Developmental Psychology
1991, Vol. 27, No. 3, 398-406
Copyright 1991 by the American Psychological Association, Inc.
0012-1649/91/$3.00
Cognitive Addition: Strategy Choice and Speed-of-Processing
Differences in Gifted, Normal, and Mathematically Disabled Children
David C. Geary and Sam C Brown
University of Missouri at Columbia
This study assessed strategy choice and information-processing differences in gifted, normal, and
mathematically disabled third- or fourth-grade children. Fourteen gifted, 12 normal, and t 5 math
disabled (MD) children solved 40 simple addition problems. Strategies, and their solution times,
used in problem solving were recorded on a trial-by-trial basis, and each was classified in accordance with the distributions of associations model of strategy choices. Group differences were
evident for the developmental maturity of the strategy mix and for the rate of verbal counting. The
gifted group showed the most mature distribution of strategy choices, followed by the normal and
MD groups. In terms of speed of processing, the gifted group showed a verbal counting rate that
was at adult levelsand less than 50% of the rate of counting for the two remaininggroups, but group
differences were not evident in the rate of retrieving answers from long-term memory. The results
were interpreted within the context of the strategy choice model and suggested that a single dimension spanned group differences in the level of mastery of early numerical skills: the maturity of the
long-term memory organization of basic facts. Finally, implications for the mental and strategic
processes contributing to academic achievement are discussed.
Academic success is often defined by performance on standardized achievement measures. These tests, however, do not
provide information about the cognitive processes or problem
solving strategies that likely contribute to academic achievement. One approach to the study of mental and strategic processes contributing to academic skills is to compare groups of
children, who vary in achievement levels, on tasks for which a
developmental progression of problem solving strategies is evident and for which the problem solving solution times can be
decomposed into basic information processes. Simple addition
is such a task. Elementary school children will typically use a
variety of strategies, ranging from counting fingers to retrieving
facts from long-term memory, to solve addition problems (Baroody, 1987; Carpenter & Moser, 1984; Groen & Resnick, 1977).
The processes by which children choose a problem solving
strategy, such as counting as opposed to memory retrieval, are
well delineated within a more general theoretical model of cognitive development: the distributions of associations model of
strategy choices (Siegler, 1986; Siegler & Shrager, 1984). Moreover, although any single child might rely on two or more strategies to solve a set of addition problems, the developmental matu-
rity of the child's strategy mix can be inferred from the strategy
choice model and adult performance in addition (Siegler & Jenkins, 1989). Finally, for each of the various strategies, the associated solution times can be decomposed into more basic elementary operations that might index, for example, rate of
counting or memory retrieval (Geary & Burlingham-Dubree,
1989; Siegler, 1987). Thus, the comparison of performance differences of children of varying achievement levels on a simple
addition task should enable inferences to be drawn regarding
the mental and strategic processes contributing to basic numerical skills. The present study followed this approach and compared groups of third- or fourth-grade mentally gifted, academically normal, and math disabled (MD) children on the strategies, and associated solution times used to solve simple addition
problems.
In terms of identifying the mental and strategic processes
that contribute to the academic success of gifted children, there
is a dearth of theory-driven research (Sternberg & Davidson,
1986). In fact, to the best of our knowledge, no previous studies
of mental addition with gifted children have been conducted.
Studies that have been conducted in cognitive domains other
than arithmetic indicate that gifted children differ from their
lower ability peers on several dimensions (Horowitz & O'Brien,
1985; Keating & Bobbitt, 1978; Marr & Sternberg, 1986; Siegler
& Kotovsky, 1986; Sternberg & Davidson, 1986). Germane to
this study are experiments that have examined differences in
the types of problem solving strategies employed by gifted, relative to nongifted, children and in the rate with which basic
information processes are executed. These studies indicate that
gifted children show developmentally mature strategy approaches to problem solving tasks; that is, gifted children use
problem solving strategies similar to those employed by older
nongifted children (Siegler & Kotovsky, 1986). Moreover, the
rate with which gifted children execute basic elementary pro-
This research was partially supported by a summer research fellowship from the Research Council of the University of Missouri at Columbia and by grants from the Weldon Spring Foundation and
AFOSR-88-0239. We would like to thank the students and teachers at
Wyman Elementary school for their cooperation, with special thanks
to Don Brown and Susan Bates. Finally,we would like to thank Mona I.
Brown and llona H. Klosterman for their assistance with data collation, and Mark Ashcraft, Peter Frensch, Jeffrey Bisanz, and three anonymous reviewers for comments on an earlier draft.
Correspondence concerning this article should be addressed to David C. Geary, Department of Psychology, 210 McAlester Hall, University of Missouri, Columbia, Missouri 65211.
398
COGNITIVE ADDITION
cesses appears to be faster relative to lower ability peers (Keating & Bobbitt, 1978).
An absence of theory-driven research is also evident in the
learning disabilities area. Nevertheless, several studies of older
(at least in third grade) math disabled children suggest that a
basic deficit involves the use o f developmentally immature
problem solving strategies; that is, these children often use strategies more commonly employed by younger academically normal children (Geary, Widaman, Little, & Cormier, 1987; Goldman, Pellegrino, & Mertz, 1988; Svenson & Broquist, 1975). To
illustrate, in a related study Geary (1990) compared the distribution of strategies employed by first- or second-grade M D and
normal children to solve a set of addition problems. The MD,
relative to the normal, children tended to use an immature
counting strategy and made numerous counting-related errors.
The methodologies used in these previous studies, however, did
not allow for a precise assessment of the rate of information
processing comparing older M D and normal children. Thus,
when M D and normal children do use the same strategy, it is
not clear whether they differ in the rate of executing the basic
processes comprising that strategy.
In all, previous research suggests that groups of gifted, normal, and M D children might differ in the developmental maturity of the strategy mix employed to solve a set of addition
problems and in the rate with which basic processes, such as
verbal counting, are executed. As noted earlier, elementary
school children may use a variety of strategies to solve a set of
addition problems. Thus, the present study followed the strategy choice model to represent the strategy mix of the gifted,
normal, and M D groups and to provide a theoretical foundation for the interpretation o f group differences in strategy
choices.
Strategy Choice Model
Four basic strategies for solving addition problems have been
identified within the strategy choice model (Siegler, 1986;
Siegler & Robinson, 1982; Siegler & Shrager, 1984). Three are
visible or audible overt strategies and are termed as follows: (a)
counting fingers--children use their fingers to physically represent the problem integers and then count their fingers to reach a
sum; (b) fingers--children use their fingers to represent the
integers but do not visibly count them before giving an answer;
and (c) verbal counting--children count audibly or move their
lips as if counting implicitly. The fourth strategy reflects the
retrieval of an addition answer from long-term memory (Siegler
& Shrager, 1984).
The specific strategy chosen for problem solving is governed
by the peakedness of the distribution o f associations between a
problem and all potential answers to that problem. With a
peaked distribution, "the preponderance o f associative
strength is concentrated on a single answer (the peak of the
distribution)" (Siegler, 1988a, p. 834), whereas for a flat distribution, the associative strength is distributed among several
potential answers. The more peaked the distribution of associations, the more readily an answer can be retrieved from longterm memory and therefore the more likely the retrieval strategy will be used to solve the presented problem. In addition to
peakedness of the distribution of associations, the actual strat-
399
egy chosen for problem solving is also governed by a confidence
criterion. The confidence criterion represents an internal standard against which the child gauges confidence in the correctness of the retrieved answer, and the rigor of this criterion appears to vary from child to child (Siegler, 1988a).
According to the strategy choice model, the problem solving
process begins with the child's first setting two parameters: (a)
the confidence criterion and (b) a search length criterion, which
indicates the maximum number of retrieval attempts a child
will make before choosing an alternative strategy. Retrieval is
then attempted and continues as long as the value o f the confidence criterion exceeds the associative strength o f each retrieved answer and as long as the number of searches does not
exceed the value o f the search length parameter. If no answer
exceeds the value of the confidence criterion and the value of
the search length parameter is exceeded, then the child will
resort to the use of a backup strategy (Siegler, 1983). Here, older
children will typically employ the verbal counting strategy, although some children might occasionally use the fingers strategy or the counting fingers strategy as a backup (Siegler, 1987).
Counting Algorithms
If counting is required to solve an addition problem, then
children, by the first grade, will typically use the rain, or counting-on, procedurewhen using either the counting fingers strategy or the verbal counting strategy (Carpenter & Moser, 1984;
Fuson, 1988; Geary, 1990; Goldman, Mertz, & Pellegrino, 1989;
Groen & Parkman, 1972; G r o e n & Resnick, 1977; Siegler,
1987). With the rain procedure, the solution of a problem, such
as 9 + 6, begins with stating the cardinal value of the larger
integer (i.e., 9) and then counting in a unit-by-unit fashion a
number of times equal to the value of the smaller or minimum
(min) integer (i.e., 6) until a sum is obtained. An alternative but
developmentally less mature strategy that appears to be occasionally used by older M D children involves counting from zero
in a unit-by-unit fashion a number of times equal to the cardinal
value of both the augend (i.e., 9) and the addend (i.e., 6; Goldman et al., 1988; Svenson & Broquist, 1975). This latter strategy
has been termed the sum, or counting-all, procedure(Baroody,
1987; Carpenter & Moser, 1984; Siegler, 1987).
Developmental Progression of Addition Strategies
The use of increasingly mature problem solving strategies
cannot simply be characterized by the substitution of one strategy, such as memory retrieval, for another less mature strategy,
such as counting (Ashcraft, 1982). Rather, "development involves changes in the mix of existing strategies as well as construction of new ones and abandonment of old ones" (Siegler &
Jenkins, 1989, p. 27). A kindergarten child might use each of
the above-described strategies to solve a set of addition probl e m s - f o r example, counting, whether verbal counting or
counting fingers, by means of either the min or the sum strategies (Baroody, 1987; G e a r y & Burlingham-Dubree, 1989;
Siegler, 1987). By the second grade, the sum strategy has typically been abandoned, and when counting is required, it is
usually by means of the verbal counting, rather than the counting fingers, strategy (Carpenter & Moser, 1984; Siegler, 1987).
400
DAVID C. GEARY AND SAM C BROWN
Thereafter, the frequency with which t h e verbal c o u n t i n g strategy is used declines, a n d the frequency o f m e m o r y retrieval
increases. So, by t h e sixth grade a c a d e m i c a l l y n o r m a l c h i l d r e n
a p p e a r to use the retrieval strategy to solve the majority o f
simple a d d i t i o n p r o b l e m s (Ashcraft & F i e r m a n , 1982; G e a r y et
al., 1987; Kaye, Post, Hall, & D i n e e n , 1986). In all, the developm e n t a l m a t u r i t y o f the strategy mix c a n b e indexed by the
frequency with which answers c a n b e correctly retrieved from
long-term memory.
The Present Study
T h e present study extends research in b o t h the gifted a n d
M D areas o n two d i m e n s i o n s . First, t h e assessment o f p r o b lem-solving strategies is b a s e d u p o n a well articulated theoretical m o d e l o f cognitive d e v e l o p m e n t - - t h a t is, the strategy choice
m o d e l (Siegler, 1986; Siegler & Shrager, 1984). Thus, the strategy choice m o d e l provides the theoretical f r a m e w o r k for the
i n t e r p r e t a t i o n o f p e r f o r m a n c e differences c o m p a r i n g gifted,
n o r m a l , a n d M D c h i l d r e n (Geary, 1990; Siegler, 1988a). Second,
because p r o b l e m solving strategies, a n d t h e i r associated solution times, are recorded o n a trial-by-trial basis, a very precise
assessment o f the rate o f executing basic i n f o r m a t i o n processes,
such as rate o f retrieving facts from l o n g - t e r m memory, is obt a i n e d a n d c o m p a r e d across groups.
Method
Subjects
The subjects were selected from two elementary schools that served a
rural working-class population. The sample included a total of 41
third- or fourth-grade mentally gifted, academically normal, or MD
students. The gifted group consisted of 4 male and 10 female third- (n =
4) or fourth- (n = 10) grade children with a mean age of119 months
(SD = 8.8). The normal group consisted of 5 male and 7 female fourthgrade students with a mean age of 125 months (SD = 5.5). The MD
group consisted of 10 male and 5 female fourth-grade students with a
mean age of 130 months (SD = 8.4). 1 The difference in mean age,
comparing the gifted and MD groups, was reliable (p < .05), but no
other group differences were reliable (ps > .05).
At the time of the study, subjects included in the MD group were
receiving Chapter I (a federally funded program for low-achieving children) remedial education services in mathematics. Inclusion in this
program required a score on a standard mathematics achievement test
that was below the 46th national percentile ranking (scores are more
typically below the 30th percentile). Chapter 1 services involved 20
min per day, 5 days per week, of specialized instruction in number
concepts and mathematical procedures. All of the MD subjects attended general education courses for most of their school day, although
many of these children also received remedial services in reading.
Children who were placed in special day classes (i.e., those children
who spent the entire school day in remedial classes) were not included
in the sample. These children were excluded as potential subjects because children in such classes often show an array o fsocial/behavioral,
cognitive, and perhaps neuropsychological deficits (Sutaria, 1985),
which would make data interpretation difficult.
The gifted children were selected from the school district's enrichment program. Inclusion in this program required a grade-point average of 4.0 (on a 0 to 4 scale) for the previous academic year and a score
at or above the 97th percentile on at least one section of the Science
Research Associates survey of basic skills. Descriptive information for
performance on the Science Research Associates survey of basic skills,
from the current academic year and obtained from school files, is presented in Table 1. Here, it can be seen that the gifted group showed the
highest percentile ranking on both the mathematics and reading
achievement measures, followed in turn by the normal and MD
groups. Group differences were assessed by means of the Bonferroni t
procedure and revealed all differences for both the mathematics and
reading measures to be reliable (ps < .05).
Stimufi
The experimental stimuli consisted of 40 pairs of vertically placed
single-digit integers. Stimuli were constructed from the 56 possible
nontie pairwise combinations of the integers 2 to 9 (a tie problem is,
e.g., 2 + 2, 4 + 4). The frequency and placement of all integers were
counterbalanced. That is, each integer (2 to 9) appeared five times as
the augend and five times as the addend, and the smaller value 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.
Apparatus
The addition problems were presented at the center of a 30-cm ×
30-cm video screen controlled by an IBM PC-XT microcomputer. A
Cognitive Testing Station clocking mechanism ensured the collection
of reaction times (RTs) with _+ l-ms accuracy. The timing mechanism
was initiated with the presentation of the problem on the video screen
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
video screen for a 1,000-ms duration, followed by a 1,000-ms period
during which the screen was blank. 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
Each subject was tested individually in a quiet room. The subjects
were asked to solve 40 addition problems, preceded by 8 practice problems. The problems were presented one at a time on the video screen,
and subjects were encouraged to use whatever strategy made it easiest
for them to obtain the answer. Equal emphasis was placed on speed and
accuracy. For 28 subjects the experimental session was conducted
about 2 months before the administration of the achievement measures; for the remaining 13 subjects the experimental session was conducted 1-4 weeks after the administration of the achievement measures.
During the experimental session, the answer and strategy used to
solve each problem was recorded by the experimenter and classified as
one of the four strategies described by Siegler and Robinson (1982): (a)
counting fingers, (b) fingers, (c) verbal counting, or (d) memory re-
A previous study (Siegler, 1988a) found no gender differences in the
overall distribution of strategy choices or in strategy characteristics.
Thus, the different gender composition of the three groups should not
substantively impact the results of this study. Moreover, the third- and
fourth-grade gifted children did not differ in their overall distribution
of strategy choices. The third-grade gifted children used the retrieval
strategy on 86% of the trials as compared to 89% for the fourth-grade
gifted children.
COGNITIVE ADDITION
Table 1
Descriptive Information for Achievement Test Scores
Math
disabled
Normal
Gifted
Subject
M
SD
M
SD
M
SD
Mathematics
Reading
19.5
23.5
12.2
19.7
63.4
62.5
25.6
28.2
95.5
95.8
5.0
2.0
N
15
12
14
Note. Tabled values refer to mean national percentile ranking for the
associated section of the Science Research Associates survey of basic
skills.
trieval. The counting fingers and verbal counting trials were further
classified in accordance with the specific algorithm used for problem
solving. That is, the trials were classified as rain (based on counting
only the smaller value integer) or sum (based on counting both integers). Finally, after each trial, subjects were asked to describe how
they got the answer. Several previous studies have demonstrated that
children can accurately describe arithmetic problem-solvingstrategies
if they are asked immediately after the problem is solved (Siegler, 1987,
1989). Comparisons of the child's description and the experimenter's
initial classification indicated agreement between the experimenter
and the subject on 96% of the trials. For those trials on which the
experimenter and the subject disagreed, the strategy was classified on
the basis of the child's description.
Results
For the sake of clarity, the results with brief discussion will be
presented in two major sections, followed by a more general
discussion of the results and their implications. In the first section, analyses of group differences in the distribution and characteristics (e.g., error rate) of strategy choices will be presented.
The second section presents a componential analysis of the RT
data designed to assess potential group differences in the rate of
executing the various arithmetical processes, for example, rate
of verbal counting.
Strategy Choices
Table 2 presents the group-level characteristics of addition
strategies. Inspection of Table 2 reveals that verbal counting
and memory retrieval were the primary strategy choices of children in each of the three groups (Siegler, 1987); the counting
fingers strategy was used occasionally, and the fingers strategy
was never used. 2 Initial univariate analyses of variance (ANOVAs) indicated reliable differences across groups: For the counting fingers, F(2, 38) = 4.58, p < .05; verbal counting, F(2, 38) =
9.77, p < .0005; and retrieval, F(2, 38) = 18.26, p < .0001,
strategies. Bonferroni t tests indicated that the gifted group
used the counting fingers strategy reliably less often than did
the MD group (p < .05); no other group differences were reliable for this strategy (ps > .05). The gifted group also employed
the verbal counting strategy less frequently than did the two
nongifted groups (ps < .05), but no reliable difference was
found between the normal and MD groups for the overall use of
401
the verbal counting strategy (p > .05). Finally, all group differences were reliable for the retrieval strategy (ps < .05).
A second set of univariate F tests indicated no reliable differences in the frequency of errors for the counting fingers, verbal
counting, or retrieval strategies (ps > .25). Additional analyses
(Z tests), however, revealed that the proportion of retrieval
errors for the gifted group was reliably lower than for both the
normal (z = 2.23, p < .05) and MD (z = 1.94, p < .06) groups.
A final univariate ANOVA indicated reliable group differences (across the counting fingers and verbal counting strategies) in the overall frequency with which the sum strategy was
used for problem solving, F(2, 38) = 4.57, p < .05. Here, the sum
strategy was used reliably less often by the gifted (less than 0.5%
of the counting trials) and normal (less than 0.5%) groups than
by the group of MD children (ps < .05), who used the sum
strategy on 12% of the counting fingers trials and 6% of the
verbal counting trials.
In summary, the primary difference in comparing the gifted
group with the two nongifted groups was in terms of the frequency with which memory retrieval was used to solve the presented problems. The frequent use of this strategy, of course,
resulted in a lower proportion of counting trials for the gifted
group relative to the nongifted groups. Moreover, the gifted
group not only used the retrieval strategy more frequently than
did the children comprising the two remaining groups, but they
also had a lower proportion of retrieval errors. Differences
comparing the normal and MD groups were less pronounced:
Both groups relied rather heavily on counting to solve many of
the presented problems, although the normal children were
more likely to use the memory retrieval strategy than were the
MD children. Finally, the MD group distinguished itself from
the two remaining groups in terms of the relatively frequent use
of the sum counting procedure, although this algorithm was
used on a minority of counting trials.
Sum counting trials. From the next set of analyses we sought
to determine what factors might have contributed to the relatively frequent use of the sum counting procedure by the MD
group. Here, we computed correlations among variables that
represented the frequency with which the sum algorithm was
employed by this group (and the associated error rates) and
variables that indexed problem characteristics (e.g., problem
difficulty, which can be indexed by the value of the correct sum
or product; Siegler, 1988b; Washburne & Vogel, 1928). For these
correlations, the individual problem, rather than the individual
subject, served as the unit of analysis. The results indicated that
the sum procedure was more likely to be employed when problems consisted of two large-value integers (which was indexed
by the problem's product; r = .42, p < .01). The frequency of
errors produced by using the sum procedure was significantly
correlated with increasing values of the correct sum (r = .32, p <
.05). These correlations suggest that the sum procedure tended
to be used by the MD children to solve more difficult problems
2One of the gifted children used a decomposition strategy (Siegler,
1987) to solvesix of the presented problems. For example, for the problem 9 + 8, the child stated that he "took the 8, subtracted l, added the l
to the 9, and then added 7 and 107'Due to the small number of decomposition trials, these were excluded from any of the analyses.
DAVID C. GEARY AND SAM C BROWN
402
Table 2
Characteristics of Addition Strategies
Mean % of trials on which
strategy used
Percentage of errors
Mean reaction time
Mean % of trials on which
min strategy used
Strategy
MD
Normal
Gifted
MD
Normal
Gifted
MD
Normal
Gifted
MD
Normal
Gifted
Counting fingers
Verbal counting
Retrieval
15
41
44
10
29
61
2
10
88
4.6
3.7
2.0
4,0
2,6
1,5
3.2
2.0
1.3
5
11
9
6
13
11
6a
Il
2
88
94
--
100
100
--
100
100
--
Note. Mean reaction time (RT) excluded error and spoiled trials. The percentages also exclude those spoiled trials for which a strategy could not be
classified. For the math disabled group, mean RT was based on min trials, to make solution times comparable to the two remaining groups. MD =
math disabled.
a This value represents only a single trial and is therefore not likely representative of the error rate of gifted children in general.
and that the frequency o f counting errors increased as the n u m ber o f required incrementations increased (Geary, 1990; Siegler,
1987; Siegler & Shrager, 1984).
Counting trials. For each o f the three groups, the frequency
o f counting trials (across the counting fingers and verbal counting strategies) was correlated with the problem's correct sum,
which, as noted above, provides a reliable indicator o f the difficulty o f the problem (Siegler & Shrager, 1984; Washburne &
Vogel, 1928). These analyses indicated that the frequency o f
counting trials increased with an increase in the value o f the
correct sum for both the n o r m a l (r = .60, p < .000 l) and the M D
(r = .54, p < .001) groups. The value o f these two correlation
coefficients did not differ reliably (z = 0.14, p > .25). These
results suggest that the normal and M D subjects used counting
to solve more difficult problems. For the gifted group, the frequency o f counting trials was not reliably correlated with the
sum variable (r = - . 16, p > .25), suggesting that the use o f a
counting procedure was not related to problem difficulty for
these subjects. The value o f this coefficient (i.e., - . 16) differed
reliably from the r values for both the n o r m a l group (z = 1.69,
p < .05, one-tailed), and the M D group (z = 1.68, p < .05, onetailed).
Retrieval trials. For each o f the three groups, the frequency
o f retrieval trials was correlated with the value o f the problem's
correct sum. These analyses mirrored the above and indicated
that the frequency o f retrieval trials decreased with an increase
in the value o f the correct sum for both the normal (r = - . 6 4 ,
p < .001 ) and the M D (r = - . 6 0 , p < .001 ) groups. 3 The value o f
these two correlation coefficients did not differ reliably (p >
.25). These results suggest that both the normal and M D children used the retrieval strategy to solve relatively easy problems
but counted to solve more difficult problems. For the gifted
group, the frequency o f retrieval trials was not significantly
correlated with the sum variable (r = - . 0 6 , p > .25), suggesting
that the retrieval strategy tended to be used to solve both difficult and easy problems. The difference between this coefficient
and the r values for the normal and M D groups was marginally
reliable (ps < .07). This finding strengthens the argument that
the gifted children were more accurate than the nongifted children in the use o f the retrieval strategy because they tended to
use this strategy, rather than counting, to solve more difficult
problems. In other words, the error rates o f the normal and M D
children would more likely have been even higher had they
used the retrieval strategy to solve the same range o f problems
as did the gifted children.
Retrieval trials and academic achievement. In terms o f the
strategy choice model, skilled performance is represented by
the error-flee retrieval o f basic information from long-term
m e m o r y (Siegler, 1986). Consistent with this view was the finding that the most reliable group difference in the overall distrib u t i o n o f strategy choices was for the retrieval strategy. As
noted earlier, the gifted g r o u p used this strategy m o r e frequently and showed a significantly lower proportion o f retrieval
errors than did subjects c o m p r i s i n g the n o r m a l and M D
groups. Moreover, the normal group employed the retrieval
strategy more frequently than did the M D group. On both theoretical and empirical grounds it might then be expected that the
overall frequency with which the retrieval strategy was correctly
used might be related to the earlier described group differences
in mathematical achievement levels (Geary & Burlingham-Dubree, 1989; Siegler, 1988a). This hypothesis was supported by
the finding that the frequency o f correct retrieval trials was
strongly correlated with performance on the mathematics section o f the Science Research Associates survey o f basic skills
(r = .73, p < .0001). This result also indicates that performance
on the simple addition task is clearly related to performance on
more general and complex measures o f mathematical skills.
Componential Analysis
T h e c o m p o n e n t i a l analyses were designed to d e t e r m i n e
whether the rate o f executing the various problem solving processes, such as verbal counting, differed across groups. All o f
these analyses were based on correct RT trials but excluded
trials on which the voice-operated relay was triggered by responses other than the answer (e.g., coughing) or on which the
3 The correlations between the sum variable and (a) the frequency of
counting trials and (b) the frequency of retrieval trials are not symmetrical because of missing data. Strategy choice data were missing because
the decomposition strategy was used on the trial or because the voiceoperated relay was triggered (e.g., by a cough) before a strategy could be
executed. In all, this missing data represented less than 5.3% of the
data.
COGNITIVE ADDITION
initial response did not trigger the relay.4 Process models for
addition were fit to average RT data by means of regression
techniques for each of the three groups. Here, RTs were analyzed separately for verbal counting trials (when the min procedure was used) and retrieval trials. Because not all subjects used
the same strategy to solve all problems, the matrix of RTs from
which these averages were computed necessarily contained
missing data. In this circumstance, the resulting regression
equation could be biased in some way. This bias would likely
result in some increase in the variability of the regression estimates and would attenuate the goodness of fit of the overall
model. Nevertheless, this procedure seems preferable to the
alternative of averaging all RTs across different strategies
(Siegler, 1987, 1989).
For each group, the min variable was used as the independent measure in the analysis of verbal counting trial RTs (Geary,
1990; Siegler, 1987). Retrieval trial RTs were correlated with
variables that are often used to represent the long-term memory network of addition facts--that is, the square of the correct
sum (Ashcraft & Battaglia, 1978), the problem's product (Geary,
Widaman, & Little, 1986; Miller, Perlmutter, & Keating, 1984),
and with various indices that may be used to represent the
associative strength in long-term memory between any given
simple addition problem and its correct answer. Specifically,
associative strength was indexed by the probability of correctly
retrieving an answer (these values were obtained from Siegler,
1986), a variable that represented the ranked difficulty of the
problems (Hamann & Ashcraft, 1985; Wheeler, 1939), the percentage of children who mastered each problem on a learning
task (Wheeler, 1939), and variables that reflected the frequency
with which simple addition problems were presented in kindergarten, first-, second-, and third-grade mathematics textbooks
(Hamann & Ashcraft, 1986).
Verbal counting strategy. The min variable showed a modest
to strong zero-order correlation with verbal counting strategy
RTs across the three groups. The resulting regression equations
are presented in the top portion of Table 3. Here, the intercept
Table 3
Statistical Summaries o f Regression Analyses
Equation
R
F
df
MS e
Verbal counting strategy
Math disabled
RT = 1,674 + 519 (min)
Normal
RT = 927 + 420 (min)
Gifted
RT = 1,241 + 206 (min)
.775
57.29
1, 38
729
.822
79.43
1, 38
501
.506
8.94
1, 26
617
.425
8.37
1, 38
279
.498
12.53
1, 38
206
Retrieval strategy
Normal
RT = 1,250 + 542 (AS)
Gifted
RT = 1,119 + 490 (AS)
Note. All models significant at p < .01. Min refers to the cardinal
value of the smaller integer; AS refers to associative strength (1 minus
the probabilityo fretrieving the correct answer from Siegler, 1986, p. 8);
RT = reaction time.
403
terms (e.g., 1,674 for the math-disabled group) theoretically represent a combined estimate for the rate o f encoding digits and
for the rate of the strategy selection and the answer production
processes (e.g., Campbell & Clark, 1988; McCloskey, Caramazza, & Basili, 1985). The regression weight for the min variable provides an estimate of the counting rate per incrementation.
To determine if the counting rate and the rate of executing
the processes subsumed by the intercept term differed across
groups, the statistical procedures described by R i n d s k o p f
(1984; Cohen & Cohen, 1983) were followed. These analyses
revealed that the estimate for the rate of implicit counting (i.e.,
the raw regression weight for the rain variable) was reliably
lower for the gifted group as compared with both the normal,
F(1, 64) = 7.08, p < .01, and MD, F(1, 64) = 9.77, p < .01,
groups. The normal group showed a reliably lower intercept
term than did the M D group, F(I, 76) = 4.42, p < .05. No other
group differences were reliable (ps > .05). In all, these procedures revealed one easily interpretable result: The gifted group
showed a significantly faster verbal counting rate than did either of the two nongifted groups. In fact, the estimated counting rate of 206 ms per incrementation is in the adult range o f
150 to 250 ms per incrementation (Landauer, 1962).
Memory retrieval strategy. For each of the three groups, no
single variable clearly provided the best representation of retrieval trial RTs. For the gifted and normal groups, the probability of correct retrieval (i.e., AS for associative strength; Siegler,
1986) showed the most consistent and readily interpretable pattern of results. To avoid negative raw regression weights, a modified AS variable (i.e., 1 - AS) was used to represent retrieval trial
RTs for these two groups. The resulting regression equations for
the normal and gifted groups are presented in the bottom portion of Table 3.
The use of procedures identical to those described for the
verbal counting strategy indicated that the value of the intercept term and the regression weight for the AS variable did not
differ reliably for the gifted and n o r m a l groups ( p s > .25).
These findings indicate that the duration o f the encoding, strategy selection, and answer production processes for retrieval
trials and the retrieval rate did not differ significantly for the
gifted and normal children.
Next, the value of the intercept terms across the counting and
retrieval strategies (e.g., 1,241 and 1,119, respectively, for the
gifted group) were compared within groups by using the earlier
described procedures (Rindskopf, 1984). These procedures indicated that the value of the intercept term comparing counting
and retrieval trials did not differ reliably for either the gifted,
F(1,64) < 1, ns, or the normal, F(I, 76) = 2.25, p > . 10, groups.
These results indicate that the rate of executing the processes
subsumed by the intercept term did not differ reliably across
the counting and retrieval strategies for either the gifted or normal groups.
4 With the use of the verbal counting strategy, some children audibly
counted. During this count, if the voice-operated relay was triggered
before the child stated the final answer, then the solution time was
"spoiled" and not used in any of the RT analyses. Less than 2% of the
verbal counting trials were spoiled in this manner.
404
DAVID C. GEARY AND SAM C BROWN
No equation is presented for the M D group because average
RTs for correct retrieval trials showed no consistent or interpretable correlation with any of the previously described variables.
In short, the solution times for correct retrieval trials for the
M D group, unlike those for the two remaining groups, were
largely unsystematic. This finding suggests that the AS variable, which adequately represented retrieval times for the gifted
and normal groups, did not adequately reflect the long-term
m e m o r y representation or the retrieval process for the M D
group.
Discussion
The present study provided a detailed comparison o f gifted,
normal, and M D children in terms of the distribution of strategies and solution times used to solve simple addition problems.
The experiment was designed so that group differences could
be interpreted within the context of a theoretical model of cognitive development: the strategy choice model (Siegler, 1986;
Siegler & Shrager, 1984). Indeed, the combination of variations
in the peakedness of the distributions of associations and the
rigor of the confidence criterion can easily accommodate group
differences in strategy choices, the proportion o f retrieval
errors, and the relation between problem difficulty and the use
of a counting procedure. In theory, peaked distributions of associations, where the associative strength is concentrated on a
single answer, should produce a high proportion of correct retrieval trials, independent of the rigor o f the confidence criterion. On the other hand, for less peaked distributions o f associations, the stringency o f the confidence criterion can have a
substantial impact on strategy choices. For these problems, a
rigorous confidence criterion should produce many counting
trials. This is so because the associative strength o f the retrieved
answers would not often exceed the value of the confidence
criterion, and thus would make necessary the use o f a backup
counting strategy. Less peaked distributions o f associations
combined with a less rigorous confidence criterion, however,
should result in relatively few counting trials and a relatively
high proportion of retrieval errors (Siegler, 1988a).
In the present study, the observation of a high proportion of
retrieval trials and a low proportion o f retrieval errors by the
gifted group could be parsimoniously explained by very peaked
distributions o f associations. For the gifted group, the retrieval
trial and retrieval error data, in themselves, do not allow for
strong inferences to be made about the rigor of the confidence
criterion because very peaked distributions o f associations
could account for both the high proportion of retrieval trials
and the low proportion o f retrieval errors. The low proportion
o f retrieval trials and the relatively high proportion of retrieval
errors for the M D group, however, suggest both relatively flat
distributions of associations and a lenient confidence criterion.
The confidence criterion of the normal children also appeared
to be relatively lenient, but the distributions of associations
were likely more peaked relative to the M D group, but less
peaked compared with the gifted group.
The distributions of associations and confidence criterion
parameters can also be used to represent the relationship between problem difficulty and the use of a backup counting
strategy. Because problems with two large-value integers (e.g.,
7 + 9) tend to be presented to children less frequently than
problems with smaller value integers ( H a m a n n & Ashcraft,
1986; Siegler & Shrager, 1984), it is likely that any problem with
such large value integers will have a relatively flat distribution
of associations. In short, these problems are difficult because
the associative strength o f any retrieved answer will not likely
exceed the value o f the confidence criterion. Thus, subjects resort to the use of a backup counting strategy to solve more
difficult problems. The finding that normal and M D children
tended to count to solve more difficult problems is consistent
with the argument that the underlying distributions of associations were less peaked for these children relative to gifted children. lndeed, for the gifted group, there was no relation between the use of a counting strategy and problem difficulty.
This pattern of results could be produced if the gifted children
retrieved correct answers for problems on which they subsequently counted but did not state the retrieved answers because
of a rather stringent confidence criterion (Siegler, 1988a). If so,
then the children in the gifted group may have had a more
rigorous confidence criterion than the children in the two nongifted groups.
In all, the gifted group appeared to have the most peaked
distributions o fassociations followed in turn by the normal and
M D groups. The gifted group, relative to the nongifted groups,
also appeared to have a more stringent confidence criterion; the
rigor of the confidence criterion appeared to be about the same
for the normal and M D groups. The combination of these two
parameters should have produced a very developmentally mature strategy mix for the gifted group, followed in turn by the
normal and M D groups. Indeed, for the gifted group, the proportion o f retrieval trials as well as the low error rate approached adult levels (Svenson, 1985). On the other hand, the
M D children, presumably because o f relatively flat distributions o f associations, appeared to have difficulty retrieving
correct addition answers and therefore had to rely on a counting
strategy to solve the majority o f the presented problems. Thus,
the M D group showed a developmentally immature distribution of strategy choices (Geary, 1990; Geary et al., 1987; Goldman et al., 1988; Groen & Resnick, 1977). The maturity of the
strategy mix for the normal group fell in between that of the
gifted and M D children.
Overall, the developmental maturity of the strategy mix appears to be directly related to the distributions of associations
parameter. That is, as the child masters basic domain-specific
facts, the distributions of associations parameters appears to
become increasingly important, and the confidence criterion
less important, in determining the developmental maturity of
the strategy mix. If so, then a fundamental parameter spanning
all achievement groups might be the development of the longterm memory organization of basic addition facts (Ashcraft,
1982; Brown, 1975). In this view, one feature underlying the
superior performance of the gifted group on the experimental
task and on the achievement measures might have been a nearly
adult-like long-term memory organization of basic facts, although the rate o f fact retrieval was relatively slow compared
with that o f adults (Miller et al., 1984). The performance of the
M D group might also be explained in terms of the development
of the long-term memory network of basic facts, but for these
children the network of facts might be immaturely or even ab-
COGNITIVE ADDITION
normally organized. This view is consistent with the finding
that the M D group showed a relatively low proportion of retrieval trials and an unsystematic pattern of solution times for
correct retrieval trials. These results are also similar in some
respects to the pattern of deficits found for younger M D children who failed to improve in mathematics skills despite remedial education (Geary, 1990). In all, this pattern of results suggests that the strength of the association between a problem and
its correct answer increases in value more slowly in M D as
compared with normal and gifted children.
Regardless of the aforementioned information, group differences were also evident in the rate of information processing.
The gifted group showed a verbal counting rate that was within
the adult range and less than 50% of the counting rate estimated
for the two nongifted groups; however, the gifted and normal
groups did not differ in the rate of retrieving facts from longterm memory, The finding o f group differences for counting
rate but not for retrieval rate indicates that speed of processing
in itself might not be a characteristic that distinguishes gifted
from nongifted children, contrary to current views of general
intelligence (Jensen, 1982; but see Keating & MacLean, 1987).
Rather, the results suggest "that asymptotic levels of mastery of
these processes exist, and that the main difference between
children of different IQs is in the rate at which they reach these
asymptotic levels" (Siegler & Kotovsky, 1986, p. 422). In this
view, the gifted group included children who, with practice,
rather quickly achieve, relative to nongifted children, asymptotic levels for rate of executing basic operations. This dimension,
however, does not appear to be a primary factor underlying a
learning disability in mathematics. In the present study and in a
related study o f a mathematical disability in younger children
(Geary, 1990), M D children did not differ substantively from
academically normal peers in rate o f processing.
Finally, the results of the present study suggest that the developmental maturity o f the strategy mix appears to be a primary
determinant of individual differences among children in performance on academic achievement measures, although other
studies have found that rate of executing basic operations (e.g.,
rate of fact retrieval) also appears to contribute to individual
differences on traditional ability measures (Geary & Burlingham-Dubree, 1989; Keating & Bobbitt, 1978). Moreover,
rate of processing by adults, as opposed to strategy differences,
appears to be a primary determinant of individual differences
in domain-specific skills (Geary & Widaman, 1987; Sternberg
& Gardner, 1983). Thus, the relative contribution of speed of
processing and strategic variables to performance on domainspecific mental ability and achievement tests might vary across
development and skill levels. That is, early in skill development,
strategic parameters might be the primary determinant of ability differences, whereas with mastery of the skill and therefore a
lessened need for strategy choices, speed of processing might be
the primary determinant of performance differences on traditional ability measures (Geary & Burlingham-Dubree, 1989).
In summary, a single cognitive dimension appeared to span
the gifted, normal, and M D groups: the maturity of the longterm memory organization of basic addition facts. Nevertheless, the gifted group's advantage in the speed of executing the
backup counting strategy was also more likely to contribute to
their strong performance on the mathematics achievement
405
measure. The long-term memory dimension, however, appeared to be the primary factor underlying the observed group
differences on the mathematics achievement test, the developmental maturity of the strategy mix, and the asymmetric solution times for correct retrieval trials for the M D group. Moreover, this long-term memory organization dimension might
also be related to a learning disability in mathematics in firstand second-grade children (Geary, 1990). A second dimension,
related to speed of processing, appeared to distinguish the
gifted from the two nongifted groups. However, this dimension
did not seem to be directly related to a learning disability in
mathematics. More precisely, the rate of executing basic elementary operations appeared to reach asymptotic levels more
quickly in gifted, relative to nongified, subjects, whether they
are academically normal or math disabled children. Nevertheless, this study did not provide an exhaustive comparison of
different achievement groups, because gifted and M D children
might also differ from academically normal peers on other cognitive dimensions, such as metamemory (Kurtz & Weinert,
1989) or insight (Davidson, 1986). Further studies are needed to
determine how various cognitive dimensions (e.g., speed of processing or metamemory) might differentially contribute to differences between achievement groups on tasks of varying complexity and in domains other than addition.
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Received May 14, 1990
Accepted September 14, 1990 •