JOURNAL
OF EXPERIMENTAL
CHILD
PSYCHOLOGY
54, 3722391(1992)
Counting Knowledge and Skill in Cognitive Addition:
A Comparison of Normal and Mathematically
Disabled Children
DAVID
C.
GEARY.
C.
CHRISTINE
BOW-THOMAS,
AND
YUHONG
YAO
The relationship
between
counting
knowledge
and computational
skills (i.e.,
skill at counting to solve addition problems)
was assessed for groups of first-grade
normal and mathematically
disabled (MD)
children.
Twenty-four
normal and I3
MD children were administered
a series of counting tasks and solved 40 computeradministered
addition problems.
For the addition
task, problem-solving
strategies
were recorded
on a trial-by-trial
basis. Performance
on the counting
tasks suggested that the MD children
were developmentally
delayed in the understanding
of essential and unessential
features of counting
and were relatively
unskilled
in
the detection
of certain forms of counting
error.
On the addition
task, the MD
children
committed
many more computational
errors and tended to use developmentally
immature
counting
procedures.
The immature
counting
knowledge
of
the MD children.
combined
with their relatively
poor skills at detecting
counting
errors. appeared
to underlie
their poor computational
skills on the addition task.
cl IYY? Acadcmlc Prcw. Inc
Suggestions
for future research arc presented.
The development of arithmetic skills involves the acquisition of problem-solving procedures and the development of memory representations
for basic numerical facts (Ashcraft, 1992; Siegler. 1986; Temple, 1991).
For the domain of arithmetic, procedures would include the use of counting algorithms, rules (N + 0 = N), carrying, and so on, to complete
problem solving (e.g., Baroody, 1983; Hamann & Ashcraft, 1985; Widaman, Geary, Cormier, & Little, 1989). The development of memory
representations for basic facts leads to the use of direct memory retrieval
We would like to thank the students, teachers,
and principals
at Parkade
and Blue Ridge
elementary
schools for their support and cooperation.
with special thanks to Linda Coutts,
Supervisor
of Elementary
Mathematics.
Columbia
Public Schools.
We would also like to
thank Michelle
Owen and Mary O’Brien
for their assistance with data collection:
Tim Cross
for programming
one of the experimental
tasks; and Mark Ashcraft,
Peter Frensch.
and an
anonymous
reviewer
for comments
on earlier drafts. Yuhong
Yao is now at Stanford
University.
Correspondence
and requests for reprints
should be sent to David C. Geary.
Department
of Psychology,
210 McAlester
Hall. University
of Missouri,
Columbia.
MO 6.521 I.
372
0022~0965/92
$5.00
CopyrIght 0 lYY2 by Academic Preu. Inc
All nyhts of rrproduction
m any form revnu!
COUNTING
KNOWLEDGE
373
for the solution of simple problems (e.g., 3 + 5 = 8; Siegler, 1986). The
relatively poor skills of mathematically
disabled (MD) children: that is,
children who show a delay in the acquisition of basic mathematical skills,
appear to involve both procedural and memory retrieval deficits (Geary,
1990; Geary, Widaman, Little, & Cormier, 1987; Svenson & Broquist,
1975).
The memory retrieval deficits of MD children are reflected in less use
of direct retrieval, relative to normal children, to solve simple arithmetic
problems and a high proportion of errors when an answer is retrieved
(Fleischner, Garnett, & Shepherd, 1982; Garnett & Fleischner, 1983;
Goldman, Pellegrino, & Mertz, 1988). Moreover, when an answer is
correctly retrieved from long-term memory, the solution times are highly
unsystematic, unlike those associated with direct memory retrieval in normal children (Geary & Brown, 1991; Geary, Brown, & Samaranayake,
1991). Mathematically
disabled children are also less skilled than normal
children in the use of counting, or computational,
procedures to solve
arithmetic problems, especially in Grade 1.’ when using a computational
procedure to solve an arithmetic problem, MD children commit many
more errors than normal children (Geary, 1990; Geary et al., 1991) and
often use developmentally
immature strategies. The high frequency of
computational
errors and use of immature computational
algorithms
largely disappears by Grade 2 (Geary & Brown, 1991; Geary et al., 1991).
while the memory retrieval deficits are still evident well into the elementary school years (Garnett & Fleischner, 1983; Geary et al., 1991; Goldman et al. 1988). These findings indicate that the computational
and
memory retrieval deficits of MD children show divergent developmental
patterns.
The finding that the computational
skills of MD children quickly approach the skill level of normal children, while retrieval skills do not,
suggests that different mechanisms might contribute to the computational
and retrieval deficits of MD children. In fact, this pattern suggests that
the factors underlying the poor computational
skills of MD children might
involve a developmental delay, whereas the retrieval deficits might involve
a developmental
difference or a more fundamental deficit (Ashcraft, Yamashita, & Aram, 1992; Goldman et al., 1988). Given the apparently
different mechanisms contributing
to the computational
skill and longterm memory representation deficits of MD children, it seems likely that
the computational
skills of MD children can be fruitfully studied independent of any more fundamental
memory retrieval deficit. Thus, the
’ In this article, counting
knowledge
refers to knowledge
of how to count objects and
counting
refers to the act of counting
(unless referenced
with min or sum counting)
these
objects.
whereas
computational
skill refers to skill in the use of counting
procedures
to
solve arithmetic
oroblems.
374
GEARY,
BOW-THOMAS,
AND
YAO
present experiment focused only on the computational skills of first-grade
MD children. In particular, the experiment tested the hypothesis that
developmentally
immature counting knowledge contributes to the relatively poor computational
skills of MD children. A proposal regarding
the potential relationship between counting knowledge and computational
skills follows a brief overview of conceptual and empirical research on
the development of addition skills and the acquisition of counting knowledge.
Skill Development
in Addition
Skill development in arithmetic proceeds on at least two dimensions,
strategic and speed-of-processing. When solving an addition problem,
children typically attempt to retrieve the answer directly from long-term
memory (see Siegler, 1986; Siegler & Shrager, 1984). If a satisfactory
answer cannot be retrieved, then the child might invoke a second retrieval
attempt, or simply guess, by stating any number in working memory.
More typically, though, when a satisfactory answer cannot be retrieved,
the child resorts to a backup strategy to complete problem solving (Siegler,
1986). Here, first-grade children most frequently count on their fingers
or count verbally(Geary,
1990; Siegler, 1987). If counting, whether on
fingers or verbally, is required to solve the problem, then children typically
use the min or counting-on procedure (Carpenter & Moser, 1984; Groen
& Parkman, 1972). With the min procedure, the solution of a problem,
such as 5 + 4, begins with stating the cardinal value of the larger valued
integer (i.e., 5) and then counting in a unit-by-unit fashion a number of
times equal to the value of the min, for minimum,
integer (i.e., 4) until
the sum is obtained (Groen & Parkman, 1972). Other forms of backup
strategy might involve looking at their fingers, but not counting them, to
help to remember the answer, or decomposing the problem into simpler
problems (Siegler, 1987).
Developmentally
less mature computational strategies involve counting
both integers (the sum or counting-all algorithm) or counting the larger
valued integer (the max algorithm).
When they count, first-grade MD
children tend to use either the min or sum algorithm, whereas normal
first-grade children almost always use the min algorithm (Geary. 1990;
Geary et al., 1991). When first-grade MD children do use min counting
to solve an addition problem it is usually for problems with smaller valued
integers (e.g., 3 + 2), whereas the sum algorithm tends to be employed
to solve more difficult problems (e.g., 5 + 7). The difficulty of a problem
can be indexed by the problem’s sum (Ashcraft, 1992). For the MD
children described by Geary et al. (1991), for instance, the frequency of
sum counting was positively correlated with the problem’s sum (r(38) =
.35, p<.O5). The most common computational error for both normal and
MD children involves undercounting by 1 (Geary, 1990; Siegler & Ro-
COUNTING
KNOWLEDGE
375
binson, 1982). For example, the problem 3 + 4 might be solved by
counting-on from the 4 but stating 4, 5, 6, rather than 4, 5, 6, 7 (Baroody,
1984). In other words, the cardinal value for the larger integer is (counted
twice, first as representing the value of the larger integer and then as the
first digit in the min count. In all, for the domain of addition, the computational skills of first-grade MD children can be described as being
developmentally
delayed relative to their normal peers (Goldman et al.,
1988); that is, the performance of these children is similar to that of
younger academically normal children. This delay manifests itself in frequent computational
errors and frequent use of the sum algorithm.
Counting
Knowledge
Gelman and Gallistel (1978) proposed that counting by preschool children is governed by five implicit principles: one-to-one correspondenceone and only one word tag (e.g., “one,” “two”) is assigned to each
counted object; the stable order principle-the
order of the word tags
must be invariant across counted sets; the cardinality principle-the
value
of the final word tag represents the quantity of items in the counted set;
the abstraction principle-bjects
of any kind can be collected together
and counted; and the order-irrelevance
principle-items
within a given
set can be tagged in any sequence. The principles of one-to-one correspondence, stable order, and cardinality define the “how to count” rules.
In this view, knowledge of these principles precedes and therefore governs
the acquisition of counting procedures (Gelman & Meek, 1983; Greeno,
Riley, & Gelman. 1984).
An alternative view is that counting is first done by rote, that is, without
an understanding of counting principles, and that the child gradually induces the essential and unessential features of counting (Briars & Siegler,
1984; Fuson, 1988; Wynn, 1990). Briars and Siegler, for instance, argued
that the one essential feature of counting is the word/object correspondence rule, “given a correctly ordered list of number words, assigning one
and only one number word to each object during the counting is both
necessary and sufficient to determine a set’s cardinality”
(p. 60,R). The
word/object correspondence rule subsumes the one-to-one correspondence and order-irrelevance
principles proposed by Gelman and Gallistel
(1978), although knowledge of the stable order principle is also implied
by this rule.
Briars and Siegler (1984) also described four common but unessential
features of counting: standard direction--counting
starts at one of the end
points of an array of objects; adjacency-a consecutive count of contiguous
objects; pointing--counted
objects are typically pointed at but only once;
and start at an end-counting
precedes from left to right. By 5 years of
age, most children know that the word/object correspondence rule is an
essential feature of counting but many of these children also believe that
376
GEARY,
BOW-THOMAS.
AND
YAO
adjacency and start at an end are also essential features of counting. The
counting knowledge of these children is thus rigid and incomplete, a
finding which is inconsistent with the principles-first argument of Gelman
and Gallistel (1978) ( see also Wynn, 1990). Regardless of whether conceptual knowledge precedes or is induced from counting, the extent of
this knowledge appears to vary with the number of items that must be
counted. For instance, even if young children appear to understand cardinality for small set sizes (e.g., 3), they might not understand cardinality
for larger set sizes (e.g., 11; Briars & Siegler, 1984; Gelman & Meek,
1983).
Counting
Knowledge
and Computational
Skills
Ohlsson and Rees (1991) recently proposed that for the domain of
arithmetic, counting knowledge influences the development of computational skills. More specifically, if the execution of a computational
algorithm (e.g., the min procedure) violates principled knowledge, then the
algorithm will be modified, or “repaired,”
so that subsequent executions
of the procedure conform to conceptual rules. In this view, computational
skills reflect skill at executing procedures and principled knowledge, with
counting knowledge providing a standard against which procedural performance is evaluated. The development of computational
skills will be
dependent upon both counting knowledge and skill at detecting violations
of counting principles. Presumably, even with adequate counting knowledge, if a child is not skilled at detecting procedural errors. then the
procedure will not be modified so as to conform to conceptual rules. Thus,
regardless of whether it is implicit or induced, counting knowledge should
influence a child’s skill at using counting procedures to solve arithmetic
problems.
In other words, before an incorrectly executed computational procedure
can be repaired. the child must first detect the
(e.g., undercounting)
incorrect execution. The detection of the procedural error, as well as the
appropriate modification of the procedure, requires counting knowledge
(Ohlsson & Rees, 1991). Moreover, the adoption of a developmentally
mature counting algorithm, such as the min strategy. would more likely
require developmentally
mature counting knowledge. In other words, the
modification of the sum procedure to the min procedure would seem to
require an understanding of both essential and unessential features of
counting, across the range of values counted (Siegler OIr Jenkins, 1989).
For example, if the child believes that counting must always start from
1, then this child will not likely abandon the sum procedure. In the present
study, we explored the relationship between counting knowledge and
computational skills and sought to determine if a developmental delay in
the acquisition of counting knowledge contributes to the relatively poor
computational
skills of MD children.
COUNTINGKNOWLEDGE
377
METHOD
Subjects
The subjects were selected from two elementary schools which served
a working-class population. The original sample included 20 academically
normal first-grade children and 18 first-grade children who had been referred by their kindergarten teachers for remedial services in mathematics
and/or reading. A referral for services in mathematics was based on a
failure to master basic numerical skills during kindergarten. These skills
include counting, object classification (e.g., based on size and shape), and
knowledge of the principles of cardinality and one-to-one correspondence.
Due to financial constraints, none of these children received remedial
services in mathematics,
although many of these children did receive
remedial services in reading.
Many children who are classified at the end of the kindergarten year
as needing remedial education appear to be misclassified, that is, many
of these children do not actually show a learning deficit (e.g., Geary,
1990). Thus, in addition to the experimental tasks, the arithmetic section
of the Wide Range Achievement Test (WRAT; Jastak & Jastak, 1978)
was administered to all children. In this way, an independent measure of
mathematics achievement was obtained. Unfortunately,
the information
obtained by using the WRAT was not especially useful, as the performance
of many of the children was at ceiling level. Nevertheless, the test did
indicate that all but one of the children could count without error from
1 to 15. The single error involved skipping one number. To obtain more
reliable information on mathematics achievement levels, the mathematics
subtests of the Woodcock-Johnson
Tests of Achievement-Revised
(WJR; Woodcock & Johnson, 1989) were administered toward the end of
first grade (all testing was done between April 4 and May 3). ‘The WJR provides national percentile rankings for calculation and applied mathematics skills, as well as a broad composite index of mathematics achievement. One child moved before the WJ-R could be administered so his
data were excluded from all of the analyses.
Following the standard cut-off of the 45th national percentile ranking
for referral to Chapter 1 remedial education programs (a federally funded
program for low-achieving children), those children who scored below the
46th national percentile on the composite index were identified as MD.
All other children were considered academically normal. Using this criterion, 13 (6 female, 7 male) children were identified as MD and 24 (10
female, 14 male) children as normal. For the MD group, scores ranged
from the 2nd to 42nd percentile, although only two children scored above
the 40th percentile (both at the 42nd percentile). For the normal group,
scores ranged from the 47th to 97th percentile, with 18 of the 24 children
scoring above the 60th percentile. The mean ages of 7.4 (SD = 0.4) and
378
GEARY.
BOW-THOMAS.
TABLE
DESCRIPTIVE
INFORMATION
area
Applied
Calculahon
Composite
Scows
SCORM
disabled
Normal
.~I)
M
SD
33
31
77
‘75
I0
Ii
71
66
71
72
15
I6
13
are national
Ttsr
M
N
Note.
YAO
I
FOR ACHIEVEMENT
Math
Mathematics
AND
percentile
21
rankings.
7.2 (SL) = 0.5) years for the MD and normal groups, respectively,
did
not differ reliably, F(1, 35) = 1.26, p > .25. Mean national percentile
rankings across the Applied, Calculation, and Composite indexes for the
MD and normal groups are displayed in Table 1. Inspection of Table 1
reveals that the MD children showed a lower mean percentile ranking
than the normal children on each of the three measures (ps < .OOl).
Experimental
Tasks
Cuurzting tasks. Each child was administered
45 counting trials. The
goal was to assess the child’s skill at detecting violations of each of the
three how to count rules (Gelman & Meek, 1983), although the tasks
also enabled a determination
of the child’s understanding
of the essential
and some of the unessential features of counting (Briars & Siegler, 1984).
For each trial, the subject was presented with a row of 8, 12, or 16
alternating red and blue chips. The chips were then counted by a puppet
who was learning to count (Briars & Siegler, 1984; Gelman & Meek,
1983). The child’s task was to determine if the puppet’s count was OK
or not OK and wrong.
The one-to-one correspondence
task consisted of 18 trials, 6 trials for
8,
12, and 16). For each set size, the 6
each of the three set sizes (i.e.,
trials included two correct counts, two pseudo errors, and two incorrect
counts. One pseudo error involved counting the red chips first and then
counting the blue chips, whereas the other pseudo error involved counting
the blue chips first and then the red chips. The pseudo error counts also
enabled a determination
of whether the child understood that adjacency
was an unessential feature of correct counting (see Briars & Siegler, 1984).
One of the incorrect counts involved double-counting
the first chip,
whereas the second error involved double-counting
the last chip.
The stable order task consisted of 15 trials, 5 trials for each of the three
set sizes. Two of the trials were correct counts, whereas the 3 remaining
trials involved reversing a word tag (e.g., 4. 6, 5), skipping a word tag
(e.g.. 4. 6. 7). or counting with random word tags (e.g., 2, 6. 1).
COUNTING
KNOWLEDGE
379
The cardinality task consisted of 12 trials, 4 trials for each of the three
set sizes. Two of the trials were correct counts, whereas the 2 remaining
trials involved the puppet stating a word tag one more than the: correct
count, or one less than the correct count. To illustrate, for the set size
of 8, the puppet counted, “1, 2, 3, 4, 5, 6, 7, 8,” and then stated. “There
are 9 chips.”
Finally, if the child were to guess on all trials, then the chance level
of performance for all three tasks would be .50. On measures combining
across all types, if the child understood one of the counting types, for
example correct counts, and guessed for all other counting types, then
chance level of performance would be greater than SO.
Addition tusk. The addition stimuli consisted of 40 pairs of vertically
placed single-digit integers (e.g., 5 + 2). Stimuli were constructed from
the 56 possible nontie pairwise combinations of the integers 2 to 9 (a tie
problem is, e.g., 2 + 2). The frequency and placement of all integers
were counterbalanced.
That is, each integer appeared 5 times as the
augend and 5 times as the addend, and the smaller valued integer appeared
20 times as the augend and 20 times as the addend. No repetition of
either the augend or the addend was allowed across consecutive problems.
The addition problems were presented at the center of a 30 >( 30-cm
CRT controlled by a PC-XT microcomputer.
A Cognitive Testing, Station
clocking mechanism ensured the collection of reaction times (RTs) with
? 1 ms accuracy. The timing mechanism was initiated with the presentation
of the problem on the CRT and was terminated via a Gerbrands G1341T
voice-operated relay. The voice-operated relay was triggered when the
subject spoke the answer into a microphone connected to the relay.
For each problem, a READY prompt appeared at the center of the
CRT for a lOOO-ms duration, followed by a lOOO-ms blank screen. Then,
an addition problem appeared on the screen and remained until the subject
responded. The experimenter
initiated each problem presentation sequence via a control key.
Procedure
All subjects participated
in three experimental
sessions. The three
counting tasks were administered during one of the sessions. The WRAT
and cognitive addition task were administered during the second session,
while the WJ-R was administered during the final session. The order of
participation in sessions one and two was counterbalanced across subjects
(all testing for sessions one and two was completed between November
12 and March 26). For session one, the order of administration
of the
counting tasks was also counterbalanced
across subjects. Within each
counting task, the order of trial administration
was randomly determined.
All subjects were administered the WJ-R after completion of the counting
380
GEARY.
BOW-THOMAS,
AND
YAO
and cognitive addition tasks. Each subject was tested individually and in
a quiet room at the school site.
The procedures for the three counting tasks followed Gelman and Meek
(1983), although, as noted above, the procedures also enabled an assessment of the child’s understanding of some of the essential and unessential features of counting described by Briars and Siegler (1984). Here,
the subjects were asked to detect violations of a counting principle (i.e.,
one-to-one correspondence, stable order, and cardinality), while monitoring a puppet who was learning to count. The instructions were the
same as those described by Gelman and Meek, except that the gender of
the puppet was always the same as that of the child. For each task, the
subject was presented with a single row of 8. 12, or 16 alternating red
and blue chips. The child’s task was to monitor the puppet’s count and
determine whether the count was OK or not OK and wrong. During the
count, if the child looked away from the puppet or if the child did not
appear to be attending to the count, then the experimenter would stop
and state, “Be sure to pay attention,”
and then readminister that trial.
Before the first task was administered, each child monitored two correct
practice counts.
For the addition task, the subjects were asked to solve the 40 addition
problems, preceded by 8 practice problems, presented one at a time on
the CRT. Subjects were encouraged to use whatever strategy made it
easiest for them to obtain the answer, although equal emphasis was placed
on speed and accuracy of responding. During the addition task, the answer
and strategy used to solve each problem were recorded by the experimenter and classified as one of the earlier described strategies: (a) counting
fingers, (b) fingers (i.e., looking at their fingers to help to remember the
answer), (c) verbal counting, (d) decomposition, or (e) memory retrieval.
After each trial, the subjects were asked to describe how they arrived at
the answer. Several previous studies have demonstrated that children can
accurately describe problem-solving
strategies in arithmetic, if they are
asked immediately
after the problem is solved (Siegler, 1987, 1989).
Based on subject descriptions, the counting tingcrs and verbal counting
trials were further classified in accordance with the specific algorithm used
for problem solving. That is, the trials were classified as min, based on
counting only the smaller valued integer, or sum/max, based on counting
both integers or the larger valued integer. Finally. the child’s description
was compared to the experimenter’s
initial classification (e.g., verbal
counting or retrieval) and indicated agreement between the experimenter
and the subjects on 96% of the trials. Disagreements typically occurred
for trials on which there was no indication of verbal counting, e.g., no
lip movements. For these trials, the experimenter
scored the trial as
retrieval but the subject described a counting or decomposition process.
COUNTING
MEAN
TABLE
2
PERCENTAGE OF CORRECT IDENTIFICATIONS
One-to-one
correspondence
Group
Math disabled
Normal
Note.
381
KNOWLEDGE
The values
Stable
ON COUNTING
TASKS
order
Cardinality
8
12
16
8
12
16
8
1’-
16
76
89
73
92
69
90
95
97
91
96
91
Y4
YX
99
96
‘)‘J
x7
97
8. 12, and 16 refer
to set size.
For those trials on which the experimenter and the subject disagreed, the
strategy was classified based on the child’s description.
RESULTS
For clarity of presentation, the results with brief discussion are presented
in three major sections, followed by a more general discussion of the
results and their implications.
In the first section, analyses of group differences for performance on the counting tasks are presented, followed
by a presentation of the results for the addition task. The final section
presents results for the relationship between variables representing counting knowledge and computational
skills.
Counting
Tasks
For the counting tasks, the dependent measure was the number of trials
correctly identified, that is, the number of correct and pseudo-error counts
identified as correct and the number of incorrect counts identified as
incorrect. The percentage of correct identifications across groups, tasks,
and set size are displayed in Table 2. The data displayed in Table 2 were
analyzed by means of a 2 (group; MD and normal) x 3 (task; one-toone correspondence, stable order, and cardinality)
x 3 (set size; 8, 12,
and 16) mixed-design analysis of variance (ANOVA),
with group as a
between-subjects factor and task and set size as within-subjects factors.
The results indicated reliable main effects for group, F( 1, 35) =z 12.15,
p < .Ol, task, F(2, 70) = 33.59, p < .OOl, and size, F(2, 70) = 3.69,
p < .05, as well as a reliable task x group interaction, F (2, 70) = 7.13,
p < .Ol. The size x group, task x size, and task x size x group
interactions were not reliable (ps > .lO).’
’ The data displayed
in Table 2 were positively
skewed.
To ensure that the skewed
distributions
did not affect our results. following
Stevens (1986),
the square roots of the
proportions
displayed
in Table 2 were submitted
to an arcsine transformation
to normalize
the distributions.
The transformed
data were then submitted
to a mixed-design
ANOVA
382
GEARY,
BOW-THOMAS,
TABLE
MEAN
PERCENTAGE
OF CORRECT
IDENTIFICATIONS
CORRESPONDENCE
AND
3
ACROSS TRIAL
TASK
One-to-one
Group
Correct
Math disabled
Normal
Note.
Correct,
and error
refer
49
x4
to trial
TYPES ON ONE-TO-ONE
correspondence
Pseudo
96
YX
pseudo,
YAO
Error
73
9n
types.
The main effect for size was due to slightly more correct identifications
for the set size of 8 (92%) than for the set size of 16 (89%). The reliable
task x group interaction was due to fewer correct identifications by the
MD children relative to the normal children, on the one-to-one correspondence task, F(1, 35) = 14.13, p < .OOl, but no group difference for
performance on the stable order, F( 1, 35) < 1, or cardinality, F(1, 35)
= 3.28, p > .05, tasks. For the one-to-one correspondence task, the
group difference in number of correct identifications was reliable for each
of the three set sizes (ps < .0.5). The performance levels of the MD and
normal children for each of the count types (i.e., correct, pseudo error,
and error), across set size, for the one-to-one correspondence task is
displayed in Table 3. The frequency of correct identifications for this task
was analyzed by means of a 2 (group) x 3 (type; correct, pseudo error,
and error) mixed-design ANOVA.
The results indicated reliable main
effects for group, F(1, 35) = 14.47, p < ,001, and type, F(2, 70) =
16.42, p < .OOl, as well as a reliable group x type interaction, F(2, 70)
= 4.93, p < .Ol. The interaction was due to group differences in the
number of correct identifications for the pseudo error, F(1, 35) = 9.61,
p < .Ol, and error, F(1, 35) = 6.17, p < .Ol, trials, but no reliable
group difference for correct trials, F (1, 35) < 1.
Individualprotocols.
To explore further the relatively poor performance
of the MD group on the one-to-one correspondence task, individual protocols for the pseudo-error and error trials for each of the MD children
were examined. Examination
of these protocols indicated that 5 of the
13 MD children appeared to believe that adjacency was a necessary feature
of counting, in that they stated that five or six of the six pseudo-error
trials were incorrect. Four of the remaining children appeared to be unsure
of the status of pseudo-error trials, as indicated by stating that about l/2
(i.e., two, three, or four trials) of the pseudo-error trials were correct
and yielded the same pattern of results that was found for the nontransformed
data, that
is. reliable main effects for group, task, and size. as well as a reliable group x task interaction
(ps > .OS). The remaining
interactions
were not reliable (ps > .05).
COUNTING
TABLE
CHARACTERISTICS
Mean % of
trials in which
strategy was
used
383
KNOWLEDGE
4
OF ADDITION
..__~
Mean reaction
time in seconds
STRATEGIES
Percentage of
errors
Mean % of
trials in which
min strategy
was used
Strategy
MD
Normal
MD
Normal
MD
Normal
MD
Normal
Counting fingers
Fingers
Verbal counting
Retrieval
57
Oh
33
9
44
2
37
17
7.4
8.0
5.8
1.4
3.7
56
23
9
24
19
31
69
91
-
6.2
-‘.
57
66
62
Note. MD, Math disabled. Mean reaction time (RT) excluded error and spoiled RTs.
For the counting fingers and verbal counting trials. mean RT was based on min trials, to
make the solution times comparable across groups. Columnar totals for the strategy percentages may not sum to 100 due to rounding.
’ For the MD and normal groups, 0.4 and 0.2%, respectively, of the trials were classified
as decomposition.
h The actual percentage was 0.4.
’ There were not enough correct retrieval trials to produce a meaningful estimate.
and l/2 were incorrect. The 4 remaining children always indicated that
the pseudo-error trials were correct. As for the error trials, 76% of the
errors occurred when the first chip was double-counted.
Summary. The results for the one-to-one correspondence task suggest
that MD children as a group show an immature understanding of some
basic number concepts, relative to their academically normal peers. The
results also suggest that the group difference in counting knowledge does
not reside in the understanding of a particular principle (Gelman & Meek,
1983) but rather in the understanding of essential and unessential features
of counting (Briars & Siegler, 1984) and in skill in detecting doublecounting errors. Specifically, the counting knowledge of MD children
appears to be rather rigid and developmentally
immature, as reflected in
the tendency of many of the MD children to behave as if adjacency was
an essential feature of correct counting. Moreover, MD children appear
to be less skilled than normal children in detecting certain forms of counting error, particularly if the error is committed at the beginning of the
count. The high error rate at the beginning, rather than the end, of the
count suggests that the failure to detect the double-counting
errors might
have been due to a memory failure rather than to poor conceptual knowledge.
Addition Task
Table 4 presents group-level characteristics of addition strategies. Consistent with previous studies (Geary, 1990; Geary et al., 1991; Siegler,
384
GEARY.
BOW-THOMAS,
AND
YAO
1987), both the normal and MD children relied primarily on counting to
solve the addition problems. In fact, univariate ANOVAs
indicated that
the normal and MD groups did not differ reliably in the frequency with
which the counting fingers, F(1, 35) < 1, or verbal counting, F(1, 35) <
1, strategies were used for problem solving. Nevertheless,
the normal
children did rely on direct retrieval for problem solving more frequently
than the MD children, F( 1, 35) = 4.13, p < .05. The normal children
also committed fewer errors than the MD children for the counting fingers,
F(1, 29) = 9.77, p < .Ol, verbal counting, F( 1. 28) = 7.68, p < .Ol.
and retrieval, F(1, 19) = 9.19, p < .Ol, strategies. Finally, the normal
children used the min algorithm more frequently than the MD children
for both the counting fingers, F( 1, 27) = 11.64, p < .Ol, and verbal
counting, F(1, 20) = 4.23, p = ,053, strategies.
Problem dificulty and use of counting (computational)
strategies. Previous research has shown that the use of direct retrieval decreases in
frequency as the difficulty of the problem increases (e.g., Siegler & Shrager, 1984). This is so because answers are less likely to be retrieved from
long-term memory for difficult problems than for less difficult problems.
As noted earlier, for addition. the difficulty of the problem can be indexed
by the sum of the augend and addend (Ashcraft.
1992). Thus, the frequency with which direct retrieval is used for problem solving should
decrease with an increase in the value of the sum of the problem. In the
present study, the frequency of retrieval trials decreased with an increase
in the value of the sum for the normal group, r(38) = - .70, p < .OOl,
but not for the MD group, r(38) = - .14, p < .2S. The difference in
the value of these two coefficients was marginally reliably, z = - 1.89,
p < .06.
Thus, consistent with many previous studies (e.g., Geary et al.. 1991;
Siegler & Shrager, 1984), for the normal group, direct retrieval appeared
to be used to solve simpler problems and backup computational
(i.e.,
finger counting and verbal counting) strategies to solve more difficult
problems. For the MD group, the failure to find a relationship between
the frequency of retrieval trials and problem difficulty, combined with the
high proportion
of retrieval errors, suggests that the MD children were
frequently guessing when they stated a retrieved answer. In other words,
rather than resort to a backup computational
strategy if they were not
certain of the accuracy of a retrieved answer, the MD children appeared
to have stated any answer that might have been retrieved, regardless of
the potential accuracy of that answer (see Siegler, 1988).
Summary. Of particular interest in the current study was group differences in computational
skills. Although the normal and MD children did
not differ in the frequency with which the counting fingers and verbal
counting strategies were used for problem solving, they did differ in skill
at executing these strategies. Consistent
with previous studies of first-
COUNTING
KNOWLEDGE
385
grade MD children (Geary, 1990; Geary et al., 1991), the MD children
in the present study committed more than twice the computational
errors
of the normal children and used the min algorithm less frequently.
Relationship between Counting Knowledge and Computational
Skills
The question addressed in this final section is whether the immature
knowledge of counting features of MD children contributed to their relatively poor computational
skills evident on the addition task. Recall that
the analyses of the performance on the counting tasks suggested that the
MD children as a group were developmentally
delayed in their understanding of counting features and were relatively poor at detecting counting errors when these errors occurred at the beginning of the count. A
rigid conceptual understanding of counting might impede the adoption of
counting algorithms that deviate from the standard count-all, or sum,
strategy. If so, then an index of immature counting knowledge (i.e., the
frequency of incorrect pseudo-error trials in this study) should be correlated with the frequency with which the min algorithm was used to solve
addition problems.
If skill at detecting counting errors and developmentally
mature counting knowledge contribute to skill at repairing incorrectly executed computational procedures, as suggested by Ohlsson and Rees (1991). then
the frequency of incorrect pseudo-error trials and frequency of doublecounting errors should be correlated with the frequency of computational
errors on the addition task. To test these hypotheses, the frequency of
correct identifications for pseudo-error trials was correlated with the frequency of min counting (across the counting fingers and verbal {counting
strategies), and a composite index (the sum of the frequency of correct
double-counting and pseudo-error trials) was correlated with the frequency
of computational
errors (again, across the counting-fingers and1 verbalcounting strategies).
The results indicated that, across groups, the frequency of correct identifications for pseudo-error trials was reliably correlated with use of the
min algorithm, r(35) = .47, p < .Ol. Moreover, the score for the composite index was reliably correlated with the frequency of computational
errors, r(3.5) = - .44, p < .05, but was unrelated to the frequency of
retrieval errors, r(35) .X3, p > .50. A t test for dependent correlations
indicated that the value of these two coefficients (i.e., - .44 and .18)
differed reliably, t(34) = 2.57, p < .05. Nevertheless, the coefficient for
the latter correlation should be interpreted with caution, given ,the relatively small number of retrieval trials.
Another potential indicator of counting knowledge is the number of
correct counts identified as incorrect. Briars and Siegler (1984), however,
found that 3-, 4-, and 5-year-old children only rarely identified correct
counts as incorrect. This result suggests that errors on correct counts do
386
GEARY,
BOW-THOMAS.
AND
YAO
not occur often and therefore are not likely to be a sensitive measure of
individual differences in knowledge of counting features. Indeed, in this
study, correct counts were rarely identified as incorrect by the normal
and MD children. The frequency of correct counts that were identified
as incorrect was not correlated with either the frequency of computational
errors (r(36) = - .25, p > .lO) or with the use of the min algorithm
(r(34)
= .02, p > .50).
Finally, Analysis of Covariance (ANCOVA)
procedures indicated that
controlling for group differences in skill at detecting double-counting
errors and frequency of pseudo-error trials eliminated the group difference
in the frequency of computational
errors, F( 1, 33) = 1.21, p > .25.
Controlling for the group difference in number of correct identifications
on pseudo-error trials also eliminated the group difference in the frequency
with which the min algorithm was used to solve the addition problems,
F(1, 32) = 3.42, p > .OS.
Summary. Consistent with the conceptual presentation of Ohlsson and
Rees (1991), the results suggest that a mature understanding of essential
and unessential features of counting is related to the adoption of the
developmentally
mature min algorithm for solving addition problems and.
along with skill at detecting counting errors, the accuracy with which
computational procedures can be executed. Moreover, these analyses support the argument that for MD children a developmental
delay in the
understanding of counting features and relatively poor skills at detecting
certain forms of counting error contribute to the poor computational skills
of these children in the domain of arithmetic.
DISCUSSION
The present study enabled an assessment of the relationship between
counting knowledge and computational
skills for the domain of addition
(i.e., skill at using counting procedures to solve addition problems). Ohlsson and Rees (1991) argued that counting knowledge provides a standard
against which the execution of computational procedures is evaluated (see
also Siegler & Jenkins, 1989). If the execution of a computational
procedure violates principled knowledge, then the procedure is modified so
that with subsequent executions it conforms to counting knowledge. The
conceptual formulation presented by Ohlsson and Rees implies that computational skills should be related to the developmental
maturity of the
child’s counting knowledge and to the child’s skill at detecting when
counting has violated this principled knowledge. The finding that the
frequency of computational errors in the domain of addition was correlated
with an index representing skill at detecting counting errors and the maturity of the child’s knowledge of counting features provides empirical
support of this position. This argument is bolstered by the finding that
this same index was not correlated with the frequency of retrieval errors.
COUNTING
KNOWLEDGE
387
The analyses of the relationship between variables representing counting
knowledge and computational
skills also suggests a relationship between
knowledge of counting features and use of the min, or counting-on, procedure (Carpenter & Moser, 1984; Groen & Parkman, 1972). More precisely, the analyses suggested that the adoption of the min procedure
might require an understanding of the essential and unessential features
of counting (Briars & Siegler, 1984). If so, then why did the MD children
sometimes use the sum procedure and at other times use the min procedure
to solve addition problems? As noted earlier, the child’s performance and
presumably counting knowledge varies with the size of the counted set.
So, the understanding of the cardinality of small set sizes does not necessarily indicate that the child will understand the concept of cardinality
when applied to larger set sizes.
Similarly, the min procedure tends to be employed to solve problems
with smaller valued integers and the sum procedure tends to be used to
solve problems with larger valued integers (Geary, 1990; Geary et al.,
1991). Thus, if counting knowledge varies with set size, then the expectation would be that the min procedure would first be adopted for the
solution of easy problems (e.g., 2 + 1) and only later for the solution
of more difficult problems (e.g., 8 + 9; Geary, 1990). This argument
must be considered a working hypothesis, however, because the experimental tasks used in the current study did not allow for a determination
of the child’s knowledge of, for instance, the unessential features of counting for relatively small set sizes (e.g.. 3). Thus, it might be the case that
MD children understand that adjacency is unessential when counting small
set sizes. Moreover, the counting tasks did not enable a complete assessment of the understanding of all of the unessential counting features
described by Briars and Siegler (1984). Finally, the work of Siegler and
Jenkins (1989) suggests that if knowledge of counting features is related
to the adoption of the min algorithm, then this knowledge is more likely
necessary but not sufficient for the abandonment of the sum procedure.
The current study also enabled a comparison of the counting knowledge
of MD and normal children and allowed for an assessment of the relationship between the counting knowledge of MD children and their relatively poor computational
skills. Recall that MD children commit many
more computational
errors than normal children and show a delay in the
adoption of the min procedure (Geary, 1990; Geary et al., 1987; Geary
et al., 1991; Goldman et al., 1988; Svenson & Broquist, 1975). As a
group, the MD children in the current study appeared to be developmentally delayed in the understanding of essential and unessential features
of counting. Many of the 7-year-old MD children assessed in this study
appeared to believe that adjacency was an essential feature of correct
counting, much like the 5-year-old children described by Briars and Siegler
(1984). The MD children were also less skilled than the normal children
388
GEARY.
BOW-THOMAS,
AND
YAO
in detecting certain types of counting errors, in particular, double-counting
errors that occurred at the beginning of the count. Failure to detect this
type of counting error can be especially detrimental to modifying incorrect
min counts. because the most common computational
error during min
counting involves double-counting
the larger valued integer (Geary, 1990;
Siegler & Shrager, 1984). In fact, a developmental
delay in the understanding of counting features and relatively poor skills at detecting doublecounting errors appeared to mediate the poor, relative to their normal
peers, computational
skills of the MD children. Nevertheless,
this too
should be considered a working hypothesis, as the entire range of unessential counting features described by Briars and Siegler (1984) was not
assessed in the current study.
Nevertheless,
the current study can be used to address the issue of
whether the functional counting skills of children are better represented
by Gelman and Gallistel’s (1978) principles-first
model or by the induction
model proposed by Briars and Siegler (1984). Recall that Gelman and
Gallistel argued that the counting behavior of children is governed by
implicit knowledge
of the one-to-one correspondence,
stable order, and
cardinality principles. Briars and Siegler, on the other hand, argued that
children first count by rote and then gradually induce the essential and
unessential features of counting. Within the framework
provided by Gelman and Gallistel, performance
on the cardinality
task should be dependent upon knowledge of the principles of one-to-one correspondence
and stable order. Thus, the poor performance
on the one-to-one correspondence task found in the current study should have affected performance on the cardinahty task. This was not the case. This result combined
with the finding that a feature of Briar and Siegler’s induction model (i.e.,
adjacency) most sharply differentiated
the counting skills of MD and
normal children indicates that the current results are more consistent with
the induction than the principles-first
model.
Regardless, other evidence supports the position that human infants
have a basic ability to detect the numerosity
of arrays. That is, infants
appear to be able to extract quantity information
from the environment
for small set sizes (Starkey.
Spelke, & Gelman, 1990). The findings of
Starkey et al. are consistent with Gelman and Gallistel’s (1978) view that
young children have an implicit understanding of basic counting principles.
Perhaps the numerical abstraction skills of infants reflect an innate bias
to attend to quantitative features in the environment.
It is not at all clear,
however, how such an innate orientation to quantitative features influences
more functional skills, such as those reflected in counting behavior, though
it is reasonable to suspect such a link (Siegler, 1991). The numerical
abstraction skills of infants may in fact provide the basis for early counting
skills, but our data suggest that the concepts associated with the functional
skills of children are, for the most part, induced.
COUNTING
389
KNOWLEDGE
In summary, the current study provided empirical support for the hypothesis that the link between counting knowledge and computational
skills (Ohlsson & Rees, 1991) resides in the developmental
maturity of
the child’s principled knowledge and the child’s skill at detecting when
the execution of computational
procedures violates this knowledge. The
results also suggest that the poor computational
skills of first-grade MD
children are related to a developmental
delay in the understanding of
essential and unessential features of counting (Briars & Siegler, 1984) and
relatively poor skills at detecting certain forms of counting error. Finally,
the present study suggests that a comparison of normal and MD children
across the entire range of essential and unessential counting features described by Briars and Siegler( 1984) and across small and large set sizes
as related to computational
skills would provide additional, and perhaps
more, valuable information on the relationship between principled knowledge and procedural skills. Such a study might also provide more explicit
information on the development of conceptual knowledge in MD children.
REFERENCES
Ashcraft.
M.
H. (1992).
Cognitive
arithmetic:
A review
of data
and theory.
Cognition.
in
pre.S.
Ashcraft.
M. H., Yamashita.
T. S.. & Aram,
D. M. (1992). Mathematics
performance
in
left and right brain-lesioned
children.
Brain and Cognition,
19, 208-252.
Baroody,
A. J. (1983). The development
of procedural
knowledge:
An alternative
explanation for chronometric
trends of mental arithmetic.
Developmenfal
Review.
3, 225230.
Baroody,
A. J. (1984). The case of Fehcia: A young child’s strategies for reducing
memory
demands
during mental addition.
Cognition
and Instruction.
1, 109Y116.
Briars, D., & Siegler, R. S. (1984). A featural analysis of preschoolers’
counting
knowledge.
Developmental
Psychology,
20, 607-618.
Carpenter,
T. P., & Moser. J. M. (1984). The acquisition
of addition and subtraction
concepts
in grades one through
three. Journal for Research in Mathematics
Education,
15, 179202.
Fleischner,
J. E., Garnett,
K., & Shepherd.
M. J. (1982). Proficiency
in arithmletic
basic
fact computation
of learning
disabled
and nondisabled
children.
Focus on Learning
Problems
in Mathematics,
4, 47-56.
Fuson, K. C. (1988).
Children’s
counting
and concepts of number.
New York:
SpringerVerlag.
Garnett,
K., & Fleischner,
J. E. (1983).
Automatization
and basic fact performance
of
normal
and learning disabled
children.
Learning
Disabilities
Quarterly,
6, 2213-230.
Geary.
D. C. (1990). A componential
analysis of an early learning
deficit in mathematics.
Journal of Experimental
Child Psychology,
49, 363-383.
Geary,
D. C., & Brown.
S. C (1991).
Cognitive
addition:
Strategy
choice and speed-ofprocessing
differences
in gifted. normal,
and mathematically
disabled children.
Developmental
Psychology,
27, 398-406.
Geary,
D. C., Brown,
S. C, & Samaranayake,
V. A. (1YYl). Cognitive
addition:
A short
longitudinal
study of strategy choice and speed-of-processing
differences
in normal and
mathematically
disabled children.
Developmental
Psychology,
27, 787-797.
Geary,
D. C.. Widaman.
K. F., Little,
T. D.. & Cormier,
P. (1987). Cognitive
addition:
390
GEARY,
BOW-THOMAS,
AND
YAO
Comparison
of learning disabled and academically
normal elementary
school children.
Cognitive
Development,
2, 249-269.
Gelman,
R.. & Gallistel,
C. R. (1978).
The child’s understanding
of number.
Cambridge,
MA: Harvard
Univ. Press.
Gelman,
R.. & Meek, E. (1983). Preschoolers’
counting:
Principles
before skill. Cogrz:nition,
13, 343-359.
Goldman,
S. R., Pellegrino,
J. W., & Mertz,
D. L. (1988).
Extended
practice of basic
addition facts: Strategy changes in learning disabled students,
Cognition
and Instruction,
5, 223-265.
Greeno,
J. Cl.. Riley, M. S., & Gelman,
R. (1984). Conceptual
competence
and children’s
counting.
Cognitive
Psychology.
16, 94-193.
Groen.
G. J.. & Parkman,
J. M. (1972).
A chronometric
analysis of simple addition.
Psychological
Review, 79, 329-343.
Hamann,
M. S., & Ashcraft.
M. H. (1985). Simple and complex
mental addition
across
development.
Journal
of Experimental
Child Psychology.
40, 49-72.
Jastak, J. F., & Jastak. S. (1978).
Manual
for the Wide Range Achievement
Test: 1978
Revised Edition.
Wilmington,
DE: Jastak Associates,
Inc.
Ohlsson,
S.. & Rees. E. (1991). The function
of conceptual
understanding
in the learning
of arithmetic
procedures.
Cognition
and Instruction,
8, 1033179.
Siegler, R. S. (1986). Unities across domains in children’s
strategy choices. In M. Perlmutter
(Ed.), Perspectives for intellectual
development:
Minnesota
symposia on child psychology
(Vol. 19. pp. 1-48). Hillsdale,
NJ: Erlbaum.
Siegler. R. S. (1987). The perils of averaging
data over strategies:
An example from children’s
addition.
Journal of Experimental
Psychology:
General,
116, 250-264.
Siegler, R. S. (1988). Individual
differences
in strategy choices: Good students, not-so-good
students.
and perfectionists.
Child Development.
59, 833-851.
Siegler, R. S. (1989). Hazards
of mental chronometry:
An example
from children’s
subtraction.
Journal of Educational
Psychology,
81, 497-506.
Siegler, R. S. (1991). In young children’s
counting.
procedures
precede principles.
Edurational
Psychology
Review, 3, l2?- 135,
Siegler, R. S., & Jenkins,
E. (1989). How children discover new strategies. Hillsdale.
NJ:
Erlbaum.
Siegler, R. S.. & Robinson,
M. (1982). The development
of numerical
understandings.
In
H. Reese & L. P. Lipsitt
(Eds.),
Advances
in child development
and behavior
(Vol.
16, pp. 241-312).
New York:
Academic
Press.
Siegler, R. S., & Shrager,
J. (1984).
Strategy
choice in addition
and subtraction:
How do
children
know what to do? In C. Sophian (Ed.).
Origins of cognitive
skills (pp. 229293). Hillsdale,
NJ: Erlbaum.
Starkey.
P., Spelke. E. S., & Gelman,
R. (1990). Numerical
abstraction
by human infants.
Cognition,
36, 97-127.
Stevens. J. (1986). Applied
multivariate
statistics for fhe social sciences. Hillsdale.
NJ: Erlbaum.
Svenson.
0.. & Broquist,
S. (1975).
Strategies
for solving simple addition
problems:
A
comparison
of normal
and subnormal
children.
Scandinavian
Journal
of Psychology,
16, 143-151.
Temple,
C. M. (1991). Procedural
dyscalculia
and number
fact dyscalculia:
Double
dissociation in developmental
dyscalculia.
Cognitive
Nemopsychology,
8, 155-176.
Widaman,
K. F., Geary. D. C., Cormier,
P.. & Little, T. D. (1989). A componential
model
for mental addition.
Journal of Experimental
Psychology:
Learning,
Memory,
and Cognition, 15, 898-919.
COUNTING
Woodcock,
R. W.,
Revised. Allen,
Wynn,
RECEIVED:
K. (1990).
November
& Johnson,
M. B. (1989).
Woodcock-Johnson
TX: DLM
Teaching
Resources.
Children’s
understanding
of counting.
Cognition,
20, 1991;
REVISED
391
KNOWLEDGE
May
14, 1992
Tests of Ach,ievement36, 155-193.