THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2001, 54A (2), 599–611
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 developmental models of arithmetic problem solving assume that any algorithmic solution of a given problem results in an association of the two operands and 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 operands in adults. Thus, the postulate of a necessary and automatic storage of operands–answer associations in memory when young children solve additions by algorithmic strategies needs to be qualified.
Simple arithmetic operations like additions and subtractions are practised very early in childhood 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 might even be expected that 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:
[email protected] Ó 2001 The Experimental Psychology Society http://www.tandf.co.uk/journals/pp/02724987.html DOI:10.1080/02724980042000291
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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 paradox: An early acquired and frequently used algorithm does not always appear to lead to a systematic 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 operands are 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 knowledge in memory initially results from the implementation of algorithmic 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 the answer and its cognitive cost should lead to a reduction in the level of activation of the 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 concurrent activation of transitory results, which induces a resource tradeoff (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 encountered in recognizing the operands either after their addition or subtraction or after their simple comparison with a third number (Wickens, Moody, & Dow, 1981). We assume that the processes required by the solution to two-digit number operations 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 recognition task in which operands were presented as targets among distractors after the solution of either an 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. Several procedures are 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
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and 20 + 8, respectively, add 20 to 30, temporarily store this transitory result (50), then compute or retrieve 7 + 8, and finally add 15 to 50. Other decompositions are possible but all of them lead to a shift of the 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 computation induces a memory decay of the initial operands, and the need to maintain intermediary results reduces the amount of resources available for each item, including the operands. Thus, the algorithmic solution of operations should degrade the memory traces of operands. In contrast, a comparison problem (e.g., decide whether 31 lies between 37 and 28) makes it necessary 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 activation is, the quicker and more probable the retrieval. Thus, both the accuracy and the speed of recognition of operands after performing the task (i.e., either an operation or a comparison) 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: additions, 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 participants 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 operands in the case of comparisons. 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). According 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-four undergraduate students from the University of Burgundy took part in this experiment.
Material A total of 96 pairs of two-digit numbers were used. In order to optimize the probability of an algorithmic 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. Appendix 1). These 96 pairs of numbers were assigned using a circular permutation to the 24 experimental 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 participant 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 second operands, respectively, the former always being larger than the latter in order to permit the subtraction 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
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subtracting 1 or 2 from the operands. As far as comparisons are concerned,n3 was larger 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 between n1 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 order to prevent participants from adopting a systematic and active strategy of memorization of the operands with a view to their subsequent recognition, we added 192 fillers by twice presenting each of the 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 corresponded 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 keyboard. The type of response, the time of presentation of n1 and n2, and the 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 whether or not they had seen this number among the first two presented numbers by pressing the same keys used for the response to the problem. The type of response (“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 because 39 + 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. The 288 trials (96 experimental trials and 192 fillers) were randomly presented.
Results The rates of correct responses to the problems were high (.897, .941, and .978 for additions, subtractions, and comparisons, respectively), providing evidence that participants paid sufficient attention to the problems that preceded the recognition task. Among the 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,
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OPERANDS–ANSWER ASSOCIATIONS IN LTM TABLE 1 a Reaction times 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 Subtraction Comparison a
0.936 0.958 0.979
0.111 0.070 0.047
1,821 1,757 1,429
457 487 402
0.825 0.889 0.983
0.177 0.154 0.064
1,898 1,862 1,373
650 511 366
In ms.
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 interaction 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 either an incorrect response to the problem (76 trials, see earlier) or a non-recognition (46, 27, and 4 trials for additions, subtractions, and comparisons, respectively) were discarded, resulting in 999 retained trials. A mean RT was calculated for each participant in each of the six experimental conditions: 3 (types of problem) × 2 (types of target). An ANOVA with the same design as the previous one was performed on these mean RTs (Table 1). As we predicted, the recognition of the targets was faster after comparisons (1,401 ms) than after additions (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 observed for the rate of correct recognitions, the RTs for the recognition of n1 and n2 did not differ significantly (1,669 and 1,711 ms, respectively), and the interaction was not significant, Fs < 1.
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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 between comparisons and operations. However, this was not the case. If the effect 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 numbers (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 between operations and comparison was 439 ms for the false problems (1,819 and 1,358 ms for 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 targets 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 memory 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 comparisons 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 judgements about the odd/even status of the answer to reject false answers in verification tasks (Krueger, 1986): for example, seeing the addition of two even numbers produce an odd answer 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 processed. The second is that the better performances in recall for comparisons than for operations 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
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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 proposed 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 similar 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 longer a number was presented on screen, the harder its subsequent 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. TABLE 2 a Times of autorepresentation of the operands n1 and n2 and the b proposed answer for the three types of problems
Type of number
Type of problems ———————————————————————– Addition Subtraction Comparison —————– —————– —————– M SD M SD M SD
n1 n2 Answer
1,549 3,154 1,636
a
835 839 580
1,549 3,197 1,844
159 1,581 614
1,298 1,397 1,895
453 724 420
In ms. For comparisons, “answer” refers to the third number presented, the participants being asked whether their “answer” was between n1 and n2 or not. b
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Discussion The results clearly highlight the fact that the retrieval of operands from memory is slower and more difficult after solving an operation (addition or subtraction) than after performing a comparison. This suggests that the algorithmic solution of additions or subtractions does not necessarily 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 decomposition of 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 strategy 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 participants 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 increase both the number of instances stored in memory and consequently 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 by the computer simulations. Anderson and Lebiere (1998) invoke the same process. Thus, all of the models assume that when an algorithm has produced an answer, the operands are still present in working memory and available to be associated with this answer in
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long-term memory. The direct retrieval strategy would only fail because of insufficient practice. 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 pronounced the longer the solution 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 concurrent 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 for 8 + 4 requires an attentional focusing on 8 and then the monitoring 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 dropped because three different 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 the algorithm goes on. 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 counting all 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 number of digits a child can memorize and recall while performing counting—is only 1.65 in 6-year-old and 2.3 in 7-yearold children (Case et al., 1982), and that the memorization of an additive number fact 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 operands and to weakening of the strength of this association. These effects should be especially pronounced in children for whom algorithmic solutions are highly demanding and time consuming, and the speed of memory decay in short-term memory is higher than 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 of verbal representation in an auditory verbal word frame (Dehaene, 1992), there is much evidence of discrepancies and dissociations between procedural and declarative knowledge.
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The neuropsychological literature describes relatively circumscribed forms of impairment in either the execution of calculation procedures or the retrieval of arithmetic facts, both in adults suffering from acquired dyscalculia (Cohen & Dehaene, 1994; Warrington, 1982) or in children presenting a developmental dyscalculia (Sokol, Macaruso, & Gollan, 1994; Temple, 1991). In the same way, studies on learning-disabled children have shown that they have difficulty in recalling basic additive arithmetic facts (Fleischner, Garnett, & Shepherd, 1982; Garnett & Fleischner, 1983). It has been observed that, when solving single-digit additions, learning-disabled children in 4th and 6th grade rely on counting strategies whereas normal-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 counting 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, resulting 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, our results could contribute to accounting for some effects concerning the 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 should be stronger and a correct encoding of the associations less probable for large than for small operands. This phenomenon could partially account for the size effect, which is particularly strong for additions—that is, additions 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 problems suffer from more interference (Campbell & Graham, 1985; Zbrodoff, 1995), are associated 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 problems 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 solved faster and more accurately than others, suggesting
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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 association 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 association 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 intermediate outputs, which damage the memory traces of operands. Psychological models and computer simulations should take account of these constraints and their relative impact depending on the subject’s developmental level and the time needed to implement different algorithmic strategies.
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OPERANDS–ANSWER ASSOCIATIONS IN LTM
APPENDIX 1 List of the 96 pairs of numbers used 26 27 27 27 28 28 28 28 28 29 29 29 29 29 29 29 32 33 33 34 34 34 35 35 35 35 36 36 36 36 36 37
15 14 15 16 13 14 15 16 17 12 13 14 15 16 17 18 19 18 19 17 18 19 16 17 18 19 15 17 18 19 25 14
37 37 37 37 37 37 37 38 38 38 38 38 38 38 38 38 38 38 39 39 39 39 39 39 39 39 39 39 39 39 39 39
15 16 18 19 24 25 26 13 14 15 16 17 19 23 24 25 26 27 12 13 14 15 16 17 18 22 23 24 25 26 27 28
42 43 43 44 44 44 45 45 45 45 46 46 46 46 47 47 47 47 47 48 48 48 48 48 48 49 49 49 49 49 49 49
19 18 19 17 18 19 16 17 18 19 15 17 18 19 14 15 16 18 19 13 14 15 16 17 19 12 13 14 15 16 17 18
611
