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Identifying Strategies in Arithmetic With the Operand Recognition
Paradigm: A Matter of Switch Cost?
Catherine Thevenot, Caroline Castel,
and Juliette Danjon
University of Geneva
Michel Fayol
Clermont Université and Centre national
de la recherche scientifique
Determining adults’ and children’s strategies in mental arithmetic constitutes a central issue in the
domain of numerical cognition. However, despite the considerable amount of research on this topic,
the conclusions in the literature are not always coherent. Therefore, there is a need to carry on the
investigation, and this is the reason why we developed the operand recognition paradigm (ORP). It
capitalizes on the fact that, contrary to retrieval, calculation procedures degrade the memory traces of the
operands involved in a problem. As a consequence, the use of calculation procedures is inferred from
relatively long recognition times of the operands. However, it has been suggested that recognition times
within the ORP do not reflect strategies but the difficulty of switching from a difficult task (calculation)
to a simpler one (recognition). In order to examine this possibility, in a series of 3 experiments we
equalized switch-cost variations in all conditions through the introduction of intermediate tasks between
problem solving and recognition. Despite this neutralization, we still obtained the classical effects of the
ORP, namely longer recognition times after addition than after comparison. We conclude that the largest
part of the ORP effects is related to different strategy use and not to difficulty-related switch costs. The
possible applications and promising outcomes of the ORP in and outside the field of numerical cognition
are discussed.
Keywords: numerical cognition, arithmetic, addition, algorithmic procedures, reconstructive strategies
Strategies in mental arithmetic have been studied by cognitive
psychologists for many years, and this has led to an impressive
numbers of studies. Yet, and quite surprisingly, the conclusions
reached in the literature are not always coherent, and it is difficult
to draw a clear picture of what exactly happens when adults and
children solve simple problems. Even the strong consensus that
simple additions (involving operands from 1 to 4) are solved
through retrieval from memory has recently been challenged (Bar-
rouillet & Thevenot, 2013;Campbell, Chen, & Maslany, 2013;
Fayol & Thevenot, 2012). Another example of the lack of consis-
tency in the literature is the considerable variability reported across
studies in the estimation of adults’ strategies for simple subtrac-
tion. In LeFevre, DeStefano, Penner-Wilger, and Daley (2006),
participants reported retrieval on 81% of easy problems and on
42% of harder problems, whereas participants in Experiment 3 of
Seyler, Kirk, and Ashcraft (2003) reported using retrieval on 97%
of easy problems and on 66% of harder problems. Moreover,
Seyler et al. reported that 30% of their participants used retrieval
exclusively, whereas, in LeFevre et al.’s study, only 6% of partic-
ipants reported using retrieval on all trials. One possible explana-
tion for this great variability was given by Kirk and Ashcraft
(2001), who stated that “the use of verbal protocols . . . is
potentially problematic, . . . and great care must be taken in
collecting reports for them to qualify as valid reflections of adult
performance” (p. 174). For Kirk and Ashcraft, verbal report col-
lection is indeed not always trustworthy in order to identify the
arithmetic strategies used by individuals. They noted that auto-
matic mental processes, such as the retrieval of number facts in
memory, might not be accessible to consciousness; hence, the
difficulty for participants to report accurately their strategy (e.g.,
Ericsson & Simon, 1993). Moreover, the requirement of verbal
report may alter the mental processing that normally occurs. For
Kirk and Ashcraft, participants may solve the problems atypically,
because they are expected to report whether they solved them by
remembering or by using a procedural strategy. Finally, it is
possible that instructions to verbally report behaviors reveal the
experimental hypothesis and affect the participants’ strategy. Kirk
and Ashcraft’s concerns can be summed up by Baumeister, Vohs,
and Funder’s (2007) clever statement that “people have not always
done what they say they have done, will not always do what they
say they will do, and often do not even know the real causes of the
things they do” (p. 347). As a matter of fact, discrepancies between
verbal reports and response time (RT) data have indeed been
revealed in studies with children (De Smedt, Torbeyns, Stassens,
Ghesquière, & Verschaffel, 2010) and adults (Thevenot, Castel,
Fanget, & Fayol, 2010).
This article was published Online First October 20, 2014.
Catherine Thevenot, Caroline Castel, and Juliette Danjon, Faculty of
Psychology and Education, University of Geneva; Michel Fayol, Labora-
toire de Psychologie Sociale et Cognitive (LAPSCO), Clermont Université,
and Centre national de la recherche scientifique.
This work was supported by an FNS (Fonds National Suisse) Project
Grant 100013-140611/1.
Correspondence concerning this article should be addressed to Catherine
Thevenot, FAPSE, University of Geneva, 40, bd du Pont D’Arve, Geneva,
Switzerland CH-1205. E-mail: catherine.thevenot@unige.ch
This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.
Journal of Experimental Psychology:
Learning, Memory, and Cognition
© 2014 American Psychological Association
2015, Vol. 41, No. 2, 541–552
0278-7393/15/$12.00 http://dx.doi.org/10.1037/a0038120
541
Other misleading conclusions in the literature could be the
consequence of hazardous interpretations drawn from individuals’
solution times for arithmetic problems. LeFevre, Sadesky, and
Bisanz (1996) stressed that, as already mentioned for children
(Siegler, 1989), averaging solution latencies across trials that in-
volve different procedures can result in misleading conclusions
about how adults solve problems. Moreover, it was admitted that
short solution times are associated with retrieval strategies,
whereas longer solution times are more likely to reflect the use of
more demanding counting procedures (e.g., LeFevre et al., 1996).
However, and as suggested in the past by Baroody (1983,1984;
see also Baroody & Varma, in press), Fayol and Thevenot (2012)
showed that routinized counting procedures in high-skilled arith-
metic solvers (see also Campbell et al., 2013) can be as fast as
arithmetic fact retrieval. Of course, these new findings can
strongly question prior conclusions of the literature based on
classic measures. Therefore, we agreed with Kirk and Ashcraft’s
(2001) conclusion that “we must find more appropriate methods of
determining the frequency of nonretrieval strategy use among
adults” (p. 174), and this is the reason why we developed the
operand recognition paradigm (ORP).
The rationale of the ORP is to infer the strategies used by
individuals in arithmetic from the time they take to recognize the
operands of a problem after they have solved it. This rationale is
motivated by the fact that algorithmic procedures degrade the
memory traces of the operands, whereas they remain intact after a
solution process by retrieval of the result from long-term memory
(Thevenot, Barrouillet, & Fayol, 2001). Indeed, suppose that an
adult has to solve 27 !18. It is quite unlikely that he or she
retrieves the result from memory. Different procedures are avail-
able, but all of them require the operands to be decomposed or
transformed. Some adults could fully decompose 27 and 18 into
20 !7 and 10 !8, respectively, add 10 to 20, temporarily store
the intermediate result (30), then compute or retrieve 7 !8, and
finally add 15 to 30. Adults (e.g., Lemaire & Arnaud, 2008) and
children (e.g., Fuson et al., 1997) could prefer to transform or
decompose only one of the two operands (i.e., 27 !18 "27 !20
(47) – 2 or 27 !10 (37) !8). It is also possible to mentally
visualize the operands in columnar spatial arrangements and to add
the units and then the 10s (Heathcote, 1994). A wide variety of
decompositions or transformations are possible, but all of them
lead to a shift of the attention from the operands to their compo-
nents or to the partial results that are reached during solving. In
contrast, suppose that an adult has to solve 3 !2. He or she will
most probably directly retrieve the result (5) from memory without
having to defocus his or her attention from the operands or, in
other words, without having to decompose them or reach sub-
results. Therefore, the level of activation of the operands is nec-
essarily lower after algorithmic procedures than after retrieval
because of the necessary sharing of attentional resources between
the operands, their components, and the intermediate results to be
reached through the solving process (Anderson, 1993). Accord-
ingly, we observed that it is more difficult for adults to recognize
two-digit operands after their addition or subtraction than after
their simple comparison with a third number, a task that makes it
necessary to keep the numbers in memory without any transfor-
mation. On the contrary, no such difference was observed when
very simple addition problems were under study (e.g., 3 !2).
Thus, contrasting the relative difficulty that adults encounter in
recognizing operands after either their involvement in an arithme-
tic problem or their simple comparison with a third number can
allow us to determine if the arithmetic problem has been solved by
an algorithmic procedure or by retrieval of the result from mem-
ory: If operands are more difficult to recognize after the operation
than after their comparison, we can assume that an algorithmic
procedure has been used. On the contrary, if the difficulty is the
same in both conditions, then the operation has most probably been
solved by retrieval, a fast activity that does not imply the decom-
position of the operands. Note that comparing recognition times of
the operands after the arithmetic operation to a baseline corre-
sponding to the comparison condition avoids the drawing of an
arbitrary time line between recognition times that would reflect
retrieval and recognition times that would reflect computation.
Whenever recognition times differ from the baseline, we can
conclude that a procedure was used.
So far, the ORP has been applied to the study of mental addition
in young adults (Thevenot et al., 2001;Thevenot, Fanget, & Fayol,
2007), in children (Fanget, Thevenot, Castel, & Fayol, 2011), and
in elderly individuals (Thevenot, Castel, Danjon, Fanget, & Fayol,
2013). Overall, we showed that retrieval strategies for simple
problems are less frequent than what is generally assumed in the
literature and that retrieval rates are highly dependent on partici-
pants’ skills and characteristics. Moreover and crucially, the su-
periority of the ORP over verbal reports in terms of reliability was
demonstrated in a study devoted to mental subtraction in adults
(Thevenot et al., 2010). We showed that for problems of medium
difficulty (i.e., constructed with a two-digit minuend up to 17 and
a one-digit subtrahend), lower arithmetic achievers reported to
have used retrieval in 50% of the problems, whereas higher achiev-
ers reported 45% of retrieval. However, when the same partici-
pants were tested with the ORP, these results were not corrobo-
rated. Indeed, recognition times of the operands were higher after
subtraction than comparison in low achievers but similar between
those conditions in higher skilled arithmetic solvers. Then, the
ORP demonstrated that lower and higher skilled adults obviously
use different strategies to solve medium subtraction, whereas ver-
bal reports failed to reveal this differential pattern.
In addition to the fact that the ORP does not rely on verbal
reports or solution times, another of its advantages is that it does
not draw the attention of participants to the object of the study.
This avoids the bias that revealing the experimental goal of the
research can impact participants’ strategy (Kirk & Ashcraft, 2001).
Moreover, the fact that, within the ORP, no verbal responses are
required from participants makes it very suitable for brain-imaging
studies that investigate mental arithmetic strategies. Indeed, even if
some techniques have been developed in order to reduce or even
eliminate the effects of body movements on brain-imaging record-
ing (see Dieler, Tupak, & Fallgatter, 2012, for a review), these
techniques entail other advantages of more classical methods in-
cluding a compact experimental setting (Moriai-Izawa et al.,
2012). Finally, the strength of the ORP has also been supported by
the fact that we ensured that the recognition task did not alter the
mental processing that normally occurs in a more ecological situ-
ation. To test this, we presented the problems with and without the
recognition task and showed that solution times were the same in
the two conditions (Thevenot et al., 2010).
Then, the ORP could have constituted a good additional tool for
the study of strategies in arithmetic, but doubts about its validity
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542 THEVENOT, CASTEL, DANJON, AND FAYOL
have been expressed by Metcalfe and Campbell (2010,2011). We
therefore needed to address Metcalfe and Campbell’s concerns in
order to figure out whether previous conclusions drawn from the
ORP could be considered reliable. Furthermore, we had to deter-
mine whether the ORP could still be considered as a valid method
of investigation.
According to Metcalfe and Campbell (2010), the results we
obtained with the ORP could be attributable to switch costs from
easy to more difficult tasks rather than to individuals’ strategies.
Indeed, because addition and subtraction are more difficult than
comparison, and because it takes longer to switch from a difficult
task to the subsequent one than from an easier task to the subse-
quent one (i.e., easier to switch from comparison to recognition
than from addition to recognition; e.g., Arbuthnott, 2008), the
longer operand recognition times after a calculation procedure than
after a simple comparison would be due to differential costs when
switching to the recognition task. In order to directly test their
alternative interpretation, Metcalfe and Campbell (2011) decided
to compare ORP performance after addition and after multiplica-
tion. The authors based their line of reasoning on the assumption
that addition relies more on procedure than multiplication does
(e.g., Dehaene, Piazza, Pinel, & Cohen, 2003) and that, at the same
time, multiplication is supposed to be slightly more difficult than
addition. Indeed, studies testing simple addition and multiplication
indicate a small overall RT and accuracy advantage for addition
(e.g., Campbell, 1994). Therefore, comparing recognition times
after addition and multiplication seemed to be ideal to disentangle
the two possible interpretations of the ORP effects: If the ORP is
sensitive to the use of procedural strategies, recognition of the
operands should be slower following addition than following mul-
tiplication. In contrast, if the ORP is rather sensitive to difficulty-
related switch costs, the recognition of the operands should be
slower following multiplication than following addition.
In fact, it was difficult to disentangle the strategy use and the
switch-cost interpretations from Metcalfe and Campbell’s (2011)
results because they failed to observe any difference in recognition
times after addition or multiplication. However, using regression
analyses, they found that problem difficulty was a better predictor
of operand recognition times than were self-reported procedure
rates. This favored their interpretation that the difficulty of the task
is responsible for recognition time durations. Nevertheless, and as
acknowledged by Metcalfe and Campbell themselves, another part
of their results was more in favor of the interpretation initially
formulated following the ORP rationale. Indeed, the rate of correct
recognition of operands was higher after multiplication than after
addition, which indicated that it was more difficult to recognize the
operands after addition. Because, as stated above, addition is
supposed to be slightly less difficult than multiplication, the inter-
pretation that recognition times are modulated by the difficulty of
the task is not supported by this part of Metcalfe and Campbell’s
results. On the contrary, because addition is supposed to be solved
more often by procedures than is multiplication, an interpretation
of the longer recognition times of the operands in terms of differ-
ent strategy use is more plausible.
Then, arguments in favor of and against a detrimental role of
switch costs within the ORP have been provided so far, and we are
left with an unclear picture of the mechanisms that really account
for the effect variations associated with recognition times. The
purpose of the present article is to bring out more definite conclu-
sions about the possible switch-cost contamination of previous
interpretations. Therefore, in the first experiment, we developed
Metcalfe and Campbell’s (2010) idea and included a parity task
just before the recognition task within the ORP. By doing so, we
aimed to neutralize the variations of switch costs related to the
difficulty of the immediately preceding task within a trial. If
recognition times remained significantly different between addi-
tion and comparison after the parity task, this difference would be
difficult to interpret in terms of differential switch costs. More-
over, this manipulation would allow us to determine precisely the
time cost of the switch in the original paradigm by comparing
response times in the parity task after addition and comparison.
Experiment 1
Method
Participants. Thirty-five undergraduate students (Mage "20
years, 30 females) at the University of Geneva took part in this
experiment for course credit.
Materials. All experimental trials consisted of five numbers
presented sequentially: the first operand, the second operand, a
proposed answer, the parity judgment number, and a recognition
target (see Figure 1). Twenty-four pairs of numbers were used in
order to construct the problems. The pairs of numbers were ran-
domly chosen from among pairs that respected the constraints
described below. The problems were categorized as a function of
Figure 1. Example of trial sequences in Experiment 1.
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543
STRATEGIES IN MENTAL ARITHMETIC
their size and divided into three types: small, medium, and large.
Problems involving 0 and 1 as well as tie problems were excluded
from the material. Small problems were constructed with pairs of
numbers between 2 and 8; their sums never exceeded 10, and their
differences were larger than 1. This allowed for a comparison with
a third number. Medium problems were constructed with pairs of
numbers between 4 and 9; their sums were always greater than 10,
and their differences were larger than 1. Finally, large problems
were constructed with pairs of numbers between 13 and 47; their
sums fell between 41 and 63, and their differences were larger than
10. In order to increase the probability of an algorithmic proce-
dure, their addition always required carrying over a number, and
neither the operands nor their sums ended with a 0.
Each pair of numbers was presented both in the addition and in
the comparison conditions. In the addition condition, participants
had to decide if a third number corresponded to the sum of the first
two numbers. In the comparison condition, they had to decide if
this third number fell between the first two numbers. Each pair of
numbers in each condition (addition vs. comparison) was pre-
sented twice, once associated with a third number eliciting a “yes”
response (the sum of the two operands for addition problems or a
number falling between them for comparisons) and once associ-
ated with a third number eliciting a “no” response. In order to
encourage calculation rather than approximation, erroneous an-
swers for addition problems were close to the correct answers (i.e.,
#1). For comparisons, the number eliciting a “no” answer was, in
half of the cases, the largest operand plus 1, and in the other half,
the smallest operand minus 1. The number eliciting a “yes” answer
was chosen randomly within the interval determined by the two
numbers presented.
After this decision task, a parity judgment task had to be
performed on a fourth number presented within the trial. This
number corresponded to a one-digit number after small and me-
dium problems and to a two-digit number after large problems. In
half of the cases, this number corresponded to an even number, and
in the other half, to an odd one.
Finally, for the recognition task, each of the experimental trials
was associated with a fifth number (i.e., a target) that could
correspond to the first operand, the second operand, the first
operand plus or minus 1, or the second operand plus or minus 1. Of
course, to make the recognition time comparison possible, the
target was always the same in the addition and the comparison
conditions for a specific number pair. In order to avoid any
interference effect, numbers could not be repeated within a trial
when the recognition target was different from the operands.
Each participant was therefore presented with 192 trials: 8 pairs
of numbers $3 problem sizes (small, medium and large) $2
types of problems (addition and comparison) $2 responses on the
third number (yes or no) $2 responses on the fifth number (yes or
no). In order to familiarize the participants with the task, 20
warm-up trials were presented before the experimental phase.
Procedure. The experiment was controlled by E-prime soft-
ware (Schneider, Eschman, & Zuccolotto, 2012). All information
was self-paced, and there was no deadline for responding. Each
trial began with the presentation of a letter that indicated the type
of problem to be solved (Afor addition or Cfor comparison).
When participants pressed the space key, the letter was replaced by
the first number of the pair (first operand). By pressing the same
key again, participants removed this number from the screen and
displayed the second number (second operand), and then the third
one (proposed answer). When the third number was displayed
onscreen, participants were asked to give their answer by pressing
one of two labeled keys on the keyboard (yes or no). The “yes”
response was required when the third number either corresponded
to the sum of the first two numbers in the addition condition or lied
between them in the comparison condition. Then, for the parity
judgment task, the fourth number appeared onscreen along with
the question “odd or even?” The response was given by pressing
one of the two labeled keys. Finally, the fifth number was dis-
played onscreen along with the question “have you seen?” By
pressing the yes or no key, participants had to judge whether they
had seen this number among the first two numbers within the trial.
This last response displayed a next-trial signal (see Figure 1).
Responses and response times were recorded by the computer.
Results
In order to give a complete picture of participants’ behaviors
and before the presentation of the results in relation to our crucial
measures (i.e., solution times in the parity task and recognition
times of the operands), the rates of correct responses in the
different tasks are presented.
Rates of correct responses.
Addition and comparison tasks. A 2 (type of problem: addi-
tion or comparison) $3 (size of problem: large, medium, or small)
analysis of variance (ANOVA), with the two factors as repeated
measures, was performed on the rates of correct responses in the
addition and comparison tasks (see Table 1). The rates of correct
responses did not differ as a function of the type of problem (Fs%
1; [.92 for both addition and comparison]). However, the rates of
correct responses differed as a function of the size of problems,
F(2, 68) "16.33, &p
2".32, p%.001, (.88, .93, and .96 for large,
medium, and small problems, respectively). Finally, there was an
interaction between the type and the size of problems, F(2, 68) "
8.90, &p
2".21, p%.001, showing that additions were associated
with lower rates of correct responses than comparisons but only for
large problems, F(1, 34) "4.56, &p
2".12, p".04, and not for
medium (F%1) or small, F(1, 34) "3.72, p".06, ones. This last
Table 1
Rates of Correct Responses Associated With the Different Tasks in Experiment 1
Problem type
Addition or comparison Parity judgment Recognition
Large Medium Small Large Medium Small Large Medium Small
Addition .85 .94 .98 .97 .97 .98 .77 .84 .81
Comparison .91 .92 .93 .98 .97 .99 .87 .89 .91
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544 THEVENOT, CASTEL, DANJON, AND FAYOL
result is quite close to the admitted level of significance, but in this
condition, and contrary to large problems, additions are associated
with higher rates of correct responses than are comparisons. In
order to ensure that our results were reliable and based on trials
wherein participants paid sufficient attention to the operands,
incorrect trials were not considered for the analyses of response
times.
Parity judgment task. As attested by the high rates of correct
responses, participants did not experience difficulty in performing
the parity judgment task. Descriptive data are presented in Table 1,
but ceiling effects precluded meaningful factorial analyses.
Recognition of the operand task. A 2 (type of problem:
addition or comparison) $3 (size of problem: large, medium,
or small) ANOVA, with the two factors as repeated measures,
was performed on the rates of correct recognition (see Table 1).
Operands were better recognized after comparison (.89) than
after addition (.81), F(1, 34) "18.00, &p
2".35, p%.001. There
was also an effect of the size of problems, F(2, 68) "3.57,
&p
2".10, p".03. The operands involved in small (.86) and
medium (.87) problems were better recognized than were op-
erands involved in large problems (.82). However, there was no
interaction between the type and the size of problems, F(2,
68) "1.05, p".36.
As in our previous studies (e.g., Thevenot & Oakhill, 2006), and
as it is seen below, the rates of correct responses in the recognition
task are never in contradiction to those for recognition times,
which attests to the absence of a speed–accuracy trade-off. How-
ever, this variable is a less sensitive and informative measure than
are recognition times. This is the reason why some effects can
appear on this last measure but not on recognition rates; hence, the
lack of significant interaction between the type and the size of
problems.
Response times.
Parity judgment task. Besides the 7% of trials for which
participants gave a wrong answer in the addition or comparison
task, the 2% of incorrect trials in the parity task were also removed
from the analyses. In order to determine whether differential
switch costs were observed in the parity task as a function of the
difficulty of the preceding task, we performed a 2 (type of prob-
lem: addition or comparison) $3 (size of problem: large, medium,
or small) ANOVA, with the two factors as repeated measures, on
response times (see Figure 2).
Parity RTs differed as a function of the size of problems, F(2,
68) "45.84, &p
2".57, p%.001 (1,531, 1,336, and 1,305 ms for
large, medium, and small problems, respectively). They were
shorter for medium than large problems, F(1, 34) "46.27, &p
2"
.58, p%.001, but similar for medium and small ones, F(1, 34) "
2.77, p".11. Moreover, response times were higher after addition
(1,423 ms) than after comparison (1,359 ms), F(1, 34) "9.42,
&p
2".22, p".004. Furthermore, and as already observed by
Metcalfe and Campbell (2010), there was an interaction between
these two variables, F(2, 68) "5.53, &p
2".14, p".006: Parity
judgment times were influenced by the nature of the preceding task
(addition vs. comparison) when large and medium problems were
presented, F(1, 34) "9.98, &p
2".23, p".003, and F(1, 34) "
11.30, &p
2".25, p".002, respectively, but not when small
problems were involved (F%1). More precisely, it took 120 ms
longer for participants to judge parity after large addition than
large comparison, 81 ms longer after medium addition than me-
dium comparison, and only 11 ms shorter to judge parity after
small addition than small comparison.
Recognition of the operand task. In addition to the trials
already removed from the previous analysis, 15% of the trials in
which participants failed to recognize the operands were also
excluded. Five participants for whom there were no more data in
one or more experimental conditions were discarded from this
analysis, which was therefore conducted on data from 30 partici-
pants (4,983 trials). In order to determine whether recognition
times differed as a function of the task despite the fact that
differential switch costs were neutralized, an ANOVA with the
same design as the previous one was conducted on the mean
recognition times of the operands (see Figure 3).
Mean recognition times of the operands differed as a function of the
size of problems, F(2, 58) "27.31, &p
2".48, p%.001 (1,642, 1,468,
and 1,433 ms for large, medium, and small problems, respectively).
Recognition times were faster after small than medium problems, F(1,
29) "5.08, &p
2".15, p".04, and faster after medium problems than
large ones, F(1, 29) "24.36, &p
2".46, p%.001. Moreover, the
recognition times were higher after an addition (1,617 ms) than after
acomparison(1,412ms),F(1, 29) "25.36, &p
2".47, p%.001. More
interestingly, there was an interaction between these two variables,
F(2, 58) "10.27, &p
2".26, p%.001. Exactly as in our previous
studies (e.g., Thevenot et al., 2001,2007,2013),recognitiontimesof
the operands were longer after large addition than large comparison,
F(1, 29) "28.03, &p
2".49, p%.001, and larger after medium
addition than medium comparison, F(1, 29) "11.23, &p
2".28, p"
.002. However, the nature of the task did not have significant effect on
small problems, F(1, 29) "1.96, &p
2".06, p".17. More precisely,
it took 371 ms longer for participants to recognize the operands after
large addition than large comparison, 158 ms longer after medium
addition than medium comparison, and only 83 ms longer to recog-
nize the operands after small addition than small comparison.
1000
1200
1400
1600
1800
2000
2200
Large Medium Small
Addi!on
Comparison
Mean response times (in ms)
Figure 2. Mean response times (in ms) and 95% confidence intervals
(indicated by the error bars) in the parity task as a function of the task and
the size of problems in Experiment 1. Note that 95% confidence intervals
are calculated by dividing the standard deviation of a specific condition by
'N and by multiplying the result by the tvalue at the .05 level.
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545
STRATEGIES IN MENTAL ARITHMETIC
Discussion
In this first experiment, we modified the original ORP and
introduced a parity judgment task just before the recognition task.
This manipulation aimed at neutralizing the variations of switch
costs related to the difficulty of the immediately preceding task
(addition or comparison). First, we replicated Metcalfe and Camp-
bell’s (2010) results and showed that parity RTs depended on the
preceding task (i.e., 120 and 81 ms longer after large and medium
additions than after large and medium comparisons, respectively).
These variations necessarily reflect differential switch costs across
conditions because the parity judgment task was strictly the same
in both conditions and the only factor that varied was the nature of
the preceding task. Second and crucially for our purpose, we also
showed that despite the introduction of the parity task within the
ORP design, we replicated the results of our previous experiments
on the recognition task (e.g., Thevenot et al., 2001;Thevenot et al.,
2007;Thevenot et al., 2013). Indeed, as in our prior studies,
recognition times of the operands were still longer after large
addition than after large comparison (!371 ms) and longer after
medium addition than after medium comparison (!158 ms). In
contrast, for small problems, recognition times did not differ as a
function of the type of problems. This is very close to the results
we obtained using the original paradigm, wherein the recognition
took place just after the numerical task (!377 ms and !202 ms for
large and medium problems, respectively, in Thevenot et al., 2007,
Experiment 1).
These first results are then quite convincing in showing that
differential switch costs between conditions cannot by themselves
account for the effects observed using the ORP. Then, an inter-
pretation of the differences in recognition times that we classically
observe within the ORP in terms of difference of strategy seems
the one to be retained. However, before definitely advancing such
conclusion, it is important to ensure that the differences in recog-
nition times are not simply due to the fact that it takes longer to
perform an addition than a comparison. Indeed, it could be that the
time elapsed between the presentation of the operands and the
recognition task is longer for addition than for comparison; hence,
the higher difficulty to recognize the operands in the first than in
the second condition. In order to examine this possibility, we
added up the self-presentation times of the third number (i.e.,
decision for comparison or verification for addition) and the fourth
number (parity task) within a trial. By doing so, it was possible to
compare the time elapsed between the disappearance of the second
operand and the apparition of the probe for recognition in both
conditions as a function of the size of problems. It turns out that
this duration was similar in the addition and comparison conditions
for large problems (3,278 and 3,228 ms for addition and compar-
ison, respectively, F%1) and even longer for comparisons than
additions for medium (3,012 and 2,571 ms, respectively), F(1,
34) "7.91, &p
2".19, p".008, and small (2,877 and 2,311 ms,
respectively), F(1, 34) "54.46, &p
2".62, p%.001, problems.
Therefore, difficulty in recognition times for the second operand
cannot be due to longer delays for addition than comparison. In
fact, and contrary to an interpretation in terms of delay, the
operands are more difficult to recognize in the condition wherein
the time elapsed between the second operand and the recognition
probe is shorter. Note that this reasoning does not hold for the
recognition of the first operand, for which the time elapsed be-
tween the presentation and the recognition (i.e., self-presentation
of the second operand !decision for comparison or verification
for addition !fourth number for the parity task) is necessarily
longer in the addition than in the comparison condition, F(1, 34) "
10.72, &p
2".24, p".002. Still, our results on recognition times
cannot be attributable to this effect because, as in all our previous
studies (Fanget et al., 2011;Thevenot et al., 2001,2007,2013),
recognition times following an addition were longer for the second
operand (1,656 ms) than the first one (1,465 ms), F(1, 34) "11.35,
&p
2".25, p".002. Therefore, and contrary to an explanation of
our results in terms of longer time elapsed between the operand
and the recognition probe, we show here that recognition times of
the operands are longer when the duration between its presentation
and the decision is shorter.
Yet, before definitely concluding that strategy use rather than
switch costs are responsible for the ORP effects, it has to be
verified that difficulty-related switch costs are not carried over
until the end of the trial but only circumscribed to the switch
between the problem-solving task (third number) and the parity
task (fourth number). In fact, it is quite unlikely, because the
differences in recognition times after addition or comparison are
noticeably larger than the differences in the parity task. Indeed,
whereas parity judgment times, which reflect pure switch costs,
were 120 ms and 81 ms longer after large and medium additions,
respectively, than after large and medium comparisons, subsequent
differences in recognition times were of 371 ms and 158 ms,
respectively, in the same conditions. Therefore switch costs rep-
resented only 40% of response times within the recognition task.
As already demonstrated above, the only interpretation that we can
provide concerning the remaining 60% of the effects is that they
result from different strategy use. Still, we thought that a more
direct test was needed in order to ensure that differential switch
costs due to the unequal difficulty of the preceding problem could
1000
1200
1400
1600
1800
2000
2200
Large Medium Small
Addi!on
Comparison
Mean recognition times (in ms)
Figure 3. Mean recognition times (in ms) of the operands and 95%
confidence intervals (indicated by the error bars) as a function of the task
and the size of problems in Experiment 1.
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546 THEVENOT, CASTEL, DANJON, AND FAYOL
not contaminate the recognition task that followed the parity task.
To test this, we asked participants to perform a simple numerical
judgment task (i.e., Is this number smaller than 5?) just after the
problem solution (addition or comparison). The numbers presented
for this numerical judgment task were strictly the same after
addition and comparison, and, following the same rationale as in
the previous experiment, we expected the task to neutralize the
effect of differential switch costs on the subsequent task. This
subsequent task could correspond to either the classic recognition
task or a parity task. If, as we assumed in the first experiment, the
numerical task presented just after the problem neutralized switch
costs, then we should not observe any difference in RTs in the
subsequent parity task, depending on the problem (addition or
comparison). In that case, and if we still replicated the results of
our previous experiments on recognition times, it would be im-
possible to attribute the effects to differential switch costs. The
validity of the ORP as a good indicator for individuals’ strategies
would therefore be proven. On the contrary, if differences in the
parity task were still observed after the numerical judgment task as
a function of the preceding problem (addition or comparison), it
would show that differential switch costs were carried over until
the end of the trial. This would challenge previous conclusions
drawn from the ORP.
Experiment 2
Method
Participants. Thirty-seven undergraduate students (Mage "
20 years, 35 females) at the University of Geneva took part in this
experiment for course credit.
Materials. All experimental trials consisted of five numbers
presented sequentially. Half of the trials were constructed from the
following sequence: first operand, second operand, proposed an-
swer, numerical judgment task, and recognition task. The recog-
nition task was replaced by a parity task for the other half of the
trials (see Figure 4). In this experiment we chose to use only large
problems involving numbers between 28 and 67 because, in the
previous experiment, it was in this condition that the largest
differential switch costs were observed. Switch costs following
large addition were therefore the more likely to contaminate RTs
until the end of the trials.
Exactly as in our previous experiments (e.g., Thevenot et al.,
2001), each pair of numbers was presented both in the addition and
in the comparison conditions, and the proposed answer had the
same characteristics as in Experiment 1. After the problem-solving
task, participants had to perform a numerical judgment task on a
fourth number. This number corresponded to a one-digit number,
and participants had to determine if it was smaller than five.
Finally, after this numerical judgment task, a fifth number was
presented. In the recognition condition, this number could corre-
spond to the first operand, the second operand, the first operand
plus or minus 1, or the second operand plus or minus 1. Exactly the
same numbers as in the recognition condition were presented in the
parity condition. In half of the trials, this number corresponded to
an even number, and in the other half, to an odd one.
Each participant was presented with 64 trials: 4 pairs of
numbers $2 type of problems (addition or comparison) $2
responses on the third number (yes or no) $2 responses on the
fourth number (yes or no) $2 responses on the fifth number
(yes or no). In order to familiarize the participants with the task,
10 warm-up trials were presented before the experimental
phase.
Procedure. In order to facilitate the task for participants
and to avoid any confusion, trials associated with a recognition
or a parity task were presented in different blocks. The order of
blocks was counterbalanced across participants. For both
blocks, and as in the previous experiment, each trial began with
the presentation of a letter that indicated the type of problem to
be solved. When participants pressed the space key, the letter
was replaced by the first number of the pair (first operand). By
pressing the same key again, participants removed this number
from the screen and displayed the second number (second
operand), and then the third one (proposed answer). When the
third number was displayed onscreen, participants were asked
to give their answer by pressing one of two labeled keys on the
keyboard (yes or no). Then, the fourth number appeared on-
screen for the numerical judgment task, and participants had to
press the yes key when it corresponded to a number smaller than
5 and the no key when it corresponded to a number larger than
Figure 4. Examples of trial sequences for the recognition task (a) and the parity task (b) in Experiment 2.
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547
STRATEGIES IN MENTAL ARITHMETIC
5. Finally, the fifth number was displayed onscreen. For the
recognition task, participants had to judge whether they had
seen this number among the first two numbers within the trial
by pressing the yes or no key. In the parity condition, partici-
pants had to judge whether the fifth number was odd or not by
pressing the same keys (see Figure 4). Responses and response
times were recorded by the computer.
Results
Rates of correct responses.
Addition and comparison tasks. An ANOVA with the type of
problem (addition or comparison) as a repeated measure was
performed on the rates of correct responses in the addition and
comparison tasks (see Table 2). They did not differ as a function
of the type of problem (F%1 [.92 for both addition and compar-
ison problems]). In order to ensure that our results were reliable
and based on trials wherein participants paid sufficient attention to
the operands, incorrect trials were not considered for further anal-
yses.
Numerical judgment and parity tasks. The rates of correct
responses were very high for these two tasks, which precluded
meaningful factorial analyses. Descriptive data are presented in
Table 2.
Recognition of the operand tasks. An ANOVA with the type
of problem (addition or comparison) as a repeated measure was
performed on the rates of correct responses in the recognition task
(see Table 2) and revealed that the operands were better recog-
nized after comparison (.95) than after addition (.89), F(1, 36) "
5.07, &p
2".12, p".03.
Response times.
Numerical judgment task. Besides the 8% of trials for which
participants gave a wrong answer to the addition or the compari-
son, 1% of trials where participants failed to perform the numerical
judgment task (i.e., smaller than 5?) were removed from the
analyses. In order to determine whether differential switch costs
were observed in the numerical judgment task as a function of the
difficulty of the preceding task, we performed an ANOVA with the
type of problem (addition or comparison) as a repeated measure on
response times. They were higher after addition (1,293 ms) than
after comparison (1,187 ms), F(1, 36) "7.58, &p
2".17, p".009.
Therefore, as in the previous experiment, and as already observed
by Metcalfe and Campbell (2010), the numerical judgment task
was influenced by the difficulty of the preceding task.
Parity and recognition of the operand tasks. In order to test
the possibility that differential switch costs could be carried over
until the end of the trial, a 2 (type of problem: addition or
comparison) $2 (last task: parity or recognition) ANOVA with
the two factors as repeated measures was performed on response
times (see Figure 5). Response times were longer for the recogni-
tion (1,306 ms) than for the parity (950 ms) task, F(1, 36) "35.52,
&p
2".50, p%.001. Moreover, they were longer after addition
(1,184 ms) than comparison (1,072 ms), F(1, 36) "4.66, &p
2".11,
p".04. More importantly, there was an interaction between these
two factors, F(1, 36) "6.58, &p
2".15, p".02. Whereas in the
parity task, RTs did not differ as a function of the type of problem
preceding the task (944 and 956 ms for addition and comparison,
respectively, F%1), the difference in RTs after addition (1,424
ms) and comparison (1,189 ms) was still significant for the rec-
ognition task, F(1, 36) "4.66, &p
2".11, p".04.
Discussion
This experiment was conducted in order to ensure that introduc-
ing a buffer task before the recognition within the ORP properly
neutralizes differential switch costs due to the unequal difficulty of
the problem-solving tasks preceding this recognition (i.e., addition
or comparison). To test this, we asked participants to perform a
numerical judgment task (Is this number smaller than 5?) just after
the addition or the comparison. The numerical judgment task was
followed by either the classical recognition task of the operands or
a parity task. We clearly showed that performance on the parity
task did not depend on the difficulty of the preceding task. Indeed,
parity solution times did not differ significantly (only 11 ms) as a
function of the difficulty of the main task (i.e., addition or com-
parison) when it was performed after the buffer task. Therefore,
the numerical judgment task acts as a good neutralizer for differ-
ential switch costs. In contrast, and crucially, the same results as in
our previous experiments (e.g., Thevenot et al., 2001) were ob-
served for recognition performance of the operands. Indeed, rec-
ognition times were significantly longer after large addition than
after large comparison.
Therefore, the results of this experiment strongly support the
fact that switch costs do not affect the tasks following the neutral-
Table 2
Rates of Correct Responses Associated With the Different Tasks
in Experiment 2
Problem type
Addition or
comparison
Numerical
judgement
Parity
judgment Recognition
Addition .92 .98 .99 .89
Comparison .92 .99 .99 .95
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
Parity Recogni!on
Addi!on
Comparison
Mean response times (in ms)
Figure 5. Mean response times (in ms) and 95% confidence intervals
(indicated by the error bars) in the parity and recognition tasks following
the numerical judgment task in Experiment 2.
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548 THEVENOT, CASTEL, DANJON, AND FAYOL
izer. However, there is still a possibility that needs to be addressed
before this conclusion can be stated definitely. In this second
experiment, we decided to introduce the numerical judgment task
before the parity task because we already knew that the parity task
could suffer from differential switch costs. Indeed, an absence of
effects on a new task would have been difficult to interpret as
evidence for an absence of carried-over switch costs because it
could have been argued that the characteristics of the new task
made it insensitive to differential switch costs. Still, because the
parity judgment task was not kept in the same position as in
Experiment 1, we can conclude that the numerical judgment task
neutralizes switch costs but we cannot conclude categorically that
the parity task neutralizes them. Then, there is still a slight possi-
bility that the results of the Experiment 1 were due to carried-over
switch costs. Therefore, we ran a third experiment wherein the
parity judgment task played the role of the buffer task and pre-
ceded either the numerical judgment task or the recognition task.
Moreover, and again to be closer to the methodology used in
Experiment 1, the three categories of problems (i.e., small, me-
dium, and large problems) were presented to participants.
Experiment 3
Method
Participants. Twenty-five undergraduate students (Mage "
21 years, 20 females) at the University of Geneva took part in this
experiment for course credit.
Materials and procedure. All experimental trials consisted
of five numbers presented sequentially. Half of the trials were
constructed from the following sequence: first operand, second
operand, proposed answer, parity task, and recognition task. The
recognition task was replaced by the numerical judgment task for
the other half of the trials. The categories of problems and the pairs
of numbers used to create those categories were the same as in
Experiment 1. All other aspects of the procedure were the same
except that the order of the parity judgment task and the numerical
judgment task were reversed compared with the procedure fol-
lowed in Experiment 2.
Results
Rates of correct responses.
Addition and comparison tasks. A 2 (type of problem: addi-
tion or comparison) $3 (size of problem: large, medium, or small)
ANOVA, with the two factors as repeated measures, was per-
formed on the rates of correct responses in the addition and
comparison tasks (see Table 3). The rates of correct responses did
not differ as a function of the type of problem (F%1 [.93 for both
addition and comparison]). However, they differed as a function of
the size of problems, F(2, 48) "10.88, &p
2".31, p%.001 (.91,
.92, and .96 for large, medium, and small problems, respectively).
Finally, there was no interaction between the type and the size of
problems, F(2, 48) "1.42, p".25. Incorrect trials were not
considered for the analyses of response times.
Parity task and numerical judgment tasks. The rates of cor-
rect responses were very high for those two tasks, which precluded
meaningful factorial analyses. Descriptive data are presented in
Table 3.
Recognition of the operand task. A 2 (type of problem: ad-
dition or comparison) $3 (size of problem: large, medium, or
small) ANOVA, with the two factors as repeated measures, was
performed on the rates of correct responses in the recognition task
(see Table 3). The operands were better recognized after compar-
ison (.93) than after addition (.84), F(1, 24) "32.76, &p
2".58, p%
.001. However, there was no effect of the size of the problem and
no interaction of this factor with the type of problem (both Fs%
1).
Response times.
Parity task. Besides the 7% of trials for which participants
gave a wrong answer to the addition or the comparison, 2% of
trials where participants failed to perform the parity task were
removed from the analyses. In order to determine whether differ-
ential switch costs were observed in the parity task as a function of
the difficulty of the preceding task and of the size of problems, a
2 (type of problem: addition or comparison) $3 (size of problems:
large, medium, or small) ANOVA with the two factors as repeated
measures was performed on response times. They were higher
after addition (1,301 ms) than after comparison (1,160 ms), F(1,
24) "21.47, &p
2".47, p%.001. Moreover, there was an effect of
the size of problems, F(2, 48) "14.29, &p
2".37, p%.001, with
1,310, 1,222, and 1,160 ms for large, medium, and small problems,
respectively: Response times associated with large problems were
higher than those associated to medium problems, F(1, 24) "8.11,
&p
2".25, p".009, which, in turn, were higher than those
associated to small problems, F(1, 24) "5.45, &p
2".19, p".03.
Finally, there was an interaction between the type and the size of
problems, F(2, 48) "6.39, &p
2".21, p".003. This interaction
showed that the effect of the type of problem was significant for
large, F(1, 24) "26.70, &p
2".53, p%.001, and medium, F(1,
24) "9.12, &p
2".28, p".008, problems but not for small ones,
F(1, 24) "2.07, p".16. More precisely, it took 226 ms longer for
participants to judge parity after large addition than large compar-
ison, 150 ms longer after medium addition than medium compar-
ison, and only 45 ms shorter to judge parity after small addition
than small comparison. These results perfectly replicate those of
Experiment 1 and show, in accordance with Metcalfe and Camp-
bell (2010), that the parity task was influenced by the difficulty of
the preceding task.
Table 3
Rates of Correct Responses Associated With the Different Tasks in Experiment 3
Problem type
Addition or comparison Parity judgment Numerical judgment Recognition
Large Medium Small Large Medium Small Large Medium Small Large Medium Small
Addition .91 .91 .97 .98 .98 .98 .97 .97 .96 .84 .85 .83
Comparison .91 .93 .95 .98 .98 .98 .96 .98 .95 .94 .93 .94
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549
STRATEGIES IN MENTAL ARITHMETIC
Numerical judgment and recognition tasks. In addition to the
trials already removed, 7% of additional trials for which partici-
pants failed to answer correctly were eliminated from the analysis.
In order to test the assumption that differential switch costs could
be carried over until the end of the trial, a 2 (type of problem:
addition or comparison) $2 (last task: numerical judgment or
recognition) $3 (size of problems: large, medium, or small)
ANOVA with the three factors as repeated measures was per-
formed on response times (see Figure 6).
Response times were longer for the recognition task (1,206 ms)
than for the numerical judgment task (856 ms), F(1, 24) "37.08,
&p
2".61, p%.001, and longer after addition (1,098 ms) than
comparison (964 ms), F(1, 24) "29.38, &p
2".55, p%.001. On
the contrary, there was no effect of the size of problems, F(2,
48) "1.17, p".32 (1,011, 1,037, and 1,045 ms for large,
medium, and small problems, respectively). More importantly,
there was an interaction between the type of problem and the last
task, F(1, 24) "22.89, &p
2".49, p%.001. Whereas in the
numerical judgment task, RTs did not differ as a function of the
type of problem preceding the task (864 and 847 ms for addition
and comparison, respectively, F%1), the difference in RTs after
addition (1,332 ms) and comparison (1,080 ms) was still highly
significant for the recognition task, F(1, 24) "50.40, &p
2".68,
p%.001. This last effect was observed for large, F(1, 24) "11.35,
&p
2".32, p".002; medium, F(1, 24) "41.49, &p
2".63, p%.001;
and small, F(1, 24) "28.54, &p
2".54, p%.001, problems.
Discussion
In this experiment, we replicated Metcalfe and Campbell’s
(2010) results for the third time and confirmed that, within the
ORP, the difficulty of a numerical task (i.e., parity task) is influ-
enced by the difficulty of the preceding task (i.e., addition or
comparison). However, we also confirmed that introducing a buf-
fer task before the recognition properly neutralizes differential
switch costs. This was done by using the parity task instead of the
numerical judgment task as the buffer task. Crucially, despite this
neutralization, the classical ORP effects on the recognition task
were still observed. This was shown whatever the size of prob-
lems. Therefore, taken together, the results of Experiments 2 and
3 cast out the possibility of carried-over differential switch costs in
Experiment 1. This strongly supports our conclusion that differ-
ential switch costs between conditions cannot by themselves ac-
count for the effects observed using the ORP.
Interestingly, in this experiment, and contrary to what we ob-
served in the Experiment 1 of this article and to the results we
obtained in the past (Thevenot et al., 2007), the operands were
better recognized after comparison than addition, even for small
problems. We can conclude that some of our participants failed to
retrieve the sum of such simple problems. This encourages us to
continue our investigations on individual differences in arithmetic
problem solving.
General Discussion
This research was conducted in order to assess the role of switch
costs in the interpretation drawn from the effects observed using
the operand recognition paradigm. As explained in the introduc-
tion, the ORP could be a precious tool in order to identify adults’
and children’s strategies in mental arithmetic and could constitute
a useful alternative to and additional method of investigating
verbal report or solution time collections. However, Metcalfe and
Campbell (2010,2011) rightly pointed out a possible confound
inherent to the ORP design. The authors interpreted longer recog-
nition times of the operands after problem solving than after
simple comparison as the consequence of differential switch costs
rather than as the consequence of operand decompositions. Be-
cause we know that it is more demanding to switch from a difficult
to an easy task than the contrary, the ORP effects could indeed
have been due to the fact that the recognition task takes place after
either a simple task (comparison) or a more complex one (mental
arithmetic problem). In order to verify this interpretation, we
adapted Metcalfe and Campbell’s (2010) idea and introduced a
parity judgment task just before the recognition task. In doing so,
we aimed to neutralize the effects of differential switch costs after
comparison or addition by inserting the same task in both condi-
tions just before the recognition of the operands. Despite this
neutralization, we still observed longer recognition times after
addition than after comparison. In a second and a third experiment,
we ensured that differential switch costs could not be carried over
after the neutralizer task and until the end of the trial.
The results of these experiments led to the conclusion that
differential switch costs cannot by themselves account for the
effects observed in the past using the ORP. The only interpretation
Figure 6. Mean response times (in ms) and 95% confidence intervals (indicated by the error bars) in the
numerical judgment and recognition tasks following the parity task in Experiment 3.
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550 THEVENOT, CASTEL, DANJON, AND FAYOL
left for the ORP effects is therefore that longer recognition times
of the operands are due to their decomposition in working mem-
ory. Because only algorithmic procedures and not retrieval require
such decompositions, arithmetic problem strategies used by indi-
viduals can be inferred from recognition times of the operands.
Nevertheless, before affirming that precise conclusions can be
drawn from the ORP about individuals’ arithmetic strategies, it is
important to ensure that not only rough categories of problems can
be studied but that a detailed account of strategies can also be
inferred from our data. The results reported in our studies show
that large and medium problems are solved through reconstructive
strategies, but the analyses that we usually perform cannot deter-
mine whether all problems within a category are solved using this
strategy. If some problems do not follow this general pattern, it is
very important to identify them. In order to verify that a problem
by problem analysis is possible and reliable within the ORP, we
randomly isolated one problem by category and examined whether
recognition of its operands differed as a function of the task. For
Experiment 1, and in perfect accordance with our general pattern
of results, the operands involved in the large problem “47 !16”
were recognized more slowly after addition (1,822 ms) than com-
parison (1,397 ms), F(1, 18) "8.09, &p
2".31, p".01. The same
result was obtained for the medium problem “9 !5” (1,853 ms
and 1,315 ms for addition and comparison, respectively), F(1,
18) "8.79, &p
2".33, p".008. On the contrary, operand
recognition times did not differ between addition (1,337 ms) and
comparison (1,470 ms) for the small problem “5 !3” (F%1).
Following the rationale of the ORP, we can therefore conclude that
47 !16 and 9 !5 are solved through reconstructive strategies,
whereas the result of 5 !3 is retrieved from memory. In order to
assess the reliability and robustness of these conclusions, we
benefited from the fact that the same pairs of numbers were used
in Experiments 1 and 3, and we conducted the same analyses from
the set of data in Experiment 3. Again, the problems 47 !16 and
9!5 led to longer recognition times of the operands after addition
than after comparison (1,707 and 1,140 ms for 47 !16 and 1,531
and 1,098 for 9 !5 for addition and comparison, respectively),
F(1, 20) "5.81, &p
2".23, p".03, and F(1, 20) "4.45, &p
2".18,
p".048, but the difference between addition (1,433 ms) and
comparison (1,265 ms) was not significant for the small problem
5!3, F(1, 20) "1.29, p".027. These results show that problem
by problem analyses are possible and trustworthy using the ORP,
and a precise picture of individuals’ behaviors can be drawn from
this paradigm.
The design and results associated with the original ORP are
therefore reliable, and future research based on its rationale inside
and outside the numerical domain can be conducted confidently. In
fact, pattern, word, frame, or gesture recognition could be required
of participants whenever it is difficult to determine whether they
relied on automated procedures run in working memory or on
knowledge stored in long-term memory. We have already sug-
gested adapting the ORP in order to study chess players’ strategies
(Thevenot et al., 2010). Grand masters are supposed to have
memorized many patterns corresponding to configurations of
pieces (Simon & Gilmartin, 1973). In order to verify this assump-
tion and to study strategies in less-expert players, a recognition
task of a specific chessboard configuration could be proposed to
participants just after they play a move. When a configuration and
its associated move have been stored in long-term memory, no
mental transformation of the current configuration has to be done
before playing. On the contrary, if the player has to play from a
configuration that has not been stored in long-term memory, sev-
eral possibilities have to be mentally represented before the next
move. As a consequence, and following the rationale of the ORP,
it should be quicker to recognize a pattern of pieces when the last
move was strongly associated with a memorized configuration
than when either the configuration was not stored or the move was
not associated with it. The ORP could also be adapted in the
domain of reading. We know that words can be read either using
a lexical path allowing words to be recognized as wholes or using
a sub-lexical path when words are recognized through grapheme–
phoneme correspondence (e.g., Coltheart, 1985;Ziegler et al.,
2008). Of course grapheme–phoneme correspondence is per-
formed quickly in expert readers, and it might be difficult to
determine exactly which one of the lexical or sub-lexical strategies
has been used by individuals. An adaptation of the ORP could be
useful to address this question. Adults, normal developing chil-
dren, or children with dyslexia could be presented with a word
recognition task after reading. Within the sub-lexical procedure, a
word is necessarily decomposed into grapheme and phoneme
components, whereas the word does not suffer from any decom-
position when processed lexically. Then, and following the ORP
rationale, it should take longer to recognize a word after sub-
lexical than lexical processing.
To conclude, we have demonstrated that the ORP constitutes a
reliable method of investigation in the domain of mental arithme-
tic. Therefore, previous conclusions that subtraction and addition
solutions do not rely on retrieval as often as what was assumed in
the literature can be trustworthy. Future investigations in the
domain of numerical cognition using the ORP will be possible to
carry out confidently, and it will be especially interesting to study
multiplication. As for other arithmetic operations, the ORP could
reveal that, contrary to the present consensus (e.g., Dehaene et al.,
2003), multiplication problems are not always solved through
retrieval of the answer from memory. Finally, as just explained,
adaptations of the ORP to other domains of human cognition also
seem of potential interest to answer questions that are harder to
address with more classical methods of investigation. Still, in the
domain of arithmetic, standard methods are necessary to answer
questions that cannot be addressed using the ORP. Indeed, the
ORP cannot reveal the specific procedural steps taken by individ-
uals throughout the solving process. When such steps are time
consuming enough to reach consciousness, and participants do not
feel the social desirability to report a specific strategy (see Yackel
& Cobb, 1996, in regard to children), verbal reports can be useful.
When it is not the case, the ORP can provide precious data from
which individuals’ strategies can be inferred.
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Received May 31, 2013
Revision received June 24, 2014
Accepted June 25, 2014 !
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