophisticated approaches, such as
the directional (i.e. ordinal) one, and proceed to true
addition and subtraction. Feigenson (1999) tested infants
ranging from 12 to 18 months of age in a discrimination
learning task involving the ordinal relationship between
numbers. She found that infants in this age range were
capable of learning the correct rule (look at the bigger
number or look at the smaller number). Hauser, Feigenson,
Carey and Mastro (1999) also found similar results using
10-month-olds in a procedure where they searched to
retrieve either one or two cookies. This evidence suggests
that by 10 months of age, infants may be able to reason
about the events using the more complex, directional
method. The studies by Uller et al. (1999) also seem to
suggest certain changes in processing by 10 months of age.
In conjunction with the issue of infant addition and
subtraction, we believe that the experiments presented
here raise a more general and important issue. One
200
Leslie B. Cohen and Kathryn S. Marks
should be cautious about attributing sophisticated
cognitive processes to young infants when simpler processes will suffice. The fact that infants, particularly
younger infants, sometimes prefer familiarity in these
tasks is not an accident or fluke. Familiarity preferences
have been reported repeatedly since the early 1970s (e.g.
Greenberg, Uzgiris & Hunt, 1970; Rose, Gottfried,
Mellow-Carminar & Bridger, 1982; Wetherford & Cohen,
1973). As we mentioned previously, Hunter and Ames
(1988) provide an excellent summary of this older literature. In addition, recent studies are also beginning to
report the same familiarity effect with 4- and 5-monthold infants in tasks similar to those used in additionsubtraction studies. Bogartz, Shinskey and Schilling
(2000) and Schilling (2000) both found that in object
permanence tasks, in which one object repeatedly
appeared and disappeared behind an occluder, 5-monthold infants, for a time, also preferred familiar events.
Cashon and Cohen (2000) reported the same effect with
8-month-old infants in an animated version of the
events. The point is that under some circumstances,
familiarity preferences are real, even predictable. Studies
that rely on assessing infant visual preferences without
first habituating infants should add appropriate controls
to rule out familiarity preferences as a possible explanation. Even studies that do habituate infants to a criterion
but include non-habituators along with habituators should
make certain their findings do not result from the nonhabituators who may still have a lingering familiarity
preference (e.g. Cashon & Cohen, 2000; Roder et al., 2000).
Based upon the evidence presented in the present
three experiments, Wynn’s (2000) claims notwithstanding, we believe it is still an open question as to whether
5-month-old infants can actually add or subtract. Just as
we mentioned in the introduction regarding research on
young infants’ ability to subitize or to do cross-modal
matching based upon number, the evidence is still in dispute. When certain abilities are attributed to young
infants, simpler mechanisms can sometimes account for
the data. Clearly, further research is needed to delineate
infants’ understanding of quantity and their development of numerical knowledge. Until that research
reveals convincing evidence of infants’ numerical competence, we believe caution and parsimony are the best
principles to follow when trying to understand the
development of infants’ abilities.
Acknowledgements
This research was supported in part by NIH grant HD23397 to the first author from the National Institute of
Child Health and Human Development. The first experi© Blackwell Publishers Ltd. 2002
ment presented in this article was based upon a master’s
thesis by Kathryn S. Marks at the University of Texas.
Portions of the first two experiments also were presented
at the 2000 meeting of the International Conference on
Infant Studies (Marks & Cohen, 2000). We would like to
express our appreciation to Christina Bailey and Tanya
Sharon for their assistance on this project and to Elizabeth Chiarello and Cara Cashon for their careful reading of the manuscript and their many suggestions for
improving it.
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Received: 20 November 2000
Accepted: 23 July 2001