Journal of Experimental Psychology:
Learning, Memory, and Cognition
2003, Vol. 29, No. 6, 1339 –1352
Copyright 2003 by the American Psychological Association, Inc.
0278-7393/03/$12.00 DOI: 10.1037/0278-7393.29.6.1339
Elementary Subtraction
Donald J. Seyler
Elizabeth P. Kirk
Cleveland State University
Florida State University
Mark H. Ashcraft
Cleveland State University
Four experiments examined performance on the 100 “basic facts” of subtraction and found a discontinuous “stair step” function for reaction times and errors beginning with 11 – n facts. Participants’
immediate retrospective reports of nonretrieval showed the same pattern in Experiment 3. The degree to
which elementary subtraction depends on working memory (WM) was examined in a dual-task paradigm
in Experiment 4. The reconstructive processing used with larger basic facts was strongly associated with
greater WM disruption, as evidenced by errors in the secondary task; this was especially the case for
participants with lower WM spans. The results support the R. S. Siegler and E. Jenkins (1989)
distribution of associations model, although discriminating among the alternative solution processes
appears to be a serious challenge.
signature finding in these studies (see Ashcraft, 1995) is the
problem size or problem difficulty effect: As the numbers in a
problem grow larger, so too do reaction times (RTs) and error
rates. For instance, RTs and errors will be larger for 9 ⫹ 7 or 8 ⫻
6 than for 5 ⫹ 3 or 4 ⫻ 2. Furthermore, the slope and shape of the
problem size effect have been used as evidence for a variety of
important—and sometimes contradictory— explanations of mental
arithmetic performance, including counting-based (Groen & Parkman, 1972) versus retrieval-based (Ashcraft & Battaglia, 1978)
performance in addition, a developmental shift from counting to
retrieval in addition (Ashcraft & Fierman, 1982) and multiplication
(Cooney, Swanson, & Ladd, 1988), an interference-based explanation of performance in multiplication (Campbell, 1987), and a
mixture of retrieval and reconstructive processing (LeFevre,
Sadesky, & Bisanz, 1996; Siegler & Shrager, 1984). Thus, all
serious theoretical work in this area has addressed the problem size
effect as the first and most important empirical phenomenon to
explain.
Strangely, little of the accumulated research has focused on
elementary subtraction, the basic facts of subtraction taught almost
universally in second grade. There have been several developmental studies of simple subtraction (e.g., Robinson, 2001; Siegler,
1987; Woods, Resnick, & Groen, 1975; see Fuson, 1992, for a
review), and there is the well-known work on the “bugs” in
children’s learning of the complex subtraction algorithm for borrowing (VanLehn, 1990). There is also increasing use of subtraction in studies of neural representation and functioning, with
provocative evidence that subtraction and addition may rely on
different neural substrates than multiplication (e.g., Cohen & Dehaene, 2000; Lee, 2000). However, few studies have examined the
basic processing characteristics of simple subtraction among normal adults, and none have discussed the results at more than a
moderate level of detail. Given the importance of the shape of the
problem size effect and the absence of detailed information in the
literature about this effect in subtraction, our main purpose here
Mathematical cognition has become a popular research topic in
the past 30 years. The appeal of the area is obvious: A rich body
of knowledge about arithmetic and the number system is acquired—with some difficulty— during early education and forms
the basis for higher mathematics, problem solving, and math
reasoning. This acquired knowledge amplifies the simpler understanding of numerosity and comparison evidenced by lower mammals (e.g., Meck & Church, 1983), infants (Wynn, 1992), and
adults (Whalen, Gallistel, & Gelman, 1999) in nonverbal counting
or comparison tasks. Despite the apparent ease with which we
decide that 2 is smaller than 3 or that 6 ⫻ 8 ⫽ 48, the literature
attests to the complexity of processing involved in such seemingly
elementary tasks (e.g., Ashcraft, 1995; Banks, 1977; Dehaene &
Cohen, 1995; Geary, 1994).
Within the area of mental arithmetic, considerable attention has
been devoted to the study of simple addition and multiplication,
especially processing of the 100 “basic facts” of addition and
multiplication (0 ⫹ 0/0 ⫻ 0 up through 9 ⫹ 9/9 ⫻ 9). The
Donald J. Seyler and Mark H. Ashcraft, Department of Psychology,
Cleveland State University; Elizabeth P. Kirk, Department of Psychology,
Florida State University.
Preliminary reports on Experiments 1, 2, and 4 were presented, respectively, at the annual meeting of the Midwestern Psychological Association,
May 1999; at the annual meeting of the Psychonomic Society, November
2001; and at the annual meeting of the Midwestern Psychological Association, May 2002. Experiment 4 was conducted as part of Donald J.
Seyler’s master’s thesis, submitted to Cleveland State University, and was
partially funded by a Council of Graduate Departments of Psychology/
American Psychological Foundation Research Scholarship awarded to
Donald J. Seyler. We are grateful to Candice Coker Morey for assistance
in data collection for Experiment 3.
Correspondence concerning this article should be addressed to Mark H.
Ashcraft, Department of Psychology, Cleveland State University, Cleveland, Ohio 44115. E-mail: m.ashcraft@csuohio.edu
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SEYLER, KIRK, AND ASHCRAFT
1340
was simply to conduct and present foundation work on the processing characteristics of simple subtraction.
Two published studies that examined subtraction do deserve
mention here. The most thorough testing of adults’ subtraction
(Campbell & Xue, 2001) was part of a large project on all four
arithmetic operations that focused on cross-cultural comparisons
of Canadian and Chinese participants. Campbell and Xue found a
significant problem size effect among all samples and across all
four arithmetic operations, as well as a significant increase in
reports of nonretrieval across problem sizes. Likewise, Geary,
Frensch, and Wiley (1993) examined simple and complex subtraction but were especially interested in younger versus older adult
samples and the performance characteristics of the various processing strategies these participants reported. Geary et al. found
that two strategies, along with retrieval, were common in subtraction: addition reference (e.g., referring to 3 ⫹ 6 ⫽ 9 to solve 9 ⫺
3) and counting down (e.g., for 8 ⫺ 2, implicitly counting down
“eight, seven, six”). They also noted that trials reported as involving the use of reconstructive strategies were considerably slower
and more error prone than those reported as retrieval trials (see
also Campbell & Xue’s, 2001, Canadian sample).
Our intent in the present studies was to provide additional
information on such findings in the context of a full presentation of
results on adults’ subtraction. Given the high level of familiarity
and fluent retrieval processing characteristic of simple addition
(e.g., Ashcraft, 1995; but see LeFevre et al., 1996) and the inverse
mathematical relationship between addition and subtraction, the
most straightforward prediction for adults’ subtraction is that it
will resemble addition performance in all essential respects. We
expected a monotonically increasing RT function across increasing
problem sizes, with mean RTs for small problems in the 800 –
1,000-ms range and mean RTs for large problems in the 1,300 –
1,400-ms range (these values were derived from Campbell & Xue,
2001, Table 2, Canadian sample).
Experiments 1 and 2
Experiment 1 presented the 100 basic facts of subtraction to a
sample of college adults. Participants were shown problems such
as 15 ⫺ 8, in column format, and were timed as they produced
their answers verbally. Because the results were surprising, we
conducted the study a second time, 2 years later, using a different
experimenter. The results were very similar, so the two experiments are reported together.
Method
Participants
Twenty-six participants, recruited from undergraduate psychology
classes at Cleveland State University, received course credit for taking part
in Experiment 1; 20 participants were tested in Experiment 2. Participants
completed paper-and-pencil tests of math and test anxiety, followed by a
computer-based test of simple subtraction. Against the remote possibility
that administering the anxiety tests first had in some way aroused participants’ suspicions and influenced their performance, the subtraction task
preceded the math anxiety instrument in Experiment 2. Sessions lasted
approximately 40 min.
Apparatus
The experimental task was performed on ordinary laboratory personal
computers through the use of the Micro Experimental Laboratory software
package (Version 2.01; Psychology Software Tools, Pittsburgh, PA;
Schneider, 1988). The software presented all task instructions on the screen
and recorded RTs automatically through a microphone and voice key
apparatus. At the end of each trial, the experimenter keyed in the participant’s verbal response to the subtraction problem, and the software scored
responses for accuracy.
Stimuli
In a subtraction problem (m ⫺ s ⫽ r), the first value is the minuend (m),
the top number in a column-format problem. It is the large number being
subtracted from. The value being subtracted is the subtrahend (s), and the
answer in the problem is termed the remainder (r) or difference. The basic
facts of subtraction are the inverses of the 100 basic addition facts. Using
the same labels, the inverse of m ⫺ s ⫽ r would be s ⫹ r ⫽ m, augend (s,
the first operand) plus addend (r, the second operand) equals sum (m).
The basic facts of subtraction can be divided into nearly equal-sized
groups of “small” and “large” problems by separating the problems into
those with one- versus two-digit minuends (0 –9 vs. 10 –18); under this
scheme, there are 55 small problems and 45 large problems. For our initial
analyses, this was the partition that defined small versus large problems.
Because we tested the inverses of the 100 basic addition facts, all subtraction problems had one-digit subtrahends and remainders, and the smallest
remainder (correct answer) was 0.
Procedure
Participants were tested individually, with the experimenter present
throughout the experimental session. Participants were provided a brief
explanation of the experiment, completed the math and test anxiety scales
and the subtraction task, and were then debriefed and excused; in Experiment 2, subtraction preceded the math anxiety test, and the test anxiety
instrument was dropped. We administered 10 practice trials, with the
problems drawn randomly from the 100 basic facts, to familiarize participants with the articulation-based RT procedure and the need to avoid
extraneous vocalizations (e.g., “um” and “uh”). The instructions placed
equal emphasis on speed and accuracy of responding. Each participant
responded to all 100 basic subtraction facts, in an order randomly determined by the software. A short rest break was provided to participants
halfway through the experimental trials. On each trial, participants saw a
problem in column form, centered on the screen, and stated the answer out
loud.
Results
Spoiled trials, on which the voice key did not register the
participant’s original vocalization or an extraneous noise triggered
the key, accounted for 129 of the observations (4.9% of the total
possible observations). An additional 41 scores (1.6% of the total
possible) were eliminated when they exceeded the p ⬍ .01 criterion for extreme scores (Dixon, 1953). Extreme scores were
screened by examining all of a participant’s correct trial RTs
within a problem size category. The comparable values in Experiment 2 were 5.4% for spoiled trials and 1.5% for outliers.
Chronometric Measures
The grand mean for correct RTs was 1,136 ms. The participants’
average RT was 908 ms for small problems and 1,363 ms for large
problems; these values were remarkably similar to those reported
by Campbell and Xue (2001): 947 ms and 1,386 ms for small and
large problems, respectively. The 455-ms problem size effect was
significant in a simple within-subjects analysis of variance
(ANOVA), F(1, 25) ⫽ 74.24, MSE ⫽ 36,246 (unless otherwise
ELEMENTARY SUBTRACTION
reported, all significant effects reported here achieved at least the
p ⬍ .05 level of significance). The overall error rate was 8.4%.
Mean error rates were 2.7% for small problems and 14.2% for
large problems, a significant difference, F(1, 25) ⫽ 60.99, MSE ⫽
28.02. These rates were somewhat higher than those reported by
Campbell and Xue for subtraction (3.1% and 7%, respectively, for
small and large problems).
In Experiment 2, small problems averaged 873 ms, with 1.6%
errors, and large problems averaged 1,351 ms, with 11.1% errors.
Both problem size effects were significant: RTs, F(1, 19) ⫽ 48.62,
MSE ⫽ 47,048.1, and errors, F(1, 19) ⫽ 19.49, MSE ⫽ 4,607.
We examined the problem size effect further, as a function of
the different operands in the problem, in an attempt to go beyond
the relatively crude two-level ANOVA factor. When RT was
examined by remainder, a relatively ordinary problem size effect
was apparent, with RT increasing in linear fashion from the
800-ms to the 1,400-ms level; remainder correlated .63 with RT
for the 100 problems. A similar increase was found as a function
of subtrahend, which correlated .42 with problem RT. Regression
analysis revealed a stronger relationship between RT and minuend
(r ⫽ .74); when the minuend was included in the regression model,
adding subtrahend or remainder contributed only 2% to the variance accounted for. Quite surprisingly, the RT pattern for minuend
was not a continuously increasing, monotonic function. Instead, a
pronounced jump in RT was apparent, beginning with minuend 11.
Fully 25 of the 26 participants showed the stair-step or discontinuity pattern in their RTs, 20 of them at minuend 11 and the
remaining 5 at either 12 or 13. Computing the difference in RT for
these adjacent minuends (e.g., 10 vs. 11), the smallest increase for
the 25 participants was 172 ms, and the largest was 2,071 ms. The
same pattern was obtained in Experiment 2, in which all 20
participants showed the discontinuity. Figure 1 displays mean RT
profiles across minuends for both experiments.
Error rates in both experiments (Figure 2) also revealed the
stair-step function, with a major discontinuity at minuend 11 (not
1341
Figure 2. Mean error percentages, by minuend, for the elementary subtraction facts from Experiments 1 and 2.
surprisingly, the same stair-step function was also found with
standard deviations: from 200 –300 ms for small problems but
from 600 –900 ms for large problems). The substantial jump in
these measures, occurring at the same point in the minuends,
signals an important change in underlying mental processing. One
likely interpretation is that performance on the larger subtraction
facts involves a mixture of processing operations. In other words,
relative uniformity of mental processes typical with smaller problems may have given way at the larger problems to a mixture of
solution processes, similar to LeFevre et al.’s (1996) interpretation
of patterns in simple addition.1
The tent-shaped frequency distribution of problems across minuends means that problems with minuends 9 and 10 are the most
frequent in the basic subtraction facts, with minuends 8 and 11 the
next most frequent, and so forth. This rules out the possibility that
the discontinuity at minuend 11 might be due to reduced sample
size or some other statistical artifact. On the other hand, there was
a pronounced decrease at minuend 18 for both performance measures, in both experiments, surely because only one basic subtraction fact has this minuend (18 – 9 ⫽ 9). This would reduce the
stability of the mean from a statistical standpoint, but it would also
reduce the uncertainty of responding from the participants’ standpoint (i.e., minuend 18 does not participate in several subtraction
facts the way that minuends 11 and 12 do).
Error Analysis
As shown in Table 1, errors were most frequently within 2
numbers of the correct answer for both small and large problems.
Incorrect answers wrong by more than 2 became very infrequent,
although there was a slight increase in answers wrong by 4 or more
within small problems. Inspection showed that these errors were
almost invariably of two types: repetition of either the minuend or
1
Figure 1. Mean reaction times (RTs), by minuend, for the elementary
subtraction facts from Experiments 1 and 2.
Although not as pronounced, the problem size effect in LeFevre et al.’s
addition data (see especially their Figures 1 and 2) demonstrates a discontinuity, as do their unpublished subtraction data (J. LeFevre, personal
communication, October 24, 2002).
SEYLER, KIRK, AND ASHCRAFT
1342
Table 1
Frequencies and Percentages of Errors in Experiments 1 and 2,
Tallied Separately for Small (n ⫽ 55) and Large (n ⫽ 45)
Problems
Problem size
Small
Answers wrong by ⫹/⫺
1
2
3
4
5
6
Total
Large
No.
%
No.
%
27
7
3
9
3
3
52
51.9
13.5
5.8
17.3
5.8
5.8
100
126
66
14
8
3
1
218
57.8
30.2
6.4
3.7
1.4
0.5
100
subtrahend (e.g., 9 – 6 ⫽ 9 or 12 – 9 ⫽ 9) or what seemed to be
a counting up or down association to the subtrahend (e.g., 11 – 3 ⫽
4 or 12 – 9 ⫽ 8). We judged the three instances of “zero errors”
(e.g., 4 – 0 ⫽ 0) to be repetition errors instead of confusion with
the zero rule from multiplication, because no other errors seemed
to stem from confusion with other arithmetic operations.
Correlation Findings
For both experiments, we correlated RTs, error rates, standard
deviations, and the two anxiety measures; we also included each
participant’s RT difference between small and large problems (i.e.,
the size of each participant’s stair step). Because only the customary patterns were observed (e.g., r ⫽ .84 between RTs and standard deviations and r ⫽ .33 between math and test anxiety), we do
not consider these results further.
Summary
There is a strong problem size effect in subtraction, just as in the
other arithmetic operations, although the size of the effect is larger
than in addition or multiplication. The shape of the problem size
effect, however, is dramatically different from the customary finding of a continuous, monotonic function. All dependent variables
examined here showed a pronounced discontinuity at minuend 11,
signaling a substantial shift in processing between smaller and
larger subtraction facts. Smaller facts, with short latencies and low
error rates and variability, probably rely quite heavily on simple
retrieval from long-term memory. Larger facts, in contrast, seem
not to be processed in such a homogeneous fashion. Instead, they
seem to call forth a mixture of processes, probably with relatively
greater reliance on reconstructive strategies. To our knowledge,
every study that has examined strategy-based trials separately has
shown them to be slower and far more prone to error than retrieval
(e.g., Campbell & Xue, 2001; Geary et al., 1993; LeFevre et al.,
1996; Siegler, 1987). Interestingly, however, none of the published
studies has noted both the unusually shaped problem size effect
and the degree of discontinuity that we report here. As an archival
resource, we present problem-by-problem results (RTs, error rates,
and standard deviations) in the Appendix.2
Experiment 3
Our interpretation of the stair-step function in Experiments 1
and 2 is that participants were relying on strategies for problem
solution on some appreciable portion of the trials, rather than
simpler retrieval from long-term memory. To test this directly, we
had participants in Experiment 3 respond in the standard timed
subtraction task and then provide immediate verbal reports on their
solution processes. We expected to find reconstructive strategies
such as counting and transformation being used far more heavily
on the larger facts, corresponding to the elevated portion of the
problem size function beginning at minuend 11. An earlier study
(Kirk & Ashcraft, 2001) examined many of the methodological
challenges involved in collecting valid, reliable math strategy
reports. We concluded there that math strategy research could
provide valuable insight into the cognitive processing of basic
arithmetic but that research involving verbal strategy reports must
be done in accordance with recommended methodology (see
Ericsson & Simon, 1993, for a detailed review of these issues) and,
in particular, under neutral experimental instructions.
There is an extensive literature on the use of verbal reports in
cognitive tasks (e.g., see Crutcher, 1994; Ericsson & Simon, 1993;
Payne, 1994; Russo, Johnson, & Stephens, 1989; Wilson, 1994)
that reflects a general consensus on two main sources of concern
when such reports are used: veridicality and reactivity (for a
thorough discussion of these issues in the context of a math task,
see Kirk & Ashcraft, 2001). Veridicality concerns whether verbal
reports accurately reflect underlying cognitive processes. Ericsson
and Simon (1993) cited two types of cognitive processes that
cannot be tapped by verbal reports: thoughts that either are not
already in or cannot be easily transferred into a verbal code and
processes that are unmediated and highly automated. Only information that enters short-term memory and is attended can be
reported; thus, verbal reports are informative only for sequential,
time-ordered processes (Payne, 1994). Note that if two processes
occur in parallel, such as relatively rapid retrieval from long-term
memory along with a slower, strategy-based solution (e.g., Ashcraft, 1992; Compton & Logan, 1991), the verbal report will likely
describe the slower process, even though the faster process governs the overt, timed response. The converging time and accuracy
measures included here were intended to address these issues.
Reactivity, an issue of potentially greater concern, refers to the
possibility that the requirement to give verbal reports may alter
participants’ thought processes. For instance, having to generate a
verbal report might induce participants to rely on a slower but
more easily described process for problem solution or result in
2
We are grateful to an anonymous reviewer for reminding us that many
other processes, such as encoding and retrieval, vary in their time to
completion as a function of numerical magnitude and will therefore be part
of the increasing RT profile across minuends (see Banks, 1977, for example, on magnitude, distance, and congruity effects). Indeed, as noted by
several authors, even the frequency with which we encounter numbers,
both as digits and as words, varies inversely with magnitude (Kučera &
Francis, 1967). Thus, problems with larger minuends here also tend to have
larger subtrahends and remainders (r ⫽ .707 for both correlations with
minuend), and processes not directly related to subtraction will contribute
to latencies and errors. On the other hand, magnitude and distance effects
tend to be small relative to the problem size effects found here. The
answers that participants generated in the present tasks varied from 0 –9, a
range showing tremendous variation in occurrence frequency (Kučera &
Francis, 1967) but only minimal variation in RT and errors here; word
frequency differences for values 11 through 18 are minimal, although these
minuends showed large effects here.
ELEMENTARY SUBTRACTION
participants feeling obligated to report a strategy they had not used
(i.e., a situation in which participants respond to the demand
characteristics of the experiment). Kirk and Ashcraft (2001) demonstrated that biased instructions could have a similar, reactive
effect on verbal reports and performance. Awareness of the need to
report may also consume processing resources and may bias participants to trade speed for accuracy (Russo et al., 1989). Any of
these effects should cause discrepancies between verbal reports
and other performance measures. Consistent with the methodology
used in Kirk and Ashcraft (2001), we included a silent control
condition here as a direct check on such reactivity effects. We also
gave participants appropriate training in providing verbal reports,
along with experimental instructions designed to avoid bias and
demand characteristics.
Method
Participants
Forty-five participants from undergraduate psychology courses at Florida State University were tested in exchange for credit in their classes.
Materials
Participants completed the same demographic information sheet and
math anxiety scale used in Experiments 1 and 2. They also completed a
standard math fluency measure (French, Ekstrom, & Price, 1963) to assess
their ability to work on basic arithmetic operations with both speed and
accuracy. Participants performed the same simple subtraction task used in
the first two experiments and gave verbal reports after each trial.
Procedure
The experiment consisted of two sessions, spaced 1 week apart. In
Session 1, 30 participants were randomly assigned to the verbal report
condition, and the remaining 15 were assigned to a silent control condition.
They were given the instructions and verbal report training described
subsequently, after which they completed the subtraction task and ancillary
instruments. They reported to the laboratory the following week for Session 2, designed to explore verbal reporting characteristics under instructions emphasizing either speed or accuracy, with half of the participants
assigned to each emphasis. The 15 silent control participants performed
silently again, under the same instructions as their first session (i.e., equal
emphasis on speed and accuracy). The ancillary measures were also readministered in the second session, and the order of computer and ancillary
measures was counterbalanced across sessions.
Instructions and Verbal Report Training
Immediately before the subtraction task, all participants were given
instructions and practice in how to provide concurrent and retrospective
verbal reports. The report training instructions (Ericsson & Kirk, 2000,
2001) were designed to facilitate valid, reliable verbal reports on short RT
stimuli. When originally developed and tested on the basic addition facts,
they yielded persuasive evidence of veridicality and reliability. For example, intercoder reliability was high, there were no significant differences
between reporting and control groups on either latencies or errors, and
there were discrepancies among untrained participants between the strategies they reported and their RT and accuracy profiles, as compared with
consistent, converging patterns of reports, RTs, and accuracy among
trained participants (Ericsson & Kirk, 2001).
The experimenter conducted report training by reading explanatory
instructions and asking participants to answer practice questions with
concurrent, think-aloud reports. On some problems, participants were also
1343
immediately asked to give a retrospective report of the exact thoughts they
had from stimulus onset until answering the practice question. Some
practice questions could be answered by direct retrieval, and some required
multistep solutions. Report training averaged 15–25 min in Session 1 and
5–10 min on an abbreviated set of instructions in Session 2. To avoid
biasing participants’ reports, no examples of strategies or reports were
provided during training.
The practice problems emphasized two points. First, questions solved by
direct access demonstrated that it was acceptable to have nothing to report
beyond the problem answer. Second, questions solved by multistep solutions helped participants learn to distinguish between reports of actual
thoughts, on the one hand, and summaries, explanations, or other nonveridical accounts of their thinking, on the other. That is, participants
learned that a valid retrospective report needed to resemble their concurrent
report of the problem. Emphasizing these points addressed the most serious
threats to validity: reporting processes that did not occur, summarizing or
interpreting processes that did occur, and altering routine problem-solving
processes to have something to report.
After this training, report participants were told that they would be
giving only retrospective reports on the main experiment. Control participants were told that they would not be giving verbal reports at all but
would instead repeat the problem and answer on each trial. All participants
were told that they should solve the subtraction problems as quickly and
accurately as possible and then report the answer out loud. Verbal reports
were tape recorded for later transcription.
Verbal Report Coding
When participants reported only the problem answer, the trial was scored
as “retrieval.” So-called “addition reference” solutions indicated that the
problem was solved with reference to the corresponding basic addition
fact; for example, in the case of 15 – 8, a participant might respond “7 ⫹
8 ⫽ 15, so 7.” Because the participant had to first retrieve the subtraction
problem answer, 7, to formulate the addition fact 7 ⫹ 8, these trials were
also scored as “retrieval.” A report was scored as “counting” if a participant
reported an ascending or descending counting sequence (e.g., for 11 – 3,
counting down “10, 9, 8” and then stating “8” as the answer; see Woods et
al., 1975). A report was scored as “transformation” if a participant reported
changing the problem to a simpler, known problem (e.g., for 15 – 8,
decrementing both values by 5 and then solving 10 – 3 ⫽ 7). Other solution
methods, including “I don’t know,” were scored as “other.”
Results
Session 1
The overall error rate was 3.2%, and this rate did not vary as a
function of report group. Another 4.4% of the trials were spoiled
as a result of inaccurate firing of the voice key. We altered our
procedures for detecting and eliminating extreme scores to ensure
that infrequent minuend values, especially when examined separately by reported solution method, would have sufficient observations. In brief, if a participant’s mean RT at a particular minuend
fell outside of a critical range—the group’s mean ⫾2.5 SDs at that
minuend—then the participant’s individual RTs were screened. If
a trial RT was extreme, it was excluded, a new critical range was
computed, and the eliminated score was replaced by the upper
bound of the new critical range. Only 17 scores (0.4% of valid
scores) were replaced through the use of this procedure. Finally, 4
participants were missing data for minuend 0, 1, or 18, so we
substituted the group’s upper bound at the minuend in question for
the missing score. Despite the errors, spoiled trials, and extreme
scores, fully 92.5% of the possible observations were available for
analysis.
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SEYLER, KIRK, AND ASHCRAFT
Consistent with prediction, requiring verbal reports did not alter
subtraction performance to any detectable degree. There were no
significant report group effects on RTs or standard deviations. The
similarities between the RT profiles of the silent control and verbal
report groups suggested that reactivity was not an issue in our
results. The stair step between minuends 10 and 11 in the problem
size effect was even greater than in the first two experiments, a
mean RT increase of 850 ms; in addition, the effect was significant
across all minuends, F(18, 756) ⫽ 48.81, MSE ⫽ 240,460. Standard deviations were in the 200 –300-ms range for small problems
but ranged from 650 – 800 ms for large problems.
Mean RTs for the large facts were also somewhat longer than
those in Experiments 1 and 2, ranging between 1,789 and 2,147
ms. This difference was most likely attributable to the sample.
Because there were no differences between report groups, the
longer RTs cannot be attributed to having participants give verbal
reports, a tendency that is sometimes observed in report data
(Ericsson & Simon, 1993).
As found in the first two experiments, the error rate increased
sharply at minuend 11; the main effect of minuend on errors was
significant, F(18, 756) ⫽ 2.58, MSE ⫽ 109.06. Overall, error rates
varied between 1% and 3% on the smaller subtraction facts but
ranged from 4%– 8% on those at minuend 11 and beyond. There
was also a Minuend ⫻ Report Group interaction, F(36, 756) ⫽
1.68. This interaction was probably attributable to participants in
one of the report conditions who had an error rate of 12% at
minuend 11, as compared with rates of 2% to 4.4% among the
other groups. Inspection of individual participants’ errors revealed
that the error pattern was general for the condition rather than
being attributable to a subset of inaccurate individuals.
Overall, participants reported using a strategy other than retrieval on 15% of all trials (8% used counting, 6% used transformation, and less than 1% used another type of strategy). The types
of strategies reported were generally consistent with those described by Geary et al. (1993). More interestingly, the verbal
reports also showed the discontinuous pattern between small and
large facts found in the other performance measures. Reported
strategy use ranged between 0.7% and 7.7% for small problems
(M ⫽ 3.2%) but ranged between 28% and 46% for large problems
(M ⫽ 33.4%), across-minuend, F(18, 504) ⫽ 18.21, MSE ⫽
431.64. At the critical adjacent pair of minuends, reported strategy
use jumped from 2.8% at minuend 10 to 34% at minuend 11.
Figure 3 displays percentages of reported nonretrieval strategies
across minuends; frequencies of retrieval and nonretrieval reports
on a problem-by-problem basis are shown in the Appendix. Interestingly, the nonretrieval percentages obtained here were considerably smaller than those reported by Campbell and Xue (2001;
27% for small problems and 58% for large problems).
With such a substantial percentage of performance attributed to
nonretrieval, it is possible that the problem size effects found in
Experiments 1 and 2 were largely or even completely due to slow,
effortful strategies. In like manner to LeFevre et al.’s (1996)
procedure, we reanalyzed RTs from the verbal report groups here,
examining only those trials classified as “retrieval.” Figure 4
presents the results of this analysis, the top curve showing the
problem size effect on RT for all trials and the lower curve
showing the effect for retrieval-only trials. Although a considerable part of the now-familiar stair-step function is clearly due to
slower strategies, there just as clearly remains a prominent increase
in the RT function when retrieval-only trials are examined: F(18,
Figure 3. Mean percentages of trials reported as solved by a strategy
other than retrieval, by minuend: Experiment 3.
504) ⫽ 20.35, MSE ⫽ 144,172, for the retrieval-only analysis
across minuends. Furthermore, the differences in RTs between the
two curves were significant according to a simple ANOVA of
difference scores (overall RT minus RT for retrieval-only trials),
F(18, 504) ⫽ 9.17, MSE ⫽ 227,113. Processing through reconstructive strategies was significantly slower than retrieval, as is
commonly reported, and clearly accounted for part of the stair-step
increase in RTs across problem size. There was nonetheless a
substantial “residual stair step”—the increase in RTs that remained
after nonretrieval trials had been excluded—that cannot be easily
attributed to such strategies.
We have several reasons to believe in the validity and reliability
of these verbal report data. First, intercoder reliability for strategy
coding was .98, suggesting that there was little doubt as to how
particular reports should be categorized. Second, 30% of our
reporting participants claimed to have used retrieval exclusively,
and another 20% reported using retrieval on at least 90% of their
trials. Reactivity to the report requirement (e.g., Russo et al., 1989)
was apparently not widespread, if it occurred at all. Finally, there
were no significant differences between the report and silent
control groups in terms of RTs, standard deviations, or error rates.
The absence of such differences, the high intercoder reliability,
and the converging evidence for the familiar stair-step pattern
across minuends together support the veridicality of the verbal
strategy reports.
Session 2
As noted previously, Session 2 was an exploratory session in
which the usefulness of verbal reports across repeated testing was
investigated, along with a manipulation of speed versus accuracy
emphasis in the instructions. The Session 2 data are not directly
comparable to the results presented so far because of the inherent
practice effect of multiple sessions; thus, we limit our discussion of
this session to two general characteristics.
First, regardless of report condition, all three groups once again
showed the familiar stair-step function at minuend 11, on both
RTs, F(18, 756) ⫽ 48.85, MSE ⫽ 201,831, and errors, F(18,
756) ⫽ 3.69, MSE ⫽ 84.99. These results clearly rule out lack of
ELEMENTARY SUBTRACTION
1345
eliminated from consideration. We pursue the source of this “residual stair step” in the General Discussion section.
Experiment 4
Figure 4. Mean reaction times (RTs), by minuend, on elementary subtraction facts for all correct trials and trials reported as “retrieval”: Experiment 3.
recent practice as an explanation for the discontinuous function
found in Experiments 1–3. Second, verbal strategy reports also
showed the discontinuity, with reports of strategy use for problems
at minuends 10 and 11 jumping from 3.4% to 41% in the speeded
condition and from 4% to 34% in the accuracy condition. Thus, the
same marked increase in strategy reports for minuends 11–17 was
found in both sessions. As a result, lack of recent practice is also
an inadequate explanation of participants’ reported reliance on
strategies in Session 1. Participants’ use of reconstructive strategies on roughly one third of the larger subtraction facts appears to
reflect their normal problem-solving procedures and does not
appear to be some short-lived effect due to relative inaccessibility
or lack of practice.
Correlation Findings
All variables showed high test–retest correlations across sessions (e.g., r ⫽ .89 for RTs, r ⫽ .81 for strategy use, and r ⫽ .90
for math anxiety scores). The analysis supported the general findings that strategy use is associated with slower performance (e.g.,
r ⫽ .60 for Session 1 RTs and strategy use) and that slower
performance is associated with lower math fluency scores (e.g.,
r ⫽ ⫺.61 for Session 1 RTs and math fluency scores). As found in
studies of simple addition and multiplication (e.g., Faust, Ashcraft,
& Fleck, 1996), the correlations between math anxiety and performance on basic subtraction facts were either weak (r ⫽ .40 for
error rates in Session 1) or nonsignificant. No other important
outcomes were observed in the correlations, so this analysis is not
pursued further here.
Summary
The elevated portion of the problem size effect evident in RTs
and errors corresponded exactly with the self-reported increase in
the use of strategies for problem solution. The problem size effect
in basic subtraction, according to the subsequent analyses, was
partially a function of slower, reconstructive strategy use, although
it retained a distinctive stair-step shape when strategy trials were
Given that strategy-based solutions to simple arithmetic problems are considerably slower and more error prone than retrievalbased solutions (e.g., Geary et al., 1993; LeFevre et al., 1996), it
seems especially likely that these solutions would place a particularly heavy load on WM. We tested this hypothesis in Experiment 4.
Two methods for testing the involvement of WM are in general
use. In the individual-differences approach, participants are given
a WM span assessment and are then tested on some task of interest
in a between-groups manipulation of span. Group differences on
the task are then interpreted as due to differences in WM capacity
assessed by span tests. In the dual-task approach, performance on
a primary task is examined while participants perform a concurrent
secondary task, one known to place demands on WM. As the
difficulty of the secondary task increases, performance on the
primary task is expected to suffer to the extent that it depends on
WM capacity for successful execution, or, likewise, secondary task
performance begins to suffer as the primary task increases in
difficulty. Thus, in general, this approach looks for an interaction
between task (dual vs. control) and processing difficulty (where
difficulty pertains to the primary task, the secondary task, or both).
We elected to use both approaches here (see Rosen & Engle,
1997, for a comparable design). That is, we performed a dual-task
experiment and also obtained WM span scores for the participants,
categorizing them into low, medium, and high memory span
groups. Specifically, we tested subtraction (primary task) concurrently with performance on a short-term memory letter recall task
(secondary task) of graded difficulty. Large subtraction facts, those
identified as frequently being performed with reconstructive strategies, were expected to suffer particularly when the secondary task
placed heavy demands on WM. That is, if the strategies used
frequently for large subtraction facts indeed rely on conscious,
WM-intensive processing, then performance on those facts should
suffer under heavy memory loads due to the secondary task;
alternatively, trials that load WM most heavily in the secondary
task should suffer the greatest interference from subtraction if
subtraction relies on WM resources. Furthermore, we expected
groups formed on the basis of memory span to be differentially
affected by the graded manipulations of difficulty in the subtraction and letter recall tasks. On the other hand, if the strategies
reported for larger problems are relatively well practiced and
automatic, as might be expected after years of practice (e.g.,
Siegler & Jenkins, 1989), then there might be only minimal interference from a demanding secondary task and little or no effect of
the categorization into low, medium, or high span groups.
Method
Participants
Thirty lower level undergraduates from the Cleveland State University
psychology department participant pool received course credit for taking
part.
Instruments
Participants were assessed with the Listening Span (L-span) task of the
Salthouse–Babcock (1990) WM capacity test. This task requires the par-
1346
SEYLER, KIRK, AND ASHCRAFT
ticipant to hold and then recall an increasing amount of information in WM
while simultaneously processing test information; thus, it requires both
storage and processing operations to occur concurrently. After completing
the span assessment, participants were tested on the basic facts of subtraction in the same production task format as in the first three experiments.
Procedure
In the L-span task, the participants were presented sentences orally,
beginning with 3 one-sentence sets, then 3 two-sentence sets, and so forth.
In the case of the one-sentence sets, participants heard the sentence and
then answered a question based on its meaning (e.g., The boy ran with the
dog. Who ran? The boy). The question (e.g., Who ran?) was presented in
a multiple-choice format on the front of the answer sheet. To complete the
trial successfully, participants had to answer the question correctly (The
boy) and then recall the final word of the sentence (dog), writing this word
on the opposite side of the answer sheet. If at least two trials were
completed successfully at the one-sentence level, the participant moved to
the two-sentence level. At this level, two sentences were presented successively, each with a question to be answered, after which the final words
of both sentences had to be recalled. As before, if at least two of the three
trials at this level were completed correctly, the participant was tested with
three sentences, and so on. If the participant passed the three-sentence test
but failed at the four-sentence level, his or her WM span score was 3, the
hardest level at which the participant succeeded.
After completing the L-span task, participants completed the subtraction
task on the computer. The subtraction phase was divided into three conditions, the dual-task condition along with the two control conditions,
subtraction only and letters only. For all three tasks, participants were
urged to respond rapidly while maintaining accuracy; equal emphasis was
placed on subtraction accuracy and letter recall accuracy.
Each of the 36 dual-task trials consisted of three main events. After a 1-s
fixation point, the letter set was presented (Event a), and participants were
encouraged to name the letters aloud once; the 2-, 4-, and 6-letter sets were
shown for 1.5, 2.5, and 3.5 s, respectively. The letters were then removed
from the screen, and after a 1-s interval the subtraction problem was
presented (Event b), centered on the screen. Participants responded verbally, as in the previous experiments. After a 1-s delay, the words “RECALL LETTERS” appeared, prompting recall of the previously presented
letter set (Event c). The experimenter recorded letter recall data, keyed in
the participant’s response to the subtraction problem, and then triggered the
next trial.
The two component tasks were assessed separately to obtain reliable
baseline estimates for control purposes. In the subtraction-only task (36
trials), participants responded to the set of subtraction facts in regular
fashion. To control for overall verbalizations, we showed participants letter
sets (Event a) comparable to those in the dual task before the subtraction
problem (Event b) and asked them to verbalize the letter set again (Event
c) after they gave the subtraction answer. To eliminate the memory demand
of the letter task, we replaced the words “RECALL LETTERS” with the
actual letter set so that participants had only to read the letters off the
screen. Similarly, in the letters-only control condition (24 trials), participants saw and recited a 2-, 4-, or 6-letter set (Event a), saw a subtraction
problem and verbalized its answer (Event b), and then had to recall the
letter set (Event c). To eliminate the mental processing of subtraction, we
showed the answer along with the problem during Event b so that participants could merely read the answer off the screen. Thus, both control tasks
included verbalizations of the letters and subtraction answers, but selectively omitted either the secondary letter recall task or the primary subtraction task. Finally, we had to prevent participants in the letters-only task
from adopting a short-cut strategy—naming the subtraction answer very
rapidly and thereby shortening the retention interval for the letters—for
improving their letter recall. To accomplish this objective, we kept the
subtraction problem and answer on the screen for 1.5 s and required
participants to read the entire subtraction problem out loud before the
screen advanced to the recall instruction.
Letter sequences were randomly constructed by selecting 2, 4, or 6
consonants and then excluding any combinations that contained meaningful abbreviations or acronyms and reordering sequences as necessary to
avoid obvious alphabetic sequences; different letter sets were used in each
task. We randomly sampled 36 subtraction facts for use, excluding from
the pool ties and problems with a minuend of 0 or 10; the computer
software randomized the order of presentation in each task. The order of
tasks (letters only, subtraction only, or dual task) was counterbalanced
across participants. After completing the computer-based subtraction test,
participants completed the math anxiety assessment and then were debriefed and excused. Because no noteworthy results were obtained with
this assessment, the results are not considered here.
Results
Listening Span Task
Participants’ scores on the L-span task ranged from 1 to 5. For
purposes of analysis, we classified the 12 participants with scores
of 1 or 2 as low capacity, the 11 participants with a score of 3 as
medium capacity, and the 7 participants with scores of 4 or 5 as
high capacity.
Error Analysis
Examination of the subtraction errors committed in the
subtraction-only and dual-task conditions revealed a pattern that
was completely redundant with the patterns shown in Experiments
1 and 2, so this analysis is not pursued further here.
Chronometric Measures
In all three tasks, a trial was scored as an error if the subtraction
answer was incorrect or if the participant failed to recall the entire
letter sequence in order. Subtraction errors accounted for 11.5% of
trials in the subtraction-only condition and 12.9% of trials in the
dual-task condition. Letter recall errors occurred on 13.1% of the
letters-only trials but 28.9% of the dual-task trials. Despite instructions to treat both tasks as equally important, participants here
clearly demonstrated the interfering effect of concurrent processing more on the letter recall task.
An additional 4% of subtraction-only RTs and 7.2% of dual-task
RTs were spoiled as a result of extraneous noises or microphone
sensitivity difficulties. Finally, 2.9% of the subtraction-only trials
were excluded because of extreme RTs ( p ⬍ .01, Dixon’s test).
The comparable extreme score rate in the dual-task condition
(0.07%) was much lower, largely because trials with especially
long RTs commonly ended with incorrect letter recall and so had
already been dropped from the analysis.
Control Conditions
The control conditions were analyzed first, in an ANOVA with
WM span group (low, medium, or high) as a between-subjects
variable and problem size (small or large) and set size (2, 4, or 6
letters) as within-subject variables. Briefly, recall errors in the
letters-only task yielded a main effect of set size, F(2, 54) ⫽ 54.01,
MSE ⫽ 228.77; error rates were 1.1%, 8.6%, and 29.6% for the 2-,
4-, and 6-letter sets, respectively. In the subtraction-only analyses,
both correct RTs and subtraction error rates showed the familiar
stair-step pattern of problem size effects, F(1, 27) ⫽ 24.14, MSE ⫽
412,123.01, and F(1, 27) ⫽ 23.4, MSE ⫽ 88.70, respectively.
ELEMENTARY SUBTRACTION
Subtraction errors also revealed an interaction between problem
size and span group, F(2, 27) ⫽ 3.38, MSE ⫽ 88.70. The low span
group averaged 12.5% errors on small problems, as compared with
error rates of 7.5% and 4.0%, respectively, for the medium and
high capacity groups. The groups did not differ systematically on
large problems; mean error rates ranged from 12.1% to 16.7%.
Dual-Task Analyses
In the dual-task analyses, the task factor (dual vs. either subtraction only or letters only) was added to the design just described, yielding a four-factor design with one between-subjects
variable.
RT evidence. The ANOVA focusing on correct RTs compared
dual-task performance and subtraction-only performance. There
was a significant (504 ms) problem size effect, F(1, 27) ⫽ 27.97,
MSE ⫽ 772,290, and a 159-ms increase in average latencies in the
dual-task condition, F(1, 27) ⫽ 7.47, MSE ⫽ 288,339. This
slowing of RT in the dual-task condition was also observed in the
Task ⫻ Set Size interaction, F(2, 54) ⫽ 3.25, MSE ⫽ 51,646.
Subtraction performance was 70 ms slower overall in the dual-task
condition than in the subtraction-only condition when the letter set
task involved only 2 letters. However, RTs were approximately
200 ms slower for 4- and 6-letter trials in the dual task.
Errors in subtraction. The analysis of subtraction errors revealed a significant problem size effect, F(1, 27) ⫽ 22.2, MSE ⫽
201.05; error rates averaged 8.6% for small problems and 15.8%
for large problems. The Task ⫻ Set Size interaction was also
1347
significant, F(2, 54) ⫽ 3.31, MSE ⫽ 179.62, showing a somewhat
steeper increase in subtraction errors for 6-letter trials than for
either 2- or 4-letter trials. Finally, the three-way interaction among
task, problem size, and WM span group was significant, F(2,
27) ⫽ 4.28, MSE ⫽ 107.38. The source of the interaction appeared
to be a shallower problem size effect (an increase of 3.2%) for high
span participants in the dual-task condition than for either low or
medium span participants in that condition (9.7% and 9.6% increases, respectively), along with the surprisingly inaccurate performance of the low span participants in both tasks; their error
rates for small and large problems were 12.5% and 16.2%, respectively, in the subtraction-only task and 10.7% and 20.4%, respectively, in the dual task. These results are consistent with the
hypothesis that larger subtraction problems depend more heavily
on WM resources, a situation that puts low span participants at a
special disadvantage.
Errors in letter recall. The analysis of letter recall errors
revealed considerable evidence of dual-task interference (see Table 2 for the ANOVA summary). The percentage of trials with
letter recall errors increased significantly in the dual-task condition
(28.9%) relative to the letters-only condition (13.1%) and increased substantially as set sizes grew larger; error rates were
4.6%, 14.0%, and 44.5%, respectively, for the 2-, 4-, and 6-letter
sets. Span groups also differed significantly in their overall error
rates; the low span group averaged 27.5% errors overall, as compared with 21.5% for the medium span group and 14.0% for the
high span group.
Table 2
Analysis of Variance Summary for Letter Recall Errors: Experiment 4
Source
df
F
MSE
p
Between subjects
Listening span group (G)
Subjects (S) within group error
2
27
⬍.025
4.58
1,066.25
Within subjects
Task (T)
T⫻G
T ⫻ S within-group error
Problem size (P)
P⫻G
P ⫻ S within-group error
Letter set size (L)
L⫻G
L ⫻ S within-group error
T⫻P
T⫻P⫻G
T ⫻ P ⫻ S within-group error
T⫻L
T⫻L⫻G
T ⫻ L ⫻ S within-group error
P⫻L
P⫻L⫻G
P ⫻ L ⫻ S within-group error
T⫻P⫻L
T⫻P⫻L⫻G
T ⫻ P ⫻ L ⫻ S within-group error
a
Computed p value.
1
2
27
1
2
27
2
4
54
1
2
27
2
4
54
2
4
54
2
4
54
⬍.001
⬍.025
134.94
4.71
157.16
3.42
1.76
193.47
123.10
3.06
402.02
12.16
2.30
184.09
25.50
0.35
168.11
3.27
1.23
⬍.076a
ns
⬍.001
⬍.025
⬍.005
ns
⬍.001
ns
⬍.05
ns
105.12
0.46
0.66
ns
ns
133.11
1348
SEYLER, KIRK, AND ASHCRAFT
These main effects, as well as the nearly significant main effect
of problem size, were all qualified by several two-way interactions, as shown in Table 2. Problem size interacted with set size in
the analysis. A moderate (5%) problem size effect was obtained on
the 2- and 4-letter trials, with error rates in the 2% to 16% range.
On 6-letter trials, however, recall errors were quite high for both
small (45.1%) and large (43.9%) problems.
Two important double interactions were obtained with the task
variable, the Task ⫻ Problem Size interaction and the Task ⫻ Set
Size interaction; both indicated the damaging effects that subtraction exerted on letter recall as a result of competition for processing capacity. In the first interaction, shown in Figure 5, it is clear
that the burden of dual processing fell especially heavily on large
subtraction problems. This is consistent with the prediction that
larger problems will suffer more from dual-task interference, because they are performed more frequently by means of reconstructive processing that consumes WM resources. Similarly, the
Task ⫻ Set Size interaction, shown in Figure 6, showed both
consistently worse recall in the dual-task condition and stronger
interference at the larger set sizes; error rates on 2-letter trials
differed by only 6% between tasks, as compared with a 30%
difference for 6-letter trials.
Finally, two significant interactions involved the betweensubjects effect of memory span, further buttressing evidence for a
strong WM component in subtraction performance. First, the increasing interference due to heavier memory loads was characteristic of all three groups but was especially pronounced for the low
capacity group, as shown by the Span Group ⫻ Set Size interaction
depicted in Figure 7. No group differences were apparent at the
smallest memory load. But at the maximal 6-letter load, high span
participants averaged 31.2% errors, as compared with error rates of
45.8% and 56.4%, respectively, among medium and low span
participants. Clearly, low span participants experienced stronger
interference due to a heavy concurrent load on WM. Second, the
increase in errors due to the dual task was apparent in all groups
but was especially strong for the low span group (see the Task ⫻
Figure 5. Mean percentages of letter recall errors for the control versus
dual-task conditions as a function of problem size: Experiment 4.
Figure 6. Mean percentages of letter recall errors for the control versus
dual-task conditions as a function of letter set size: Experiment 4.
Span Group interaction depicted in Figure 8). As would be expected if both difficult subtraction and difficult letter recall depend
on WM capacity, concurrent processing led to greater interference,
in the form of disrupted recall of the memory sets. This interference was particularly pronounced for low span participants, especially when concurrent processing involved a heavy WM load.
Figure 7. Mean percentages of letter recall errors for the low, medium,
and high working memory capacity groups as a function of letter set size:
Experiment 4.
ELEMENTARY SUBTRACTION
1349
in Experiment 3. These interference effects demonstrate the importance of WM processing in subtraction, in that subtraction and
letter storage–recall were clearly competing for the same processing resources. Thus, elementary subtraction relies heavily on WM.
It disrupts concurrent memory load tasks, especially when those
tasks impose a heavy memory load and especially when participants of low WM capacity are tested.
General Discussion
Figure 8. Mean percentages of letter recall errors for the low, medium,
and high working memory capacity groups as a function of control versus
dual task: Experiment 4.
One final aspect of the results bears brief mention. In the most
difficult condition here, large problems performed under a 6-letter
memory load, high span participants averaged 44% errors in their
letter recall. Low span participants approached this level of inaccuracy in their letter recall in the letters-only condition, with an
error rate of 38% on 6-letter trials. That is, when they were not
under the demands of the dual task, the low span group showed
only slightly better performance than the high span participants did
under the dual-task demands (for comparison, the low span group
committed 74% errors in this condition in the dual task). This is
reminiscent of the results of Rosen and Engle’s (1997) study, in
which the demands of the dual task reduced high span participants’
performance to the level achieved by low span participants in a
single task setting. The plausible reason for our result is also
comparable to that offered by Rosen and Engle. Large subtraction
problems require considerable WM capacity, as do maintenance
and recall of a heavy memory load. The combination of the tasks
puts all groups at a disadvantage as a result of competition for
processing resources. Low memory span participants are particularly disadvantaged, however, because they have fewer processing
resources to begin with. Indeed, the letter recall task by itself
damaged low span participants’ performance, whereas for high
span participants, a comparable level of damage was reached only
when they performed the most difficult conditions concurrently.
Summary
The patterns of interference observed in Experiment 4 indicated
a heavy reliance on WM during the processing of subtraction facts,
especially those with minuends of 11 or greater. This, of course, is
the same problem size range that yielded the stair-step functions in
Experiments 1–3 and the dramatic increase in reported strategy use
We conclude with a brief discussion of three points. First,
adults’ mental subtraction seems to consist more heavily of reconstructive, strategy-based processing than either addition or multiplication (see also Campbell & Xue, 2001). Kirk and Ashcraft
(2001) found, with controlled verbal report techniques, that the
bulk of adults’ processing of addition and multiplication can be
attributed to retrieval characterized by a conventional, monotonic
problem size effect. Subtraction, however, seems to rely more
heavily on reconstructive strategies; fully one third of the large
problem trials in Experiment 3 seem to have been performed
through such strategies, along with 58% in Campbell and Xue’s
Canadian sample. Experiment 4 demonstrated an additional effect,
that solving these larger problems depends heavily on WM
capacity.
Second, Siegler’s distribution of associations model seems to be
the most accommodating theoretical framework for the present
results (e.g., Siegler & Jenkins, 1989). According to this model,
problems and strategies accrue associative strength on the basis of
experience, as do problems and answers. If a particular strategy is
repeatedly used for a particular problem, the model suggests that
the accrued strength of the problem–strategy association can easily
be higher than the direct association between problem and answer.
If so, then the more likely solution method for that problem would
be application of the strategy rather than retrieval from memory.
Such a strategy could be selected and initiated more rapidly than
retrieval for the problem, as a result of higher associative strength,
although execution of the multistep strategy would probably still
yield a higher rate of errors.
Finally, although the present results are consistent with this
theoretical position, and although they indeed suggest that strategies are common and dominant in adults’ mental representations of
larger subtraction facts, there are some difficulties with this explanation. A strong test of the Siegler and Jenkins (1989) model
would require an a priori prediction about which problem–strategy
pairs attain sufficient strength for any given participant and which
strategies are most likely for different problems. It is not clear how
to derive such predictions independent of obtained performance
data.
Furthermore, at least three competing explanations can be offered for the “residual stair step” observed in Experiment 3. First,
it might simply be that our verbal report methods did not elicit full
reporting of strategy use, although the congruence of the RT and
error patterns between reporting and silent groups argues against
this interpretation. Second, and somewhat more plausibly, the
residual stair-step pattern might reflect selection and execution of
a “retrieve then check” strategy, which would match the “addition
reference” reports we observed. Participants might retrieve an
answer to a large subtraction fact but, because of their relatively
low confidence in the answer, follow this with an answer-checking
process in which they retrieve the corresponding addition fact
SEYLER, KIRK, AND ASHCRAFT
1350
(e.g., for 15 – 8, responding “7 ⫹ 8 is 15, so 7”). Although this
would represent a genuine multistep process with implications for
RTs and errors, the “check” itself would involve another retrieval
rather than reconstructive processing (unless the corresponding
addition fact itself needed to be processed by means of a strategy;
e.g., see LeFevre et al., 1996).
A third alternative is that the verbal report technique accurately
identified the nonretrieval trials, and the residual stair step depicts
a time- and accuracy-based profile of genuine retrieval from longterm memory. Under this alternative, retrieval of larger subtraction
facts would simply be especially slow and effortful, whether as a
result of lower associative strength of problem–answer pairs,
greater competition with strategy associations, or greater competition from other, similar associative pathways (e.g., Campbell,
1987). Such an explanation has in fact been offered in a rather
different context; Rosen and Engle’s (1997) high WM capacity
participants were said to augment their automatic semantic memory search with a more controlled, deliberate, and relatively slow
retrieval process. Note, however, that it may be difficult to establish this possibility in the context of arithmetic, in which a procedural, computational solution is possible.
Although the present results do not contradict the differences
between subtraction and other operations found in cognitive neuropsychological research (e.g., Cohen & Dehaene, 2000; Lee,
2000), they also do not speak directly to such reported differences.
It is possible, or even probable, that strategy-based performance on
larger facts would be discriminable from retrieval in neuroimaging
studies, although to our knowledge such results have not yet been
reported (but see El Yagoubi, Lemaire, & Besson, 2003, for
event-related potential evidence discriminating between two strategies used in verifying complex addition equations). It is sufficient
here to report the full set of data so that other investigators may
consider these chronometric and strategy-based results more precisely in future cognitive neuroscience studies of mathematical
cognition, perhaps searching for the retrieval–strategy differences
we are reporting. In any event, it is somewhat surprising that a
simple, elementary arithmetic operation routinely taught in second
grade could show such provocative and complex patterns of performance years later, when normal college adults are tested.
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Appendix
Basic Facts of Subtraction
Subtrahend
Remainder
0
1
2
3
4
Minuend
RT
SD
% Error
RET
Non-RET
0
822
178
0
29
0
1
894
296
0
27
0
2
839
186
0
29
1
3
793
159
0
29
1
4
783
127
0
29
1
Minuend
RT
SD
% Error
RET
Non-RET
1
918
270
7
24
1
2
833
238
0
30
0
3
799
166
0
29
0
4
829
324
0
28
0
Minuend
RT
SD
% Error
RET
Non-RET
2
919
351
0
22
1
3
861
174
7
28
1
4
837
344
2
25
1
Minuend
RT
SD
% Error
RET
Non-RET
3
870
232
2
22
0
4
875
285
0
29
0
Minuend
RT
SD
% Error
RET
Non-RET
4
815
218
13
28
0
Minuend
RT
SD
% Error
RET
Non-RET
5
880
315
7
29
0
5
6
7
8
9
5
787
166
0
28
0
6
789
165
0
29
2
7
778
156
0
26
1
8
849
234
0
29
0
9
825
143
0
29
1
5
821
244
0
29
0
6
961
981
2
28
0
7
848
266
2
28
0
8
856
201
7
29
0
9
846
220
0
28
0
10
970
304
0
28
0
5
934
315
2
24
2
6
981
331
0
24
4
7
879
261
2
23
4
8
1,014
390
2
25
3
9
966
337
11
25
5
10
904
210
0
25
2
11
1,422
1,065
13
15
7
5
895
305
2
29
0
6
880
362
0
27
2
7
1,196
338
2
26
3
8
957
337
0
23
2
9
1,041
446
2
27
1
10
990
457
0
28
1
11
1,530
792
2
13
13
12
1,194
570
15
13
11
5
825
209
2
26
1
6
1,029
313
2
28
1
7
1,060
503
9
25
4
8
850
241
2
29
0
9
957
272
2
27
0
10
951
258
2
28
0
11
1,520
961
30
13
12
12
1,582
1,372
7
20
9
13
1,532
827
9
19
11
6
859
212
2
29
0
7
1,015
536
0
27
2
8
992
455
2
25
4
9
1,009
416
4
27
2
10
856
290
0
28
1
11
1,113
392
11
23
6
12
1,450
882
17
16
10
13
1,696
1,023
17
13
10
14
1,400
591
15
18
9
0
1
2
3
4
5
(Appendix continues)
SEYLER, KIRK, AND ASHCRAFT
1352
Appendix (continued)
Subtrahend
Remainder
0
1
2
3
4
5
6
7
8
9
Minuend
RT
SD
% Error
RET
Non-RET
6
881
305
4
29
0
7
940
284
0
28
0
8
1,048
292
4
24
6
9
1,201
461
4
24
6
10
1,126
513
2
26
1
11
1,337
709
2
20
6
12
1,102
467
7
26
2
13
1,614
1,179
2
17
7
14
1,764
944
15
13
14
15
1,428
527
22
18
8
Minuend
RT
SD
% Error
RET
Non-RET
7
857
315
0
28
0
8
903
248
2
30
0
9
999
393
2
22
5
10
967
523
2
15
0
11
1,763
1,105
4
16
8
12
1,322
662
9
18
8
13
1,477
664
11
19
6
14
1,079
518
9
23
4
15
1,630
1,055
22
16
11
16
1,723
1,065
17
11
12
Minuend
RT
SD
% Error
RET
Non-RET
8
854
203
4
27
0
9
840
176
0
27
2
10
912
396
0
26
2
11
1,627
936
26
12
12
12
1,515
853
15
20
8
13
1,747
806
9
15
10
14
1,618
651
13
16
12
15
1,637
891
9
18
11
16
1,050
694
7
26
1
17
1,536
798
11
15
13
Minuend
RT
SD
% Error
RET
Non-RET
9
898
301
0
28
1
10
862
183
0
29
0
11
1,400
720
9
21
6
12
1,502
809
15
21
6
13
1,900
1,026
15
17
10
14
1,578
722
17
19
10
15
1,818
967
13
17
8
16
1,951
934
26
15
9
17
1,533
706
7
14
13
18
1,162
501
4
26
2
6
7
8
9
Note. The 100 cells depicted correspond to the 100 basic facts of subtraction; the minuend is the first entry in
each cell, the subtrahend is the column header, and the remainder is the row header. Thus, the bottom-most right
cell corresponds to the problem 18 ⫺ 9 ⫽ 9. The next three entries in each cell are the mean reaction time (RT),
standard deviation, and error rate for that problem (1,162 ms, 501 ms, and 4%), averaged across all participants
in Experiments 1 and 2 (N ⫽ 46). The final two entries in each cell are the reported number of retrieval (RET)
trials for that problem and the reported number of non-RET trials, based on the 30 participants in the verbal
report groups of Experiment 3, Session 1; because of errors and occasional nonreports, these values do not
necessarily sum to 30.
Received April 2, 2002
Revision received April 7, 2003
Accepted April 11, 2003 䡲