Copyright 1996 by the American Psychological Association, Inc.
0278-7393/96/53.00
Journal of Experimental Psychology:
Learning, Memory, and Cognition
1996, Vol. 22, No. 1,216-230
Selection of Procedures in Mental Addition:
Reassessing the Problem Size Effect in Adults
Gregory S. Sadesky and Jeffrey Bisanz
Jo-Anne LeFevre
University of Alberta
Carleton University
Adults' solution times to simple addition problems typically increase with the sum of the problems
(the problem size effect). Models of the solution process are based on the assumption that adults
always directly retrieve answers to problems from an associative network. Accordingly, attempts to
explain the problem size effect have focused either on structural explanations that relate latencies
to numerical indices (e.g., the area of a tabular representation) or on explanations that are based
on frequency of presentation or amount of practice. In this study, the authors have shown that the
problem size effect in simple addition is mainly due to participants' selection of nonretrieval
procedures on larger problems (i.e., problems with sums greater than 10). The implications of these
results for extant models of addition performance are discussed.
Twenty years of research on mental arithmetic has shown
that problems involving larger numbers (e.g., 9 + 6) are solved
more slowly than problems involving smaller numbers (e.g.,
3 + 4). Surprisingly, in spite of the wealth of empirical data
and the extensive theoretical development on mental arithmetic, the problem size effect has eluded satisfactory explanation (Ashcraft, 1992; McCloskey, Harley, & Sokol, 1991;
Widaman & Little, 1992). The goal of the present research was
to test an explanation of the problem size effect in adults that
has been used to account for the arithmetic performance of
children (Ashcraft, 1992; Siegler, 1987). We hypothesized that
variability in the selection of procedures to solve simple addition
problems has a major impact on solution latencies and may
account for a substantial portion of the problem size effect.
Retrieval in Mental Arithmetic
Most current models of arithmetic solution share the assumption that adults solve simple arithmetic problems by retrieving
an answer from a network of stored associations (Ashcraft,
1992; Campbell & Oliphant, 1992; Widaman, Geary, Cormier, &
Little, 1989; Widaman & Little, 1992; reviewed by McCloskey
Jo-Anne LeFevre, Department of Psychology, Carleton University,
Ottawa, Ontario, Canada; Gregory S. Sadesky and Jeffrey Bisanz,
Department of Psychology, University of Alberta, Edmonton, Alberta,
Canada.
The research reported in this article was supported by grants from
the Natural Sciences and Engineering Research Council of Canada.
Jo-Anne LeFevre also gratefully acknowledges the support provided
by the Medical Research Council Applied Psychology Unit in Cambridge, England during the preparation of this article.
Mark Ashcraft, David Geary, Chris Herdman, Michael McCloskey,
and Monique S6n6chal provided helpful comments on a version of this
article. We are particularly grateful to Alison Kulak for providing
insights that led to the initiation of this research.
Correspondence concerning this article should be addressed to
Jo-Anne LeFevre, Department of Psychology, Carleton University,
Ottawa, Ontario, Canada K1S 5B6. Electronic mail may be sent via
Internet to jlefevre@ccs.carleton.ca.
216
et al., 1991). In accord with this view, the problem size effect
has been attributed to various aspects of the representation of
stored arithmetic knowledge. For example, Widaman and his
colleagues (Widaman et al., 1989; Widaman & Little, 1992;
Widaman, Little, Geary, & Cormier, 1992) have proposed a
structural account in which a fact (e.g., 3 + 5) activates the
area of a stored table denned by the product of the addends
(e.g., 15). This model is based on the finding that, in regression
analyses of solution latencies, the product is a good predictor
of latencies to addition problems (Miller, Perlmutter, &
Keating, 1984; Widaman et al., 1989). Similarly, Campbell and
Oliphant proposed that larger numbers are represented less
distinctively than smaller numbers so that retrieving the
answer for a large problem is more difficult than for a small
problem.
These structural accounts of the problem size effect can be
contrasted with associative accounts in which the form of the
representation is primarily based on the acquisition history of
arithmetic knowledge (Ashcraft, 1987, 1992; Campbell &
Graham, 1985). For example, Ashcraft (1987, 1992) proposed
that the strength of the connection between a problem and its
sum is stronger for small problems than for large problems
because small problems are presented and practiced more
frequently (Hamann & Ashcraft, 1986). Campbell and Graham suggested that, for multiplication facts, both frequency of
exposure and order of acquisition affect the associative strength
between problems and answers. In these models, retrieval time
is assumed to be proportional to the strength of the connection, and thus small problems are retrieved more quickly than
large problems.
An associative model of arithmetic knowledge has also been
proposed by Siegler and his colleagues (Siegler & Jenkins,
1989; Siegler & Shipley, 1995; Siegler & Shrager, 1984) to
account for the performance of children on simple addition
problems. In contrast to the models described above, Siegler's
model includes an explicit and important role for nonretrieval
procedures such as counting: Siegler and others have convincingly demonstrated that retrieval is only one of a number of
possible procedures used by young children to solve arithmetic
problems (Geary & Brown, 1991; Geary & Burlingham-
PROBLEM SIZE AND PROCEDURE SELECTION
Dubree, 1989; Siegler, 1987, 1988b, 1989; Siegler & Jenkins,
1989; Siegler & Shrager, 1984).
According to Siegler's distribution-of-associations model (Siegler, 1988b; Siegler & Jenkins, 1989; Siegler & Shrager, 1984),
problems are associated with both correct and incorrect
answers. As children practice simple arithmetic, their representations become more "peaked" in that the association between
a problem and the correct answer becomes stronger than the
associations between that problem and incorrect answers. In
this model, use of nonretrieval procedures (e.g., counting) is
one factor that contributes to the problem size effect. Essentially, nonretrieval procedures operate less accurately on large
than on small problems, and thus the distributions of associations for large problems are less peaked than those for small
problems. Siegler's model has not explicitly been applied to
adult performance: Researchers have assumed that adults
retrieve virtually 100% of the time.
The assumption that adults always retrieve on simple arithmetic problems has rarely been questioned, even by Siegler,
who stated that "adults do not typically need to use any
approach other than retrieval . . . to solve simple arithmetic
problems" (Siegler, 1989, p. 505). This assumption was consistent with interpretations of latency data for children and adults
(e.g., Groen & Parkman, 1972). For children younger than 10
years of age, response times for simple addition problems (e.g.,
3 + 4) increase linearly with the size of the augend or the
addend, whichever is smaller (the min). These data suggested
that children count on from the larger addend to obtain a sum
(the min model; Ashcraft & Fierman, 1982; Groen & Parkman, 1972; Hamann & Ashcraft, 1985). Adults also show a
significant increase in response latencies as a function of
problem size, but this increase is much smaller (e.g., 20 ms per
unit of problem size) than the effect found in children (e.g.,
400 ms per unit of problem size). Moreover, response times for
young children form a linear function of problem size, as
indexed by min; whereas response times for adults form
curvilinear functions that are fitted best by the square of the
sum or by the products of the augends and addends. These
differences in the problem size effect between adults and
young children are generally interpreted as evidence that,
unlike children, adults use fast and efficient retrieval from
memory to solve simple addition problems.
Other data support the view that adults rely on retrieval to
solve simple arithmetic problems. For example, adults and
older children show evidence of associative facilitation and
interference that is consistent with the assumption that they
are retrieving solutions from a network of stored facts (Campbell, 1987b, 1991). For example, cross-operation errors are
common among both adults and practiced children (e.g.,
3 x 4 = 7; Miller & Paredes, 1990; Miller et al., 1984),
suggesting that the two operations are stored in closely linked
networks (Geary, Widaman, & Little, 1986). There is also
evidence that arithmetic facts are activated without intention
in nonarithmetic tasks (Ashcraft & Stazyk, 1981; LeFevre,
Bisanz, & Mrkonjic, 1988; LeFevre & Kulak, 1994; LeFevre,
Kulak, & Bisanz, 1991) and that multiplication facts are
subject to facilitation through priming (Campbell, 1987b, 1991;
Koshmider & Ashcraft, 1991). Thesefindingssupport the view
217
that retrieval is the dominant procedure for adults and older
children.
Retrieval-based accounts of the problem size effect, however, do not account for all of the available data. For example,
structural models of arithmetic knowledge, such as the model
proposed by Widaman and colleagues (Widaman et al., 1989;
Widaman & Little, 1992), do not account for the effects of ties
(e.g., 3 + 3; McCloskey et al., 1991). Ties are consistently
solved more quickly than nontie problems and show substantially smaller effects of problem size (Miller et al., 1984). To
account for the effect of ties, Campbell and Oliphant (1992)
proposed that information about ties is represented separately
from information about nonties. In contrast, associative models can account for the effect of ties without postulating
separate representations: It is assumed that ties are practiced
more frequently than nonties (Siegler & Shrager, 1984; cf.
Baroody, 1994).
Associative models of problem size have also been criticized,
however. The problem size effect in these models has been
related to the amount of difficulty that children have learning
individual problems (Wheeler, 1939), the frequency with
which problems are presented in textbooks (Hamann &
Ashcraft, 1986), and the order of acquisition of specific facts
(Graham & Campbell, 1992). There is no clear empirical
support, however, for the assumption that differences in
acquisition experience could be preserved in the years between
initial learning and adulthood (McCloskey et al., 1991; Widaman & Little, 1992). Further, in a recent compilation of the
frequency of addition facts across Grades 1-6, cumulative
frequency of occurrence was not strongly related to structural
indices of problem size such as product (Ashcraft & Christy,
1995). Thus, neither structural nor associative models provide
satisfactory accounts of the problem size effect (Ashcraft,
1992; McCloskey et al., 1991).
An Alternative Approach
We propose that adults, like children, use both retrieval and
nonretrieval procedures to solve simple addition problems
(see also Baroody, 1994). Siegler has consistently found that
children as young as 4 years of age use a variety of procedures
and that this variability obtains at least through Grade 2
(Siegler, 1987) with addition problems and Grade 4 with
subtraction problems (Siegler, 1989). Siegler (1987,1989) used
the variability in children's selection of procedures to show
that averaging response latencies across procedures can result
in misleading conclusions about how children solve problems.
For example, averaging across trials in which different procedures were used leads to the conclusion that young children
use the min procedure (count up from larger addend) on most
simple addition trials (e.g., Ashcraft & Fierman, 1982). However, Siegler (1987) found that kindergarten, Grade 1, and
Grade 2 children reported only using the min procedure on
36% of the trials. On 35% of the trials they used retrieval, and
other procedures accounted for the remaining 29% of the
trials. In spite of this variability in procedure selection across
trials, the min model accounted for 74% of the variance on
averaged trials in Siegler's (1987) study. Because latencies
were longer and more variable on min trials than on other
218
L E F E V R E , SADESKY, AND BISANZ
trials, averaging across trials resulted in min being the best
predictor (see Siegler, 1987, for details). Thesefindingsmake
it clear that regression analyses in which latencies are averaged
across trials that involve different procedures can produce
misleading conclusions about performance.
The problems inherent in averaging across trials in which
different solution procedures are used might also apply to
adult performance if adults, like children, use a variety of
solution procedures. If adults do not retrieve on all trials, and
if the problem size effect reflects variability in selection of
procedures, then models with a strictly representational basis
for the problem size effect may need to be modified. Thus, the
assumption that adults use only retrieval to solve simple
addition deserves thorough evaluation.
Interestingly, Groen and Parkman (1972) did suggest that
the problem size effect they observed in adults could reflect the
combination of a few counting trials with a majority of retrieval
trials. Ashcraft and Stazyk (1981) tested Groen and Parkman's
claim that adults count on 5% of the trials by removing the
slowest 5% and then recalculating the problem size effect.
They predicted that the problem size effect would disappear if
counting was occurring on that 5%. They found, however, that
the problem size effect was still significant. Ashcraft and
Stazyk had to remove the slowest 50% of the trials before the
latencies were no longer predicted by problem size. Ashcraft
and Stazyk concluded that it was unlikely that adults would
count on such a high proportion of trials, but they did not
consider the possibility that adults might use other nonretrieval procedures to solve simple addition problems.
A few researchers have used the more direct approach of
asking participants to report their solution procedures in
combination with chronometric data to examine simple arithmetic performance. The results of these studies challenged the
strong assumption that adults always use retrieval on simple
addition problems. For example, Svenson (1985) asked participants to indicate how sure they were that they used retrieval on
simple addition problems. Svenson found that participants
were sure they had used retrieval on only 78% of trials.
Svenson did not ask participants to describe their procedures,
however, and the performance of individual participants was
not analyzed.
Geary and his colleagues (Geary, Frensch, & Wiley, 1993;
Geary & Wiley, 1991) have also found that some adults do not
always use retrieval to solve simple arithmetic problems. Geary
and Wiley found that university students (i.e., younger than 30
years of age) reported using retrieval procedures on 88% of
addition problems; older adults (older than 60 years) rarely
used procedures other than retrieval. Similarly, Geary et al.
found that younger adults used retrieval on only 71% of
subtraction trials. Thus, adults may not always use retrieval to
solve simple arithmetic problems.
Self-reports of procedure use have been criticized (see
Russo, Johnson, & Stephens, 1989) on the basis that such
reports may be inaccurate and may change the pattern of data.
Although Siegler has demonstrated the usefulness of selfreports with children (see Siegler, 1987, 1989; cf. Cooney &
Ladd, 1992), it is possible that adults' performance is less
amenable to accurate introspection because adults' arithmetic
performance is much faster than that of children. Self-reports
have successfully been used in a similar domain, however, by
Logan and his colleagues (Compton & Logan, 1991; Logan &
Klapp, 1991). Compton and Logan used self-reports of procedure use to clarify patterns of latency data in an alphabet
arithmetic task. In alphabet arithmetic, participants are given
problems such as A + 3 = D and are required to verify them
either by counting along the alphabet string or by memory
retrieval (once they have learned specific combinations).
Self-reports of procedure use (i.e., counting vs. memory
retrieval) conformed well to patterns of latency data, even
after substantial practice (Logan & Klapp, 1991). The results
supported the validity of using self-report data to study
memory processes (Compton & Logan, 1991).
In the present research, we examined adults' use of retrieval
and nonretrieval procedures on simple addition problems by
collecting trial-by-trial reports of procedure use (Siegler, 1987,
1988b, 1989; Siegler & Shrager, 1984). In contrast to previous
research in which similar methods were used with adults
(Geary & Wiley, 1991; Svenson, 1985), we addressed the issue
of how selection of procedures varies across participants as
well as across problems. Furthermore, few researchers have
addressed the issues of validity and reactivity in adults'
solution descriptions (cf. Cooney & Ladd, 1992; Russo et al.,
1989). We compared our results to those of previous researchers to illuminate these issues.
We hypothesized that adults use a variety of procedures to
solve simple addition problems. University students were
presented with simple addition problems on a computer screen
and were asked to generate the answers. Response times were
recorded, and the experimenter prompted participants to
describe how they solved each problem. We expected to find
both individual differences and problem differences in the
frequency of use of various procedures. Participants also
completed a paper-and-pencil arithmetic test that was designed to provide an independent measure of fluency in solving
simple arithmetic problems.
Method
Participants
Sixteen undergraduate students (8 male and 8 female) participated
in this experiment to partially fulfill a course requirement. The
students' median age was 19:10 (years:months), with a range of 18:3 to
37:11.
Materials
Simple addition problems. The problem set was composed of the
100 combinations of addends 0-9 and augends 0-9. Each participant
saw one of four pseudorandom orders of these 100 problems. Problems were ordered with the constraints that no addend, augend, or
sum were repeated on consecutive trials, and no problem and its
inverse (e.g., 2 + 3 and 3 + 2) appeared in the same half of the
problem set.
Arithmetic fluency test. After finishing the computer trials, participants completed the Addition test and the Multiplication test of the
Subtraction-Multiplication Test from the French kit (French, Ekstrom, & Price, 1963). The Addition test consists of two pages of
three-term addition problems (total 120). The Multiplication test
consists of two pages of one- by two-digit multiplication problems
219
PROBLEM SIZE AND PROCEDURE SELECTION
(total 60). Arithmetic fluency was defined as the total number of
problems solved correctly on both tests. This measure of arithmetic
fluency reflects an individual's ability to quickly and accurately execute
addition and multiplication procedures (including retrieval) on multidigit problems.
two different procedures on the same trial, memory aids (e.g., a poem),
and visualization. Examples of these procedures are given in Appendix A.
Procedure
Accuracy
Each participant was individually tested in a single session lasting
not more than SO min. Before the session began, the participant was
told that the purpose of the study was to determine what sorts of
procedures adults use when solving simple arithmetic problems. An
example, as well as some possibilities, was offered:
Participants rarely made errors. Of the 1,600 trials, 5
responses were invalid because of equipment or experimenter
errors, and 17 responses were incorrect. Thus, errors were too
infrequent to be analyzed. This 1% error rate is similar to
those reported in other production experiments, including the
2% error rate reported by Miller et al. (1984) and the rate of
3% reported by Geary and Wiley (1991) for their young adults.
What do people do when asked to add 9 + 6? You could just
remember the answer, 15. It just sort of pops into your head. You
could figure the answer out by counting. You think 9, and then 10,
11,12,13,14,15. You could figure it out using a special trick. You
could remember that 10 + 6 = 16, so 9 + 6 has to be just one less.
Or you could solve it in some other way.
Participants were told that each trial was composed of two parts.
First, they were to say, as quickly and accurately as possible, the
answer to an addition problem that appeared on a video monitor.
Second, they were to describe how they had solved the problem (i.e.,
whether they had remembered, counted, or used a special trick or
other procedure). Participants were encouraged to report as much as
possible about how they solved the problem, and the experimenter
noted their responses. Ten practice problems were presented, participants were again reminded to answer as quickly and accurately as
possible; the 100 experimental problems were then presented. The
practice problems for each list were randomly selected from the
problems in the second half of that list. A short break was allowed after
50 problems.
Participants initiated each trial by saying "go" after afixationpoint
(an asterisk) appeared on the monitor. Following a 100-ms tone and a
700-ms blank interval, the problem appeared and timing was initiated.
When presented at a distance of 0.6 m, the problem subtended a visual
angle of 0.33° vertically and 1.15° horizontally. The participant's vocal
responses activated a voice key, and each distinct latency was counted
and recorded. In the few cases in which the participant made more
than one response, the experimenter keyed in the count for the
intended response and the corresponding latency was recorded.
Following the response, the cue "Remember or Strategy?" appeared
on the screen to prompt participants to report the procedure that they
had used.
After the computer trials, the paper-and-pencil arithmetic tests
were administered with the standard instructions. Participants were
given 2 min per page for each of the tests. Total time to complete these
tests was about 10 min.
Classification of Self-Reports
Participants' self-reports were used to classify procedures. Five
categories were used: retrieval, counting, transformation, zero rule,
and other. Retrieval was inferred when participants reported remembering or knowing the answer without any additional processing. When
participants mentioned any sort of mental incrementation, the selfreport was classified as counting. A procedure in which an individual
reportedly changed the presented problem to take advantage of
familiar arithmetic facts was classified as a transformation (referred to
elsewhere as decomposition [e.g., Siegler, 1987] or as a derived-facts
procedure [e.g., Putnam, deBettencourt, & Leinhardt, 1990]). A
zero-rule classification was given only if the addend or augend was zero
and the participant reported that any number added to zero is that
same number. Other procedures included guessing, the reporting of
Results
Selection of Procedures
As shown in Table 1, the four substantive categories accounted for over 98% of participants' reported procedures.
For each participant we computed the frequency, median
latency, and standard deviation of latencies for each procedure. Means are presented in Table 1 along with information
about the prevalence of use among individuals. Retrieval was
the fastest and most common procedure, but frequency of use
was highly variable across subjects and, overall, well below
100%. In fact, only 2 participants reported using retrieval on
all trials. Transformation and counting procedures were somewhat slower than retrieval but were used by most participants
and accounted for 25% of all self-reports. Interestingly, the
reported 9% use of counting was close to the 5% use of
counting estimated by Groen and Parkman (1972).
Transformations included both decomposition, by using a
known fact (Siegler, 1987), and conversion of addition to
multiplication. Examples of the most common transformations
are shown in Appendix A. The majority of the transformations
reported by our participants (65%) involved the use of facts
that summed to 10. For example, 4 + 7 was transformed to 7 +
3 + 1, and 8 + 5 was transformed to 8 + 2 + 3. On problems
that included a 9, an alternative procedure involved changing
the 9 to 10, adding the remaining addend to 10, then subtracting 1. For example, 9 + 7 was transformed to 10 + 7 — 1. On
19% of transformation trials a tie was used as the known fact,
usually when the addend and augend differed by one (e.g.,
6 + 7 became 6 + 6 + 1). On 13% of the trials a multiplication
Table 1
Prevalence and Latencies of Procedures Used
on Addition Problems
Procedure
Trials
(%)
No. of
people3
Frequency
ofuse b (%)
Retrieval
Transformation
Counting
Zero rule
Other
71.2
16.5
8.6
2.2
1.5
16
33-100
1-37
2-26
1-17
1-8
13
9
4
Latency
SD
M
749 120
1,094 301
985 378
854 134
1,386 990
"Number of participants who used the procedure at least once.
••Represents a range across those individuals who used the procedure
at least once.
220
L E F E V R E , SADESKY, AND BISANZ
fact was used (e.g., 6 + 5 became 2 x 5 + 1). Only 4% of trials
involved some other fact. Thus, 84% of transformation procedures involved the retrieval of a well-known fact (i.e., ties or
sums to 10) in combination with an adjustment that reflected
the initial transformation.
Although transformation procedures involved direct retrieval of a stored association, this retrieval was only a
subcomponent of processing. The frequency with which transformations were used is inconsistent with the assumption that
direct retrieval alone is used to solve simple addition problems
(Ashcraft, 1987; Campbell & Oliphant, 1992; Widaman &
Little, 1992). In the future, therefore, an important distinction
needs to be made between direct retrieval of an association
from memory and the more complex procedures that include
retrieval, or a series of retrievals, as part of processing.
The zero rule was reported on 12% of the trials in which one
digit was zero. The results also provide evidence pertaining to
the issue of the relative latencies of rules versus retrieval (i.e.,
Ashcraft, 1982; Baroody, 1983): In this case, rule use was
reasonably fast and efficient. Other procedures were reported
very infrequently. Standard deviations of latencies were relatively low for the retrieval and zero-rule procedures, which are
presumed to be fairly uniform in execution. Latencies were
more variable for counting, a procedure that, depending on the
problem, could involve more or fewer increments. Similarly,
the categories labeled transformation and other included procedures that varied considerably in computational complexity
(see Appendix A), and, as would be expected, the corresponding standard deviations were high.
Selection of procedures was compared across subjects.
Siegler (1987) found that 99% of young children used two or
more procedures on addition problems. In this study, 81% of
the participants used two or more of the counting, retrieval,
and transformation procedures, and 63% used three or more
procedures. This result extends Siegler's conclusions about the
variability of procedure selection to adults. Over half (56%) of
the participants used counting at least once, and 81% used a
transformation procedure at least once. This variety in procedure selection supports the view that the importance of
retrieval has been overemphasized in models of adult performance. Furthermore, there are significant individual differences in the frequency of procedure use (see Appendix B for
frequency of use across individuals). This variability across
subjects has important implications for models of arithmetic:
The modal model (i.e., that adults always retrieve) applied to
only a few of our participants.
Relative latencies of the various procedures were examined
for individual participants. In comparing latencies for retrieval, transformations, and counting, retrieval was the fastest
of the three procedures for 15 of the 16 participants. For 3 of
the 4 participants who used the zero rule, rule use was faster
than retrieval. Finally, for 6 of the 8 participants who used both
procedures, counting was faster than transformations. Thus,
the pattern of relative speed across procedures for individuals
was similar to that in the grouped data (see Table 1).
On the basis of the data in Table 1 and Appendix B, we can
conclude that adults use a variety of procedures to solve
single-digit addition problems. This finding contrasts sharply
with the common assumption that adults always, or nearly
always, directly retrieve addition facts from memory. Transformation procedures involve retrieval of an intermediate sum
but not direct retrieval of an answer to the presented problem.
It is also clear from our data that use of procedures varies
substantially among individuals: Some participants used nonretrieval procedures on a majority of problems.
Analysis of the Problem Size Effect
Problem size and averaged solution latencies.
One concern
with self-reports is that they may change the pattern of
performance (i.e., they may show reactivity; Russo et al., 1989).
To address this issue, we compared our results to those of
other investigators by using the same analyses as have frequently been reported in the literature. Note that none of
these other studies included self-reports of procedure use. Six
structural variables were used to predict solution latencies on
all problems. Regression analyses were calculated by using
minimum addend (min), correct sum (sum), sum squared,
product (prod), the Wheeler (1939) difficulty variable, and a
variable (Greater 10) that coded whether the sum was greater
than 10 (cf. Siegler, 1987). The Wheeler difficulty index
reflects children's errors in learning simple addition facts and
has been used in research with adults as a measure of
associative strength (e.g., Ashcraft, 1987). As shown in Table 2,
for the complete data set (i.e., including ties and zero-addend
problems), each of these structural predictors accounted for a
significant percentage of the variance in latencies. The best
predictor was Greater 10, followed closely by product (Widaman et al., 1989) and the Wheeler difficulty measure (Ashcraft,
1987). Next, ties and zero-addend problems were removed
from the data set, and the regressions were recalculated on all
remaining problems. Again, all of the structural variables
accounted for a significant percentage of the variance. The two
best predictors were product and sum squared (Ashcraft &
Stazyk, 1981), but sum and Greater 10 accounted for almost as
much variance as the two "best" predictors. Thus, consistent
with other investigations of mental arithmetic, latencies increased with problem size as indexed by these structural
variables.
Table 2
Structural Predictors of Latency
Subsets of data
All data
Predictor
r
2
Minimum
Sum
Sum2
Product
Wheeler
Greater 10
.46
.49
.53
.54
.55
.57
a
Reduced set
b
Slope
r
2
Slope
39.1
23.2
1.3
5.0
3.5
213.6
.49
.55
.59
.58
.48
.56
49.4
30.1
1.5
5.9
4.1
210.2
Zeros 0
2
r
Slope
.08
.14
-3.97
-0.05
.00
-0.21
Ties"
r2
Slope
.47
.47
.53
.53
.33
.66
10.7
10.7
0.3
1.2
1.1
73.8
Note. All squared correlations (r2) are significant (ps < .05), except
those for the Zeros subset.
a
Number of problems in the regressions was 100. 'The number of
problems in the regressions was 79. T h e number of problems in the
regressions was 19. dThe number of problems in the regressions was 10.
221
PROBLEM SIZE AND PROCEDURE SELECTION
Comparison of the percentage of variance accounted for by
the various structural variables in this experiment with the
values reported by Ashcraft and Stazyk (1981)1 and by Miller
et al. (1984)2 indicated that there is substantial agreement
across studies. For example, min accounted for 60% and 50%
of the variance, respectively, in those studies, as compared
with 47% in the present study. Similarly, sum squared accounted for 60%, 57%, and 53% of the variance in latencies in
Ashcraft and Stazyk, Miller et al., and the present study,
respectively. The best predictor identified by Miller et al. was
product, which accounted for 57% of the variance in latencies,
compared with 54% in the present study. Slope values reported in the literature are also similar to those in the present
study. For example, Miller et al. reported a slope of 3.02 for
latencies on addition problems when product was used as the
predictor. Ashcraft and Stazyk reported a slope of 2.2 for
addition latencies with sum squared as the predictor (see also
Ashcraft, 1987). Thus, the results of the regression analyses
indicated that solution latencies in this experiment showed
very similar problem size effects as have been reported
elsewhere in the literature. Self-reports apparently did not
change the pattern of results.
We analyzed zero-addend and tie problems separately
because of suggestions that these problems are solved differently than other problems (e.g., Ashcraft & Stazyk, 1981;
Miller et al., 1984). On zero-addend problems, none of the
structural predictors accounted for a significant percentage of
the variance in solution latencies. Participants generally either
retrieved the answer or used the zero rule (i.e., n + 0 = n). On
ties, however, the structural variables predicted a significant
percentage of the variance. The slope of the problem size
effect on nontie problems, however, was approximately five
times larger than that on tie problems (for the product
variable). Thus, although ties did show a significant problem
size effect, it was small compared with that found on other
(nontie) problems (see Miller et al., 1984 for a similar result).
In summary, when averaged across procedures, our data are
comparable to those reported by a number of other investigators (e.g., Ashcraft & Stazyk, 1981; Miller et al., 1984; Siegler,
1987; Widaman et al., 1989). We found that latencies generally
increased as a function of problem size, that the nonlinear
predictors, sum squared and product, accounted for more
variance than min or sum (a result typical of adult performance), and that the problem size effects for zero-addend and
tie problems were different than for nontie problems.
Problem size and selection of procedures.
According to
extant models, problem size effects in adults are due either to
the structure of the mental network (e.g., tabular models;
Widaman & Little, 1992) or to a function of network strength
as determined by frequency of presentation, amount of practice, and interference from competing associations (Ashcraft,
1987, 1992; Campbell, 1987a; Campbell & Graham, 1985;
Graham & Campbell, 1992; Siegler & Shrager, 1984). Alternatively, the problem size effect may be a result of differential
selection of procedures across problems. The proportion of
use of each of the three most common procedures (retrieval,
counting, and transformations) is shown in Figure 1 as a
100
<D
p.
70
R
60
T
C
50
Retrieval
Transformations
Counting
40
T-T
30
20
10
10
-
OS BH*
/ ^C^C--'-'
T--T-.T^- T ~-T-
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
SUM
Figure 1. The percentage of problems solved by each of the retrieval
(R), transformation (T), and counting (C) procedures as a function of
the sum of the problems.
function of the sum.3 It is clear that procedure use varied
systematically with problem size. First, retrieval was the most
frequently used procedure on problems with sums less than 10.
Second, the use of transformation procedures increased dramatically on problems with sums greater than 10. Third,
although counting procedures were used across problem size,
frequency of use was higher on problems with mins of one, two,
or three (16%) than on problems with mins larger than three
(7%, excluding ties; see Appendix C). Thus, procedure selection was associated with particular problem characteristics.
Further examination of the data in Figure 1 suggests that the
nonlinear relation between problem size and problem latency,
and hence the enhanced predictiveness of sum squared or
product relative to the other structural variables, is closely
linked to the pattern of retrieval use. The best predictor of
latencies on the complete data set was a variable that simply
represented whether the sum was greater than 10 (as shown in
Table 2). Because latencies on transformation trials were
slower than on retrieval trials (Table 1), it is likely that the
problem size effect is strongly influenced by latencies on
nonretrieval trials. To explore this hypothesis, latencies for
retrieval, transformation, and counting trials were separately
analyzed, as described below.
When latencies were separately analyzed for retrieval,
counting, and transformation trials, the best structural predictors on retrieval trials accounted for substantially less variance
than on all trials (see Table 3). Similarly, on transformation
trials none of the structural variables were particularly good
1
The data reported by Ashcraft and Stazyk (1981; Experiment 1)
were based on the performance of 20 participants in a verification task.
The reported squared correlation values include the 90 nontie problems.
2
The data reported by Miller et al. (1984) were based on the
performance of 6 participants in a production task. The reported
squared correlation values include the 90 nontie problems.
3
Sum was chosen as the variable in the graphs because it has a
reasonably straightforward relation to the individual problems (Geary
& Wiley, 1991) and, thus, the graphs are easily interpreted.
222
L E F E V R E , SADESKY, AND BISANZ
Table 3
Regression Analyses of Latencies for the Different Procedures
All trials
Data set/
predictor
Retrieval
trials
r2
Slope
t2
.49**
.47**
.54**
.57**
23.2
39.1
5.0
213.6
.18 #H *
.22 *i *
.22 • i *
.12 *x
.55** *
.49** *
.58** *
.56**
30.1
49.4
5.7
210.2
.08 **
.07 *i
.09
.03
Transformation
trials
Counting
trials
Slope
r2
Slope
r2
Slope
7.7
14.6
1.7
52.2
.61 *
.72 **
.70 ** *
.66 ** *
131.8
263.8
28.7
997.6
.14***
.07
.II*11
.24*'
34.4
41.5
5.2
302.2
602.0
10.0
1.2
26.9
.60 • • *
.71 * *
.69 * *
.65 * *
135.6
264.7
28.7
986.5
.16*'
.15*' *
.15*' *
.23*' *
42.8
67.8
6.7
310.2
11
Complete
Sum
Min
Product
Greater 10
Reduced6
Sum
Min
Product
Greater 10
Note. Min = minimum; r2 = squared correlation.
a
n = 100. bn = 72.
**p < .05. ***p < .01.
predictors of latencies, although the best predictor was whether
the sum was greater than 10. Separation of trials by reported
procedure use suggests that averaged latencies provide a
misleading picture of performance (Siegler, 1987, 1989). The
latency data for all trials and for retrieval trials only are plotted
in Figure 2 for the complete data set (100 addition problems).
These data are consistent with the hypothesis that use of
nonretrieval procedures on larger problems contributes substantially to the problem size effect. Indeed, although latencies
on retrieval trials still increased with problem size, the increase
was not particularly systematic. Moreover, on the reduced data
set (ties and zero addends omitted), the percentage of variance
accounted for on retrieval trials by the best structural variable
was less than 10% as compared with almost 60% on all trials.
Given that many researchers have studied only the nontie,
nonzero problems (e.g., Geary & Wiley, 1991), this finding has
important consequences for interpreting reported effects of
problem size.
1100
o All Trials
• Retrieval Trials Only
1000
On counting trials, the best structural predictor of latencies
was the minimum addend (Table 3). This result is consistent
with the participants' claims that they used counting on these
trials. As shown in Figure 3, the increase in latencies with min
was dramatic, especially as compared with latencies on retrieval trials. Note that the scale on this figure is different from
that of Figure 2 and that the latencies are plotted as a function
of min rather than of sum. As shown in Table 3, the coefficient
associated with the min regression was consistent with the
interpretation that participants counted on these trials. The
estimate of 264 ms per count was similar to other estimates of
well-practiced internal counting; whereas the coefficients associated with min on retrieval and transformation trials were
small and not easily associated with a plausible rate of internal
counting.
The results for tie problems support the view that ties are
generally solved with retrieval. Self-reports indicated that
participants either retrieved the solutions to these problems or
2500
o Counting
• Retrieval
2000
900
1500
800
700
1000
600 -
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
SUM
Figure 2. Median latencies collapsed across subjects for all trials
(open circles) and for trials in which retrieval was used (closed
squares). Ties were omitted. Also shown are best fitting regression
lines in which sum was used as the predictor: The dotted line is for all
trials, and the dashed line is for retrieval trials only.
500
1 2
3 4 5 6 7
Minimum Addend
8
9
Figure 3. Median latencies collapsed across subjects on counting
trials (open circles), and for comparison, on retrieval trials (closed
squares) as a function of the minimum addend. Also shown are best
fitting regression lines in which min was used as the predictor.
PROBLEM SIZE AND PROCEDURE SELECTION
used a multiplication transformation (e.g., 4 + 4 = 2 x 4 ) .
The slope of the regression equation for latencies on ties (with
use of product as the structural variable) was equivalent to that
on retrieval trials alone for nonties (1.2 ms). Removing the
nonretrieval trials for tie problems had little effect on the
pattern of performance. Thus, ties are solved differently than
nontie problems: Ties are primarily solved with retrieval;
whereas nontie problems are solved with a variety of procedures.
In summary, the present results suggest that averaging
across procedures results in misleading conclusions about
performance (Siegler, 1987). When retrieval trials alone were
considered, the problem size effect was substantially reduced.
The small amount of variance accounted for by problem size
reduced its status as a major variable to a noticeable but minor
factor in retrieval latencies. Of importance was the finding that
retrieval was used less than 50% of the time on some problems
(e.g., 8 + 9 and 9 + 8; see Appendix C for the percentage of
procedure use on individual problems). Thus, the trials in
which adults did not retrieve exerted a significant influence on
latencies.
Veridicality and Reactivity
The present results provide some support for the assumption that participants can accurately report their procedure use
across trials (cf. Russo et al., 1989). First, the latency data
conformed fairly well to expectations that were based on the
nature of the self-reported procedures. In general, procedures
that should have been faster were faster, arid procedures that
should have been more variable were more variable (see Table
1). Second, latencies were very strongly related to structural
variables only for counting trials and not for retrieval or
transformation trials (see Table 3). Moreover, the coefficient
for min on counting trials represented a rate of internal
counting that is entirely plausible. If participants merely had
reported solution procedures randomly, then patterns of
slopes and squared correlations should have been similar for
all three types of trials. If participants had only discriminated
fast ("retrieval") from slow trials and labeled the latter either
counting or transformation on a random basis, then slopes and
squared correlations should have been similar for counting and
transformation trials. The finding that the coefficients for the
counting trials differed from those for the comparably slow
transformation trials, as well as those for the faster retrieval
trials, is consistent with the view that participants discriminated and reported use of these different procedures accurately. Third, the finding that latencies (averaged across
procedures) showed similar patterns as in previous studies
supports the conclusion that asking participants to report their
procedures did not substantially alter their performance (see
also Compton & Logan, 1991; Logan & Klapp, 1991). In
summary, the pattern of latencies was such that we feel
confident that our participants were reasonably accurate (i.e.,
veridical) at reporting their procedures and that their performance was not influenced systematically (i.e., was not reactive)
by the requirement to provide self-reports.
223
Analyses of Individual Differences
Data from the performance of individual participants are
rarely analyzed separately. As described above, data are
usually averaged across individuals, even when procedures are
analyzed separately (Siegler, 1987). The same criticisms apply
to averaging across individuals, however, as to averaging across
procedures. Averaged data may not describe the performance
of any individual person. We have included analyses on
averaged data primarily to illustrate the similarities between
our data and those reported by other researchers. The procedure of averaging across subjects is, however, statistically
unsound (Lorch & Myers, 1990). As illustrated by Lorch and
Myers, averaging across subjects has two unfortunate consequences. If there is any interaction between subjects and items
(i.e., if participants vary in their latencies across items), then
averaging results in inflated alpha levels and inflated estimates
of the percentage of variance accounted for, as outlined below.
Problem size and individual differences. To examine the
problem size effect for individuals, we conducted regression
analyses on correct trials separately for each participant by
using product as the structural variable.4 The regression
coefficients and squared correlation values for individuals are
shown in Table 4. Overall, the percentage of variance accounted for by product was lower for individual participants
than for averaged data, with a mean of 22% across subjects as
compared with 54% in the averaged data (Table 3). The mean
slope value across subjects (6.9 ms) was statistically significant,
<(15) = 4.94, p < .05, and was similar to that for averaged data
(5.0 ms). As noted by Lorch and Myers (1990), estimates of
population coefficients with averaged data are not seriously
compromised by the averaging procedure, although estimated
standard errors would be incorrect. Thus, averaging across
subjects in this study resulted in substantially inflated estimates of the relation between problem size and solution
latencies.
Retrieval latencies and individual differences. We were par-
ticularly interested in the influence of nonretrieval processing
on patterns of latencies. Recall that, for averaged data,
squared correlations and slope values decreased when retrieval trials were separately analyzed. The same result occurred in the data of individuals. Although problem size (as
indexed by product) continued to predict latencies, <(15) =
7.37, p < .05, its influence was reduced: Mean percentage of
variance accounted for by product decreased to 15% (from
22%), and mean slope was reduced to 3.5 ms (from 6.9 ms).
Thus, the change in the pattern of latencies observed in
averaged data when retrieval trials were eliminated was not
due to some artifact of averaging across individuals. This
analysis indicated that, on retrieval trials alone, problem size
accounts for a relatively small percentage of the variance in
solution times. The finding contrasts sharply with the result
that problem size accounts for over half the variance in
solution latencies when latencies are averaged across subjects
and across procedures.
4
The analyses were very similar when sum was used as the predictor;
however, product was chosen because it is typically the best predictor
of latencies for adults (Ashcraft, 1987; Miller et al., 1984).
224
L E F E V R E , SADESKY, AND BISANZ
Table 4
Individual Regression Results on All Trials and Retrieval-Only
Trials Using Product and Percentage Use of
Nonretrieval as Predictors
All trials3
Retrieval
only2
Retrieval
onlyb
Participant
r2
Slope
rz
Slope
r2
Slope
01
13
15
09
02
16
07
14
06
03
10
05
04
11
12
08
.086*
.070*
.279*
.379*
.091* •
.456*
.006
.190*
.216*
.284*
.553*
.009
.383*
.181*
.140*
.236* *
0.7
3.3
2.3
7.2
3.4
9.0
0.5
6.8
16.3
10.9
18.1
1.2
7.7
3.4
4.8
14.1
.086* "
.070*
.265* •
.377*
.086* '
.214*
.053c
.051*
.149* 1
.168* "
.192* k
.067* "
.141*
.248* 1
.062* k
.082
0.7
3.3
1.7
7.0
4.0
4.1
0.8
3.2
5.5
4.3
6.9
1.1
3.9
3.6
3.7
2.5
.069* "
.040*
.176* '
.399* '
.135* "
.368*
.063* *
.001
.147*
.175* "
.117*"
.010
.111* •
.115*
.024
.118°
0.6
2.6
1.4
7.9
6.5
7.0
0.8
0.8
7.9
4.6
7.5
0.4
2.8
3.5
3.3
4.3
Note. Participants are ordered according to percentage use of retrieval as shown in Appendix B; r 2 = squared correlation.
a
Predictor variable was product. b Predictor variable was nonretrieval
use. 'Marginally significant a t p < .06.
"p < .05.
trials were removed; whereas for skilled participants, the
amount of variance predicted by product was similar for all
trials as compared with retrieval-only trials (10% vs. 12%).
Similarly, slopes for less skilled participants decreased from
10.5 ms to 4.6 ms; whereas for skilled participants slope
estimates were stable (2.1 ms vs. 2.2 ms). Thus, for both skilled
and less skilled participants, problem size continued to predict
latencies but to a much more modest degree than was
indicated by latencies averaged across procedures. Even on
retrieval trials alone, however, the difference in slopes between more and less skilled participants was significant,
f(14) = 8.82, p < .05. Less skilled participants showed larger
effects of problem size than more skilled participants.
The results of the analyses of individual participants indicated that averaging across individuals and procedures creates
a misleading picture of performance. First, averaging resulted
in hugely inflated values of squared correlations for the
problem size effect. Second, problem size effects for individual
participants were strongly influenced by the presence of
nonretrieval trials, even though most individuals used retrieval
on the majority of trials. Third, problem size effects varied with
arithmetic fluency. More skilled individuals consistently used
retrieval and showed relatively small effects of problem size.
Less skilled individuals showed larger effects of problem size,
but these effects were heavily influenced by the presence of
nonretrieval trials.
Discussion
Individual differences and arithmetic fluency. We also exam-
ined the relations between arithmetic fluency and solution
latencies. More skilled individuals are assumed to retrieve
arithmetic facts quickly and efficiently (Ashcraft, Donley,
Halas, & Vakali, 1992; Kaye, 1986; Kaye, deWinstanley, Chen,
& Bonnefil, 1989; LeFevre & Kulak, 1994). Consistent with
this hypothesis, there was a significant correlation between
fluency and percentage use of retrieval: Skilled individuals
used retrieval more frequently than did less skilled individuals
(r = .52, p < .05). The three individuals who used retrieval
most frequently (98%, 100%, and 100% of the time) were in
the more skilled group. In fact, their fluency performance
ranked first, second, and fourth, respectively (see Appendix B).
Problem size effects also varied with arithmeticfluency.For
example, the correlation betweenfluencyand squared correlations on all trials approached significance (r = —.47,/? = .064),
suggesting that problem size was a less important predictor of
latencies for more skilled than for less skilled participants.
When only retrieval trials were considered, however, the
correlation betweenfluencyand squared correlations dropped
to -.04, indicating that skilled and less skilled participants
showed similar patterns of performance when only retrieval
trials were considered. These results suggest that selection of
procedures is a factor in individual differences in addition
performance, even among relatively skilled adults (e.g., Siegler, 1988a).
To further examine the relation between arithmetic fluency
and patterns of performance, we divided participants into
more and less skilled groups on the basis offluencyscore.5 For
less skilled participants, the amount of variance accounted for
by product decreased from 32% to 16% when nonretrieval
The most important variable in the area of mental arithmetic is problem size: Problems with larger numbers are solved
more slowly than problems with smaller numbers. Given its
persistence across a variety of tasks and participant groups, the
problem size effect is equivalent in importance to that of the
frequency effect in the study of word recognition. Certainly any
complete theory of mental arithmetic must include an account
of the problem size effect. Our results indicate that averaging
across participants and procedures has resulted in inflated
estimates of the importance of the problem size effect. For
individual participants, selection of procedures appears to
account for a substantial proportion of the problem size effect
on simple addition problems, afindingthat is inconsistent with
theories of mental arithmetic that do not include a role for
solution procedures other than direct retrieval of facts. These
results, plus the observation that problem size effect is quite
modest on problems in which retrieval is used, have important
implications for theories of mental arithmetic in adults.
The Problem Size Effect on All Problems
Our findings were simple. On problems with sums greater
than 10, transformation procedures were used almost as
frequently as direct retrieval. Retrieval was consistently faster,
5
The mean arithmetic fluency score was 71.2. The less skilled group
included 9 participants, with a mean fluency score of 56.6 (SD = 7.0),
range of 48-65. The more skilled group included 7 participants, with a
mean fluency score of 89.6 (SD = 8.4) and a range of 77-99.
PROBLEM SIZE AND PROCEDURE SELECTION
however. Hence, averaging retrieval and transformation trials
resulted in relatively slow latencies on problems with sums
greater than 10. Participants showed fast and accurate retrieval of sums up to 10, and they rarely used transformations
on these problems. This finding presumably reflects the centrality of 10 in the base-ten number system. In contrast, counting
was used most often on problems with minimum addends or
augends of one, two, or three, and latencies on counting trials
were predicted best by the number of counts that were
required. Thus, our data showed clear relations between
problem characteristics and procedure selection that had
important implications for latencies.
These results might be considered unremarkable if the
participants had been children (Bisanz, Morrison, & Dunn,
1995; Geary & Brown, 1991; Geary, Brown, & Samaranayake,
1991; Siegler, 1987; Siegler & Shrager, 1984). Our participants,
however, were relatively skilled university students. Moreover,
careful examination of the literature revealed other findings
consistent with our own. Geary and Wiley (1991) also used a
self-report method with students who solved simple addition
problems. Their participants reported nonretrieval procedures
on 12% of trials and also used nonretrieval procedures
increasingly as a function of problem size. Although Geary and
Wiley did not explicitly consider differences in the problem
size effect on retrieval versus nonretrieval trials, they found a
squared correlation of .30 for the relation between problem
size (indexed by product) and latencies on retrieval trials.6
Although not as small as in our study, this value is considerably
smaller than those reported in the literature for all trials when
latencies were averaged across procedures. Similarly, Svenson
(1985) also reported an attenuation in the slope of the problem
size effect on retrieval trials (from 23 ms per increment to 4 ms
per increment). In accord with the present results, the findings
reported by Svenson and Geary and Wiley are consistent with
the view that problem size and nonretrieval solutions are
strongly related. We also found that selection of procedures
varies across individuals as well as across problems.
Theorists have assumed that retrieval is the dominant, if not
the sole, procedure used by adults and, thus, that any nonretrieval trials would not substantially affect the overall pattern
of results. This view has led to models that neglect or downplay
the importance of the link between selection of procedures
and problem solutions (e.g., Ashcraft, 1982,1987; Campbell &
Oliphant, 1992; Widaman et al., 1989; Widaman & Little,
1992). The presentfindings,therefore, have important implications for extant models of how arithmetic facts are processed.
In most models, the problem size effect serves as an index of
how retrieval occurs by means of spreading activation in the
mental network (e.g., Ashcraft, 1987; Campbell & Oliphant,
1992; Widaman et al., 1989). We found, however, that direct
retrieval of an answer occurs infrequently on some problems
(see Appendix C). Given that performance on such problems
contributes substantially to the problem size effect, this index
is not likely to provide a veridical reflection of retrieval
processes. Thus, models of mental arithmetic must incorporate
a link between selection of procedures and performance
(Bisanz & LeFevre, 1990; Siegler & Shipley, 1995). Our results
do not imply that there is no network but rather that indices of
problem size on trials averaged across procedures do not
directly reflect processing within the network.
225
The need to incorporate processes other than fact retrieval
in models of mental arithmetic has previously been identified
(e.g., Baroody, 1994), but the relation between fact retrieval
and nonretrieval processes is rarely described explicitly. One
exception is a model by Logan and his colleagues (Compton &
Logan, 1991; Klapp, Bodies, Trabert, & Logan, 1991; Logan &
Klapp, 1991), which was designed to account for changes in
performance of adults on a novel, alphabetic arithmetic task as
a function of extended practice. According to this model, the
process of direct retrieval can operate in parallel with nonretrieval procedures. Whether the answer is produced by retrieval or nonretrieval procedures depends on which process is
completed more quickly. Another exception is a series of
models developed by Siegler and his colleagues (e.g., Siegler,
1988b; Siegler & Jenkins, 1989; Siegler & Shrager, 1984) to
account for age-related changes in the diversity of solution
methods used by children. In the most recent version of this
model (Siegler & Shipley, 1995), associations are assumed to
exist between a problem (e.g., 8 + 9) and possible answers
(e.g., 16,17, and 18) and also between a problem and various
nonretrieval procedures. Whether a child first attempts to
solve the problem with fact retrieval or with a nonretrieval
procedure depends on the relative strengths of the corresponding problem-answer and problem-procedure associations:
Stronger associations are more likely to be activated and used
to generate a solution.
The models proposed by Logan (Compton & Logan, 1991;
Klapp et al., 1991; Logan & Klapp, 1991) and by Siegler
(Siegler, 1988b; Siegler & Jenkins, 1989; Siegler & Shrager,
1984) differ in many ways, but both can, in principle, accommodate the finding that adults frequently use nonretrieval procedures. In the case of Logan's model, the failure of most adults
to rely entirely on retrieval could result from insufficient
practice or from the use of procedures that are extremely
efficient. Similarly, in Siegler and Shipley's (1995) model, the
associative strength between a problem and a nonretrieval
procedure could exceed that between the problem and an
answer if experience with the problem is limited or if the
nonretrieval procedure is especially efficient and accurate. In a
simulation of the effects of extended practice, however, their
model eventually used retrieval on 95% of all trials, a level
greatly exceeding that of most of our adult participants. Either
such extensive practice is not common among adults or the
model requires some modification to prevent retrieval from
being used so frequently. One such modification would be to
ensure that the strength of problem-procedure associations is
maintained relative to retrieval when a procedure is successfully executed, thus contributing to persistence in using that
procedure on future trials. A second modification would be to
include a greater variety of nonretrieval procedures during
acquisition. In Siegler and Shipley's simulation of development, only two nonretrieval procedures (sum and min) were
available. Notably absent were transformation procedures,
which were numerous among our adults and which are not
uncommon among children (Putnam et al., 1990; Siegler,
1987). Providing a broader range of fast and accurate proce6
Geary and Wiley (1991) used a subset (n = 40) of the 56 nontie
combinations of the digits from 2-9, so their problem set is not directly
comparable to that used in our study.
226
LEFEVRE, SADESKY, AND BISANZ
dures (and, ultimately, the conceptual capacity to construct
these procedures) might contribute toward continued use of
these procedures, even as problem-answer associations become increasingly strong. Neither of these changes is particularly dramatic, and so our results seem consistent with the
basic architecture of Siegler and Shipley's model.
Thus, adults' frequent use of nonretrieval procedures presumably could be accommodated in both models with appropriate assumptions about the learning history of the individual
(practice with particular problems and procedures), about the
relative efficacy of constructed procedures in the course of
their development, and about how these factors influence
associative strengths (Siegler) or speed (Logan). To account
for the pervasive problem size effect, it must also be assumed
that smaller problems are practiced more frequently than
larger problems and that nonretrieval procedures are relatively more efficacious for larger than for smaller problems.
The plausibility of any such assumptions would have to be
evaluated empirically in simulations or in research with children or novices.
The Problem Size Effect on Retrieval Trials
Although the problem size effect appears to be due primarily to the selective use of nonretrieval procedures and is
severely reduced when nonretrieval latencies are omitted, it
does not disappear completely (Tables 3 and 4). The issue,
then, is how to explain this (reduced) problem size effect on
retrieval trials. We do not propose a definitive explanation, but
we suggest four possibilities, none of which necessarily excludes the others.
First, retrieval latencies may reflect the structural characteristics of a network of problem-answer associations. Existing
hypotheses about the structure of this network (e.g., Campbell &
Oliphant, 1992; Widaman & Little, 1992) may apply, but they
will require considerable reformulation in view of the fact that
the problem size effect is so small on retrieval trials. In revising
these hypotheses, it will be important to avoid the use of
variables that index internal structure, such as product, that
have very limited psychological plausibility (Ashcraft, 1992;
McCloskey et al., 1991; Siegler & Shipley, 1995).
The second possibility is that retrieval latencies may simply
reflect the strength of problem-answer associations or the
speed of retrieving answers that, in turn, depends primarily on
frequency of exposure to specific facts (Ashcraft, 1987;
Ashcraft & Christy, 1995; Campbell & Graham, 1985; Hamann
& Ashcraft, 1986). Although frequency may be important early
in acquisition (see Siegler & Shrager, 1984), in a recent and
thorough study by Ashcraft and Christy (1995),7 the frequency
with which addition facts are presented in a mathematics
textbook series (Grades 1-6) was essentially uncorrelated with
problem size (r = .12). The measure of frequency derived from
Ashcraft and Christy was moderately correlated with latencies
on retrieval trials in the present study (r = .23, p < .05),
however, and predicted a significant 3% of variance in retrieval
latencies, even after product was partialed out. Thus, frequency of presentation may contribute to a complete model of
arithmetical representation, but on its own it does not provide
a compelling account of problem size effects in retrieval of
addition facts.
The third possibility is based on the notion that retrieval
latencies may reflect the use of nonretrieval procedures in the
course of development. More specifically, a history of frequently using nonretrieval procedures for particular problems
may result in longer latencies on the (infrequent) occasions
when retrieval is used for those problems. That is, the
probability of using nonretrieval procedures may influence
latencies on problems solved with retrieval. Empirical support
for this somewhat counterintuitive proposition is evident in the
data of individual participants. We calculated the probability,
across all subjects, of using a nonretrieval procedure for each
problem (see the Probability of Retrieval section in Appendix
C) and used this variable to account for variability in retrieval
latencies across problems. We hypothesized that solution
latencies would be slower on problems that were retrieved less
often as compared with those that were retrieved more often.
This hypothesis was based on the assumption that variability in
procedure selection may reduce the peakedness of the retrieval distribution for a given problem (Siegler, 1987; Siegler &
Shipley, 1995; Siegler & Shrager, 1984).
As compared with the structural predictors, percentage of
nonretrieval was not a particularly good predictor for data
averaged across subjects, predicting only 1% of the variance
for all trials and 2% for the reduced set. The picture is very
different, however, when we analyzed the latencies of individual participants by using percentage of nonretrieval as the
predictor. We compared these results with those obtained with
product as the predictor, as shown in Table 4. For 15 of the 16
participants, product was a significant predictor of latencies on
retrieval trials, accounting for a mean of 15% of the variance in
latencies. Similarly, percentage of nonretrieval was a significant predictor of retrieval latencies for 13 of the 16 participants, accounting for a mean of 13% of the variance in
latencies. Thus, percentage use of nonretrieval, as estimated
on the basis of data from all participants, was almost as
powerful of a predictor of retrieval latencies for individual
participants as the most commonly used structural predictor.
In contrast, frequency of presentation from Ashcraft and
Christy (1995) was significantly correlated with retrieval latencies for only 4 of the 16 participants.
Unlike product and other structural predictors, the probability of using nonretrieval procedures appears to have considerable psychological plausibility as a predictor of retrieval
latencies, at least in the sense that the relation can be
accommodated without major alterations to some existing
models. In Siegler and Shipley's (1995) model, for example,
the rinding that the probability of using a nonretrieval procedure is related to latencies on retrieval trials can be understood as a consequence of how individuals attempt to solve
problems for which the answer is not well learned. More
specifically, when retrieval of a poorly learned problemanswer association is used to generate a response, it is likely
that multiple retrieval attempts are involved, rather than a
single retrieval attempt. The explanation for this inference
requires some elaboration.
In Siegler and Shipley's (1995) model, as noted earlier,
problems are associated with varying strengths to answers and
7
For multiplication facts, frequency of exposure was correlated with
problem size.
PROBLEM SIZE AND PROCEDURE SELECTION
to nonretrieval procedures, and the probability of retrieving an
answer or activating a nonretrieval procedure is proportional
to these strengths. If the strength of a problem-answer
association is somewhat weaker than the strength of a problemprocedure association, then chances are that the nonretrieval
procedure will be used to solve a given problem. This selection
is probabilistic, however, and on the assumption that all
problem-answer associations are not zero, retrieval will be
used on at least some occasions. In this model, one characteristic of problem-answer associations that are not well learned is
that the distribution of associative strengths is relatively flat
rather than peaked. Consequently, the chance of retrieving an
acceptable answer on the first retrieval attempt (i.e., an answer
that exceeds an internal confidence criterion) is not high; if the
answer is retrieved at all, it is likely that multiple retrieval
attempts will be required (see Siegler & Shipley, 1995; Siegler &
Shrager, 1984).
Thus, if a problem is presented that has weak problemanswer associations and relatively strong problem-procedure
associations, then a nonretrieval procedure is likely to be used.
On the occasions when retrieval happens to be used to answer
such a problem, the latency to produce the answer is likely to
involve multiple retrieval attempts and hence is likely to be
slower than for problems in which an answer is likely to be
retrieved in a single attempt. Consequently, a positive correlation across problems should be observed between (a) the
probability of solving a problem with a nonretrieval procedure
and (b) latency on a problem when retrieval is used. Precisely
this pattern was observed in the data for most individual
participants.
According to this view, the relation between indices of
problem size and latency is likely to be epiphenomenal. In
Siegler's (Siegler & Shipley, 1995; Siegler & Shrager, 1984)
models, for example, the strength of a problem-answer association depends in part on the accuracy with which answers were
produced when a nonretrieval procedure was used. When
learning addition facts, children often use counting-based
procedures to solve problems, and associations between problems and answers are strengthened when correct answers are
generated. Counting errors are assumed to be more likely on
problems with larger addends or augends because these
problems require more counting, and so the strengthening of
problem-answer associations is more likely to be delayed on
these problems than on problems with smaller addends and
augends. Thus, indices of problem size, such as product, are
correlated with latency primarily because of the way in which
knowledge of arithmetic facts is acquired and not because
these indices reflect structural variables that influence solution
directly. Indeed, a simulation of learning in Siegler and
Shipley's (1995) model generated substantial problem size
effects indexed by product, even though no structural features
reflecting product are represented in the model.
In summary, our data are consistent with the notion that
acquisition history, as preserved both in the distributions of
associations for problems and answers and in the relative
strength of nonretrieval procedures, accounts for the small
problem size effect found in retrieval latencies. The framework
described by Siegler and Shipley (1995) provides a psychologically plausible account of the relation between nonretrieval
processing and retrieval processing and of the patterns of
227
latencies on problems averaged across all procedures. We also
expect that this model will be useful for understanding
performance on multiplication problems. Research has shown
that latencies on addition and multiplication problems are
closely related (Geary et al., 1986; Miller et al., 1984). Thus,
multiplication performance may also be influenced by the
pattern of procedure selection across problems. Although the
overall frequency of retrieval may be greater on multiplication
than on addition trials (Hubbard, LeFevre, & Greenham,
1994), the course of acquisition of multiplication facts may be
similar to that for addition (Siegler, 1988b). For example, in
multiplication, ties and facts involving multiples of five may
thoroughly be learned (e.g., Campbell & Graham, 1985) and
function as "known" facts in transformation solutions in the
way that the sums-to-ten and ties do in addition. Further, there
is some evidence that multiple retrievals are common in
multiplication performance and influence latencies (Thibodeau, 1993). Thus, we expect that nonretrieval use will be
important for understanding performance in multiplication.
Finally, the link between probability of nonretrieval procedures being used and latency on retrieval trials may arise from
errors of self-report. More specifically, if participants have a
tendency to report use of retrieval when in fact they have used
a nonretrieval procedure, and if this false-positive rate is
constant across problems, then falsely reported retrieval latencies would be more likely to occur for problems on which
nonretrieval procedures are used more commonly. Given the
correlation between problem size and probability of using
nonretrieval procedures (i.e., r = .71 with product as an index
of problem size), the resulting problem size effect for retrieval
latencies is not surprising. Although our data generally are
consistent with the notion that self-reports were veridical, this
final possibility cannot be ruled out conclusively. If this
alternative is correct, however, the likely conclusion is that the
venerable problem size effect for retrieval latencies is even less
substantial than we propose. Resolution of this issue will
require additional research.
In summary, although there are a number of alternative
explanations for the problem size effect on retrieval trials,
most can be linked to the influence of nonretrieval procedures
either concurrently or during acquisition. Thus, in combination with thefindingthat nonretrieval trials have a substantial
effect on latencies on trials that are averaged across procedures, the analyses of retrieval-only trials suggest that attention to nonretrieval processing is critical to understanding
simple arithmetic processes. These results have implications
for theories of simple arithmetic, as discussed above, and for
the study of arithmetic performance when methodologies are
used in which latencies are averaged across diverse procedures.
Concluding Remarks
One striking finding in the present research is the richness
and variability of the nonretrieval alternatives. Although a
variety of studies have shown that adults use many procedures
when solving problems (e.g., LeFevre & Bisanz, 1986), finding
such variability on a task as basic as simple addition emphasizes that diversity is the rule in cognitive processing (see also
228
L E F E V R E , SADESKY, AND BISANZ
Bisanz & LeFevre, 1990; Siegler & Shipley, 1995). Models that
do not have a role for such diversity will never be adequate.
Ashcraft (1992) reviewed 20 years of research on simple
arithmetic and concluded that, in spite of the advances made
in the field, the central question had not been answered: Why
is there a problem size effect? The results of our research
suggest that an important part of the answer is related to the
role of nonretrieval processing in simple arithmetic. Implausible though it might seem, adults do continue to use procedures other than direct retrieval to solve simple addition
problems.
References
Ashcraft, M. H. (1982). The development of mental arithmetic: A
chronometric approach. Developmental Review, 2, 213-236.
Ashcraft, M. H. (1987). Children's knowledge of simple arithmetic: A
developmental model and simulation. In J. Bisanz, C. J. Brainerd, &
R. Kail (Eds.), Formal methods in developmental psychology: Progress
in cognitive development research (pp. 302-338). New York: SpringerVerlag.
Ashcraft, M. H. (1992). Cognitive arithmetic: A review of data and
theory. Cognition, 44, 75-106.
Ashcraft, M. H., & Christy, K. S. (1995). The frequency of arithmetic
facts in elementary texts: Addition and multiplication in Grades 1-6.
Journal for Research in Mathematics Education, 26, 396-421.
Ashcraft, M. H., Donley, R. D., Halas, M. A., & Vakali, M. (1992).
Working memory, automaticity, and problem difficulty. In J. I. D.
Campbell (Ed.), The nature and origins of mathematical skills (pp.
301-329). Amsterdam: Elsevier Science.
Ashcraft, M. H., & Fierman, B. A. (1982). Mental addition in third,
fourth, and sixth graders. Journal of Experimental Child Psychology,
33, 216-234.
Ashcraft, M. H., & Stazyk, E. H. (1981). Mental addition: A test of
three verification models. Memory & Cognition, 9, 185-196.
Baroody, A. J. (1983). The development of procedural knowledge: An
alternative explanation for chronometric trends of mental arithmetic. Developmental Review, 3, 225-230.
Baroody, A. J. (1994). An evaluation of evidence supporting factretrieval models. Learning and Individual Differences, 6, 1-36.
Bisanz, J., & LeFevre, J. (1990). Strategic and nonstrategic processing
in the development of mathematical cognition. In D. F. Bjorklund
(Ed.), Children's strategies: Contemporary views of cognitive development (pp. 213-243). Hillsdale, NJ: Erlbaum.
Bisanz, J., Morrison, F. J., & Dunn, M. (1995). Effects of age and
schooling on the acquisition of elementary quantitative skills.
Developmental Psychology, 31, 221-236.
Campbell, J. I. D. (1987a). Network interference and mental multiplication. Journal of Experimental Psychology: Learning, Memory, and
Cognition, 13, 109-123.
Campbell, J. I. D. (1987b). Production, verification, and priming of
multiplication facts. Memory & Cognition, 15, 349-364.
Campbell, J. I. D. (1991). Conditions of error priming in number-fact
retrieval. Memory & Cognition, 19, 197-209.
Campbell, J. I. D., & Graham, D. J. (1985). Mental multiplication skill:
Structure, process, and acquisition. Canadian Journal of Psychology,
39, 338-366.
Campbell, J. I. D., & Oliphant, M. (1992). Representation and
retrieval of arithmetic facts: A network-interference model and
simulation. In J. I. D. Campbell (Ed.), The nature and origin of
mathematical skills (pp. 331-364). Amsterdam: Elsevier Science.
Compton, B. J., & Logan, G. D. (1991). The transition from algorithm
to retrieval in memory-based theories of automaticity. Memory &
Cognition, 19, 151-158.
Cooney, J. B., & Ladd, S. F. (1992). The influence of verbal protocol
methods on children's mental computation. Learning and Individual
Differences, 4, 237-257.
French, J. W., Ekstrom, R. B., & Price, I. A. (1963). Kit of reference tests
for cognitive factors. Princeton, NJ: Educational Testing Service.
Geary, D. C , & Brown, S. C. (1991). Cognitive addition: Strategy
choice and speed-of-processing differences in gifted, normal, and
mathematically disabled children. Developmental Psychology, 27,
398-406.
Geary, D. C , Brown, S. C , & Samaranayake, V. A. (1991). Cognitive
addition: A short longitudinal study of strategy choice and speed-ofprocessing differences in normal and mathematically disabled children. Developmental Psychology, 27, 787-797.
Geary, D. C , & Burlingham-Dubree, M. (1989). External validation of
the strategy choice model for addition. Journal of Experimental Child
Psychology, 47, 175-192.
Geary, D. C , Frensch, P. A., & Wiley, J. G. (1993). Simple and
complex mental subtraction: Strategy choice and speed-of-processing differences in younger and older adults. Psychology and Aging, 8,
242-256.
Geary, D. C , Widaman, K. R., & Little, T. D. (1986). Cognitive
addition and multiplication: Evidence for a single memory network.
Memory & Cognition, 14, 478-487.
Geary, D. C , & Wiley, J. G. (1991). Cognitive addition: Strategy
choice and speed-of-processing differences in young and elderly
adults. Psychology and Aging, 6, 474-483.
Graham, D. J., & Campbell, J. I. D. (1992). Network interference and
number-fact retrieval: Evidence from children's alphaplication.
Canadian Journal of Psychology, 46, 65-91.
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.
Hamann, M. S., & Ashcraft, M. H. (1986). Textbook presentations of
the basic arithmetic facts. Cognition and Instruction, 3, 173-192.
Hubbard, K. E., LeFevre, J., & Greenham, S. L. (1994, June).
Procedure use in multiplication by adolescents. Poster presented at the
annual meeting of the Canadian Society for Brain, Behaviour, and
Cognitive Science, Vancouver, British Columbia, Canada.
Kaye, D. B. (1986). The development of mathematical cognition.
Cognitive Development, 1, 157-170.
Kaye, D. B., deWinstanley, P., Chen, Q., & Bonnefil, V. (1989).
Development of efficient arithmetic computation. Joumalof Educational Psychology, 81, 467-480.
Klapp, S. T., Boches, C. A., Trabert, M. L., & Logan, G. D. (1991).
Automatizing alphabet arithmetic: II. Are there practice effects
after automaticity is achieved? Journal of Experimental Psychology:
Learning, Memory, and Cognition, 17, 196-209.
Koshmider, J. W., & Ashcraft, M. H. (1991). The development of
children's mental multiplication skills. Journal of Experimental Child
Psychology, 51, 53-89.
LeFevre, J., & Bisanz, J. (1986). A cognitive analysis of number series
problems: Sources of individual differences in performance. Memory
& Cognition, 14, 287-298.
LeFevre, J., Bisanz, J., & Mrkonjic, L. (1988). Cognitive arithmetic:
Evidence for obligatory activation of arithmetic facts. Memory &
Cognition, 16, 45-53.
LeFevre, J., & Kulak, A. G. (1994). Individual differences in the
obligatory activation of addition facts. Memory & Cognition, 22,
188-200.
LeFevre, J., Kulak, A. G., & Bisanz, J. (1991). Individual differences
and developmental change in the associative relations among
numbers. Journal of Experimental Child Psychology, 52, 256-274.
Logan, G. D., & Klapp, S. T. (1991). Automatizing alphabet arithmetic: I. Is extended practice necessary to produce automaticity?
229
PROBLEM SIZE AND PROCEDURE SELECTION
Siegler, R. S., & Jenkins, E. A. (1989). How children discover new
strategies. Hillsdale, NJ: Erlbaum.
Siegler, R. S., & Shipley, E. (1995). Variation, selection, and cognitive
change. In G. Halford & T. Simon (Eds.), Developing cognitive
competence: New approaches to process modelling (pp. 31-76).
Hillsdale, NJ: Erlbaum.
Siegler, R. S., & Shrager, J. (1984). Strategy choices in addition and
subtraction: How do children know what to do? In C. Sophian (Ed.),
Origins of cognitive skills (pp. 229-293). Hillsdale, NJ: Erlbaum.
Svenson, O. (1985). Memory retrieval of answers of simple additions as
reflected in response latencies. Acta Psychologica, 59, 285-304.
Thibodeau, M. T. (1993). Automatic activation of multiplication facts
and procedure use on multiplication problems: Evidence for ADAMM.
Unpublished master's thesis, University of Alberta, Edmonton,
Alberta, Canada.
Wheeler, L. R. (1939). A comparative study of the difficulty of the 100
addition combinations. Journal of Genetic Psychology, 54, 295-312.
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.
Widaman, K. F., & Little, T. D. (1992). The development of skill in
mental arithmetic: An individual differences perspective. In J. I. D.
Campbell (Ed.), The nature and origins of mathematical skills (pp.
189-253). Amsterdam: Elsevier Science.
Widaman, K. F., Little, T. D., Geary, D. C , & Cormier, P. (1992).
Individual differences in the development of skill in mental addition:
Internal and external validation of chronometric models. Learning
and Individual Differences, 4, 167-213.
Journal of Experimental Psychology: Learning, Memory, and Cognition,
17, 179-195.
Lorch, R. F., Jr., & Myers, J. L. (1990). Regression analyses of
repeated measures data in cognitive research. Journal of Experimental Psychology: Learning, Memory, and Cognition, 16, 149-157.
McCloskey, M., Harley, W., & Sokol, S. M. (1991). Models of
arithmetic fact retrieval: An evaluation in light of findings from
normal and brain-damaged subjects. Journal ofExperimental Psychology: Learning, Memory, and Cognition, 17, 377-397.
Miller, K. F., & Paredes, D. R. (1990). Starting to add worse: Effects of
learning to multiply on children's addition. Cognition, 37, 213-242.
Miller, K. F., Perlmutter, M., & Keating, D. (1984). Cognitive
arithmetic: Comparison of operations. Journal of Experimental
Psychology: Learning, Memory, and Cognition, 10, 46-60.
Putnam, R. T., deBettencourt, L. U., & Leinhardt, G. (1990).
Understanding of derived-facts strategies in addition and subtraction. Cognition and Instruction, 7, 245-285.
Russo, J. E., Johnson, E. J., & Stephens, D. L. (1989). The validity of
verbal protocols. Memory & Cognition, 17, 759-769.
Siegler, R. S. (1987). The perils of averaging over strategies: An
example from children's addition. Journal of Experimental Psychology: General, 116, 250-264.
Siegler, R. S. (1988a). Individual differences in strategy choices: Good
students, not-so-good students, and perfectionists. Child Development, 59, 833-851.
Siegler, R. S. (1988b). Strategy choice procedures and the development of multiplication skill. Journal of Experimental Psychology:
General, 117, 258-275.
Siegler, R. S. (1989). Hazards of mental chronometry. An example from
children's subtraction./o(OTUi/o/£(iu<;ariona/ftryc/io/c^v, 81, 497-506.
Appendix A
Procedures Used on Addition Problems
Procedure
Retrieval
Counting
Transformation
Examples of self-reports
Procedure
Examples of self-reports
"I remembered it. I knew it."
"I counted. I counted in my head."
"I used my fingers."
"It's just one more than . . . "
"5 + 6 is just 5 + 5 + 1" (using a tie).
"9 + 6 is 9 + 1 + 5" or "4 + 9 is 10 + 4 - 1" (through
10).
"6 + 5 is 2 x 5 - 1" (using multiplication).
Zero rule
"Anything plus zero is that same number."
"Zero again."
"I guessed."
"I counted, but then I remembered the answer."
"2 plus any even number is the next even number."
"I visualized the numbers 8 and 7."
"I visualized the numbers and then counted."
Other
Appendix B
Arithmetic Fluency Scores and Frequency of Procedure Use (Percentage)
for Each Participant
Frequency of use
Participant
Fluency
Retrieval
Counting
Transformation
Rule
Other
01
13
15
09
02
16
07
14
06
97
90
98
56
77
65
95
54
49
100
100
98
90
84
79
70
70
69
—
—
—
2
8
_
—
1
2
4
—
5
1
2
18
(Appendixes continue on next page)
4
16
18
15
30
12
1
230
L E F E V R E , SADESKY, AND BISANZ
Appendix B (continued)
Frequency of use
Participant
Fluency
Retrieval
Counting
Transformation
03
10
05
04
11
12
08
65
57
80
65
89
48
50
68
68
61
—
23
16
6
26
21
16
32
1
10
37
26
31
30
57
47
46
33
Rule
Other
12
17
Note. Participants are ordered according to frequency of use of retrieval. Dashes represent 0% use of a
procedure by a participant.
Appendix C
Probability of Use of Retrieval, Counting, and
Transformation Procedures
Addend
Augend
0
1
2
3
0
1
2
3
4
5
6
7
8
9
.88
.88
.94
.94
.88
.88
.87
.88
.88
.88
.81
.88
.69
.81
.69
.81
.81
.69
.81
.75
.88
.87
.81
.75
.75
.75
.88
.56
.81
.75
.94
.81
.81
.94
.81
1.00
.40
.53
4
5
6
7
8
9
.94
.75
.73
.81
.75
.87
.50
.60
.56
.19
.88
.69
.87
.81
.81
.56
.75
.44
.44
.44
.94
.75
.75
.94
.46
.54
.50
.81
.50
.25
.94
.75
.94
.88
.75
.63
.56
.38
.38
.40
.25
.27
.81
Retrieval
.75
.69
.88
.81
.81
1.00
.75
.75
.88
.56
.50
.33
.50
.56
.53
.69
.50
.88
.27
Counting
0
1
2
3
4
5
6
7
8
9
(.12) (.19) (.19) (.19) (.12) (.06) (.12) (.06) (.06) (.12)
(.12) .06 .25
.25 .19 .25 .31 .25 .25 .25
(.06) .31 —
.06 .06 .20 .07 .19 .06 .31
—
—
—
—
(.06) .19 .13
.06 .14 .06
—
—
—
—
(.12) .31 .19
.15 .13 .19
—
—
—
—
(.06) .13 .25
.15 .07 .06
—
—
—
—
(.12) .19 .19
.06 .06 .13
—
(.12) .31 .19
.13 .07 .06 —
.06 .06
—
—
(.12) .25 .13
.07 .13 .06 .13 —
(.12) .25 .13
.07
.13 .06 .06 .06 —
—
Transformation
0
1
2
3
4
5
6
7
8
9
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
.12
—
—
—
—
.06
—
.31
.19
.13
.06
.13
.40
.40
.06
—
.25
.25
.13
.25
.38
.53
—
—
—
—
.06
.25
.13
.44
.33
.38
.69
.19
.13
.44
—
.38
.23
.44
.19
.38
.69
.36
.31
.40
.25
.38
.12
.67
.25
.50
.44
.50
.06
.38
.44
.56
.47
.69
.73
.19
Note. Numbers in parentheses represent the probability of zero rule.
Dashes represent zero probability of use.
Received January 19,1994
Revision received January 30,1995
Accepted January 31,1995