Academic Paper Writing Service http://StudyHub.vip/Algorithmic-Solution-Of-Arithmetic-Prob 👈

1. Algorithmic solution of arithmetic problems
and operands–answer associations in
long-term memory
Catherine Thevenot and Pierre Barrouillet
University of Burgundy, Dijon, France
Michel Fayol
Blaise Pascal University, Clermont-Ferrand, France
Many developmentalmodels of arithmetic problem solving assume that any algorithmic solution
of a givenproblemresults in an association of the two operandsand the answer in memory (Logan
& Klapp, 1991; Siegler, 1996). In this experiment, adults had to perform either an operation or a
comparison on the same pairs of two-digit numbers and then a recognition task. It is shown that
unlike comparisons, the algorithmic solution of operations impairs the recognition of operandsin
adults. Thus, the postulate of a necessary and automatic storage of operands–answer associations
inmemory whenyoungchildren solve additionsbyalgorithmic strategies needsto bequalified.
Simple arithmetic operations like additions and subtractions are practised very early in child-
hood and used frequently throughout life. Developmental research into arithmetic skills has
shown that young children solve these operations by counting one by one, with or without
external cues, for example, using their fingers (Baroody, 1987; Carpenter & Moser, 1983;
Fuson, 1982). Later in development, these operations, especially additions, are thought to be
solved using a strategy for the direct retrieval of the answer associated with the operands. For
example, 3 + 4 would trigger the retrieval of 7 from memory without any need to count
(Ashcraft & Battaglia, 1978; Ashcraft & Fierman, 1982; Ashcraft & Stazyk, 1981; Siegler &
Shrager, 1984). Thus, the strategies would evolve with practice from algorithmic computing
to direct retrieval from memory (Barrouillet & Fayol, 1998; Siegler, 1996).
Considering both the early acquisition, from age 3 or 4 onwards, and the frequent use of
these operations, it might be expected that adults would systematically retrieve the answers
from memory, at least for the simplest problems. Accordingly, Siegler’s computer simulation
of addition solving manifests an increasing and finally systematic use of the retrieval strategy
(Siegler& Shipley,1995;Siegler& Shrager,1984).It mighteven be expectedthat many adults
would retrieve answers larger than 18 (9 + 9). Indeed, the arithmetic operations used in
Requests for reprints should be sent to Pierre Barrouillet, Université de Bourgogne,LEAD–CNRS, Faculté des
Sciences Gabriel, 6 Bld Gabriel, 21000 Dijon, France. Email: barouil@satie.u-bourgogne.fr
Ó 2001 The Experimental Psychology Society
http://www.tandf.co.uk/journals/pp/02724987.html DOI:10.1080/02724980042000291
THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2001, 54A (2), 599–611

2. everyday life often make it necessary to process numbers larger than 9. Now, it appears that
many adults do not systematically use the retrieval strategy and that even the more expert
among them are sometimes prone to use back-up strategies to solve 7 + 9, for example
(Lefevre, Sadesky, & Bisanz, 1996). The aim of this article is to shed some light on this para-
dox: An early acquired and frequently used algorithm does not always appear to lead to a sys-
tematic and automatic retrieval of the answer from memory. This fact could be due to weak
associations between certain operands and the answer, which reduce the chances of a quick
and reliable retrieval from memory (Anderson, 1993; Anderson & Lebiere, 1998; Siegler &
Shrager, 1984).
The storage of associations between operands and answer requires the simultaneous
attentional processing of these three numbers (Logan, 1988; Logan, Taylor, & Etherton,
1996). Reaching the answer while the operandsare held in working memory would lead to the
storage of a chunk of knowledge (Anderson, 1993), or a memory instance (Logan, 1988).
Recurrent solutions of the same problem would either strengthen the associative links in this
chunk (Anderson, 1993) or increase the number of available instances in memory (Logan,
1988). In both cases, the strengthening of associations in the chunk or the constitution of new
memory instances available for retrieval would depend on the amount of attentional resources
simultaneously allocated to the three numbers (i.e., the two operands and their associated
answer).
The storage of this knowledgein memory initially results from the implementation ofalgo-
rithmic strategies (Logan, 1988; Siegler, 1996). Now, these strategies are demanding and
rather slow, at least for young children (Siegler, 1996). The time needed by the algorithm to
reach theanswer and its cognitive costshouldlead to a reductionin the level ofactivation ofthe
operands. This decrease in activation would result both from a memory decay phenomenon,
which damages memory traces (Towse & Hitch, 1995; Towse, Hitch, & Hutton, 1998), and
from the necessary concurrentactivation oftransitory results, which inducesa resource trade-
off (Anderson, 1993). As a consequence, when the algorithm reaches the answer, the memory
traces of operands could be too damaged to ensure correct storage in long-term memory. This
phenomenon should be more pronounced for large operands, because they result in a higher
number of processing steps and longer solution times. Thus, subjects would have to count
again when confronted once again with the same problem.
This hypothesis has been tested by contrasting the relative difficulty that adults encoun-
tered in recognizingthe operandseitherafter theiraddition or subtraction orafter theirsimple
comparison with a third number (Wickens, Moody, & Dow, 1981). We assume that the pro-
cesses required by the solution to two-digit numberoperations in adults are akin to those used
by children to solve simple problems (i.e., some decomposition of at least one of the two
operands and a step-by-step process that makes it necessary to keep intermediate results in
working memory). The accessibility of operands in memory has been tested using a recogni-
tion task in which operands were presented as targets among distractors after the solution of
eitheran operation (i.e., addition or subtraction) or a comparison task. The participants had to
judge whether a given presented number was involved or not in the task they had previously
performed.
Suppose that an adult has to perform 37 + 28. Severalproceduresare available but most of
them require the subject to decompose the operands because it is quite improbable that the
answer would be available in memory. Some subjects could decompose 37 and 28 into 30 + 7
600 THEVENOT, BARROUILLET, FAYOL

3. and 20 + 8, respectively, add 20 to 30, temporarily store this transitory result (50), then com-
pute or retrieve 7 + 8, and finally add 15 to 50. Other decompositions are possible but all of
them lead to a shift ofthe attention from the operands(37 and 28) to their components(30 and
7, and 20 and 8, respectively). As we noted earlier, the time needed by algorithmic computa-
tion induces a memory decay of the initial operands, and the need to maintain intermediary
results reducesthe amountofresources available for each item, including the operands. Thus,
the algorithmic solution of operations should degrade the memory traces of operands. In con-
trast, a comparison problem (e.g., decide whether 31 lies between 37 and 28) makes it neces-
sary to keep the numbers in memory without any transformation.
According to Anderson (1993), both the latency and the probability of correct retrieval are
exponential functions of the level of activation of knowledge: The higher this level of activa-
tion is, the quicker and more probable the retrieval. Thus, both the accuracy and the speed of
recognitionofoperandsafter performingthe task (i.e., eitheran operation or acomparison)are
indicative of the residual activation of the operands. As a consequence, operand recognition
should be easier and faster after performing a comparison than after an addition or a
subtraction.
To summarize, in this experiment we asked adults to solve three kinds of problem: addi-
tions, subtractions, and comparisons. After being informed of the kind of problem to be
solved, they were successively presented with the two operands and a third number.The par-
ticipants were asked to judge whether this third number corresponded to the sum of (or the
difference between) the two operands in the case of additions (or subtractions), or whether it
lay between the two operandsin the case ofcomparisons. The participants gave their response
by pressing a key on the computer keyboard, and then a fourth number was presented. They
had to judge whether they had previously seen this number or not (recognition task). Accord-
ing to the hypothesis that algorithmic solution damages the memory traces of operands, we
predicted longer reaction times and lower rates of correct responses in the recognition task
after solving operation (additions and subtractions) than after solving comparisons.
METHOD
Participants
Twenty-fourundergraduatestudents from the University ofBurgundytookpart inthis experiment.
Material
A total of 96 pairs of two-digit numberswere used.In order to optimize the probability of an algorith-
mic solution for operations, the addition always required a carry, the difference was always larger than
10, and neither of the two operations (addition and subtraction) had an answer ending with a 0 (cf.
Appendix1).These96pairs of numberswere assigned using a circular permutation to the 24experimen-
tal conditions resulting from a 3 (problems: addition, subtraction, comparison) × 2 (possible answers for
the problems: true or false) × 4 (types of item presented in the recognition task) design. Thus, each par-
ticipant was presented with 4 pairs of numbers in each of the 24 experimental conditions. Four types of
item were presented in the recognition task (n1, n2, n3, n4). The items n1 and n2 were the first and sec-
ond operands, respectively, the former always being larger than the latter in order to permit the subtrac-
tion of n2 from n1, and the items n3 and n4 were distractors, which differed according to the task
(operations vs. comparison). As far as operations are concerned, n3 and n4 were produced by adding or
OPERANDS–ANSWER ASSOCIATIONS IN LTM 601

4. subtracting 1 or2 from the operands.Asfar ascomparisonsare concerned,n3waslarger than n1(i.e., the
larger operand), and n4 was smaller than n2. Half of the experimental trials presented an operation or a
comparison that elicited a “yes” response (i.e., the third number was the sum n1 + n2 for additions, or
was the difference n1 – n2 for subtractions, or lay between n1 and n2 for comparisons), whereas the
remaining trials presented problems with an incorrect answer for operations or a third number that did
not lie betweenn1 and n2 for comparisons. For each of these two kinds of trial, half were followed by the
presentation of a number previously seen (n1 or n2) and the other half by a distractor (n3 or n4).
In orderto preventparticipants from adoptinga systematic andactive strategy ofmemorization ofthe
operandswith aview to their subsequentrecognition,weadded192fillers bytwice presenting eachofthe
96 experimental trials without a recognition task in them.
Procedure
The stimuli were presented on screen. Each trial began with the presentation for 1 s of a word that
indicated the type of problem to be solved (addition, subtraction, or comparison). This word was
replaced by the first 2-digit number of the pair (n1). By pressing a key on the computer keyboard, the
participant deleted this number and displayed the second operand (n2). By pressing the key again, the
participant substituted a third number for n2. He or she had to judge whether this third number corre-
sponded to n1 + n2 in the case of additions, to the difference n1 – n2 in the case of subtractions, or
whether it lay between n1 and n2 in the case of a comparison. The participants were asked to give their
response (“yes” or “no”) as quickly and as correctly as possible by pressing one of two keys on the key-
board. Thetype of response,the time of presentation of n1 andn2, andthe reaction time for the response
were recorded. For the experiment trials only, this response displayed a fourth number on screen. The
participants had to judge whetheror not they had seen this numberamong the first two presented num-
bersbypressing the same keysusedfor the response to the problem.Thetypeofresponse (“yes”or“no”)
and the reaction time were registered. This last response displayed a next-trial signal.
For example, an experimental trial might have taken the form
Addition/39/16/55/41
where the third number(55) required “yes” response because39 + 16 = 55, whereas the fourth number
(41) required a “No” response because 41 had not appeared previously in this trial. The corresponding
filler comprised the series “Addition/39/16/55”only.The288trials (96 experimentaltrials and192fill-
ers) were randomly presented.
Results
The rates of correctresponsestothe problemswere high (.897, .941,and .978 for additions,
subtractions, and comparisons, respectively), providing evidence that participants paid suffi-
cientattention tothe problemsthat precededthe recognitiontask. Amongthe 96 experimental
trials, only the 48 trials where a target (n1 or n2) was presented in the recognition task were
analysed—that is, 1,152 trials (24 participants × 48 trials). Indeed, the rate of correct rejection
of distractors (n3 and n4) was very high (.955) and did not differ from one type of problem to
another (.957, .954, and .955 for additions, subtractions, and comparisons, respectively).
Analysis of the rate of correct responses in the recognition task
Among the 1,152 trials collected, those that elicited an incorrect response to the problem
were discarded (42, 27, and 7 trials for additions, subtractions, and comparisons,
602 THEVENOT, BARROUILLET, FAYOL

5. respectively)—that is, less than 7% of the trials. A 3 (type of problem:additions, subtractions,
and comparisons) × 2 (type of target: n1 and n2) analysis of variance (ANOVA) with the two
factors as repeated measures were performed on the rate of correct recognition (Table 1). As
we predicted, the rate of correct recognition was higher after a comparison (.98) than after an
addition (.88) or a subtraction (.92), F(2, 46) = 8.24, p < .001, MSE = 0.015. There was no
significant difference between these last operations, F(1, 23) = 2.54, p = .12, MSE = 0.018.
The first operand was recognized more often than the second (.96 and .90 for n1 and n2,
respectively), F(1, 23) = 13.84, p = .001, MSE = 0.009. This effect was observed for both
additions (.94 and .83 for n1 and n2, respectively) and subtractions (.96 and .89, respectively),
but not for comparisons (.98 for n1 and n2). The Type of Problem × Type of Target interac-
tion was significant, F(2, 46) = 4.61, p < .02, MSE = 0.009. This result suggests that the
additions and subtractions had a more detrimental effect on the memory trace of the second
than of the first operand.
Analysis of the reaction times (RT) in the recognition task
This analysis concerned only the correct recognitions (hits). Among the 1,152 trials, those
that elicited eitheran incorrectresponsetotheproblem(76 trials, seeearlier) ora non-recogni-
tion (46, 27, and 4 trials for additions, subtractions, and comparisons, respectively) were dis-
carded, resulting in 999 retained trials. A mean RT was calculated for each participant in each
of the six experimentalconditions:3 (types of problem)× 2 (typesoftarget). An ANOVA with
the same design as the previous one was performed on these mean RTs (Table 1). As we pre-
dicted, the recognition of the targets was faster after comparisons (1,401 ms) than after addi-
tions (1,859 ms) and subtractions (1,809 ms), F(2, 46) = 21.72, p < .001, MSE = 140,040.
The difference between additions and subtractions was not significant, F < 1. Contrary to
what was observedfor the rate ofcorrectrecognitions,theRTs fortherecognition of n1 and n2
did not differ significantly (1,669 and 1,711 ms, respectively), and the interaction was not sig-
nificant, Fs < 1.
OPERANDS–ANSWER ASSOCIATIONS IN LTM 603
TABLE 1
Reaction times
a
and frequencies of correct recognition of the targets n1 and n2 (first and
second operands) as a function of the type of problem to be solved
Type of target
————————————————————————————————–
n1 n2
——————————————– ——————————————–
Frequency RT Frequency RT
—————– —————– —————– —————–
M SD M SD M SD M SD
Addition 0.936 0.111 1,821 457 0.825 0.177 1,898 650
Subtraction 0.958 0.070 1,757 487 0.889 0.154 1,862 511
Comparison 0.979 0.047 1,429 402 0.983 0.064 1,373 366
a
In ms.

6. Recognition task performance as a function of the status (true or false) of
the operations
It could be argued that comparisons elicited longer RTs and poorer rates of recognition
because operations required the participants to memorize more numbers than comparisons.
Indeed, although in each of the experimental conditions three numbers were presented on
screen before the target to be recognized, four numbers were involved when false operations
were proposed (i.e., the two operands, the correct answer the participant had calculated, and
the false answer proposed). This higher number of items could account for the difference
betweencomparisonsand operations.However,this was notthe case. If theeffect we observed
depended mainly on the number of items treated during problem solving, then this effect
should be larger for false than for true problems. Indeed, false problems involved four num-
bers (two operands, the calculated answer, and the proposed answer), whereas true problems
involved only three, calculated and proposed answer being the same. Actually, the size of the
effect was roughly equivalent for false and true problems. The difference in recognition times
betweenoperations and comparison was 439 ms for the false problems (1,819 and 1,358 msfor
calculations and comparison, respectively), and 407 ms for the true problems (1,850 and
1,443 ms, respectively). This latter difference was highly significant, F(1, 23) = 15.05, p <
.01. Thus,even when true problems were presented,participants took longer to recognize tar-
gets after operations than after comparisons. The difference in the rate of correct recognition
was .097 for false problems (.887 and .984 for calculations and comparison, respectively), and
.060 for true problems (.917 and .977,respectively). This latter difference was also significant,
F(1, 23) = 9,91, p < .01.
As a consequence, the hypothesis that the difference in recognition performance between
comparison and operations was due to a difference in the number of items to be kept in mem-
ory can be discarded.
Relations between RTs for the solution of problems and performance on
the recognition task
Although the results confirmed our hypothesis, two possible alternative explanations have
to be discarded. The first is that the faster and more accurate recognition of targets after com-
parisons than after operations could be due to the fact that participants used some short-cut
strategies to verify the operations. For example, it is known that participants sometimes use
judgementsabouttheodd/even status oftheanswer torejectfalse answers in verification tasks
(Krueger,1986):for example, seeing theaddition oftwo even numbersproduceanoddanswer
would let participants know that the answer is incorrect without doing the addition. Such a
strategy would lead to incorrect recognitions because the numbers had not been really pro-
cessed. The second is that the better performances in recall for comparisons than for opera-
tions could be due to a slower solution of these latter problems leading to a longer delay of
memorization and thus to weaker performance.
As far as the first point is concerned,if the failure to recognize the operands was due to the
use of short-cut strategies in solving operations, the operands would not have been processed
in these trials, resulting in shorter presentation times. In fact, the presentation times did not
differ whether the operands were subsequently recognized (1,549, 3,176, and 1,740 ms for
n1, n2, and n3, respectively) or not (1,640, 3,296, 1,814 ms, respectively), Fs < 1. Thus, we
604 THEVENOT, BARROUILLET, FAYOL

7. cannot suppose that the operations led to poorer recognition of operands than comparisons
because the former could be solved without calculation.
As far as the second point was concerned,the time required for solution (total presentation
time on screen for n1 + n2 + the proposed answer) was longer for additions and subtractions
(6,339 and 6,641 ms, respectively) than for comparisons (4,590 ms), F(1, 23) = 39.85, p <
.001, MSE = 2,990,500. However, this difference was mainly due to a longer presentation of
n2 for additions and subtractions (3,154 and 3,197 ms, respectively) than for comparisons
(1,397 ms), F(1, 23) = 49.00, p < .001, MSE = 2,065,400, whereas the presentation times of
n1 and of the proposed answer were quite similar among the three types of problem (see Table
2).
As a consequence,it is quite improbable that the differences observed in the rates and RTs
for the correct recognition of the targets were due to a longer memory retention period in the
case of additions and subtractions. Indeed, the differences in the rates of correct recognition
mainly affected n2.Note that n2 had to be maintained only during the presentation of the pro-
posed answer (the third number presented) before being recognized when it appeared as the
target in the recognition task. The presentation times of the proposed answer were quite simi-
lar for the three types of problem, and even slightly shorter for additions than for comparisons
(cf., Table 2). Thus, the better recognition of n2 targets after comparisons could not result
from shorter periods of maintenance. Furthermore, the rate of correct recognition for n2 was
lower for additions and subtractions than for comparisons, whereas the presentation times of
n2 were longer for additions and subtractions (3,154 and 3,197 ms) than for comparisons
(1,397 ms). In other words,the longera numberwas presentedon screen, the harder its subse-
quent recognition.
The same argument applies to the RTs. The faster recognition of n1 after comparisons
could result from a faster solution of these problems (and especially a shorter presentation of
n2). However, n2 targets were also recognized faster after comparisons than after operations,
whereas the periods of maintenance were quite similar for the different types of problem (see
earlier). As a consequence, the observed differences between comparisons, on the one hand,
and additions and subtractions on the other, cannot be explained by the shorter solution times
for comparisons.
OPERANDS–ANSWER ASSOCIATIONS IN LTM 605
TABLE 2
Times of autorepresentation
a
of the operands n1 and n2 and the
proposed answer
b
for the three types of problems
Type of problems
———————————————————————–
Addition Subtraction Comparison
—————– —————– —————–
Type of number M SD M SD M SD
n1 1,549 835 1,549 159 1,298 453
n2 3,154 839 3,197 1,581 1,397 724
Answer 1,636 580 1,844 614 1,895 420
a
In ms.
b
For comparisons,“answer” refers to the third number presented, the participants
being asked whether their “answer” was between n1 and n2 or not.

8. Discussion
The results clearly highlight the fact that the retrieval of operandsfrom memory is slower and
moredifficult after solving an operation (addition or subtraction)than after performing a com-
parison. This suggests that the algorithmic solution of additions or subtractions does not nec-
essarily preserve memory traces of operands, unlike comparisons that do not require any
transformation of operands. This fact suggests that even a short algorithmic solution (about
5 s, that is to say the total presentation time for n2 and the proposed answer) is sufficient to
impede the retrieval of operands from memory.
Our results are of interest for the understanding of numerical processes in adults but also
have implications for the development of numerical abilities in children. Numerical abilities
do not represent a unitary set of skills and knowledge (Dowker, 1998). Indeed, they involve
different kinds of knowledge (conceptual, declarative, and procedural) and representations.
Dehaene (1992) suggested that numerical abilities involve auditory verbal representations,
but also visual arabic and even analogue representations, each kind of representation being
used for a different purpose. For example, number comparison would be performed from an
analogue magnitude representation that does not lead to number decomposition, whereas
multi-digit operations would involve both visual arabic and verbal codes and the use of
sequential strategies leading to the decompositionof numbers.Our results are in line with this
componential approach. It could be supposed that, in many cases, participants could perform
the comparison between two-digit numbers by comparing only the decade digits. This strat-
egy would lead to a kind of decomposition and to poor recognition performance. However,
recognition was quite easy after comparison, suggesting that comparison involves an analogue
holistic representation that preserves numbers, whereas solving operations requires the par-
ticipants to decompose the numbers and retrieve verbal addition tables from memory.
Thus, our results support a modular conception of numerical abilities in adults but seem
also particularly relevant to understanding the learning of number facts in children. The
development of arithmetic skills such as problem solving is usually described as a shift from
algorithmic strategies (e.g., counting all, counting on, or min strategies) to the direct retrieval
of the answer from memory. Most of the theoretical models take it for granted that the use of
algorithms in young children leads them to memorize associations between operands and the
reached answer. Thus, Logan (1988; Logan & Klapp, 1991) assumes that each algorithmic
solution gives rise to a memory instance that links the operands with the answer. To retrieve
one of these instances would ensure an automatic solution (“automaticity as memory
retrieval”). Practice would increaseboththenumberofinstancesstoredin memory and conse-
quently the probability of retrieving one of them. As far as Siegler’s model is concerned
(Siegler & Shipley, 1995; Siegler & Shrager, 1984), associations between operands and
answers would also result from algorithmic solutions. The associations between a pair of
operands and a given answer would be strengthened each time this answer is reached. In the
end, all the simple additions (operands less than 10) would be solved by directly retrieving
their answers from memory because the strength of the associations between operands and
answers is sufficient, as testified to bythe computersimulations. Andersonand Lebiere (1998)
invoke the same process.
Thus, all of the models assume that when an algorithm has produced an answer, the
operandsare still presentin working memory and available to be associated with this answer in
606 THEVENOT, BARROUILLET, FAYOL

9. long-term memory. The direct retrieval strategy would only fail because of insufficient prac-
tice. However, our results suggest that these models underestimate certain constraints which
hamper the memorization of association process. Even in adults, and after a relatively short
solution time (about 5 s), one of the two operands was sometimes inaccessible, probably
because its memory trace was damaged (it can be considered that in more than 20% of the
addition trails, at least one operand was lost after calculation: 1–.936 * .825 in Table 1). This
blurring of the memory traces could impede the storage of associations in long-term memory.
We assume that this phenomenon could be more pronounced in young children.
Indeed, it is quite possible that this blurring is all the more pronouncedthe longer the solu-
tion is and the younger the children are. Using a double-task paradigm, Towse and Hitch
(1995) have shown that the memorization and recall of items depend heavily on the time
needed to perform a concurrent task, especially in young children. Interestingly, the concur-
rent task used by the authors was a counting task, which is akin to the processes involved in
algorithmic strategies like min or counting all, and the items to be memorized were the results
of this counting—that is, numbers (Case’s counting span task, Case, Kurland, & Goldberg,
1982). As we have already stressed, the computation needed by all the algorithmic addition
strategies implies an attentional shift from the operands to some intermediate outputs. For
example, the min strategy for8 + 4 requires an attentional focusing on 8andthenthemonitor-
ing of 4 steps forward in the number line. Thus, the number 4 must be kept in short-term
memory but as soon as the first step is achieved (i.e., 9), 8 can be droppedbecause three differ-
ent values have to be held active in memory: 9, 1 (i.e., the number of steps already performed)
and 4 (i.e., the total number of steps to be performed). Thus, the memory trace for 8 can only
fade away as thealgorithm goeson. Finally, as we observed,it could be that this memory decay
results in blurred memory traces or even in a loss of the operands, which could impede their
association with the answer in long-term memory.
The studies on addition in young children reinforce this hypothesis. Siegler (1987)
reported a mean time of 5.6 s to perform the min strategy in 5- to 7-year-old children. Less
sophisticated strategies such as countingall require up to 15 s. It is quite conceivable that after
such delays, memory traces are sufficiently blurred to impede associations in memory. It
should be remembered that the mean counting span—that is, the numberof digits a child can
memorize and recall while performing counting—is only 1.65 in 6-year-old and 2.3 in 7-year-
old children (Case et al., 1982), and that the memorization of an additive numberfact requires
the storage of three numbers.
We are not claiming that algorithmic solution prevents any association between operands
and answer in memory but only that algorithmic solution involves both time duration and
attentional shifting that lead to damage to the memory traces of operandsand to weakening of
the strength of this association. These effects should be especially pronouncedin children for
whom algorithmic solutions are highly demanding and time consuming, and the speed of
memory decay in short-term memory is higherthan in adults (Keller & Cowan, 1994; Saults &
Cowan, 1996).
Thus, it is possible that the relation between additive procedural knowledge based on
counting and declarative knowledge of number facts is not as strong and straightforward as is
usually assumed. Although both types of knowledge are probably underpinned by the same
kind ofverbalrepresentation in an auditory verbal word frame (Dehaene,1992), there is much
evidence of discrepancies and dissociations between procedural and declarative knowledge.
OPERANDS–ANSWER ASSOCIATIONS IN LTM 607

10. The neuropsychological literature describes relatively circumscribed forms of impairment in
eithertheexecutionofcalculation proceduresortheretrieval ofarithmetic facts, bothin adults
suffering from acquired dyscalculia (Cohen & Dehaene, 1994; Warrington, 1982) or in chil-
dren presenting a developmental dyscalculia (Sokol, Macaruso, & Gollan, 1994; Temple,
1991).
In thesame way, studieson learning-disabled children haveshown that theyhavedifficulty
in recalling basic additive arithmetic facts (Fleischner, Garnett, & Shepherd,1982; Garnett &
Fleischner, 1983).It hasbeenobservedthat,whensolvingsingle-digit additions, learning-dis-
abled children in 4thand6th graderelyoncountingstrategies whereasnormal-ability children
use direct retrieval from memory, and that they count more slowly and produce more errors
than normal-ability children (Geary,Brown, & Samaranayake, 1991;Geary, Widaman, Little,
& Cormier, 1987). These facts could suggest that there is a strong relationship between count-
ing procedures and the retrieval of arithmetic facts. For example, Siegler & Shrager’s (1984)
distribution of association model accounts for these phenomena assuming that erroneous
answers produced by faulty counting strategies become associated with the problems, result-
ing in flat distributions of associations. However, Ackerman, Anhalt, and Dykman (1986)
have suggested that this difficulty in automatization of number facts could stem from poor
sequential-memory abilities. Our results lend support to this hypothesis and suggest that, as
Macaruso and Sokol (1998, p. 220) pointed out, “the relationship between counting skills and
subsequent retrieval of facts is not as straightforward as one might assume”.
Finally, ourresults could contribute to accounting for some effects concerningthe solution
of additive problems in children and adults, namely the size and the tie effects. As evidenced
by Towse and Hitch (1995; Towse et al., 1998), the longer the retention period, the stronger
the memory decay. Algorithmic strategies take longer, the larger the operands are (Groen &
Parkman, 1972). Thus, the blurring of memory traces shouldbe stronger and a correct encod-
ing of theassociations less probablefor large than for small operands.This phenomenoncould
partially account for the size effect, which is particularly strong for additions—that is, addi-
tions take longer with large than with small operands (Campbell & Graham, 1985; Groen &
Parkman, 1972;Zbrodoff, 1995). This size effect is usually ascribed to the fact that large prob-
lems suffer from more interference (Campbell & Graham, 1985; Zbrodoff, 1995), are associ-
ated with more erroneous answers (Siegler, 1988; Siegler & Shrager, 1984), or are performed
less frequently (Anderson & Lebiere, 1998; Sigler, 1996) than small problems. For example,
the association between operands and answer should be stronger for small than for large prob-
lems because the former are performed earlier and more frequently than the latter. When the
retrieval strategy becomes predominant, each retrieval would strengthen the association and
in turn increase both the probability of the use of retrieval and its speed. It is undeniable that
the earlier practice and the higher frequency of small problems can at least in part account for
the size effect. However, it could be that the associations between operands and answers are
weaker for the large operands not only because they are less frequently encountered, but also
because the algorithmic solution of large problems takes longer and has a more detrimental
effect on the memory traces and the associative learning process. Indeed, Lefevre et al. (1996)
reported that large problems are more often solved by algorithmic strategies than are small
problems, even in skilled adults.
The hypothesis of a memory decay could also account for the tie effect in additions. Tie
additions(e.g., 3 + 3,4 + 4,etc.) are solvedfaster and more accurately thanothers,suggesting
608 THEVENOT, BARROUILLET, FAYOL

11. that their answer is more often retrieved from memory. The two operands are identical, a fact
that should make them easier to maintain in working memory and thus facilitate their associa-
tion with the answer because there is only one memory trace to keep active.
In conclusion, our results suggest that the postulate of a systematic and automatic associa-
tion between operands and answers cannot be endorsed as it stands. Any algorithmic strategy
implies both a temporal delay between the encoding of operands and the reaching of the
answer, and a transformation of at least one of the operands. Thus, algorithmic strategies
imply a memory decay phenomenon and an attentional shift from the operands to intermedi-
ate outputs, which damage the memory traces of operands. Psychological models and com-
putersimulations shouldtake accountofthese constraints andtheir relative impact depending
on the subject’s developmental level and the time needed to implement different algorithmic
strategies.
REFERENCES
Ackerman,P.T., Anhalt,J.M., & Dykman,R.A. (1986).Arithmetic automatizationfailure in children with attention
and reading disorders: Associations and sequela. Journal of Learning Disabilities, 19(4), 222–232.
Anderson, J.R. (1993). Rules of the mind. Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
Anderson,J.R., & Lebiere, C.(1998).Atomic componentsof thought.Hillsdale, NJ:LawrenceErlbaumAssociates,Inc.
Ashcraft, M.H., & Battaglia, J. (1978).Cognitive arithmetic: Evidence for retrieval and decision processes in mental
addition. Journal of Experimental Psychology: Human Learning and Memory, 4, 527–538.
Ashcraft, M.H., & Fierman, B.A. (1982).Mental addition in third, fourth,and fifth graders. Journalof Experimental
Child Psychology, 33, 216–234.
Ashcraft, M.H., & Stazyk, E.H. (1981).Mental addition: A set of three verification models. Memory & Cognition, 9,
185–196.
Baroody,A.J. (1987).The development ofcountingstrategies forsingle-digit addition.Journalfor Researchin Mathe-
matics Education, 18, 141–157.
Barrouillet, P., & Fayol, M. (1998). From algorithmic computing to direct retrieval. Evidence from number- and
alphabetic-arithmetic in children and adults. Memory & Cognition, 26, 355–368.
Campbell, J.I.D., & Graham, D.J. (1985).Mental multiplication skill: Structure, process, and acquisition. Canadian
Journal of Psychology, 39, 338–366.
Carpenter,T.P., & Moser, J.M. (1983).The acquisition of addition and subtractionconcepts.In R. Lesch & M. Lan-
dau (Eds.), Acquisition of mathematical concepts and processes. New York: Academic Press.
Case, R., Kurland, M., & Goldberg, J. (1982). Operational efficiency and the growth of short-term memory span.
Journal of Experimental Child Psychology, 33, 386–404.
Cohen,L.,&Dehaene, S.(1994).Amnesiaforarithmetic facts:A single case study.BrainandLanguage,47,214–232.
Dehaene, S. (1992). Varieties of numerical abilities. Cognition, 44, 1–42.
Dowker,A. (1998).Individual differences in normal arithmetic development. In C. Donlan (Ed.), The development of
mathematical skills (pp. 275–302). Hove, UK: Psychology Press.
Fleischner, J.E., Garnett, K., & Shepherd, M. (1982). Proficiency in arithmetic basic fact computation of learning
disabled and non disabled children. Focus on Learning Problems in Mathematics, 4, 47–55.
Fuson,K.C. (1982).An analysisof thecountingon solution procedurein addition.In T.P. Carpenter, J.M. Moser, &
T.A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 67–81). Hillsdale, NJ: Lawrence
Erlbaum Associates, Inc.
Garnett, K., & Fleischner, J.E. (1983).Automatization and basic fact performance of normal and learning disabled
children. Learning Disability Quarterly, 6, 223–230.
Geary, D.C., Widaman, K.F., Little, T.D., & Cormier, P. (1987). Cognitive addition: Comparison of learning dis-
abled and academically normal elementary school children. Cognitive Development, 2(3), 249–269.
Geary, D.C., Brown, S.C., & Samaranayake,V.A. (1991).Cognitiveaddition: A short longitudinal study of strategy
choice and speed of processing differences in normal and mathematically disabled children. Developmental Psy-
chology, 28(5), 787–797.
OPERANDS–ANSWER ASSOCIATIONS IN LTM 609

12. Groen, G.J., &Parkman,J.M., (1972).Achronometricanalysisofsimple addition.Psychological Review, 79,329–343.
Keller, T.A., & Cowan,N. (1994).Developmental increase in the duration of memory for tone pitch. Developmental
Psychology, 30, 855–863.
Krueger, L.E. (1986).Why 2*2 = 5 lookssowrong:On the odd-evenrule in productverification. MemoryandCogni-
tion, 14, 141–149.
Lefevre, J., Sadesky, G.S., & Bisanz, J. (1996).Selection of procedures in mental addition: Reassessing the problem
size effect in adults. Journal of Experimental Psychology: Learning, Memory, and Cognition, 22, 216–230.
Logan, G.D. (1988).Toward an instance theory of automatization. Psychological Review, 95, 492–527.
Logan,G.D., & Klapp, S.T. (1991).Automatizing alphabet arithmetic: I. Is extended practice necessary to produce
automaticity? Journal of Experimental Psychology: Learning, Memory, and Cognition, 17, 179–195.
Logan,G.D., Taylor,S.E., &Etherton,J.L. (1996).Attentionin theacquisition andexpression of automaticity.Jour-
nal of Experimental Psychology: Learning, Memory, and Cognition, 22, 620–638.
Macaruso,P., & Sokol, S.M. (1998).Cognitiveneuropsychologyand developmental dyscalculia. In C. Donlan (Ed.),
The development of mathematical skills (pp. 201–225). Hove, UK: Psychology Press.
Saults, J.S., & Cowan,N. (1996).The developmentof memory forignored speech. Journalof ExperimentalChild Psy-
chology, 63, 239–261.
Siegler, R.S. (1987). The perils of averaging data over strategies: An example from children’s addition. Journal of
Experimental Psychology: General, 116, 250–264.
Siegler, R.S. (1988).Individual differences in strategy choice: Good students, not-so-goodstudents,and perfection-
ists. Child Development, 59, 833–851.
Siegler, R.S. (1996).Emergingminds:Theprocess ofchangein children’s thinking. New York:OxfordUniversity Press.
Siegler, R.S., & Shipley, C. (1995).Variation, selection, and cognitive change. In T.J. Simon & G.S. Halford (Eds.),
Developing cognitive competence: New approachesto process modeling. Hillsdale, NJ: Lawrence Erlbaum Associates,
Inc.
Siegler, R.S., & Shrager,J. (1984).Strategychoices in addition andsubtraction:Howdochildren knowwhatto do?In
C. Sophian (Ed.), Origins of cognitive skills. Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
Sokol, S.M., Macaruso, P., & Gollan, T.H. (1994). Developmental dyscalculia and cognitive neuropsychology.
Developmental Neuropsychology, 10, 413–441.
Temple, C.M. (1991). Procedural dyscalculia and number fact dyscalculia: Double dissociation in developmental
dyscalculia. Cognitive Neuropsychology, 8(2), 155–176.
Towse, J.N., & Hitch, G.J. (1995).Is there a relationship between task demand and storagespace in tests of working
memory capacity? Quarterly Journal of Experimental Psychology, 48A(1), 108–124.
Towse, J.N., Hitch, G.J., & Hutton, U. (1998).A reevaluation of working memory capacity in children. Journal of
Memory and Language, 39, 195–217.
Warrington,E.K. (1982).The fractionationof arithmetical skills: A single case study.QuarterlyJournalofExperimen-
tal Psychology, 34A, 31–51.
Wickens, D.D., Moody,M.J., & Dow, R. (1981).The nature and timing of the retrieval process and of interference
effects. Journal of Experimental Psychology: General, 110, 1–20.
Zbrodoff,N.J. (1995).Why is 9 + 7 harder than2 + 3? Strengthand interference as explanation of the problem-size
effect. Memory and Cognition, 23, 689–700.
Original manuscript received 18 October 1999
Accepted revision received 7 June 2000
610 THEVENOT, BARROUILLET, FAYOL

13. OPERANDS–ANSWER ASSOCIATIONS IN LTM 611
APPENDIX 1
List of the 96 pairs of numbers used
26 15 37 15 42 19
27 14 37 16 43 18
27 15 37 18 43 19
27 16 37 19 44 17
28 13 37 24 44 18
28 14 37 25 44 19
28 15 37 26 45 16
28 16 38 13 45 17
28 17 38 14 45 18
29 12 38 15 45 19
29 13 38 16 46 15
29 14 38 17 46 17
29 15 38 19 46 18
29 16 38 23 46 19
29 17 38 24 47 14
29 18 38 25 47 15
32 19 38 26 47 16
33 18 38 27 47 18
33 19 39 12 47 19
34 17 39 13 48 13
34 18 39 14 48 14
34 19 39 15 48 15
35 16 39 16 48 16
35 17 39 17 48 17
35 18 39 18 48 19
35 19 39 22 49 12
36 15 39 23 49 13
36 17 39 24 49 14
36 18 39 25 49 15
36 19 39 26 49 16
36 25 39 27 49 17
37 14 39 28 49 18