The Psychological Record , 1998, 48, 325-332
INDIVIDUAL DIFFERENCES IN COGNITIVE ADDITION
MOHAMED BERNOUSSI
Universite de Nantes, France
This study addresses the problem of individual differences in
solving simple addition problems by comparing the result of group
to individual data. We found that if the group data are better
explained by the currently dominant model (retrieval of arithmetic
facts from memory), individual data are explained by different
models at the level of the process used.
The study of cognitive processes involved in solving simple arithmetic
problems has aroused much interest in the past twenty years. Several
models have attempted to account for these processes (for review see
Ashcraft, 1995; Lemaire & Bernoussi, 1991; McCloskey, Harley, & Sokol,
1991). Most research in this area uses a chronometric method. The
measurement of latencies (RTs) can be applied to the formulation of a
judgment in a verification task: The subject is asked to verify the accuracy
of an already solved problem (e.g., 3 + 4 = 7 True or False ?) or to
produce the response in a production task: The subject is asked to solve
arithmetic problems and produce the result (e.g., 3 + 4 = ?).
The first study in this area was conducted by Groen and Parkman
(1972) with children. These authors assumed that the process involved
in solving simple addition problems is based on an internal mental
counter. Using this counter, an addition problem (like 3 + 4) can be
solved in the following way.
First, the counter is set to an initial value corresponding to one of the
two digits of the problem. Subsequently, the second digit is added to the
counter by increments of one. Both the time to set the counter and the
time to increase it by one are assumed to be constant. So, the RT will
vary only as a function of the number of increments.
The results obtained by Groen and Parkman indicated that RT was
I am grateful to Dr. Angela Notari-Syverson (Washington Research Institute) for her
helpful editing work. I thank the reviewers for their helpful comments on an earlier version
of this article.
Reprint requests may be sent to Mohamed Bernoussi, Laboratoire de Psychologie,
Universite de Nantes. B.P. 81227, Chemin de la Censive du Tertre. 44312 Nantes Cedex3.
France. E-mail: [email protected]
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BERNOUSSI
best predicted by the value of the smaller digit of the problem named the
minimum (min). This suggests that children assign the biggest value to
. the counter and add the smallest value to it.
For the adults, however, two variables provided a best fit of RT: the
min variable, but also the sum of the two digits. The authors proposed a
mixed-model interpretation: Results of simple arithmetic problems are
stored in long-term memory, and adults use a direct-access process to
retrieve these facts. However, if the direct-access fails, adults may
sometimes use a counting process like children.
Ashcraft and Battaglia (1978) empirically tested the model proposed by
Groen and Parkman (1972) using a simple addition verification task. The
results of this study indicated that RTs of adults are best predicted by the
square of the correct sum (sum 2). This finding was inconsistent with the
Groen and Parkman counting and direct-access model described above.
Ashcraft and Battaglia (1978) assumed then that the used process is the
retrieval of the sum from a memory network. However, Miller, Perlmutter,
and Keating (1984) showed that the structural variable that predicted RTs of
adults was the product of the two addends. These authors hypothesized
that arithmetical facts are stored in long-term memory in the form of a
tabular network (an addition table). On each side of this network are the
entry nodes for the integers 0 through 9. The correct sum is stored at the
intersection of the entry nodes values corresponding to the two addends.
The structural variable "product" represents the surface of the network to
browse for retrieving a result. So, to retrieve the sum of 2 + 3, it is
necessary to activate the associate node of the first addend (2), then the
one partner in the second (3), and browse the surface (2 x 3 nodes) to
access to the intersection.
Since the study by Miller et al. (1984), several authors share this
conception of memory network (see Geary, Widaman, & Little, 1986;
Timmers & Claeys, 1990; Widaman, Geary, Cormier, & Little, 1989).
These studies, however, have been criticized from both conceptual
and methological standpoints. The conceptual criticisms stem from
authors who developed alternative models of memory network (see
Campbell, 1987, 1995). The methodological criticisms concern primarily
two aspects:
(1) The choice of structural variables. The structural variables are in
fact the mathematical formulation of cognitive processes (Geary, 1987)
and used to differentiate models proposed. These variables, however,
are strongly correlated with each other, and it is difficult to evaluate
statistically the unique contribution of each individual variable.
(2) The level of the analyses. Most studies used as a dependent
variable the mean RT of the group. But the utilization of the mean masks
the different strategies used by subjects (see Siegler, 1987); moreover, it
reduces the observed variance, which permits a best fit of data (see
Pellegrino & Goldman, 1989).
The present study addressed primarily this second aspect,
specifically, the problem of individual differences in these tasks.
INDIVIDUAL DIFFERENCES IN COGNITIVE ADDITION
327
Although, individual differences in solving simple arithmetic problems in
children was extensively examined (see Ashcraft, 1990; Siegler, 1988),
few studies have investigated this topic in adults (Geary & Widaman,
1987; Widaman et aI., 1989). In these studies, the authors presented a
number of psychometric and chronometric tests to subjects and then
performed confirmatory Lisrel factorial analysis (Joreskog & Sorbom,
1993). This analysis presents the advantage of representing at the same
time latent variables and the relationships between them. However, the
problem of individual differences may be approached in another way, by
comparing group data to individual data (Svenson, 1985). The latter
approach is adopted here.
Our assumption is that adults, like children, use different procedures to
solve simple addition problems (see Baroody, 1994; LeFevre, Sadesky, &
Bisanz, 1996). Therefore, individual RTs will be fit by different structural
variables, even if the group RTs are fit by only one variable. This approach
confirms results obtained by other authors (Baroody, 1994; Geary & Wiley,
1991; LeFevre et aI., 1996) by using a different method. In this sense, our
assumption is similar to theirs. To test this hypothesis, we will use a
comparison between individual and group data.
Method
Subjects
Fifty first year students in psychology participated to this experiment.
Materials
The problem set was composed of the 100 possible combinations
of addends 0-9 and augends 0-9. These combinations were presented
both with the correct sum (e.g., 3 + 4 = 7), and an incorrect sum. Fifty
problems had an incorrect sum that differed from the correct one by
±1 (small difference), and 50 differed from the correct sum by ±5
(large difference). In all, 200 additions were presented. The frequency
and placement of all integers and problems were counterbalanced,
and no addend, augend, or sum was presented on consecutive trials.
Procedures
A verification task was presented using an Atari 1040 ST
computer equipped with a monochrome monitor. Latencies (RTs)
were measured by an internal clock of the computer with a precision
of ±1 ms. Subjects were tested individually and were told that their
task was to respond "True" or "False" to the problem presented by
pressing the appropriate key. A practice set of 20 problems was
presented at the beginning of the experiment. A short break was
allowed after 50 problems. The total time to complete this experiment
was approximatively 15 minutes.
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Results
The principal objective of this study was to examine the individual
differences in the area of cognitive addition by comparing group data
(general analyses) to individual data (differential analyses); we will
present results with respect to this.
Accuracy of response. The percentage of error in this experiment
was only 2%. This rate is similar for those reported by other studies in
this area (e.g., LeFevre et aL, 1996; Miller et aL, 1984). Only RTs of
correct responses were taken into account.
General analysis. In this analysis we looked for the best structural
variable that provides the best fit of RT in the addition chronometric test.
For this, we performed a set of regression analyses using as dependent
variable the group mean RT for each problem presented and as
independent variable the structural variables described above (min, sum,
sum 2, and product). These analyses were carried out with SAS (Statistical
Analysis System) software using the R2 procedure. This procedure makes
it possible to compare simple regressions with independent variables and
to select the variable which offers the best fit of RTs.
The results indicated that the best predictive variable of RT means
was the product structural variable (see Table 1). However, even if the
product accounts for more variance, there are no significant differences
between R2s of the different structural variables. This result is similar to
those reported by other authors (see for example Geary et aL, 1986;
Miller et aL, 1984).
Table 1
Prediction of Latencies - Group Data
Slope
Structural Variable
Min
Sum
Sum 2
Product
.33
.31
.42
.50
42.5
25.6
3.2
.50
For the product variable, the regression equation is:
RT (ms) = 988.42 + 9.28 (product). R2 = .50.
F(1, 51) = 52.68, P < .0001.
This finding confirmed those obtained in the cognitive arithmetic
area (Geary et aL, 1986; Miller et aL, 1984). Moreover, a tie effect was
observed in this study. It is well known that ties (a + a) are resolved
faster than other problems. The tie problems were resolved 250 ms
faster than the other problems in this study. When we suppressed the RT
from ties of analyses, we noted an important increase of the variance
explained by the product variable. Indeed, the percentage of explained
variance rose from 50% to 78%:
INDIVIDUAL DIFFERENCES IN COGNITIVE ADDITION
329
RT (ms) = 960.98 + 13.13 (product). R2 = .78.
F(1, 33) = 116.61, P < .0001.
In conclusion, the results obtained in this experiment confirmed the
results of other studies mentioned above. It seems indeed, that the process
used by subjects in our study was a retrieval process insofar as the best
predictive variable was the product of the two addends. It remains to be
seen whether or not the differential analysis confirms this result.
Differential analysis. To conduct this analysis, we used as dependent
variable each individual's RT, and we looked for the best predictive
variable using the regression analysis as for the group data. For these
analysis, we select a model if there is only one significant predictive
variable (a significant Fwith an alpha level at least equal to .05). This set
of analyses yielded the following results (Table 2).
Table 2
Structural Prediction of Latencies - Individual Data
-------
Model
No fit
Multiple regression
Sum
Min
Sum 2
Product
Number of Subjects
5
2
15
11
6
11
Median R2
Range of R2
------------.21
.24
.22
.28
.08-.43
.07-.40
.09-.33
.09-.46
Five subjects were not adjusted to any model. No variable made a
significant contribution to the explanation of the variance.
For two subjects, only one multiple equation of regression predicted
their RT, insofar as the structural variables were considered as exclusive
and associated to a different type of particular process. In fact, these
subjects seemed to have used some strategies, or combinations of
strategies not described by the classical structural variables.
The RTs of 15 subjects adjusted to the model described by the
structural variable "sum." This variable corresponds to a direct-access/
counting process.
The RTs of 11 subjects adjusted perfectly to the min model. This
answering pattern corresponds, theoretically, to the utilization of a
counting strategy. However, we did not assume that they really used this
strategy, but rather a procedure similar to counting.
The RTs of 11 subjects adjusted to the structural variable "product,"
and those of six others to the "sum 2" variable. These two structural
variables are associated to a retrieval model.
These results showed that with exception of the subjects who did
not adjust to any model, the group was constituted of three
subgroups, each identified by the structural variable which provided
the best fit of RTs.
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To study the difference between these groups, we used a one-way
ANOVA on mean groups RTs. Results showed a significant difference
between groups, F(3, 39) = 2.85, P < .05.
1Z00 .---------------------------------------R 1000
T 800
600
m 400
s
)
ZOO
o
Sum
Sum.Z
Min
Prod
Figure 1. Mean RT for each group.
As presented in Figure 1, the sum 2 group seems to be the faster one.
Partial comparison with Fischer's PLSD revealed only one significant
difference between sum 2 and min groups on the one hand, and the product
group on the other hand. All the other differences were not significant.
Discussion
The purpose of this study was to address individual differences in
cognitive addition by comparing group data to individual data. In the area
of cognitive arithmetic, one important factor is the size effect of the
problem (Ashcraft, 1995; Geary, 1996): RTs are slower for larger
problems (8+9) than for smaller ones (2+3). The most important studies
in this domain demonstrated that adults' RTs are best fit by the product
of the two addends. They suggest that adults use a retrieval process
(Ashcraft, 1982; Geary et aL, 1986; Miller et aL, 1984).
The results obtained in this study indicated that when we use the
group data (averaging data), the best structural variable is the product.
This result is consistent with other findings in this area (e.g., Miller et aL,
1984; Widaman et aL, 1989). But when we consider individual data, we
find a large number of individual differences. Subjects seem to use
different strategies, and RTs are fitted by different structural variables.
Our findings are also consistent with other studies. Geary and Wiley
(1991) using a self-report procedure with students who solved simple
addition problems noted that their subjects used nonretrieval procedures
in 12% of trials. Recently, Geary (1996) found that American students
used a combination of counting and decomposition to solve 27% of the
problems presented. All these procedures are non retrieval ones.
LeFevre et aL (1996) reported that adults use a variety of nonretrieval
procedures to solve simple addition problems.
INDIVIDUAL DIFFERENCES IN COGNITIVE ADDITION
331
How can we explain these individual differences? According to
Geary (1994) , individual differences in cognitive arithmetic can be
approached from three perspectives: psychometric, cognitive , and
behavioral genetic. In this experiment, we can make the assumption that
at least two categories of variables can explain these differences :
psychometric factors (variation in mathematical abil ity or in IQ) and
cognitive factors (speed of processing). To test these assumptions, other
experiments are necessary.
Despite differences in methods, our results together with those of
these studies confirm that individual differences in cognitive addition are
important to explain adult performance as well as that of children
(Siegler, 1987; Siegler & Jenkins, 1989).
Taking into account these individual differences permits not only the
validation of general cognitive models (Underwood , 1975) but the
construction of models relying on both the individual and the universal
and achieves complementarity between differential cognitive psychology
and general psychology.
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