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Short-Term Memory, Working Memory, and Inhibitory Control in Children with Difficulties in Arithmetic Problem Solving
Journal of Experimental Child Psychology 80, 44–57 (2001) doi:10.1006/jecp.2000.2626, available online at http://www.idealibrary.com on
Short-Term Memory, Working Memory, and Inhibitory Control in Children with Difficulties in Arithmetic Problem Solving M. Chiara Passolunghi University of Trieste, Italy
and Linda S. Siegel University of British Columbia, Vancouver, Canada The relations between short-term memory, working memory, inhibitory control, and arithmetic word problem solution were studied in children who were poor in arithmetic problem solving (n ⫽ 23). The children were compared with a group of good problem solvers (n ⫽ 26), matched for vocabulary, age, and gender. The results corroborate the hypothesis of poor problem solvers’ general deficit in inhibitory processes. They had lower scores and made more intrusion errors in a series of working memory tasks requiring inhibition of irrelevant information. The results showed that problem solving performance is related to the ability of reducing the accessibility of nontarget and irrelevant information in memory. Span tasks that imply passive storage of information showed that poor problem solvers were impaired when they have to retain numerical information, but they did not differ from children who did not have difficulty with mathematics when the material included words. © 2001 Academic Press Key Words: working memory; word problem difficulties; inhibitory processes.
Children with specific arithmetic learning difficulties often perform arithmetic computations slowly and inaccurately, an impairment that has been attributed to limitations in working memory (Hitch & McAuley, 1991; Siegel & Linder, 1984; Siegel & Ryan, 1989; Swanson, 1993). That is, these children perform poorly on tasks in which they must manipulate or transform material while remembering information. Working memory may also contribute to children’s success in solving arithmetic word problems. Cooney and Swanson (1990) showed a positive relation between working memory and problem solution (i.e., a person’s representation of problem schemata was significantly correlated with memory span) and Swanson, Cochran, and Ewers (1990) found that children with learning disAddress correspondence and reprint requests to Maria Chiara Passolunghi, University of Trieste, Faculty of Psychology, via S. Anastasio, 12, 34100 Trieste, Italy. E-mail: [email protected]. 44 0022-0965/01 $35.00 Copyright © 2001 by Academic Press All rights of reproduction in any form reserved.
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abilities showed impaired performance in working memory test. Moreover, Swanson (1994) found that children and adults with learning disabilities had impaired performance on a working memory test (e.g., a modification of the Daneman & Carpenter’s, 1980, reading span task). However, not all investigators have found a relation between working memory and word problems performance. For example, in a study with third and fourth grade children, Swanson, Cooney, and Brock (1993) found that working memory and solution accuracy were only weakly correlated, and this correlation was not significant when the influence of reading comprehension ability was controlled. In addition Kail and Hall (1999) found that short-term memory and working memory were not consistently related to word problem performance and span tasks did not account for independent variance when reading skill was assessed. Given these inconsistent findings in the literature one aim of the present research was to examine the link between working memory and word problem performance across a range of tasks that are used to measure working memory. One possibility, first proposed by Siegel and Ryan (1989), is that mathematical disability may be not associated with a general deficit in working memory but in working memory in which remembering arithmetic information is critical to solution. Indeed, Siegel and Ryan (1989) found that the performance of children with a mathematical learning disability was similar to that of normal achievers in a working memory task involving sentence processing, but impaired on a working memory task required processing of numerical information. By examining children’s performance on working memory tasks that required processing of numerical information as well as tasks that did not, we could evaluate whether working memory impairments was general or specific to processing numerical information. A second aim was to determine whether impaired performance was specific to working memory tasks or was also seen in short-term memory tasks in which small amounts of material are held passively and then recalled without any transformation (Cantor, Engle, & Hamilton, 1991; Swanson, 1993; Vecchi & Cornoldi, 1999). In the study by Swanson (1994), the children and adults with learning disabilities had impaired performance on a working memory task but not on digit span, a common measure of short-term memory. To determine whether such impaired performance is typical, we also administered several span tasks thought to asses short-term memory. Because children who are poor problem solvers may show a selective impairment only in the recall of numerical information, some span tasks required children to remember digits but others required them to remember words. A third aim of our work was to investigate the mechanism that leads to impaired working memory. Several investigations in reading comprehension have shown that poor readers suppress information less efficiently (De Beni, Palladino, Pazzaglia, & Cornoldi, 1988; Gernsbacher, 1993; Gernsbacher & Faust, 1991; Chiappe, Hasher, & Siegel, 2000). In like manner, Passolunghi, Cornoldi, and De Liberto (1999) found that children who were poor arithmetic problem solvers had
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low scores in working memory tasks requiring inhibition of irrelevant information. Based on these findings, we hypothesized that the working memory deficit of the poor problem solvers may be related to an inability to control and to ignore irrelevant or no longer relevant information. Consequently, we expected that poor problem solvers would have a high number of intrusion errors during recall. To summarize, we examined performance on a range of working memory and short-term memory tasks by children who were either poor or good in problem solving. We included four working memory tasks: (1) the Listening Span Task, in which children listened to sets of increasing numbers of sentences, judged whether the sentences were true or false, then at the end of each set, recalled the last word of each of the sentences; (2) the Animal Dual Task (De Beni et al., 1998; Palladino & De Beni, 1999; Passolunghi et al., 1999), in which children listened to a series of words, identified the animal nouns, then remembered the last word of each series; (3) the Listening Span Completion Task, in which several sentences with the final word missing were presented and the child had to generate the missing word at the end of the sentence, than they had to repeat all the missing words from the sets; (4) the Counting Span Task, in which children counted target dots from an irregular pattern of dots, then recalled the counts of the presented patterns. The short-term memory tasks included forward and backward digit span, as well as forward and backward word span. Several outcomes were possible: if children who are poor problem solvers have a truly general deficit, their performance would be impaired on all tasks. If their deficit is limited to working memory, their performance would be unimpaired on the forward digit and word span tasks, but impaired on all others including the reverse digit span tasks, which are often assumed to measure working memory (e.g., Cornoldi & Vecchi, 2000). Finally, if children who are poor problem solvers are particularly impaired when processing numerical information, we would expect relatively poor performance on the counting span and digit span tasks. METHOD Participants The participants were 49 fourth-grade children divided into a group of 23 poor problem solvers and a group of 26 good problem solvers. These groups were formed on the basis of two screening tests (on arithmetic word problems and vocabulary abilities) administered to 280 participants of a town in northern Italy (Trieste) during the first 3 months of the fourth grade. The participants had no documented brain injury, sociocultural disadvantage, or behavioral problems. Children were included in the poor problem solver group if their score was less than 30th percentile in a standardized Italian mathematics test (Amoretti, Bazzini, Pesci, & Reggiani, 1994) and if the teachers had noted that the child had difficulty with math problems. The good problem solvers were between the 50th and the 80th percentile in the same mathematical test and were considered by their teachers to be performing at or above grade level in mathematics. Respectively, the
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mean standard score in the mathematical test was ⫺.95 (SD ⫽ .20) for the poor problem group and .84 (SD ⫽ .56) for the good problem group. This test consisted of 12 items: 8 written arithmetical word problems and 4 questions examining the manipulation of Arabic and verbal numerals (i.e., to put in ascending order a series of Arabic numbers; to write the Arabic numeral that corresponds to a verbal numeral). A typical problem was: “This Saturday Robert goes with his mother to the supermarket. His mother buys 4 hectograms of ham and spends 6,960 Liras. What is the cost of one hectogram of ham?” The two groups were matched for age (9 years 4 months), gender (poor problem solver group: 11 males and 12 females; good problem solver group: 14 males and 12 females), and their scores in the Vocabulary subtest of PMA battery (Thurstone & Thurstone, 1941). Mean scores in this test were, respectively, 54.22 (SD ⫽ 2.66) for the poor problem solvers and 55.26 (SD ⫽ 2.96) for the good problem solvers. Converted to verbal IQ, the mean scores were, respectively, 119.12 (SD ⫽ 4.23) for the poor problem solvers group and 120.73 (SD ⫽ 4.83) for the good problem solvers group. This differences was not significant (p ⬎ .22). We assessed reading comprehension ability (scores on a standardized test, Cornoldi & Colpo, 1981) for poor problem solvers and good problem solvers. The test requested the children to read a one page text and then answer a series of multiple choice comprehension questions. There was no significant difference between the score of the two groups (poor problem solvers: mean score ⫽ 7.43, SD ⫽ 1.93 (standard score ⫽ ⫺.19, SD ⫽ 1.00); good problem solvers: mean score ⫽ 8.15, SD ⫽ 1.82 (standard score ⫽ .17, SD ⫽ .98), p ⬎ .15). Children selected were individually tested with seven tasks in December: Listening Span Task, Animal Dual Task, Listening Span Completion Task, and Span Tasks. Each task was administered on a separate day in the above order. The four span tasks were administered in a single session in the following order: word span tasks (forward and backward), and digit span tasks (forward and backward). Three months later, the children received the Counting Span Task, which was added subsequently to the battery. The approximate length of a testing session was 15 min. Tasks The experimenter presented the tasks and verified that the children understood them perfectly. No feedback was given during the testing phase of each task. Listening span task. An Italian adaptation (see Passolunghi et al., 1999) of the listening span test devised by Daneman and Carpenter (1980) was used, including two series of two sentences, two series of three sentences, two series of four sentences, and two series of five sentences. For each set, there was a practice series. There were two kind of sentences: true and false. An example of true sentence was “A and B are the first two letters of the ALPHABET,” an example of false sentence was “The hen is a mammal that lives in the SEA.” The child received all sets of stimuli. Each sentence was presented at a rate of approximately 1 s per word, followed by an interval for the child’s answer
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concerning the truth of the sentence. The child was instructed to give the answer as soon as possible. His/her answer was followed by the presentation of the next sentence. At the end of each set of sentences, immediately after the verification answer for the last sentence, the child was asked to recall the last word of each sentence (in the same order of presentation) and to pay attention to avoid mentioning nonfinal words. For example, if he/she had processed the above-mentioned sentences, the correct response was “ALPHABET, SEA.” Animal dual task. The material consisted of strings of words, each string composed of four two-syllable words. The strings of words were used in place of the sentences typically used in the Listening Span Test. There were three sets, each with 2, 3, and 4 strings, for a total of 27 strings and 128 words. The words, of medium-high frequency and medium-high level of imagery value, were selected from the Cornoldi and Pra Baldi (1978) norms. Among these words, 26 were animal nouns (9 in the final position and 17 placed in the strings). An example of a string of words is the following: tetto (roof), tiger (tigre), becco (beak), ponte (bridge). The testing phase was preceded by practice items. The child received all sets of stimuli. The words were presented at an approximate rate of 1 s per word with a 2-s interval after each string. When the name of an animal was given, the child had to tap the table. At the end of each series, the child was asked to recall the last name in each string in the series in the same order of presentation. The Animal Dual task is considered a real working memory task, since the child is required to process all of the material then discharge the items that became irrelevant in order to recall the last words of each string (see De Beni et al., 1998; Palladino & De Beni, 1999; Passolunghi et al., 1999). Listening span completion task. An Italian version of the working memory sentence developed by Siegel and Ryan (1989) was used, including three series of two sentences, three series of three sentences, three series of four sentences, and three series of five sentences. In each sentence, the final word was missing. For example, one series of these sentences was: “A fish swims, a bird ________; The snow is white, the coal is ________; In the barn the farm worker milked the ________.” To minimize word-finding problems, sentences were chosen so that the missing words were virtually predetermined. The children were tested following a procedure similar to that of the Listening Span Task. Each sentence was presented aurally at a rate of approximately 1 s per word. The task for the child was to supply the missing word at the end of the sentence and to repeat all the missing words from the set, in the same order that the sentences had been presented (e.g., in the example: “flies, black, cow”). No child had any difficulty in supplying the missing words. Counting span task. This task was a version of Counting Span developed by Siegel and Ryan (1989) and was originally designed by Case, Kurland, and Goldberg (1982). The pattern used for the stimuli was a field of blue and yellow dots arranged in a randomly determined order on a card of 30 ⫻ 20 cm. Fortytwo cards were presented; there were three trials at each span level (2, 3, 4, and 5 span levels). The child had to count the yellow dots from a field of blue dots
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and then recall the counts of each set in the correct order. No child experienced any difficulty in counting the dots. To replicate the procedure used by Siegel and Ryan (1989), task administration was stopped when the child failed all three trials at one span level. The task was discontinued for only four children, making it unlikely that this procedural difference from other working memory tasks has significant implications for the results. Short-term memory tasks. The Word Span Task used an increasing number of familiar two-syllable words, from two to eight words per span length (two trials for each span length). The Digit Span Task was tested using the WISC-R Digit Span subtest (Wechsler, 1974). In the backward tasks, words and digits were recalled in the reverse order of presentation. Each task was scored separately. In the Word Span Task, the words were presented at a 1-s-per-word rate until a child made a mistake in both trials of the same span length. A span was considered to have been correct if all the nouns were recalled in their correct order. The experimenter started from a span length of two. When the repetition was correct, the experimenter moved to a higher level; when the repetition was incorrect, the child had the opportunity of trying again with another trial of the same span length. If the child failed on the second trial the testing was discontinued. The same procedure was used for the Digit Span Task. In Word and Digit Span backward tasks, the rate of stimuli presentation was the same as for the forward tasks, and children were asked to recall the items in the reverse order of presentation. RESULTS Listening Span Task The mean performance of the two groups in the Listening Span Task is presented in the first row of Table 1. Although the poor problem solvers had a lower performance than good problem solvers in the sentence verification (t(47) ⫽ 3.05, p ⬍ .05), no one children made more than 3 errors on the maximum score of 28. Therefore, we think that they performed this task as a real working memory task, in which they concurrently processed the sentences and remembered the final words. We found a significant difference between the groups, t(47) ⫽ 2.23, p ⫽ .03, in correct recall (this index indicates the number of the target words recalled in order). The poor problem solver children erroneously remembered more words embedded in the sentences than the children in the good problem solver group. The difference in the intrusions (see Table 1) was significant (t(47) ⫽ 2.05, p ⬍ .05). An intrusion error could be due either to an item of a preceding list (i.e., sentences in the case of Listening Tasks, strings of words in the case of Animal Dual Task), or to an item of the currently presented list. In order to analyze the nature of intrusion errors, we measured these different types of intrusion errors. In the Listening Span Task, the two groups did not differ in the comparison between intrusions of preceding lists and the currently presented list (see first row, Table 1, p ⬎ .11).
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TABLE 1 Mean Performance (Standard Deviations in Brackets) of the Poor Problem Solver Group and of the Good Problem Solver Group in the Listening Span Task, Animal Dual Task, Listening Completion Task, Counting Span Task, and Word and Digit Span Tasks (Forward and Backward) Poor problem solver group Correct recall
Total Intrusions intrusion current list
Intrusions preceding lists
9.65 (3.04)
4.78 (3.01)
3.87 (2.43)
.91 (1.27)
Animal Dual Task
7.52 (4.12)
5.83 (3.39)
4.69 (3.00)
1.13 (1.10)
Listening Span Compl. Task Counting Span Task Word Span Forward Word Span Backward Digit Span Forward Digit Span Backward
18.74 (5.67) 35.69 (3.72) 3.96 (.20) 3.52 (.66) 5.26 (.96) 3.48 (.73)
2.56 (2.08)
1.56 (1.51)
1.00 (1.17) 1.95 (1.61) — — — —
Correct recall 25.48 (1.31) (true/false) 2.87 3.04 (2.16) (2.34) (anim. (nonanim. intrus.) intrus.) — —
Total Intrusions Intrusions intrusion current preceding list lists
11.73 (3.43)
3.07 (2.79)
2.65 (2.91)
.42 (.75)
10.27 (4.37)
3.15 (2.26)
2.19 (1.55)
.96 (1.22)
22.23 (5.80) 38.23 (3.57) 4.08 (.48) 3.93 (.63) 5.85 (1.00) 4.11 (.77)
.85 (1.19)
.31 (.55)
.54 (.90) 1.15 (1.19) —
—
— — —
27.50 (1.03) (true/false) 1.77 1.38 (1.61) (1.36) (anim. (nonanim. intrus.) intrus.) — —
PASSOLUNGHI AND SIEGEL
Listening Span Task
—
Good problem solver group
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Animal Dual Task The results of Animal Dual Task (see the second row of Table 1) showed the same trend as the Listening Span Task: the children in the poor problem solving group recalled fewer words in the correct order compared to the good problem solver group (t(47) ⫽ 2.26, p ⬍ .03). All children were very accurate in the identification of animal words; no one made more than 1 or 2 misses. In this task, animal nouns received more stress. Our hypothesis is that this deeper encoding increased their probability to be better recalled when they are the target items (see Passolunghi et al., 1999). Focusing on the number of the target animal nouns (e.g., those in the last position) that were recalled correctly, animal nouns were better remembered than nonanimal nouns (respectively, 58% vs 49%, t(48) ⫽ 3.39, p ⫽ .0014). The poor problems solvers performed as well as good problem solvers in recalling the target items which required deeper elaboration (respectively, 55%, SD ⫽ 17.88, of correct recall of animal nouns for poor problem solvers and 60%, SD ⫽ 14.43, for good problem solvers, the difference between the two groups was not significant, p ⫽ .31). In contrast, the poor problem solvers performed at a significantly lower level in the recall of nonanimal nouns in the last positions, respectively 44% (SD ⫽ 9.95) for poor problem solvers and 54% (SD ⫽ 12.65) for good problem solvers, t(47) ⫽ 3.08, p ⫽ .0034. A 2 groups ⫻ 2 source of errors (intrusions due to the current list and intrusions due to preceding lists) ANOVA, showed a main effect of group, F(1, 47) ⫽ 10,76 p ⫽ .002, a main effect of source of errors F(1, 47) ⫽ 49,92, p ⫽ .001, due to the fact that intrusion mainly concerned items of the current list, and a significant interaction between groups and source of errors, F(1, 47) ⫽ 11,82, p ⫽ .0012, due to the fact that poor problem solvers made more intrusions from the current list. Listening Span Completion Task The good problem solvers group had higher mean number of correct recall than poor problem solvers, t(47) ⫽ 2.13, p ⫽ .03 (see the third row of Table 1). Poor problem solvers made more intrusions than the good problem solvers, erroneously recalling more words embedded in the sentences. A 2 groups ⫻ 2 source of errors ANOVA showed a main effect of groups, F(1, 47) ⫽ 12,93, p ⫽ 12,93, p ⫽ .0008, and a significant interaction between groups and source of errors, F(1, 47) ⫽ 4,31, p ⬍ .05, due to the fact that poor problem solvers made more intrusions from the current list. Counting Span Task The poor problem solvers had lower scores than the good problem solvers in the correct recall in order, t(47) ⫽ 2.43, p ⫽ .018 (see the fourth row of Table 1). We defined an intrusion as the recall of a number from the card presented prior to the current card. Poor problem solvers made significantly more intrusions than good problems solvers (see fourth row of Table 1, t(47) ⫽ 2.00, p ⬍ .05).
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Span Tasks The results of the Word and Digit Span Tasks (forward and backward) are presented in the last four rows of Table 1. A 2 (groups) ⫻ 2 (type, word vs digit) ⫻ 2 (direction, forward vs backward) ANOVA for a mixed design showed a significant main effect of group, F(1, 47) ⫽ 11.41, p ⫽ .0015, an interaction of type by group, F(1, 47) ⫽ 3.83, p ⬍ .05, due to the fact that the poor problem solver group had lower scores than the good problem solver group in the recall of numerical information, a main effect of direction (forward vs backward), F(1, 47) ⫽ 95.97, p ⫽ .0001, and an interaction of type by direction F(1, 47) ⫽ 72.62, p ⫽ .0001, due to the fact that the highest performance was in the digit forward task and the lowest performance was in the word backward task. DISCUSSION Working Memory Deficit of Poor Problem Solvers Several researchers have postulated certain cognitive processes (e.g., comprehension, integration, planning, and execution) and metacognitive abilities that are fundamental in solving a problem (Lucangeli & Passolunghi, 1995; Mayer, 1998; Nathan, Kintsch, & Young, 1992; Passolunghi, Lonciari, & Cornoldi 1996; Swanson, 1990). However, investigators have not consistently found a relation between problem performance and working memory ability. One aim of the present research was to examine the link between working memory and word problem performance across a range of tasks whose purpose was to measure working memory. We examined children’s performance on working memory tasks that required processing of numerical information as well as tasks that did not, in order to evaluate whether working memory impairments of poor problem solvers was general or specific to processing numerical information. Siegel and Ryan (1989) found that the performance of children with a mathematical learning disability was impaired only on a working memory task that required processing of numerical information. The results of this research demonstrated that the cognitive impairments of poor problem solver children are not restricted to a numerical working memory task. The Listening Span Task, Animal Dual Task, and Listening Span Completion Task are clearly verbal working memory tasks, while the Counting Span Task has a much smaller verbal component, and involves the processing of numerical information. In these tasks, poor problem solvers demonstrated less recall of relevant information and their impairment was similar in working memory tasks involving both verbal and numerical information. Therefore, the results of this study support the hypothesis of a generalized working memory deficit of poor problem solvers, assessed by tasks involving manipulation and transformation of information. In addition, we attempted to control a factor that can mediate the relations between working memory and word problem solution. We matched the groups on a measure of the verbal intelligence in order to rule out the possibility that the relations between working memory and word problem solution may be attributed
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to intellectual capacity. The results showed that even when the two groups were matched on measure of verbal intelligence, we observed poorer performance in the poor problem solver group. Moreover, we controlled the influence of reading ability. When we compared the comprehension ability of the two groups, we did not find a significant difference. This result supports the hypothesis of a specific relation between working memory and arithmetic problem solution. Impairment in Short-Term Memory Tasks A second aim of the research was to determine whether poor problem solvers’ impaired performance was specific to working memory tasks or was also extended to short-term memory tasks in which small amounts of material are held passively and then recalled without any transformation. Swanson (1994) found that the children and adults with learning disabilities had impaired performance on a working memory task but not on digit span task, a common measure of short-term memory. He hypothesized that short-term memory and working memory are independent measures, and he found that they loaded on different factors. Short-term memory tasks rely on a passive storage system and are those in which the individuals are required to retain the material to recall without performing any modification on it (see also Cantor, Engle, & Hamilton, 1991; Cornoldi & Vecchi, 2000; Vecchi, Monticelli, & Cornoldi, 1995). In contrast, working memory tasks request more active processes and are those in which information is temporarily held while being manipulated or transformed (e.g., process all the material and only successively discharge the items that become irrelevant). An implication for this assumption is that children or adults with learning disabilities may have working memory problems independent of problems in short-term memory. Our results showed that poor problem solvers had a general impairment in working memory tests, but they did not reveal an impairment in typical tests of short-term memory, i.e., span tasks, if the span concerned linguistic information. However, our data showed that children with a mathematics disability had lower performance than the normally achieving group in immediate recall of numerical information. A possible interpretation is that these children had slower access to number representations in long-term memory, which in turn may have led to slow counting and lower digit span (see also Geary, 1990, 1993; Hitch & McAuley, 1991). The preserved ability of poor problem solvers in the passive processing of verbal information does not support the hypothesis that their impairment in passive processing of numerical information is due to task difficulty. The discrepancy between Swanson’s (1994) results on digit span task and ours is probably due to the difference in the sample selection. The participants of Swanson’s (1994) study were defined as learning disabled on the basis of reading performance, while mathematics performance was not included in the selection criteria and was left to covary. In contrast, our children have a main difficulty in arithmetic word problem solutions and they and did not differ from good problem solvers in reading and vocabulary ability.
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Intrusion Errors and Inhibitory Processes A third aim of our work was to investigate the mechanism that leads to impaired working memory. Our data provided evidence that the working memory deficit of the poor problem solvers is related to an inability to control and to ignore irrelevant or no longer relevant information. Indeed, the high number of intrusion errors in the working memory tasks suggests that the poor problem solvers maintained information available in memory, which initially had to be processed, but then had to be suppressed. We found that poor problem solvers had a higher number of intrusion errors than good problem solvers in the Listening Span Task, Animal Dual Task, and Listening Span Completion Task. Moreover, poor problem solvers showed the same failure pattern (e.g., more intrusion errors) in the processing of numerical information as required by Counting Span Task. The intrusions of items of the currently presented list may be considered working memory errors due to intrusive elements still active in working memory, whereas we hypothesized that the intrusions concerning items of the preceding lists do not involve the working memory, but rather involve long-term memory. We found that poor problem solvers made more intrusion errors than good problem solvers only with respect to the items from the current lists. This result supports the hypothesis that poor problem solvers have a poorer working memory system and that their working memory failure is related to the manipulation of highly accessible information. Altogether the results of intrusion errors support the hypothesis that problem solving ability is related to the ability of reducing the memory accessibility of nontarget and irrelevant information. This hypothesis is consistent with the literature which tries to connect working memory capacity and the efficiency of inhibition mechanisms (Bjorklund & Harnishfeger, 1990; Chiappe, Hasher, & Siegel, 2000; De Beni et al., 1998; Gernsbacher, 1993; Hamm & Hasher, 1991; Passolunghi & Cornoldi, 2000; Passolunghi et al., 1999). In our view, the working memory deficit of poor problem solvers reflected problems in the central executive component of Baddley’s model (Baddeley 1986, 1996; Baddeley & Hitch, 1974). Children with mathematical disability did not have the same working memory capacity as children without mathematical disability, since poor problem solvers maintain in memory information that initially was necessary to process but then became irrelevant and had to be discharged. The results of the present study suggest that children with difficulties in the arithmetic word problem solution have a type of deficit related to a lack of flexibility in coordinating both verbal and numerical information rather an output problem. The involvement of the central executive may be related to a metacognitive problem. It must be noted that, on the basis of our data, it is not possible to decide whether the inhibition failure is a primary and causal deficit or if it is connected to other more general factors such as processing speed (see Kail & Hall, 1999) or selective attention (Tarver, Hallan, Cohen, & Kauffman, 1977). From a conceptual point of view it is difficult to disentangle the inhibition and selective
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attention hypothesis, since selective inhibition can be considered complementary to selective attention. In particular, our data are compatible with the hypothesis of Conway and Engle (1994) and Engle, Conway, Tuholski, and Shisler (1995) that individuals with a low span do not have the attentional and resource capacity necessary to inhibit irrelevant information. It might be argued that poor problem solvers may use the wrong retrieval cue to access memory or they may lack a strategy for maintaining relevant information in memory. However, poor problem solvers are as able as good problem solvers in recalling the target items whether these items required deeper elaboration. In the Animal Dual Task, the two groups did not differ in the correct recall if the to-be-recalled items are the animal nouns (i.e., the items that received more stress and deeply encoding); however, the two groups differed in the case of nonanimal nouns. The attention required for animal items had the effect of increasing their probability of being both correctly and incorrectly retrieved. It seems that this attention increased their accessibility (Craik & Tulving, 1975), but it also increased the difficulty of inhibiting them. A deeper encoding of these items may improve the recall of the poor problem solvers. In summary, the results showed that children who are poor problem solvers have a general working memory deficit: their performance was impairment both in verbal and numerical working memory tasks. However, our data showed that they had performance similar to good problem solvers in typical tests of shortterm memory, i.e., span tasks, if the span concerned linguistic information. In contrast, poor problem solvers had lower performance than normally achieving group in immediate recall of numerical information. Finally, we found that poor problem solvers had a high number of intrusion errors during recall. This result supports our hypothesis that the poor problem solvers’ working memory deficit is related to a failure of inhibitory mechanism in processing the information. REFERENCES Amoretti, G., Bazzini, L., Pesci A., & Reggiani, M. (1994). Test di matematica per la scuola dell’obbligo [Mathematics test for primary schools]. Firenze: Organizzazioni Speciali. Baddeley, A. D. (1986). Working memory. Oxford: Clarendon Press. Baddeley, A. D. (1996). Exploring the central executive. The Quarterly Journal of Experimental Psychology, 49a(1), 5–28. Baddeley, A. D., & Hitch, G. J. (1974). Working memory. In G. H. Bower (Ed.), The psychology of learning and motivation: Advances in research and theory (Vol. VIII, pp. 47–90). New York: Academic Press. Bjorklund, D., & Harnishfeger, K. (1990). The resources construct in cognitive development: Diverse sources of evidence and a theory of inefficient inhibition. Developmental Review, 10, 48–71. Cantor, J., Engle, R. W., & Hamilton, G. (1991). Short term memory, working memory and verbal abilities: How do they relate? Intelligence, 15, 229–246. Case, R., Kurland, D. M., & Goldberg, J. (1982). Operational efficiency of short-term memory span. Journal of Experimental Psychology, 33, 386–404. Chiappe, P., Hasher, L., & Siegel, L. S. (2000). Working memory, inhibitory control and reading disability. Memory and Cognition, 28, 8–17. Conway, A. R. A., & Engle, R. W. (1994). Working memory and retrieval: A resource dependent inhibition model. Journal of Experimental Psychology: General, 123, 354–373.
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Cooney, J. B., & Swanson, H. L. (1990). Individual differences in memory for mathematical story problems: Memory span and problem perception. Journal of Educational Psychology, 82, 570–577. Cornoldi, C., & Colpo, G. (1981). Prove di lettura MT [Reading comprehension test MT]. Firenze: Organizzazioni Speciali. Cornoldi, C., & Pra Baldi, A. (1978). Il valore d’immagine nei bambini: Norme per 257 nomi e usi nella ricerca applicata. Formazione e Cambiamento, 3–4, 275–304. Cornoldi, C., & Vecchi, T. (2000). Mental imagery in blind people: The role of passive and active visuo-spatial processes. In M. Heller (Ed.), Touch, representation and blindness (pp. 143–181). Oxford: Oxford Univ. Press. Craik, F. I. M., & Tulving, E. (1975). Depth of processing and the retention of words in episodic memory. Journal of Experimental Psychology: General, 14, 268–294. Daneman, M., & Carpenter, P. A. (1980). Individual differences in working memory and reading. Journal of Verbal Learning and Verbal Behaviour, 19, 450–466. De Beni, R., Palladino, P., Pazzaglia, F., & Cornoldi, C. (1998). The role of inhibition in the working memory deficits of poor readers. Quarterly Journal of Experimental Psychology, 51A, 305–320. Engle, R. W., Conway, A. R. A., Tuholski, S. W., & Shisler, R. J. (1995). A resource account of inhibition. Psychological Science, 6, 122–125. Geary, D. C. (1990). A componential analysis of an early learning deficit in mathematics. Journal of Experimental Child Psychology, 49, 363–383. Geary, D. C. (1993). Mathematical disabilities: Cognitive, neuropsychological, and genetic components. Psychological Bulletin, 114, 345–362. Gernsbacher, M. A. (1993). Less skilled readers have less efficient suppression mechanisms. Psychological Science, 4, 294–298. Gernsbacher, M. A., & Faust, M. E. (1991). The mechanism of suppression: A component of general comprehension skill. Journal of Experimental Psychology: Learning, Memory and Cognition, 17, 245–262. Hamm, V. P., & Hasher, K. (1992). Age and the availability of inferences. Psychology and Aging, 7, 56–64. Hitch, G. J., & McAuley, E. (1991). Working memory in children with specific arithmetical learning difficulties. British Journal of Psychology, 82, 375–386. Kail, R., & Hall, L. K. (1999). Sources of developmental change in children’s word-problem performance. Journal of Educational Psychology, 91, 660–668. Lucangeli, D., & Passolunghi, M. C. (1995). Psicologia dell’apprendimento matematico. [Psychology of mathematical learning]. Torino: Utet. Mayer, R. E. (1998). Cognitive, metacognitive, and motivational aspects of problem solving. Instructional Science, 26, 49–63. Nathan, M. J., Kintsch, W., & Young, E. (1992). A theory of algebra word problem comprehension and its implications for the design of learning environments. Cognition and Instructions, 4, 329–390. Palladino, P., & De Beni, R. (1999). Short term and working memory in aging: Maintenance and suppression. Aging: Clinical and Experimental Research, 11, 301–306. Passolunghi, M. C., & Cornoldi, C. (2000). Working memory and cognitive abilities in children with specific difficulties in arithmetic word problem solving. Advances in Learning and Behavioral Disabilities, 14, 155–178. Passolunghi, M. C., Cornoldi, C., & De Liberto, S. (1999). Working memory and intrusions of irrelevant information in a group of specific poor problem solvers. Memory and Cognition, 27, 779–790. Passolunghi, M. C., Lonciari, I., & Cornoldi, C. (1996). Abilità di pianificazione, comprensione, metacognizione e risoluzione di problemi aritmetici i tipo verbale. [The effects of planning, comprehension and metacognitive abilities on arithmetic word problems solution]. Età Evolutiva, 54, 36–48.
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Siegel, L. S., & Linder, B. A. (1984). Short-term memory processes in children with reading and arithmetic learning disabilities. Developmental Psychology, 20, 200–207. Siegel, L. S., & Ryan, E. B. (1989). The development of working memory in normally achieving and subtypes of learning disabled children. Child Development, 60, 973–980. Swanson, H. L. (1990). Influence of metacognitive knowledge and aptitude on problem solving. Journal of Educational Psychology, 82, 306–314. Swanson, H. L. (1993). Working memory in learning disability subgroups. Journal of Experimental Child Psychology, 56, 87–114. Swanson, H. L. (1994). Short-term memory and working memory: Do both contribute to our understanding of academic achievement in children and adults with learning disabilities? Journal of Learning disabilities, 27, 34–50. Swanson, H. L., Cochran, K., & Ewers, C. (1990). Can learning disabilities be determined from working memory performance? Journal of Learning Disabilities, 23, 59–67. Swanson, H. L., Cooney, J. B., & Brock, S. (1993). The influence of working memory and classification ability on children’s word problem solution. Journal of Experimental Child Psychology, 55, 374–395. Tarver, S. G., Hallan, D. P., Cohen, S. B., & Kauffman, J. M. (1977). The development of visual selective attention and verbal rehearsal in learning disabled boys. Journal of Learning Disabilities, 10, 491–500. Thurstone, N. L., & Thurstone, T. G. (1941). Factorial studies of intelligence. Psychometric Monographs, 2 [Italian translation PMA-Batteria Primaria di Abilità, Firenze, Organizzazioni Speciali, 1968]. Vecchi, T., Monticelli, M. L., & Cornoldi, C. (1995). Visuo-spatial working memory: Structures and variables affecting a capacity measure. Neuropsychologia, 33, 1549–1564. Vecchi, T., & Cornoldi, C. (1999). Passive storage and active manipulation in visuo-spatial working memory: Further evidence from the study of age differences. European Journal of Cognitive Psychology, 3, 391–406. Wechsler, D. (1974). Manual for the Wechsler Intelligence Scale for Children—Revised. New York: Psychological Corporation [Scala d’intelligenza Wechsler per bambini, 1987, Firenze, O.S.]. Received July 23, 1999; revised November 14, 2000; published online May 29, 2001
Short-Term Memory, Working Memory, and Inhibitory Control in Children with Difficulties in Arithmetic Problem Solving M. Chiara Passolunghi University of Trieste, Italy
and Linda S. Siegel University of British Columbia, Vancouver, Canada The relations between short-term memory, working memory, inhibitory control, and arithmetic word problem solution were studied in children who were poor in arithmetic problem solving (n ⫽ 23). The children were compared with a group of good problem solvers (n ⫽ 26), matched for vocabulary, age, and gender. The results corroborate the hypothesis of poor problem solvers’ general deficit in inhibitory processes. They had lower scores and made more intrusion errors in a series of working memory tasks requiring inhibition of irrelevant information. The results showed that problem solving performance is related to the ability of reducing the accessibility of nontarget and irrelevant information in memory. Span tasks that imply passive storage of information showed that poor problem solvers were impaired when they have to retain numerical information, but they did not differ from children who did not have difficulty with mathematics when the material included words. © 2001 Academic Press Key Words: working memory; word problem difficulties; inhibitory processes.
Children with specific arithmetic learning difficulties often perform arithmetic computations slowly and inaccurately, an impairment that has been attributed to limitations in working memory (Hitch & McAuley, 1991; Siegel & Linder, 1984; Siegel & Ryan, 1989; Swanson, 1993). That is, these children perform poorly on tasks in which they must manipulate or transform material while remembering information. Working memory may also contribute to children’s success in solving arithmetic word problems. Cooney and Swanson (1990) showed a positive relation between working memory and problem solution (i.e., a person’s representation of problem schemata was significantly correlated with memory span) and Swanson, Cochran, and Ewers (1990) found that children with learning disAddress correspondence and reprint requests to Maria Chiara Passolunghi, University of Trieste, Faculty of Psychology, via S. Anastasio, 12, 34100 Trieste, Italy. E-mail: [email protected]. 44 0022-0965/01 $35.00 Copyright © 2001 by Academic Press All rights of reproduction in any form reserved.
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abilities showed impaired performance in working memory test. Moreover, Swanson (1994) found that children and adults with learning disabilities had impaired performance on a working memory test (e.g., a modification of the Daneman & Carpenter’s, 1980, reading span task). However, not all investigators have found a relation between working memory and word problems performance. For example, in a study with third and fourth grade children, Swanson, Cooney, and Brock (1993) found that working memory and solution accuracy were only weakly correlated, and this correlation was not significant when the influence of reading comprehension ability was controlled. In addition Kail and Hall (1999) found that short-term memory and working memory were not consistently related to word problem performance and span tasks did not account for independent variance when reading skill was assessed. Given these inconsistent findings in the literature one aim of the present research was to examine the link between working memory and word problem performance across a range of tasks that are used to measure working memory. One possibility, first proposed by Siegel and Ryan (1989), is that mathematical disability may be not associated with a general deficit in working memory but in working memory in which remembering arithmetic information is critical to solution. Indeed, Siegel and Ryan (1989) found that the performance of children with a mathematical learning disability was similar to that of normal achievers in a working memory task involving sentence processing, but impaired on a working memory task required processing of numerical information. By examining children’s performance on working memory tasks that required processing of numerical information as well as tasks that did not, we could evaluate whether working memory impairments was general or specific to processing numerical information. A second aim was to determine whether impaired performance was specific to working memory tasks or was also seen in short-term memory tasks in which small amounts of material are held passively and then recalled without any transformation (Cantor, Engle, & Hamilton, 1991; Swanson, 1993; Vecchi & Cornoldi, 1999). In the study by Swanson (1994), the children and adults with learning disabilities had impaired performance on a working memory task but not on digit span, a common measure of short-term memory. To determine whether such impaired performance is typical, we also administered several span tasks thought to asses short-term memory. Because children who are poor problem solvers may show a selective impairment only in the recall of numerical information, some span tasks required children to remember digits but others required them to remember words. A third aim of our work was to investigate the mechanism that leads to impaired working memory. Several investigations in reading comprehension have shown that poor readers suppress information less efficiently (De Beni, Palladino, Pazzaglia, & Cornoldi, 1988; Gernsbacher, 1993; Gernsbacher & Faust, 1991; Chiappe, Hasher, & Siegel, 2000). In like manner, Passolunghi, Cornoldi, and De Liberto (1999) found that children who were poor arithmetic problem solvers had
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low scores in working memory tasks requiring inhibition of irrelevant information. Based on these findings, we hypothesized that the working memory deficit of the poor problem solvers may be related to an inability to control and to ignore irrelevant or no longer relevant information. Consequently, we expected that poor problem solvers would have a high number of intrusion errors during recall. To summarize, we examined performance on a range of working memory and short-term memory tasks by children who were either poor or good in problem solving. We included four working memory tasks: (1) the Listening Span Task, in which children listened to sets of increasing numbers of sentences, judged whether the sentences were true or false, then at the end of each set, recalled the last word of each of the sentences; (2) the Animal Dual Task (De Beni et al., 1998; Palladino & De Beni, 1999; Passolunghi et al., 1999), in which children listened to a series of words, identified the animal nouns, then remembered the last word of each series; (3) the Listening Span Completion Task, in which several sentences with the final word missing were presented and the child had to generate the missing word at the end of the sentence, than they had to repeat all the missing words from the sets; (4) the Counting Span Task, in which children counted target dots from an irregular pattern of dots, then recalled the counts of the presented patterns. The short-term memory tasks included forward and backward digit span, as well as forward and backward word span. Several outcomes were possible: if children who are poor problem solvers have a truly general deficit, their performance would be impaired on all tasks. If their deficit is limited to working memory, their performance would be unimpaired on the forward digit and word span tasks, but impaired on all others including the reverse digit span tasks, which are often assumed to measure working memory (e.g., Cornoldi & Vecchi, 2000). Finally, if children who are poor problem solvers are particularly impaired when processing numerical information, we would expect relatively poor performance on the counting span and digit span tasks. METHOD Participants The participants were 49 fourth-grade children divided into a group of 23 poor problem solvers and a group of 26 good problem solvers. These groups were formed on the basis of two screening tests (on arithmetic word problems and vocabulary abilities) administered to 280 participants of a town in northern Italy (Trieste) during the first 3 months of the fourth grade. The participants had no documented brain injury, sociocultural disadvantage, or behavioral problems. Children were included in the poor problem solver group if their score was less than 30th percentile in a standardized Italian mathematics test (Amoretti, Bazzini, Pesci, & Reggiani, 1994) and if the teachers had noted that the child had difficulty with math problems. The good problem solvers were between the 50th and the 80th percentile in the same mathematical test and were considered by their teachers to be performing at or above grade level in mathematics. Respectively, the
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mean standard score in the mathematical test was ⫺.95 (SD ⫽ .20) for the poor problem group and .84 (SD ⫽ .56) for the good problem group. This test consisted of 12 items: 8 written arithmetical word problems and 4 questions examining the manipulation of Arabic and verbal numerals (i.e., to put in ascending order a series of Arabic numbers; to write the Arabic numeral that corresponds to a verbal numeral). A typical problem was: “This Saturday Robert goes with his mother to the supermarket. His mother buys 4 hectograms of ham and spends 6,960 Liras. What is the cost of one hectogram of ham?” The two groups were matched for age (9 years 4 months), gender (poor problem solver group: 11 males and 12 females; good problem solver group: 14 males and 12 females), and their scores in the Vocabulary subtest of PMA battery (Thurstone & Thurstone, 1941). Mean scores in this test were, respectively, 54.22 (SD ⫽ 2.66) for the poor problem solvers and 55.26 (SD ⫽ 2.96) for the good problem solvers. Converted to verbal IQ, the mean scores were, respectively, 119.12 (SD ⫽ 4.23) for the poor problem solvers group and 120.73 (SD ⫽ 4.83) for the good problem solvers group. This differences was not significant (p ⬎ .22). We assessed reading comprehension ability (scores on a standardized test, Cornoldi & Colpo, 1981) for poor problem solvers and good problem solvers. The test requested the children to read a one page text and then answer a series of multiple choice comprehension questions. There was no significant difference between the score of the two groups (poor problem solvers: mean score ⫽ 7.43, SD ⫽ 1.93 (standard score ⫽ ⫺.19, SD ⫽ 1.00); good problem solvers: mean score ⫽ 8.15, SD ⫽ 1.82 (standard score ⫽ .17, SD ⫽ .98), p ⬎ .15). Children selected were individually tested with seven tasks in December: Listening Span Task, Animal Dual Task, Listening Span Completion Task, and Span Tasks. Each task was administered on a separate day in the above order. The four span tasks were administered in a single session in the following order: word span tasks (forward and backward), and digit span tasks (forward and backward). Three months later, the children received the Counting Span Task, which was added subsequently to the battery. The approximate length of a testing session was 15 min. Tasks The experimenter presented the tasks and verified that the children understood them perfectly. No feedback was given during the testing phase of each task. Listening span task. An Italian adaptation (see Passolunghi et al., 1999) of the listening span test devised by Daneman and Carpenter (1980) was used, including two series of two sentences, two series of three sentences, two series of four sentences, and two series of five sentences. For each set, there was a practice series. There were two kind of sentences: true and false. An example of true sentence was “A and B are the first two letters of the ALPHABET,” an example of false sentence was “The hen is a mammal that lives in the SEA.” The child received all sets of stimuli. Each sentence was presented at a rate of approximately 1 s per word, followed by an interval for the child’s answer
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concerning the truth of the sentence. The child was instructed to give the answer as soon as possible. His/her answer was followed by the presentation of the next sentence. At the end of each set of sentences, immediately after the verification answer for the last sentence, the child was asked to recall the last word of each sentence (in the same order of presentation) and to pay attention to avoid mentioning nonfinal words. For example, if he/she had processed the above-mentioned sentences, the correct response was “ALPHABET, SEA.” Animal dual task. The material consisted of strings of words, each string composed of four two-syllable words. The strings of words were used in place of the sentences typically used in the Listening Span Test. There were three sets, each with 2, 3, and 4 strings, for a total of 27 strings and 128 words. The words, of medium-high frequency and medium-high level of imagery value, were selected from the Cornoldi and Pra Baldi (1978) norms. Among these words, 26 were animal nouns (9 in the final position and 17 placed in the strings). An example of a string of words is the following: tetto (roof), tiger (tigre), becco (beak), ponte (bridge). The testing phase was preceded by practice items. The child received all sets of stimuli. The words were presented at an approximate rate of 1 s per word with a 2-s interval after each string. When the name of an animal was given, the child had to tap the table. At the end of each series, the child was asked to recall the last name in each string in the series in the same order of presentation. The Animal Dual task is considered a real working memory task, since the child is required to process all of the material then discharge the items that became irrelevant in order to recall the last words of each string (see De Beni et al., 1998; Palladino & De Beni, 1999; Passolunghi et al., 1999). Listening span completion task. An Italian version of the working memory sentence developed by Siegel and Ryan (1989) was used, including three series of two sentences, three series of three sentences, three series of four sentences, and three series of five sentences. In each sentence, the final word was missing. For example, one series of these sentences was: “A fish swims, a bird ________; The snow is white, the coal is ________; In the barn the farm worker milked the ________.” To minimize word-finding problems, sentences were chosen so that the missing words were virtually predetermined. The children were tested following a procedure similar to that of the Listening Span Task. Each sentence was presented aurally at a rate of approximately 1 s per word. The task for the child was to supply the missing word at the end of the sentence and to repeat all the missing words from the set, in the same order that the sentences had been presented (e.g., in the example: “flies, black, cow”). No child had any difficulty in supplying the missing words. Counting span task. This task was a version of Counting Span developed by Siegel and Ryan (1989) and was originally designed by Case, Kurland, and Goldberg (1982). The pattern used for the stimuli was a field of blue and yellow dots arranged in a randomly determined order on a card of 30 ⫻ 20 cm. Fortytwo cards were presented; there were three trials at each span level (2, 3, 4, and 5 span levels). The child had to count the yellow dots from a field of blue dots
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and then recall the counts of each set in the correct order. No child experienced any difficulty in counting the dots. To replicate the procedure used by Siegel and Ryan (1989), task administration was stopped when the child failed all three trials at one span level. The task was discontinued for only four children, making it unlikely that this procedural difference from other working memory tasks has significant implications for the results. Short-term memory tasks. The Word Span Task used an increasing number of familiar two-syllable words, from two to eight words per span length (two trials for each span length). The Digit Span Task was tested using the WISC-R Digit Span subtest (Wechsler, 1974). In the backward tasks, words and digits were recalled in the reverse order of presentation. Each task was scored separately. In the Word Span Task, the words were presented at a 1-s-per-word rate until a child made a mistake in both trials of the same span length. A span was considered to have been correct if all the nouns were recalled in their correct order. The experimenter started from a span length of two. When the repetition was correct, the experimenter moved to a higher level; when the repetition was incorrect, the child had the opportunity of trying again with another trial of the same span length. If the child failed on the second trial the testing was discontinued. The same procedure was used for the Digit Span Task. In Word and Digit Span backward tasks, the rate of stimuli presentation was the same as for the forward tasks, and children were asked to recall the items in the reverse order of presentation. RESULTS Listening Span Task The mean performance of the two groups in the Listening Span Task is presented in the first row of Table 1. Although the poor problem solvers had a lower performance than good problem solvers in the sentence verification (t(47) ⫽ 3.05, p ⬍ .05), no one children made more than 3 errors on the maximum score of 28. Therefore, we think that they performed this task as a real working memory task, in which they concurrently processed the sentences and remembered the final words. We found a significant difference between the groups, t(47) ⫽ 2.23, p ⫽ .03, in correct recall (this index indicates the number of the target words recalled in order). The poor problem solver children erroneously remembered more words embedded in the sentences than the children in the good problem solver group. The difference in the intrusions (see Table 1) was significant (t(47) ⫽ 2.05, p ⬍ .05). An intrusion error could be due either to an item of a preceding list (i.e., sentences in the case of Listening Tasks, strings of words in the case of Animal Dual Task), or to an item of the currently presented list. In order to analyze the nature of intrusion errors, we measured these different types of intrusion errors. In the Listening Span Task, the two groups did not differ in the comparison between intrusions of preceding lists and the currently presented list (see first row, Table 1, p ⬎ .11).
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TABLE 1 Mean Performance (Standard Deviations in Brackets) of the Poor Problem Solver Group and of the Good Problem Solver Group in the Listening Span Task, Animal Dual Task, Listening Completion Task, Counting Span Task, and Word and Digit Span Tasks (Forward and Backward) Poor problem solver group Correct recall
Total Intrusions intrusion current list
Intrusions preceding lists
9.65 (3.04)
4.78 (3.01)
3.87 (2.43)
.91 (1.27)
Animal Dual Task
7.52 (4.12)
5.83 (3.39)
4.69 (3.00)
1.13 (1.10)
Listening Span Compl. Task Counting Span Task Word Span Forward Word Span Backward Digit Span Forward Digit Span Backward
18.74 (5.67) 35.69 (3.72) 3.96 (.20) 3.52 (.66) 5.26 (.96) 3.48 (.73)
2.56 (2.08)
1.56 (1.51)
1.00 (1.17) 1.95 (1.61) — — — —
Correct recall 25.48 (1.31) (true/false) 2.87 3.04 (2.16) (2.34) (anim. (nonanim. intrus.) intrus.) — —
Total Intrusions Intrusions intrusion current preceding list lists
11.73 (3.43)
3.07 (2.79)
2.65 (2.91)
.42 (.75)
10.27 (4.37)
3.15 (2.26)
2.19 (1.55)
.96 (1.22)
22.23 (5.80) 38.23 (3.57) 4.08 (.48) 3.93 (.63) 5.85 (1.00) 4.11 (.77)
.85 (1.19)
.31 (.55)
.54 (.90) 1.15 (1.19) —
—
— — —
27.50 (1.03) (true/false) 1.77 1.38 (1.61) (1.36) (anim. (nonanim. intrus.) intrus.) — —
PASSOLUNGHI AND SIEGEL
Listening Span Task
—
Good problem solver group
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Animal Dual Task The results of Animal Dual Task (see the second row of Table 1) showed the same trend as the Listening Span Task: the children in the poor problem solving group recalled fewer words in the correct order compared to the good problem solver group (t(47) ⫽ 2.26, p ⬍ .03). All children were very accurate in the identification of animal words; no one made more than 1 or 2 misses. In this task, animal nouns received more stress. Our hypothesis is that this deeper encoding increased their probability to be better recalled when they are the target items (see Passolunghi et al., 1999). Focusing on the number of the target animal nouns (e.g., those in the last position) that were recalled correctly, animal nouns were better remembered than nonanimal nouns (respectively, 58% vs 49%, t(48) ⫽ 3.39, p ⫽ .0014). The poor problems solvers performed as well as good problem solvers in recalling the target items which required deeper elaboration (respectively, 55%, SD ⫽ 17.88, of correct recall of animal nouns for poor problem solvers and 60%, SD ⫽ 14.43, for good problem solvers, the difference between the two groups was not significant, p ⫽ .31). In contrast, the poor problem solvers performed at a significantly lower level in the recall of nonanimal nouns in the last positions, respectively 44% (SD ⫽ 9.95) for poor problem solvers and 54% (SD ⫽ 12.65) for good problem solvers, t(47) ⫽ 3.08, p ⫽ .0034. A 2 groups ⫻ 2 source of errors (intrusions due to the current list and intrusions due to preceding lists) ANOVA, showed a main effect of group, F(1, 47) ⫽ 10,76 p ⫽ .002, a main effect of source of errors F(1, 47) ⫽ 49,92, p ⫽ .001, due to the fact that intrusion mainly concerned items of the current list, and a significant interaction between groups and source of errors, F(1, 47) ⫽ 11,82, p ⫽ .0012, due to the fact that poor problem solvers made more intrusions from the current list. Listening Span Completion Task The good problem solvers group had higher mean number of correct recall than poor problem solvers, t(47) ⫽ 2.13, p ⫽ .03 (see the third row of Table 1). Poor problem solvers made more intrusions than the good problem solvers, erroneously recalling more words embedded in the sentences. A 2 groups ⫻ 2 source of errors ANOVA showed a main effect of groups, F(1, 47) ⫽ 12,93, p ⫽ 12,93, p ⫽ .0008, and a significant interaction between groups and source of errors, F(1, 47) ⫽ 4,31, p ⬍ .05, due to the fact that poor problem solvers made more intrusions from the current list. Counting Span Task The poor problem solvers had lower scores than the good problem solvers in the correct recall in order, t(47) ⫽ 2.43, p ⫽ .018 (see the fourth row of Table 1). We defined an intrusion as the recall of a number from the card presented prior to the current card. Poor problem solvers made significantly more intrusions than good problems solvers (see fourth row of Table 1, t(47) ⫽ 2.00, p ⬍ .05).
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Span Tasks The results of the Word and Digit Span Tasks (forward and backward) are presented in the last four rows of Table 1. A 2 (groups) ⫻ 2 (type, word vs digit) ⫻ 2 (direction, forward vs backward) ANOVA for a mixed design showed a significant main effect of group, F(1, 47) ⫽ 11.41, p ⫽ .0015, an interaction of type by group, F(1, 47) ⫽ 3.83, p ⬍ .05, due to the fact that the poor problem solver group had lower scores than the good problem solver group in the recall of numerical information, a main effect of direction (forward vs backward), F(1, 47) ⫽ 95.97, p ⫽ .0001, and an interaction of type by direction F(1, 47) ⫽ 72.62, p ⫽ .0001, due to the fact that the highest performance was in the digit forward task and the lowest performance was in the word backward task. DISCUSSION Working Memory Deficit of Poor Problem Solvers Several researchers have postulated certain cognitive processes (e.g., comprehension, integration, planning, and execution) and metacognitive abilities that are fundamental in solving a problem (Lucangeli & Passolunghi, 1995; Mayer, 1998; Nathan, Kintsch, & Young, 1992; Passolunghi, Lonciari, & Cornoldi 1996; Swanson, 1990). However, investigators have not consistently found a relation between problem performance and working memory ability. One aim of the present research was to examine the link between working memory and word problem performance across a range of tasks whose purpose was to measure working memory. We examined children’s performance on working memory tasks that required processing of numerical information as well as tasks that did not, in order to evaluate whether working memory impairments of poor problem solvers was general or specific to processing numerical information. Siegel and Ryan (1989) found that the performance of children with a mathematical learning disability was impaired only on a working memory task that required processing of numerical information. The results of this research demonstrated that the cognitive impairments of poor problem solver children are not restricted to a numerical working memory task. The Listening Span Task, Animal Dual Task, and Listening Span Completion Task are clearly verbal working memory tasks, while the Counting Span Task has a much smaller verbal component, and involves the processing of numerical information. In these tasks, poor problem solvers demonstrated less recall of relevant information and their impairment was similar in working memory tasks involving both verbal and numerical information. Therefore, the results of this study support the hypothesis of a generalized working memory deficit of poor problem solvers, assessed by tasks involving manipulation and transformation of information. In addition, we attempted to control a factor that can mediate the relations between working memory and word problem solution. We matched the groups on a measure of the verbal intelligence in order to rule out the possibility that the relations between working memory and word problem solution may be attributed
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to intellectual capacity. The results showed that even when the two groups were matched on measure of verbal intelligence, we observed poorer performance in the poor problem solver group. Moreover, we controlled the influence of reading ability. When we compared the comprehension ability of the two groups, we did not find a significant difference. This result supports the hypothesis of a specific relation between working memory and arithmetic problem solution. Impairment in Short-Term Memory Tasks A second aim of the research was to determine whether poor problem solvers’ impaired performance was specific to working memory tasks or was also extended to short-term memory tasks in which small amounts of material are held passively and then recalled without any transformation. Swanson (1994) found that the children and adults with learning disabilities had impaired performance on a working memory task but not on digit span task, a common measure of short-term memory. He hypothesized that short-term memory and working memory are independent measures, and he found that they loaded on different factors. Short-term memory tasks rely on a passive storage system and are those in which the individuals are required to retain the material to recall without performing any modification on it (see also Cantor, Engle, & Hamilton, 1991; Cornoldi & Vecchi, 2000; Vecchi, Monticelli, & Cornoldi, 1995). In contrast, working memory tasks request more active processes and are those in which information is temporarily held while being manipulated or transformed (e.g., process all the material and only successively discharge the items that become irrelevant). An implication for this assumption is that children or adults with learning disabilities may have working memory problems independent of problems in short-term memory. Our results showed that poor problem solvers had a general impairment in working memory tests, but they did not reveal an impairment in typical tests of short-term memory, i.e., span tasks, if the span concerned linguistic information. However, our data showed that children with a mathematics disability had lower performance than the normally achieving group in immediate recall of numerical information. A possible interpretation is that these children had slower access to number representations in long-term memory, which in turn may have led to slow counting and lower digit span (see also Geary, 1990, 1993; Hitch & McAuley, 1991). The preserved ability of poor problem solvers in the passive processing of verbal information does not support the hypothesis that their impairment in passive processing of numerical information is due to task difficulty. The discrepancy between Swanson’s (1994) results on digit span task and ours is probably due to the difference in the sample selection. The participants of Swanson’s (1994) study were defined as learning disabled on the basis of reading performance, while mathematics performance was not included in the selection criteria and was left to covary. In contrast, our children have a main difficulty in arithmetic word problem solutions and they and did not differ from good problem solvers in reading and vocabulary ability.
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Intrusion Errors and Inhibitory Processes A third aim of our work was to investigate the mechanism that leads to impaired working memory. Our data provided evidence that the working memory deficit of the poor problem solvers is related to an inability to control and to ignore irrelevant or no longer relevant information. Indeed, the high number of intrusion errors in the working memory tasks suggests that the poor problem solvers maintained information available in memory, which initially had to be processed, but then had to be suppressed. We found that poor problem solvers had a higher number of intrusion errors than good problem solvers in the Listening Span Task, Animal Dual Task, and Listening Span Completion Task. Moreover, poor problem solvers showed the same failure pattern (e.g., more intrusion errors) in the processing of numerical information as required by Counting Span Task. The intrusions of items of the currently presented list may be considered working memory errors due to intrusive elements still active in working memory, whereas we hypothesized that the intrusions concerning items of the preceding lists do not involve the working memory, but rather involve long-term memory. We found that poor problem solvers made more intrusion errors than good problem solvers only with respect to the items from the current lists. This result supports the hypothesis that poor problem solvers have a poorer working memory system and that their working memory failure is related to the manipulation of highly accessible information. Altogether the results of intrusion errors support the hypothesis that problem solving ability is related to the ability of reducing the memory accessibility of nontarget and irrelevant information. This hypothesis is consistent with the literature which tries to connect working memory capacity and the efficiency of inhibition mechanisms (Bjorklund & Harnishfeger, 1990; Chiappe, Hasher, & Siegel, 2000; De Beni et al., 1998; Gernsbacher, 1993; Hamm & Hasher, 1991; Passolunghi & Cornoldi, 2000; Passolunghi et al., 1999). In our view, the working memory deficit of poor problem solvers reflected problems in the central executive component of Baddley’s model (Baddeley 1986, 1996; Baddeley & Hitch, 1974). Children with mathematical disability did not have the same working memory capacity as children without mathematical disability, since poor problem solvers maintain in memory information that initially was necessary to process but then became irrelevant and had to be discharged. The results of the present study suggest that children with difficulties in the arithmetic word problem solution have a type of deficit related to a lack of flexibility in coordinating both verbal and numerical information rather an output problem. The involvement of the central executive may be related to a metacognitive problem. It must be noted that, on the basis of our data, it is not possible to decide whether the inhibition failure is a primary and causal deficit or if it is connected to other more general factors such as processing speed (see Kail & Hall, 1999) or selective attention (Tarver, Hallan, Cohen, & Kauffman, 1977). From a conceptual point of view it is difficult to disentangle the inhibition and selective
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attention hypothesis, since selective inhibition can be considered complementary to selective attention. In particular, our data are compatible with the hypothesis of Conway and Engle (1994) and Engle, Conway, Tuholski, and Shisler (1995) that individuals with a low span do not have the attentional and resource capacity necessary to inhibit irrelevant information. It might be argued that poor problem solvers may use the wrong retrieval cue to access memory or they may lack a strategy for maintaining relevant information in memory. However, poor problem solvers are as able as good problem solvers in recalling the target items whether these items required deeper elaboration. In the Animal Dual Task, the two groups did not differ in the correct recall if the to-be-recalled items are the animal nouns (i.e., the items that received more stress and deeply encoding); however, the two groups differed in the case of nonanimal nouns. The attention required for animal items had the effect of increasing their probability of being both correctly and incorrectly retrieved. It seems that this attention increased their accessibility (Craik & Tulving, 1975), but it also increased the difficulty of inhibiting them. A deeper encoding of these items may improve the recall of the poor problem solvers. In summary, the results showed that children who are poor problem solvers have a general working memory deficit: their performance was impairment both in verbal and numerical working memory tasks. However, our data showed that they had performance similar to good problem solvers in typical tests of shortterm memory, i.e., span tasks, if the span concerned linguistic information. In contrast, poor problem solvers had lower performance than normally achieving group in immediate recall of numerical information. Finally, we found that poor problem solvers had a high number of intrusion errors during recall. This result supports our hypothesis that the poor problem solvers’ working memory deficit is related to a failure of inhibitory mechanism in processing the information. REFERENCES Amoretti, G., Bazzini, L., Pesci A., & Reggiani, M. (1994). Test di matematica per la scuola dell’obbligo [Mathematics test for primary schools]. Firenze: Organizzazioni Speciali. Baddeley, A. D. (1986). Working memory. Oxford: Clarendon Press. Baddeley, A. D. (1996). Exploring the central executive. The Quarterly Journal of Experimental Psychology, 49a(1), 5–28. Baddeley, A. D., & Hitch, G. J. (1974). Working memory. In G. H. Bower (Ed.), The psychology of learning and motivation: Advances in research and theory (Vol. VIII, pp. 47–90). New York: Academic Press. Bjorklund, D., & Harnishfeger, K. (1990). The resources construct in cognitive development: Diverse sources of evidence and a theory of inefficient inhibition. Developmental Review, 10, 48–71. Cantor, J., Engle, R. W., & Hamilton, G. (1991). Short term memory, working memory and verbal abilities: How do they relate? Intelligence, 15, 229–246. Case, R., Kurland, D. M., & Goldberg, J. (1982). Operational efficiency of short-term memory span. Journal of Experimental Psychology, 33, 386–404. Chiappe, P., Hasher, L., & Siegel, L. S. (2000). Working memory, inhibitory control and reading disability. Memory and Cognition, 28, 8–17. Conway, A. R. A., & Engle, R. W. (1994). Working memory and retrieval: A resource dependent inhibition model. Journal of Experimental Psychology: General, 123, 354–373.
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Cooney, J. B., & Swanson, H. L. (1990). Individual differences in memory for mathematical story problems: Memory span and problem perception. Journal of Educational Psychology, 82, 570–577. Cornoldi, C., & Colpo, G. (1981). Prove di lettura MT [Reading comprehension test MT]. Firenze: Organizzazioni Speciali. Cornoldi, C., & Pra Baldi, A. (1978). Il valore d’immagine nei bambini: Norme per 257 nomi e usi nella ricerca applicata. Formazione e Cambiamento, 3–4, 275–304. Cornoldi, C., & Vecchi, T. (2000). Mental imagery in blind people: The role of passive and active visuo-spatial processes. In M. Heller (Ed.), Touch, representation and blindness (pp. 143–181). Oxford: Oxford Univ. Press. Craik, F. I. M., & Tulving, E. (1975). Depth of processing and the retention of words in episodic memory. Journal of Experimental Psychology: General, 14, 268–294. Daneman, M., & Carpenter, P. A. (1980). Individual differences in working memory and reading. Journal of Verbal Learning and Verbal Behaviour, 19, 450–466. De Beni, R., Palladino, P., Pazzaglia, F., & Cornoldi, C. (1998). The role of inhibition in the working memory deficits of poor readers. Quarterly Journal of Experimental Psychology, 51A, 305–320. Engle, R. W., Conway, A. R. A., Tuholski, S. W., & Shisler, R. J. (1995). A resource account of inhibition. Psychological Science, 6, 122–125. Geary, D. C. (1990). A componential analysis of an early learning deficit in mathematics. Journal of Experimental Child Psychology, 49, 363–383. Geary, D. C. (1993). Mathematical disabilities: Cognitive, neuropsychological, and genetic components. Psychological Bulletin, 114, 345–362. Gernsbacher, M. A. (1993). Less skilled readers have less efficient suppression mechanisms. Psychological Science, 4, 294–298. Gernsbacher, M. A., & Faust, M. E. (1991). The mechanism of suppression: A component of general comprehension skill. Journal of Experimental Psychology: Learning, Memory and Cognition, 17, 245–262. Hamm, V. P., & Hasher, K. (1992). Age and the availability of inferences. Psychology and Aging, 7, 56–64. Hitch, G. J., & McAuley, E. (1991). Working memory in children with specific arithmetical learning difficulties. British Journal of Psychology, 82, 375–386. Kail, R., & Hall, L. K. (1999). Sources of developmental change in children’s word-problem performance. Journal of Educational Psychology, 91, 660–668. Lucangeli, D., & Passolunghi, M. C. (1995). Psicologia dell’apprendimento matematico. [Psychology of mathematical learning]. Torino: Utet. Mayer, R. E. (1998). Cognitive, metacognitive, and motivational aspects of problem solving. Instructional Science, 26, 49–63. Nathan, M. J., Kintsch, W., & Young, E. (1992). A theory of algebra word problem comprehension and its implications for the design of learning environments. Cognition and Instructions, 4, 329–390. Palladino, P., & De Beni, R. (1999). Short term and working memory in aging: Maintenance and suppression. Aging: Clinical and Experimental Research, 11, 301–306. Passolunghi, M. C., & Cornoldi, C. (2000). Working memory and cognitive abilities in children with specific difficulties in arithmetic word problem solving. Advances in Learning and Behavioral Disabilities, 14, 155–178. Passolunghi, M. C., Cornoldi, C., & De Liberto, S. (1999). Working memory and intrusions of irrelevant information in a group of specific poor problem solvers. Memory and Cognition, 27, 779–790. Passolunghi, M. C., Lonciari, I., & Cornoldi, C. (1996). Abilità di pianificazione, comprensione, metacognizione e risoluzione di problemi aritmetici i tipo verbale. [The effects of planning, comprehension and metacognitive abilities on arithmetic word problems solution]. Età Evolutiva, 54, 36–48.
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Siegel, L. S., & Linder, B. A. (1984). Short-term memory processes in children with reading and arithmetic learning disabilities. Developmental Psychology, 20, 200–207. Siegel, L. S., & Ryan, E. B. (1989). The development of working memory in normally achieving and subtypes of learning disabled children. Child Development, 60, 973–980. Swanson, H. L. (1990). Influence of metacognitive knowledge and aptitude on problem solving. Journal of Educational Psychology, 82, 306–314. Swanson, H. L. (1993). Working memory in learning disability subgroups. Journal of Experimental Child Psychology, 56, 87–114. Swanson, H. L. (1994). Short-term memory and working memory: Do both contribute to our understanding of academic achievement in children and adults with learning disabilities? Journal of Learning disabilities, 27, 34–50. Swanson, H. L., Cochran, K., & Ewers, C. (1990). Can learning disabilities be determined from working memory performance? Journal of Learning Disabilities, 23, 59–67. Swanson, H. L., Cooney, J. B., & Brock, S. (1993). The influence of working memory and classification ability on children’s word problem solution. Journal of Experimental Child Psychology, 55, 374–395. Tarver, S. G., Hallan, D. P., Cohen, S. B., & Kauffman, J. M. (1977). The development of visual selective attention and verbal rehearsal in learning disabled boys. Journal of Learning Disabilities, 10, 491–500. Thurstone, N. L., & Thurstone, T. G. (1941). Factorial studies of intelligence. Psychometric Monographs, 2 [Italian translation PMA-Batteria Primaria di Abilità, Firenze, Organizzazioni Speciali, 1968]. Vecchi, T., Monticelli, M. L., & Cornoldi, C. (1995). Visuo-spatial working memory: Structures and variables affecting a capacity measure. Neuropsychologia, 33, 1549–1564. Vecchi, T., & Cornoldi, C. (1999). Passive storage and active manipulation in visuo-spatial working memory: Further evidence from the study of age differences. European Journal of Cognitive Psychology, 3, 391–406. Wechsler, D. (1974). Manual for the Wechsler Intelligence Scale for Children—Revised. New York: Psychological Corporation [Scala d’intelligenza Wechsler per bambini, 1987, Firenze, O.S.]. Received July 23, 1999; revised November 14, 2000; published online May 29, 2001