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Journal
of
Compa rative Psychology
1989,
Vol.
103,
No.
1,23-31
Copyright
1989
by the
Ame rican Psychological Association,
Inc.
0735-7036/89/S00.75
Numerical
Competence
in a
Chimpanzee (Pan
troglodytes)
Sarah
T.
Bqysen
and
Gary
G.
Berntson
Ohio State University
and
Yerkes
Regional Primate Research Center, Emory University
A
chimpanzee (Pan
troglodytes),
trained
to
count
foods
and
objects
by
using Arabic numbers,
demonstrated
the
ability
to sum
arrays
of 0-4
food
items placed
in 2 of 3
possible sites.
To
address representational
use of
numbers,
we
next baited sites with Arabic numbers
as
stimuli.
In
both cases performance
was
significantly
above chance
from
the
first
sessions, which suggests
that
without explicit training
in
combining
arrays,
the
animal
was
able
to
select
the
correct
arithmetic
sum for
arrays
of
foods
or
Arabic numbers under novel test conditions. These
findings
demonstrate that counting strategies
and the
representational
use of
numbers
lie
within
the
cognitive
domain
of the
chimpanzee
and
compare
favorably
with
the
spontaneous
use of
addition
algorithms demonstrated
in
preschool children.
Sensitivity
to
specific
quantities
has
been demonstrated
in
a
wide range
of
species, including rats, raccoons, chimpanzees,
and
human neonates (Antell
&
Keating, 1983; Boysen, 1987;
Boysen
&
Berntson, 1986b;
Capaldi
&
Miller, 1988; Davis,
1984;
Rumbaugh,
Savage-Rumbaugh,
Hopkins,
Washburn,
&
Runfeldt,
1987; Woodruff
&
Premack,
1981).
Counting,
however,
like other cognitive behaviors such
as
language,
has
often
been
defined
to
exclude nonhuman numerical compe-
tence
(Davis
&
Memmott,
1982).
The
revitalization
of
studies
of
animal cognition
has
provided
an
emerging theoretical
framework
for
examining
the
phylogenetic continuum
of
information
processing
(Hulse,
Fowler,
&
Honig,
1978;
Roit-
blat,
Bever,
&
Terrace,
1984).
Thus, studies
of
number
use in
animals
may
provide
an
evolutionary perspective
on the
emergence
of
human cognition, including numerical
and
mathematical capabilities.
The
chimpanzee (Pan
troglodytes)
shares
a
notable portion
of
genetic material with humans (Goldman,
Giri,
&
O'Brien,
1987;
Goodman, Braunitzer,
Stangl,
&
Shrank, 1983)
and has
demonstrated complex abilities with various symbol systems
(Gardner
&
Gardner, 1984; Premack, 1986; Savage-Rum-
baugh,
1986).
It is
plausible that
the
biological overlap
be-
tween
the two
species
may be
similarly
reflected
in
underlying
cognitive
processes that support such representational skills.
Thus,
the
chimpanzee
may be an
optimal animal model
for
exploring
new
dimensions
of
cognitive
function,
including
This research
was
supported
by a
grant
from
the
Ohio State
University,
Office
of
Research
and
Graduate
Studies,
and by an
ADAHMA
Small Grant
to
Sarah
T.
Boysen.
Two of our
chimpanzees
are on
loan
from
Yerkes
Regional
Primate
Research
Center
(sup-
ported
by
Base Grant
RR-00165,
Division
of
Research Resources),
which
is
fully
accredited
by the
American Association
for
Laboratory
Animal
Care.
We
wish
t o
thank Phil
and
Judy Shoup, Terry Russell,
and
Barbara
Thomson
for
their support
and
encouragement
of the
project.
Com-
ments
by
Hank Davis, Duane Rumbaugh, Martin Heesacker,
and an
anonymous
reviewer were greatly appreciated.
Correspondence concerning this article should
be
addressed
to
Sarah
T.
Boysen, Primate Cognition Project, Room
48,
Townshend
Hall, Ohio State University,
1885
Neil Avenue, Columbus, Ohio
43210.
numerical
capacities.
In
this study
we
report evidence
to
suggest
counting
and
related number reasoning abilities
in a
juvenile
chimpanzee.
Method
Subject
A
5.8-year-old
female
chimpanzee (Pan
troglodytes),
Sheba,
was
the
subject
of our
study.
She had
been cross-fostered
in a
human
home
from
4
months
to 2.5
years
of
age.
At
that time Sheba
was
obtained
from
the
Columbus Zoological Gardens
and has
since been
immersed
in
training
on a
range
of
conceptual skills, including such
tasks
as
one-to-one-correspondence, color discriminations,
the use of
colors
as
attributes, slide recognition, drawing
(Boysen,
Berntson,
&
Prentice,
1987),
cross-modal discriminations,
a
vigilance task
of
sus-
tained attention (Berntson
&
Boysen, 1987),
and
same-different
concepts.
In
addition, Sheba served
as the
subject
for a
study
of the
heart rate indexes
o f
recognition
of
humans
and
chimpanzees (Boysen
&
Berntson, 1986a,
in
press-a). Although none
of
these tasks were
directly
related
to
subsequent numerical studies (with
the
exception
of the
initial one-to-one task correspondence described below), such
early
training likely provided Sheba with
a
variety
of
conceptual
experiences that contributed
in
part
to her
subsequent acquisition
of
number concepts.
Preliminary
Number
Training
Procedure
One-to-one
correspondence
task.
The first
structured task that
Sheba learned
(at age 2.5
years)
was a
simple one-to-one correspond-
ence
task.
She was
required
to
place
one and
only
one
object
in
each
of
six
compartments
of a
divided
tray
and
readily acquired
the
skill
with
candy
and
praise
as
reinforcers.
Modified
one-to-one
correspondence.
In
this
first
phase
of
num-
bers training (begun
at 4
years
of
age), Sheba learned
to
select round
placards
to
which
a
corresponding number
of
metal disks were
affixed,
in
response
to 1, 2 , or 3
food
items (Figure
1).
During training, Sheba
and her
teacher-experimenter
typically
sat
directly across
from
one
another, which maximized
the
opportunities
for the
teacher
to
coor-
dinate Sheba's attention
to the
task. Food items were always presented
on
the
same tray
and
were placed
in a
variety
of
positions.
As the
experimenter placed
the
food
items
on the
tray
one at a
time,
she
always
counted aloud.
If
Sheba
did not
respond immediately
by
23
24
SARAH
T.
BOYSEN
AND
GARY
G.
BERNTSON
Figure
1.
Stimuli used
in
modified one-to-one correspondence task
(top)
and for
introduction
of
Arabic numbers (bottom).
selecting
one of the
three lids,
the
experimenter recounted
the
items
aloud,
touching each
one as she
counted.
If
Sheba
was
correct,
she
was
permitted
to eat the
candy
on the
tray.
For
approximately
6
weeks
during
the
initial acquisition phase,
the
experimenter wore
mirrored
sunglasses
to
discourage
Sheba's
potential attempts
to use
social cues. Once Sheba
was
responding reliably,
the
glasses were
no
longer
necessary.
After
Sheba reached criterion performance
(90%
correct
for two
successive
sessions;
Table
1),
the
round placards were
replaced with plastic placards
to
which
black Arabic numbers
had
been
affixed
(see
Figure
1),
and
training continued
until
Sheba
was
able
to
label arrays
of 1 -3
foods
(see
Table
1;
Boysen
&
Berntson,
in
press-b).
The
positions
of all
number alternatives were random
throughout both phases
of
training
in
order
to
control
for
position
preferences. Once criterion performance
had
been reached, training
was
initiated
for
number comprehension.
Number comprehension. Training
for the
comprehension
of
number symbols began with
the
presentation
of
individual numbers
1,
2, and 3 on a
video monitor, with
the
response
placards placed
in
a
horizontal
row in
front
of the
screen (Figure
2). As
with
the
labeling
tasks, positions
for all
choice stimuli were random.
The
chimpanzee
was
required
to
attend
to the
number displayed
and to
select
the
corresponding placard bearing
1, 2, or 3
disks.
If
Sheba
was
correct,
one
candy
was
placed
on
each
of the
markers
on the
placard
as the
experimenter
counted aloud.
The
subject
was
then permitted
to eat
the
candy.
The
acquisition
of
receptive (comprehension) skills with
Arabic
numbers
1,
2, and 3 is
depicted
in
Table
2.
Introduction
of 4 and 0. The
number
4 was
introduced next,
directly
as an
Arabic number
in the
labeling-task
format, followed
by
Table
1
Trials
to
Criterion
for
One-to-One
Correspondence
and
Introduction
of
Arabic
Numbers
MeasureTrials
One-to-one correspondence
Match
1
disk
150
Match
2
disks
50
Match
1 or 2
disks
975
Match
1, 2, or 3
disks
300
Introduction
of
Arabic
numbers
1,2,
and 3 325
Note.
Criterion
for all
measures
was 90%
correct
responses
for two
successive
sessions.
the
number
0
(Table
3). The
same
presentation
tray
was
always
used
to
display items
to be
counted,
and
thus
an
empty tray indicated
a 0
trial.
As the
number
4 was
introduced
and
Sheba's performance
deteriorated somewhat
(see
Table
3), she was
observed
to
begin
to
touch,
point
to, or
move
items
in the
array before
making
her final
decision.
Such motor tagging
and
partitioning have been noted
in
very
young children
in the
early stages
of
learning
to
count.
It has
been proposed that such behaviors permit children
to
keep track
of
those items already counted
and
those which remain
to be
counted
(Gelman
&
Gallistel,
1978).
These
tagging behaviors emerged spon-
taneously
in
Sheba
after
approximately
18
months
of
number train-
ing,
during
which
her
teacher
had
consistently tagged items presented
for
counting.
The
functional significance
o f
these behaviors
for
Sheba
has not yet
been determined,
but she has
continued
to
reliably
tag
items during
all
subsequent number-related tasks (Figure
3;
Boysen
&
Berntson,
in
press-b).
Numerical labeling
of
object
arrays.
Up to
this point,
all
labeling
tasks
with
numbers involved
the
presentation
of
edible arrays, either
homogeneous (arrays
of
gumdrops, M&M's,
or
grapes)
or
heteroge-
neous
combinations
of
such items.
To
determine Sheba's ability
to
generalize
number labels
to
inedible objects,
we
presented common
household items such
as a flashlight
battery, spool,
and
other junk
objects
in
collections
of
1-3
items. Sheba
had not
been asked
to
label
arrays
of
objects prior
to
this test,
nor was she
familiar
with
the
items
used.
Testing
was
completed under double-blind conditions, with
Experimenter
1
seated behind
a
barrier
with Sheba.
The
Arabic
number placards were placed
in a
horizontal row,
in
ordinal sequence
in
front
of
her.
The
numbers were
not
visible
t o
Experimenter
2, who
sat on the
other side
of the
barrier
and
presented
the
objects
on the
same tray used
in the
previous counting tasks. During each trial Sheba
came around
the
barrier, examined
the
tray,
and
returned
to the
other side
to
make
her
number selection. Once
a
choice
was
made,
Experimenter
1
announced
her
response
verbally,
and
Experimenter
2
verified whether
she was
correct
or
incorrect.
If
correct,
she
received
a
matching number
of
small candies. Sheba
was
able
to
correctly
assign
the
appropriate
Arabic number
to
13
of
15
arrays,
for an
overall
performance
of 87%
under double-blind, novel test condi-
tions. These
data
indicate
that
Sheba
was
able
to
directly
transfer
her
skills
at
labeling
homogeneous
and
heterogeneous arrays
of
edibles
to
labeling
heterogenous arrays
of
objects.
To
further
evaluate Sheba's
flexibility
with
number concepts,
in
this
study
we
examined
her
counting ability
in two
novel paradigms.
Experiment
1
Functional Counting Task
To
provide
a
more naturalistic opportunity
for
counting,
we de-
vised
a
functional counting task during which
one or two of
three
possible
food
sites
in the
laboratory were baited with oranges.
Three
separate locations were designated
as
food
sites. These sites,
a
section
of
a
tree stump,
a
stainless steel
food
bin
about
1 m off the
ground,
and a
plastic
dishpan
were
approximately
2 m
apart.
Together
with
the
starting platform, they formed
an
approximate square (Figure
4).
The
cache
of
oranges
at a
given site
was not
visible
from
either
the
start platform
(or
from
any of the
other
food
sites) when Sheba
made
her
number selection
from
Point
A
(see
Figure
4).
Sheba
was
required
to
move among
all
three sites
and
then
to
select
the
correct Arabic symbol that represented
the
total number
of
oranges hidden.
For the first
three trials
of the first
session,
the
experimenter
held
Sheba's
hand
and
walked with
her to all
three
sites,
drawing
her
attention
to the
hidden oranges
by
pointing
and
indicat-
ing
verbally.
When they returned
to the
start platform (Point
A),
Sheba made
her
choice
from
among
the
number alternatives
1-4
NUMERICAL
COMPETENCE
25
Figure
2.
Stimuli
and
position
of
chimpanzee
during
number
com-
prehension
training.
(placed
in
ordinal sequence).
O n all
subsequent
trials
Sheba
was
free
to
move
among
the
sites
as she
chose,
while
the
experimenter
sat at
the
start
platform,
facing
away
from
the
food
sites.
Sheba
usually
completed
visits
to the
three
sites
within
15-20
s. If she did not
return
to the
start
platform
within
approximately
20 s, the
experimenter
verbally
encouraged
her to do so.
When
making
her
number choice
during
training
trials,
Sheba
faced
the
experimenter,
with
the
number
alternatives
i n a row
between
them.
She was
required
to
make direct
physical
contact
with
the
number
she
chose. This
typically
entailed
touching
the
placard
with
her
left
index
finger
and
maintaining
that position
until
the
experi-
menter
acknowledged
whether
or not her
response
was
correct.
For
blind
testing
the
experimenter
sat
behind Sheba,
so
that
the
experi-
menter
could
not
initially
see her
selection,
and
Sheba
could
not see
the
experimenter
(see Figure
5).
Because
the
prior training
had
required
that
Sheba
maintain
physical
contact
with
the
selected
number
until
given
feedback
by the
experimenter,
during
blind trials
the
experimenter
waited
3-5 s
before
looking
around
in
front
of
Sheba
to see
what
choice
had
been
made.
Sheba
was
approached
b y
the
experimenter
from
either
her right
side
or her
left
side
randomly,
so
that
the
experimenter's
movements
could
not
contribute
t o
pre-
dicting
toward
which
end of the
number
sequence
the
correct response
placard
was
positioned.
Results
and
Discussion
We
had
hypothesized
that
given
her
success
on
previous
number-related tasks, Sheba could
be
taught
to
perform
the
functional
counting task. However,
her
performance during
the
initial session
was
significantly
above chance (Table
4).
Thus,
no
specific additional training
was
necessary
for
Sheba
to
generalize
t o the
numerous novel demands
of the
functional
task.
For
example, prior training
or
testing conditions
had
not
required that
the
animal
(a)
move
from
site
to
site,
(b)
attend
to
different
quantities separated
in
time
and
space,
(c)
maintain
a
representation
in
memory
of the
number
of
items
seen
(food
items were always visible during previous tasks
but
were
not
visible during response selection
in
this study);
or
(d)
select
the
correct symbol corresponding
to the
total
num-
ber of
items presented
as two
separate quantities.
Sheba's
subsequent performance
on the
functional counting
task, also presented
in
Table
4,
continued
to be
highly
signif-
icant.
The
last
four
sessions
(18-21)
used
1-4
oranges
and
were
blind,
as
described above.
Her
continued
significant
performance
under
the
blind conditions suggested that Sheba
could
not
only
sum
arrays
that
were
no
longer perceptually
available
but
that
she
could demonstrate such abilities under
novel
test conditions.
Although
Sheba
did not
receive explicit training
on
com-
bining
arrays,
her
performance
may
potentially
be
explained
through
her
application
of
previously acquired counting strat-
egies
or
addition algorithms. Growing evidence
from
studies
with
very
young children suggests
that
preschool children with
no
formal schooling
in
arithmetic
possess
some understanding
of
addition
and
subtraction (Carpenter
&
Moser,
1982;
Fuson,
1982;
Groen
&
Resnick, 1977;
Starkey
&
Gelman,
1982).
Whereas much
of the
early research
on
addition concentrated
on
number-fact problems
and
whether particular types
of
problems would facilitate
or
interfere with subsequent prob-
lem
learning, studies
of the
last decade have addressed
the use
of
counting algorithms
by
young children (Groen
&
Parkman,
1972;
Svenson
&
Broquist, 1975; Woods, Resnick,
&
Groen,
1975).
Although
the use of
algorithms
by
children
has
been
noted previously (e.g., Woody, 1931)
and
described
in
some
detail
(Ilg
&
Ames,
1951),
such potential abilities
in
preschool
children
may
have been overlooked until more recent reports
brought
to
light
the
skills
that
younger children
do
possess,
as
opposed
to
those which they lack (Gelman
&
Gallistel,
1978).
Thus, children
as
young
as
4'A
years
old
were
found
to
apply
spontaneously
a
more
efficient
algorithm than
one
taught
by
experimenters
to
solve simple addition problems (Groen
&
Resnick,
1977).
In a
related
study,
Starkey
and
Gelman
(1982)
found
that young children (particularly
4- and
5-year-olds)
would
use
counting algorithms even when objects were added
or
subtracted
from
arrays that were hidden
from
view.
This
suggests
that
the
child
may
have
an
implicit understanding
of
some properties
of
arithmetic, namely
that
addition increases
numerosity
and
subtraction decreases numerosity (Starkey
&
Gelman, 1982). Young children's addition errors typically
deviated
by +
1,
suggesting that counting errors were respon-
sible
for
deviations
from
the
correct total (Gelman
&
Gallistel,
1978;
Ilg &
Ames,
1951;
Starkey
&
Gelman, 1982).
It
seems plausible, given
her
success with counting arrays
of
foods
and
objects, that Sheba might apply
a
counting
Table
2
Acquisition
of
Number
Comprehension
Skills
Session
1
2
3
4
5
6
7
8
9
10
11
12
No. of
trials
25
(25)
21
(46)
25
(71)
20
(91)
22(113)
25(138)
17(155)
25(180)
25
(205)
25
(230)
17
(247)
20
(267)
%
of
correct
responses
44
81
76
50
91
56
59
52
68
76
82
85
Note.
Cumulative
number
of
trials
is
shown
in
parentheses.
The
overall
percentage
of
correct responses across
the
12
sessions
was
75%.
26
SARAH
T.
BOYSEN
AND
GARY
G.
BERNTSON
Table
3
Introduction
of
Numbers
0 and 4
Days
%
of
correct
responses
Trials
with
new All
number
trials
Introduction
of 0
1-4
5-8
9-12
13-16
67
78
68
100
75
75
70
66
1-4
5-8
9-12
13-16
Introduction
of 4
65
62
87
74
57
51
69
68
strategy
the
first
time
she was
confronted with
two
separate
arrays.
Sheba's
training
of
Arabic numbers
was
serial,
so
that
each
number added
to her
repertoire
met the
requirements
of
X
+
1,
with
X
being
the
previously acquired number
in the
series. Thus,
the
ordinal characteristics
of
numbers
and the
additive component
of
increasing numerosity were implicit
facets
of the
training procedures, just
as
they
are in a
typical
counting situation between
a
knowledgeable adult
and a
child
learning
to
count.
It is not
unreasonable
to
propose
that Sheba
was
able
to
count each array
and to
arrive
at a
correct total,
particularly
with
the
small numbers involved.
The
spontaneous application
of
addition algorithms
by
very
young
children provides potential validation
for
Sheba's per-
formance,
even though
it is not
possible
at
this time
to
determine
the
precise nature
of her
counting strategy.
On the
basis
of the
children's literature,
we can
hypothesize several
possible strategies that could provide Sheba with
the
correct
total
for
such problems (Fuson, 1982; Starkey
&
Gelman,
1982).
She
could,
for
example,
view
both arrays
and
begin
her
count
from
the
larger array,
a
strategy
for
which
there
is
considerable data
for
young children (Fuson,
1982;
Groen
&
Parkman, 1972; Groen
&
Resnick, 1977;
Ilg
&
Ames,
1951;
Starkey
&
Gelman,
1982).
This approach
is
known
as
count-
ing-on.
With this strategy
the
counting
may
begin
with
the
number
that
represents
the
entire
first
array (its
cardinal
number)
and
continue
until
all the
elements
of the
second
array have been enumerated (Fuson,
1982).
Thus,
for an
array
of
1
orange
and 3
oranges, Sheba could encode
1
after
viewing
the first
array
of
oranges
and
then count-on
from
1
when
encountering
the
second array
of 3
oranges
to 4,
which would
be
selected
to
represent
the
total
of the two
arrays.
A
more
efficient
version
of the
counting-on strategy
is to
begin
the
count with
the
larger
of the two
arrays rather than
the first
addend (Fuson, 1982).
In
this case Sheba would begin
her
count with
3,
representing
the
larger
of the two
arrays, con-
tinue
her
count with
the
second array
of
1,
and
arrive
at the
sum
of
4.
It
might also
be
possible
for her to use a
strategy known
as
counting
all,
a
simpler counting solution procedure
in
which
the sum is
determined
by a
count
of the
total
number
of
entities comprising
the two
addends,
in
this case,
the two
separate arrays
of
oranges (Fuson,
1982).
Thus, Sheba would
begin
her
count with
the first
array
and
continue
the
count
sequence
as she
encountered
the
second array until
she had
counted
all the
items
to
reach
a
total.
In
children
the
differ-
ences between
the
strategies
are
often
overt, because children
verbally
count-all
or
count-on
or
count
on
their
fingers.
Thus
Figure
3.
Motor
tagging
by
Sheba
of
three apples
in an
array.
NUMERICAL
COMPETENCE
27
Figure
4.
Experimental setting
for
functional counting task.
Figure
5.
Blind testing format
for
functional
and
symbolic counting
tasks with
the
experimenter positioned behind
the
chimpanzee.
the
emergence
of the
more
efficient
counting-on
strategies
can be
more directly measured than
in the
case
of a
nonverbal,
nonhuman
primate. Further study will
be
necessary
to
more
clearly
characterize
the
nature
of the
counting strategy that
Sheba
may be
using
in the
functional counting context.
Experiment
2
Symbolic
Counting Task
To
test Sheba's ability
to use
numbers
representationally,
we
replaced
the
arrays
of
oranges used
in the
functional counting task
with
the
Arabic numbers
0-4
on a
duplicate
set of
separate plastic
placards identical
to her
response placards (Figure
6). As in the
functional
task,
two of the
three possible sites were baited with
a
single
number during each trial. These included
the
following
number
pairs:
1,0;
1,
1;
1,2;
1, 3; 2, 0; 2, 2; and 3, 0. As in the
previous task,
Sheba
was
required
to
visit
all
three sites during each trial
and
then
to
select
the
Arabic number that
was the
arithmetic
sum of the two
numbers. Similar testing procedures were used, with
two
exceptions.
The
first six
trials
of the
symbolic counting task were
run
under
the
blind conditions described earlier
for the
functional task, with
the
experimenter seated behind Sheba during
her
number selection.
The
next
six
trials were
run
under double-blind conditions
in
which
the
numbers were
placed
at the
sites
by a
second
experimenter.
Thus,
the
experimenter seated with Sheba
did not
know
the
correct answer
for
a
given
trial. Subsequent blind trials were
run
under conditions
as
described above
for the
functional counting task.
Results
Sheba's performance during
the first
session
was
signifi-
cantly
above chance,
and her
performance
for
blind tests
was
also significant (Table
5).
Trial
1
performance
for
each
of the
seven
pairs
of
numbers
was
also significant (86% correct;
see
Table
5).
Overall, Sheba achieved
81
%
correct performance
(p
<
.001).
Double-blind testing
was
completed
on Day
1,
28
SARAH
T.
BOYSEN
AND
GARY
G.
BERNTSON
Table
4
Functional
Counting Task
No.
No. of of
Stimuli
sessions
trials
1-3 7 82
1-4
10 89
Blind
4 38
No. of
correct
trials
60
75
28
Probability
Correct
Chance
x
2
(!)
.73
.33
58.32*
.84 .25
166.42*
.74
.25
49.83*
*/7<.001.
and
significant
performance
was
maintained during these
and
subsequent
blind
tests
(Table
6).
A
breakdown
of
Sheba's
performance
on
different
classes
of
trials
is
given
in
Table
6.
Although
the
statistical probability
of
a
correct
response
in
these studies
was
.25,
the
deployment
of
noncounting strategies
could
have
effectively
reduced
the
response options.
Thus,
Sheba
may
have simply avoided
choosing
an
answer that
was one of the
addends.
In
view
of
this possibility, Table
6
also gives
the
more conservative
chi-
square values with adjusted expected probabilities based only
on
response options
that
were
not
among
the
addends.
As is
apparent, performance
in all
trial classes still remained
signif-
icant (Table
6).
Alternatively, Sheba
may
have inappropriately
applied
a
simple strategy entailing
the
selection
of the
next
higher
number than
the
largest addend (e.g.,
1 +2
= 3,
select
3;
1
+ 3
= 4,
select
4). The
critical comparison trials
for
evaluation
of
this
strategy
are 2 + 2 and 0 + n (n =
1,
2 or
3),
where
this
strategy would
fail.
Sheba's
performance
on
these trials, however,
far
exceeded chance, even with adjusted
expected probabilities
(16
correct
responses
in
17
trials
or
94%;
expected probabilities
=
.33),
x30,
N = 17) =
26.2.
Whereas
all of the
counting strategies discussed
for the
functional
counting task
may
also
be
potential algorithms
for
use in the
symbolic counting situation, successful application
of
any of the
strategies includes
an
important exception.
The
arrays
in
this task were presented
as
numerical representa-
tions, rather than physical entities. Sheba
was
nevertheless
able
to
provide
the
correct
total
for the
numbers represented.
General
Discussion
Our
results suggest that Sheba could
effectively
sum
arrays
of
objects
or
Arabic numbers.
It is
important
to
note
that
before
this study Sheba received
no
specific
training adding
arrays
of
food,
objects,
or
Arabic numbers,
nor had any
situation ever arisen
in
which
two
arrays
or two
numbers were
presented, either simultaneously
or
sequentially,
for her to
sum.
Rather, Sheba
had
extensive work
in
counting
a
wide
variety
of
food
items
with
Arabic numbers,
and she had a
transfer
test
of
counting arrays
of
novel objects.
Her
compre-
hension
training involved responding
to
Arabic numbers
and
selecting
the
corresponding array
from
among three
fixed
arrays
(see Figure
2). She
also
had
extensive experience with
the
number placards, used
as
choice stimuli, presented
in
both
ordinal
and
random sequence during
all
number tasks.
She
was
never, however, explicitly required
to
respond
to
combi-
nations
of
numbers
in any
task.
If
comparisons
of
Sheba's performance
are
made with
preschool children,
her
responses
may
represent
the first
evidence
for
counting that meet
the
more stringent criteria
proposed
by
Davis
and
Perusse
(1988)
for
nonhuman species.
Although
noncounting processes have been
put
forth
to ac-
count
for
numerical skills demonstrated
by
other animals,
such
as
number discriminations
by
birds, numerousness judg-
ments
by
monkeys,
and
summation
in
chimpanzees (Koeh-
ler,
1950; Rumbaugh,
Savage-Rumbaugh,
&
Hegel, 1987;
Thomas
&
Chase, 1980), none
of
these
are
adequate
to
Figure
6.
Experimental
setting
for
symbolic
counting
task.
NUMERICAL
COMPETENCE
29
Table
5
Symbolic
Counting Task
No.
Probability
No. of of No. of
correct
Stimuli
sessions trials trials Correct Chance
x
2(l)
1-4
Blind
3
4
21
20
16
17
.76
.85
.25
.25
28.94*
35.27*
'/X.001.
account
for
Sheba's
performance during
the
functional
and
symbolic counting tasks. Davis
and
Perusse proposed several
criteria that support
a
counting explanation
of
numerical
competence
for
Sheba. They
defined
counting
following
Ste-
vens
(1951),
as the
process
by
which
one
discriminates
the
absolute numerosity
of a set of
items.
The
authors noted that
unlike
subitizing,
the
direct apperception
of a
small
set of
items (von Glasersfeld,
1982),
counting
can
yield
a
number
along
an
ordered continuum
and can be
applied
in
both
logical
and
descriptive contexts. Their criteria
for
counting
incorporate
the
principles
of
cardinality,
the use of
number
tags,
and
ordinality,
the use of
these labels
in an
ordered
series; this combination
of tag and
order must
be
stable
and
reliable (Gelman
&
Gallistel, 1978).
Although
Sheba does
not
apply verbal number labels,
she
has a set of
reliable, ordered tags,
in the
form
of
plastic number
placards. These
were
learned
in
serial order
and
serve
as her
output
for
number tasks.
She
reliably applies
the
correct
Arabic number label
to
arrays
of
familiar
and
novel
foods
or
objects,
demonstrating
an
appreciation
for the
special
status
of
the final
cardinal number
of a
count series that designates
the
entire array (Gelman
&
Gallistel, 1978). Although
it has
been
assumed that
an
individual
who is
able
to
count must
by
definition
have
a n
understanding
of the
underlying ordinal
nature
of
numbers, ordinality
has
been assessed separately
in
Sheba
through
a
replication
of
Gillan's
(1981)
study
of
tran-
sitive
inference
(Boysen
&
Berntson, 1988).
In
addition
to
learning
the
ordinal relations among
a
series
of five
colored
boxes,
as in
Gillan's
study, Sheba also
was
successful
in
demonstrating
an
understanding
of the
ordinal relation
among
the
numbers
1-5,
because
she was
able
to
select
the
larger
of a
novel pair
of
numbers
(2 and 4)
presented during
blind testing. These
data
provide additional support
for the
Table
6
Response
Distribution
for
Symbolic Counting Task
-
—
No.
of No. of
correct
% of
correct
.25
Adjusted
Sum
trials trials trials probability
probability*
1
2
3
4
8
11
12
12
7
10
10
11
88
91
83
92
13.5***
22.1***
18.8***
25.0***
8.4**
14.1***
5.3*
9.2**
Note.
For
Sum,
1
comprised addends
1 + 0; 2
comprised
0 + 2 and
1
+
1;
3
comprised
0 + 3 and 1 + 2; 4
comprised
1 + 3 and 2 + 2.
*
Expected
value
adjusted
to
exclude addends
as
response options.
*p<.05.
**/><.01.
***p<.001.
proposal that Sheba
had
some understanding
of the
ordinal
characteristics
of the
numbers
she
used.
Evidence
for an
understanding
of
ordinality provides
ad-
ditional support
for a
counting explanation
of
Sheba's
emer-
gent
abilities
in the
functional
and
symbolic counting tasks.
In
an
earlier study
Ferster
(1964)
trained chimpanzees
to
associate binary numbers with small arrays,
but
these animals
could
not
later
use the
numbers
to
enumerate items.
In
contrast, Sheba learned numbers serially,
a
training procedure
that seemed
an
intuitive
way to
proceed.
In
hindsight, serial
training
was
likely
one of
several critical features
of the
procedures that contributed
to the
subsequent
flexibility
Sheba
has
demonstrated with numbers
in
novel contexts.
For
example,
during
the
modified one-to-one correspondence task
(the
first
phase
of
numbers training), only gumdrops were
used
for
counting.
The
same
foods
were used
as
Sheba made
the
transition
to
Arabic numerals, because other
food
items
created considerable confusion.
It was not
until
her
skills with
labeling
arrays
of
gumdrops were highly overtrained that
Sheba evidenced more
flexibility in
applying
the
number
labels
to
arrays
of
other
food
types,
and
then only
if
they were
presented
as
homogeneous arrays. Combinations
of
food
items again
created
confusion. Combinations
of
foods
were
gradually
introduced, until eventually
it
made
no
difference
in
Sheba's performance
if
arrays were homogeneous
or
het-
erogeneous collections
of
edibles.
Thus, Sheba initially related
the
use of
numbers
to a
very
narrow
context,
that
of
labeling
collections
of
gumdrops.
The
applicability
of
numbers
to
other items
was
gradually incorporated into
her
repertoire.
This process
may
represent,
in one
sense,
a
manifestation
of
Gelman
and
Gallistel's
(1978)
abstraction principle, that
the
how-to-count
principles
may be
applied
to any
collection
of
items. Thus, Sheba
was
able ultimately
to
apply numbers
to
heterogeneous arrays
of
novel objects; through
her
prior train-
ing
with
different
types
of
foods,
she
presumably
had
already
come
to
understand
that
anything could
be
counted.
Sheba's numerical skills, particularly
as
evidenced
by the
functional
and
symbolic counting tasks, exceed
the
criteria
for
demonstrating counting
in a
nonhuman species
and
com-
pare
favorably
with those specified
for
young children (Davis
&
Perusse, 1988; Gelman
&
Gallistel,
1978).
In
addition,
Sheba also demonstrates motor tagging
and
partitioning,
by
pointing, touching
or
moving items
to be
counted.
A
study
is
currently underway
to
evaluate
the
function
of
such motor
tags during counting.
For
example,
do
responses that
are
errors with respect
to the
number
of
items
in the
array
reflect
the
same, though incorrect, number
of
tags made
by
Sheba?
If
the
data reveal that motor tags
reflect
both correct
and
incorrect responses,
it may
suggest
that
these behaviors
func-
tion
to
partition
items
to be
counted,
similar
to the
tagging
function
described
for
young children (Gelman
&
Gallistel,
1978).
Inconsistent
use of
tagging,
on the
other hand,
may
reflect
imitation
of the
experimenter's behavior during train-
ing;
such imitation
may be
acquired through observational
learning
and may
have little
or no
functional significance.
That Sheba
has
acquired
the
ability
to
count
may
provide
a
reasonable explanation
for her
performance
on the
func-
tional counting task.
It
suggests that
she
completed each novel
trial
by
counting-on (Fuson, 1982; Groen
&
Resnick,
1977).
30
SARAH
T.
BOYSEN
AND
GARY
G.
BERNTSON
She
may
have only
had to
remember
the
number
of
oranges