Cognitive processes that underlie mathematical precociousness in young children

Journal of Experimental Child Psychology 93 (2006) 239–264 www.elsevier.com/locate/jecp

Cognitive processes that underlie mathematical precociousness in young children H. Lee Swanson ¤ Graduate School of Education, Area of Educational Psychology, University of California, Riverside, CA 92521, USA Received 2 March 2005; revised 15 August 2005 Available online 5 December 2005

Abstract The working memory (WM) processes that underlie young children’s (ages 6–8 years) mathematical precociousness were examined. A battery of tests that assessed components of WM (phonological loop, visual–spatial sketchpad, and central executive), naming speed, random generation, and Xuency was administered to mathematically precocious and average-achieving children. The results showed that (a) precocious children performed better on executive processing, inhibition, and naming speed tasks than did average-achieving children, although the two groups were statistically comparable on measures of the phonological loop and visual–spatial sketchpad, and (b) the executive component of WM predicted mathematical accuracy independent of chronological age, reading, inhibition, and naming speed. The results support the notion that the executive system is an important predictor of children’s mathematical precociousness and that this system can operate independent of individual diVerences in the phonological loop, inhibition, and reading in predicting mathematical accuracy. © 2005 Elsevier Inc. All rights reserved. Keywords: Working memory; Math precociousness; Problem solving; Phonological processing; Executive processing

Introduction This study focused on the cognitive processes that predict mathematical precociousness in young children. There have been several studies examining individual diVerences in *

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children with mathematical disabilities in the elementary grades (Geary, Hoard, Byrd-Craven, & DeSoto, 2004). Few studies, however, have analyzed math-advanced elementary school children (for a review, see Robinson, Abbott, Berninger, & Busse, 1996). Of those available, most have focused on strategy use and sequence of skill acquisition (e.g., Geary & Burlingham-Dubree, 1989; Siegler, 1988). For example, a seminal study by Siegler (1988) described strategy use in three subtypes of Wrst graders: good students, not-so-good students, and perfectionists. The perfectionists had higher standards for checking (monitoring) than did the other two groups; however, they performed as well as the other good students on other cognitive measures. Thus, outside of the fact that young precocious children are better at monitoring their work than are less precocious children, we are uncertain about other cognitive variables that may play a role in predicting math precociousness. Several studies in the literature have examined the relation between math ability and working memory (WM) capacity (e.g., Geary et al., 2004; Swanson & Beebe-Frankenberger, 2004). For the most part, this literature suggests that the more mathematically capable individuals code and scan numerical information and have better memory in general (Passolunghi & Siegel, 2001; Wilson & Swanson, 2001; however, see also Hitch & McAuley, 1991; McLean & Hitch, 1999). Dark and Benbow (1990, 1991) qualiWed this research by suggesting that individual diVerences in WM span were a function of the type of intellectual giftedness. Using intellectually precocious 13- and 14-year-olds, they found that math achievement was related to enhanced performance on a WM span task with digits and location of stimuli, whereas verbal precocity was related to enhanced performance with word stimuli (Dark & Benbow, 1991). These results suggested that math precociousness was a function of how numeric verbal stimuli (digits) were represented in the WM system. Components of working memory Although math precociousness is related to processing advantages for numerical information, we are uncertain which components of WM account for this advantage. Thus, one purpose of the current study was to identify WM components that underlie mathematical precociousness in young children. Our framework to isolate components of WM related to mathematical precociousness (as well as other domains) is Baddeley’s (1986, 1996, 2000) multicomponent model. Baddeley and Logie (1999) described WM as a limited-capacity central executive system that interacts with a set of two passive store systems used for temporary storage of diVerent classes of information: the speech-based phonological loop and the visual sketchpad. The phonological loop is responsible for the temporary storage of verbal information; items are held within a phonological store of limited duration, and the items are maintained within the store via the process of articulation. The visual sketchpad is responsible for the storage of visual–spatial information over brief periods and plays a key role in the generation and manipulation of mental images. Both storage systems are in direct contact with the central executive system. The central executive system is considered to be primarily responsible for coordinating activity within the cognitive system, but it also devotes some of its resources to increasing the amount of information that can be held in the two subsystems (Baddeley & Logie, 1999). (A recent reformulation of the model [Baddeley, 2000] also includes a temporary multimodal storage component called the episodic buVer.) This multimodal representation of WM has been found to capture adults’ performance as well as children’s performance (e.g., Alloway, Gathercole, Willis, & Adams, 2004).

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In attempting to apply Baddeley’s model to mathematical precociousness in children, we were unable to Wnd any studies that linked precociousness to a component of WM. Furthermore, the research lacks some clarity in terms of those components most important in predicting math performance. For example, studies with older children and adults have attributed mathematical skill to the phonological loop (e.g., Logie, Gilhooly, & Wynn, 1994), the executive system (e.g., Lemaire, Abdi, & Fayol, 1996), a combination of both the phonological loop and the executive system (e.g., Furst & Hitch, 2000), or a combination of both the phonological loop and the visual–spatial sketchpad (Bull, Johnston, & Roy, 1999; Lee & Kang, 2002). In this study, we hoped to clarify, at least for young children, the components of WM that best predict mathematical performance in precocious children. Prior to discussing our hypotheses about those components of WM that may underlie mathematical precociousness, some clariWcation regarding WM and short-term memory (STM) is necessary. The distinctions made between speciWc memory storage systems (e.g., the phonological loop) and the central executive system in some ways parallel the distinctions made between STM and WM. Short-term memory typically involves situations where small amounts of material are held passively (i.e., minimal resources from long-term memory are activated to interpret the task, e.g., a digit or word span task) and then are reproduced in a sequential fashion. That is, participants are asked only to reproduce the sequence of items in the order they were presented (e.g., Daneman & Carpenter, 1980). Individual diVerences on these STM measures have been primarily attributed to phonological coding and rehearsal (e.g., Willis & Gathercole, 2001). In fact, STM measures capture a subset of WM performance, the use and/or operation of the phonological loop (for comprehensive reviews, see Gathercole, 1998; Gathercole & Baddeley, 1993). In contrast, WM is referred to as a processing resource of limited capacity involved in preserving information while simultaneously processing the same or other information (e.g., Baddeley, 1986; Baddeley & Logie, 1999; Engle, Tuholski, Laughlin, & Conway, 1999; Just & Carpenter, 1992). Individual diVerences in WM capacity have been attributed to executive processing (e.g., Engle et al., 1999; Swanson, 2003) such as the ability to inhibit irrelevant information (e.g., Chiappe, Hasher, & Siegel, 2000). Although there is an emerging consensus in the literature that STM and WM are distinct, they are highly related processes (for a review, see Heitz, Unsworth, & Engle, 2005). For example, Engle and colleagues (1999) investigated the relation among measures of STM, WM, and Xuid intelligence in adults. Although they found strong correlations among the factors, they found that a two-factor model Wt the data better than did a one-factor model. Furthermore, they found that by statistically controlling the variance between STM and WM factors, the residual variance related to the WM factor was signiWcantly correlated with measures of intelligence. Namely, there was a strong link between the latent measures of WM, but not of STM, to Xuid intelligence (see also Conway, Cowan, Bunting, Therriault, & MinkoV, 2002). They concluded that the residual variance related to the WM factor corresponded to controlled attention of the central executive system. In further elaborating the distinction between STM and WM in children, Cowan (1995) emphasized the role of attentional processing in children’s development. When high demands on attentional processing are applied to the content of STM tasks, it creates WM tasks in children. Thus, because WM includes both STM and focused attention, the question arises as to whether WM as an attentional mechanism (i.e., the executive system) or WM as storage (STM) is the important source of math-related diVerences in children.

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We extended this research by investigating whether processes related to the phonological loop mediate the inXuence of WM on children’s mathematical performance. A simple version of this hypothesis states that math-precocious children are more eYcient at processing phonological information than are average-achieving children and that such enhanced processing underlies high performance on WM and math measures. For example, one of the possible reasons why WM is related to math precociousness is that mathprecocious children can name items (e.g., digits) more rapidly than can average-achieving children (e.g., Swanson & Sachse-Lee, 2001). Rapid naming enhances the eVectiveness of subvocal rehearsal processes and, hence, reduces the decay of memory items in the phonological store prior to output (for reviews, see Gathercole, 1998; Henry & Millar, 1993). These hypotheses are also consistent with a number of models suggesting that the phonological system, via the phonological loop (e.g., phonological store, subvocal rehearsal), inXuences verbatim memory capacity, which in turn supports math performance (e.g., Swanson & Sachse-Lee, 2001). For example, Logie and colleagues (1994) showed that subvocal rehearsal plays a major role in maintaining accuracy of mental calculations. Likewise, some studies have attributed individual diVerences in mathematical problem solving to the phonological system (for a review, see Swanson & Sachse-Lee, 2001). Research to date also suggests that children with high mathematical skills outperform those of average ability on measures requiring the short-term retention of ordered numerical information (e.g., Geary, Brown, & Samaranayake, 1991), signifying eYcient use of the phonological rehearsal process (Geary, Hoard, & Hamson, 1999). There are clear expectations in the aforementioned model; speciWcally that individual diVerences in mathematics are related to the phonological loop. Accuracy in mathematical computations of increasing diYculty follows automatically with improvement in STM and naming speed. In particular, if individual diVerences in WM and mathematical computations are mediated by the phonological system, then the relation between calculation and WM should be eliminated when measures of the phonological loop (e.g., STM, naming speed) are partialed out of the analysis. In contrast to the above model, the second model views math precociousness as directly related to the controlling functions of the central executive system itself. How might the executive system contribute to individual diVerences in mathematical performance? To address this question, we consider children’s performance on tasks assumed to measure executive processing free or independent of phonological processing. Several cognitive activities have been assigned to the central executive (for a review, see Miyake, Friedman, Emerson, Witzki, & Howerter, 2000), including controlling subsidiary memory systems, controlling encoding and retrieval strategies, switching attention during manipulation of material held in the verbal and visual–spatial systems, suppressing irrelevant information, and accessing information from LTM (e.g., Baddeley, 1996; Miyake et al., 2000; Oberauer, Süß, Wilhelm, & Wittman, 2003). Recent studies suggest, however, that a speciWc activity of the central executive, controlled attention, mediates WM and reasoning skills (e.g., Engle et al., 1999; Oberauer et al., 2003). Controlled attention is deWned as the capacity to maintain and hold relevant information in the face of interference or distraction (Engle et al., 1999). Two tasks that we assume are related to diVerent aspects of controlled attention are Xuency and random generation. Both tasks measure inhibition but emphasize diVerent aspects. The Xuency task requires individuals to spontaneously generate words in response to a category cue (e.g., generate animal names) or a speciWc letter cue (e.g., generate words that begin with the letter b). These tasks have been

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associated with the executive functions related to the controlled search for words (for a review, see Rende, Ramsberger, & Miyake, 2002). That is, participants are directed to activate needed information (animal names) while controlling the repetition of exemplars. In the random generation procedure, in contrast, participants are asked to keep track of the number of times items have been generated and to inhibit well-known sequences such as “1, 2, 3, 4” and “a, b, c, d.” Baddeley (1986) indicated that during random generation, the central executive acts as a Wltering device, screening out automatically generated (and therefore nonrandom) responses. This task diVers somewhat from the Xuency measure described earlier because participants must suppress rote or habitual responses (saying the letters of the alphabet in order) to complete the task quickly. Thus, during random generation, the central executive acts as a rate-limited Wltering device that Wlters out habitual responses (for a review of this measure, see Towse, 1998). There are clear expectations for this second model. Based on several studies (for a review, see Heitz et al., 2005), we predicted that the executive component of WM will predict children’s mathematical performance. We assume that the executive component of WM (also referred to as controlled attention [Engle et al., 1999]) will correlate with mathematical computational ability after accounting for variance due to STM. Another way of stating the hypothesis is that if individual diVerences in WM and mathematical computations are mediated by the executive system, then the relation between calculation and WM should be eliminated when activities related to controlled attention (e.g., inhibition) are partialed out of the analysis. Purpose and predictions In summary, the purpose of this study was to assess the components of WM that are related to mathematical precociousness. We considered two possible models: either that (a) the relationship between WM and problem solving is primarily mediated by the phonological system or that (b) executive processes operate independent of the phonological system and, therefore, contribute unique variance to mathematical performance beyond the phonological system. Measures of the phonological loop included tasks related to STM and rapid naming of letters and numbers. Measures of the executive system were modeled after Daneman and Carpenter’s (1980) WM tasks. These tasks demand the coordination of both processing and storage. Recent studies have suggested that these tasks capture at least two factors of executive processing: susceptibility to interference and manipulation of capacity (e.g., Oberauer, 2002; Whitney, Arnett, Driver, & Budd, 2001). In this study, we determined whether the executive component of WM was mediated by measures of inhibition (random generation and Xuency).

Methods Participants Children (N D 127) in Wrst, second, and third grades participated in this study. The children were selected from one public and one private school district in Southern California. Final selection was related to achievement scores and parent approval for participation. Of the 127 students selected, 70 were boys and 57 were girls. Ethnic representation of the sample was as follows: 65 Anglo, 47 Latino, 6 African American,

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6 Asian American, and 3 “other” (e.g., Native American, Vietnamese). The sample mean socioeconomic status (SES) was primarily middle class, based on either parent education or parent occupation, and varied from lower middle class to upper middle class. Means and standard deviations for the selection variables employed in this study are shown in Table 1. DeWnition of math precociousness Children were included in the mathematically precocious group if they scored 1.5 standard deviations above average on two standardized tests of mathematical computation: the Wechsler Individual Achievement Test (WIAT) and the math subtest of the Table 1 Means and standard deviations for measures as a function of mathematically precocious and average-achieving children

1. Chronological age

Precocious (n D 50)

Average (n D 77)

M

M

SD

F ratio

2

SD

7.44

0.88

7.30

0.85

Calculation skill 2. WRAT-III math (SS) 3. WRAT-III math (raw) 4. WIAT math (SS) 5. WIAT math (raw)

126.48 28.72 124.84 19.82

5.25 5.25 6.17 4.74

99.21 21.30 97.06 12.51

8.22 3.05 7.60 4.05

435.10¤ 101.76¤ 468.39¤ 86.53¤

.77 .45 .78 .41

Covariate 4. Raven (SS) 5. WRAT-III reading (SS) 6. WRAT-III reading (raw)

112.36 107.94 30.20

13.76 7.24 5.10

104.79 101.12 27.07

12.84 7.07 4.36

9.99¤ 27.75¤ 13.65¤

.07 .18 .10

37.82 43.70

11.34 11.99

44.56 50.80

12.78 14.69

9.25¤ 8.19¤

.07 .06

Short-term memory (phonological loop) 9. WISC-III Forward Digit Span 10. Pseudoword Span 11. Word Span

3.04 4.40 4.04

1.12 1.98 0.78

2.91 4.35 3.75

0.95 2.48 0.84

0.49 0.01 3.66

— — .03

Working memory (executive system) 12. WISC-III Backward Digit Span 13. Listening/Sentence Span 14. Digit/Sentence Span 15. Updating

1.66 5.56 3.94 6.64

0.65 4.24 2.84 5.40

1.33 3.20 2.80 4.32

0.69 3.44 2.68 4.34

6.74¤ 11.55¤ 5.18 8.11¤

.05 .09 .04 .06

Working memory (visual–spatial sketchpad) 16. Visual Matrix 17. Mapping

2.14 4.00

4.51 5.74

1.54 3.19

3.65 3.33

0.77 1.00

.006 .007

11.38 25.72 15.08 8.50

6.28 8.41 4.34 3.60

7.33 19.16 12.44 6.83

5.58 7.79 4.58 2.89

14.48¤ 20.44¤ 10.41¤ 8.31¤

.10 .14 .08 .06

Speed 7. Rapid Digit Naming (raw) 8. Rapid Letter Naming (raw)

Inhibition 18. Random Letter Generation 19. Random Number Generation 20. Categorical Fluency 21. Letter Fluency ¤

p < .05.

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Wide Range Achievement Test (WRAT-IIII)]. To ensure that reading ability was comparable across the two groups, children with reading scores greater or less than 1 standard deviation from the mean on the reading subtest of the WRAT-III (scores <90 and >120) were excluded. This was done because reading skills are highly correlated with math calculation skills (e.g., Bull & Scerif, 2001; Hecht, Torgesen, Wagner, & Rashotte, 2001) and, therefore, some authors have suggested that math precociousness is diYcult to evaluate as a domain-speciWc skill when such measures are highly correlated with reading (Bull & Scerif, 2001). Thus, we made attempts at selecting children whose reading scores were in the average range. However, the reader should note that it was still necessary to statistically partial out the inXuence of reading on math performance in the subsequent analysis. In the current sample, 50 children (28 boys and 22 girls) were classiWed as math precocious and 77 children (42 boys and 35 girls) were classiWed as average math achievers. There were no between-group diVerences in terms of ethnic background or gender. In terms of grade level, 38 children in Grade 1, 41 in Grade 2, and 48 in Grade 3 participated in the study. Tasks and materials The battery of tasks is described subsequently, with experimental tasks described in more detail than published and standardized tasks. Tasks were divided into classiWcation, covariate, and comparison measures. Cronbach’s  reliability coeYcients for the sample were calculated and are provided for all measures. ClassiWcation measures Mathematical computation The arithmetic subtests from the WRAT-III (Wilkinson, 1993) and the WIAT (Psychological Corporation, 1992) were administered. Both subtests required written computation to problems that increase in diYculty. Standard scores have a mean of 100 and a standard deviation of 15. Cronbach’s ’s were .92 for the WRAT-III and .93 for the WIAT. Time given to complete the tests was approximately 20 min. Covariates Children precocious in math have been found to also excel on measures of reading and intelligence (e.g., Robinson et al., 1996). To address these issues, we also administered measures of reading and Xuid intelligence. Fluid intelligence Fluid intelligence was assessed by the Raven Colored Progressive Matrices (Raven, 1976). Children were given a booklet with patterns displayed on each page and with each pattern revealing a missing piece. For each pattern, six possible replacement pattern pieces were displayed, and children were required to circle the replacement piece that best completed the pattern. After the introduction of the Wrst matrix, children completed their booklets at their own pace. Patterns progressively increased in diYculty. The dependent measure (range 0–36) was the number of problems solved correctly, which yielded a

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standardized score (M D 100, SD D 15). Cronbach’s coeYcient  was .88. Strong correlations (e.g., r D .80) have been noted with this measure and WM (e.g., Engle et al., 1999; Kyllonen & Christal, 1990). Word recognition Word recognition was assessed by the reading subtest of the WRAT-III. The task provided a list of words of increasing diYculty. The children’s task was to read the words until 10 consecutive errors occurred. The dependent measure was the number of words read correctly. Cronbach’s  for the word recognition task was .89. Comparison measures Naming speed For digit naming speed, the administration procedures followed those speciWed in the manual of the Comprehensive Test of Phonological Processing (CTOPP) (Wagner, Torgesen, & Rashotte, 2000), including the presentation of practice trials. For this task, the examiner presented participants with two printed arrays, each containing 36 digits. Participants were required to name the digits as quickly as possible, and the task administrator used a stopwatch to time participants on speed of naming. The dependent measure was the total time to name both arrays of numbers. For letter naming speed, the administration procedures followed those speciWed in the CTOPP (Wagner et al., 2000). For this task, the examiner presented participants with two arrays of 36 letters each for a total of 72 letters. Participants were required to name the letters as quickly as possible, and the task administrator used a stopwatch to time participants on speed of naming. The dependent measure was the total time to name both arrays of letters. Short-term memory measures (phonological loop) The Forward Digit Span, Word Span, and Pseudoword Span tasks were administered as measures of the phonological loop. The Forward Digit Span task required participants to recall and repeat in order sets of digits that were spoken by the examiner and that increased in number (range D 0–8 digits). The dependent measure was the highest set recalled. The technical manual of the Wechsler Intelligence Scale for Children-III (WISCIII) (Psychological Corporation, 1991) reported a test–retest reliability of .91. The Word Span and Pseudoword Span tasks were presented in the same manner as the Forward Digit Span measure. The Word Span task was used previously by Swanson, Ashbaker, and Lee (1996). The word stimuli are one-, two-, and three-syllable high-frequency words. Students are read lists of common but unrelated nouns (e.g., brother, telephone, milk, elephant) and then are asked to recall the words. Word lists gradually increase in set size, from a minimum of two words to a maximum of eight words. The Phonetic Memory task (Pseudoword Span task) (Swanson & Berninger, 1995) uses strings of one-syllable nonsense words that are presented one at a time in sets of two to seven nonwords (e.g., nik, kes, int). Prior to administering test items, children practiced saying pseudowords (e.g., des, seeg, dez). The dependent measure for all STM measures was the highest set of items retrieved in the correct serial order (range D 0–7). Cronbach’s  was .62 for the Word Span task and .82 for the Phonetic Memory task.

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Working memory span—central executive The WM tasks in this study required children to hold increasingly complex information in memory while responding to questions about the tasks. The questions served as distracters to item recall because they reXected the recognition of targeted and closely related nontargeted items. A question was asked for each set of items, and the tasks were discontinued if the question was answered incorrectly or if all items within a set could not be remembered. Thus, WM span reXected a balance between item storage and correct responses to questions. Consistent with a number of previous studies, our WM tasks required the maintenance of some information during the processing of additional information. For example, consistent with Daneman and Carpenter’s (1980) seminal WM measure, the processing of information was assessed by asking participants simple questions about the to-be-remembered material (storage plus processing demands), whereas storage was assessed by accuracy of item retrieval (storage demands only). The questions required simple recognition of new and old information and were analogous to the yes/no response feature of Daneman and Carpenter’s task. It is important to note, however, that in our tasks the diYculty of the processing question remained constant within task conditions, thereby allowing the source of individual diVerences to reXect increased storage demands. Furthermore, the questions focused on the discrimination of items (old and new information) rather than on deeper levels of processing such as mathematical computations (e.g., Hitch, Towse, & Hutton, 2001; Towse, Hitch, & Hutton, 1998). A previous study with a diVerent sample established the reliability and construct validity of the measures with the Daneman and Carpenter (1980) measure (Swanson, 1996). For this study, four WM tasks were divided into those requiring the recall of verbal information (Listening/Sentence Span task and Digit/Sentence Span task) and those requiring the recall of visual–spatial information (e.g., Visual Matrix task, Mapping task) and were selected from a standardized battery of 11 WM tasks due to their high construct validity and reliability (Swanson, 1992). A complete description of administration and scoring of the tasks was reported in Swanson (1995). A children’s adaptation of the Daneman and Carpenter (1980) measure (Swanson, 1992) was also administered. In addition, we administered the Backward Digit Span and Updating tasks. Task descriptions follow. Listening Sentence Span The children’s adaptation (Swanson, 1992) of Daneman and Carpenter’s (1980) Sentence Span task was administered. The construction and pattern of results associated with the two measures are comparable. The only diVerence was that each sentence was read to children with a 2-s pause that indicated the end of a sentence. The original Sentence Span measure was used with university students, whereas the current measure used a simpler sentence structure and reading vocabulary. As a common measure of WM (Daneman & Carpenter, 1980; Just & Carpenter, 1992), this task required the presentation of groups of sentences, read aloud, for which children simultaneously tried to understand the passage and remember the last word of each sentence. The number of sentences in the group gradually increased. After each group, participants answered a question about a sentence and then recalled the last word of the sentence in the same order in which the words had been presented. WM capacity was deWned as the largest group of ending words recalled. The mean sentence reading level was approximately Grade 3.8. The dependent measure was the highest set recalled correctly (range D 0–8) in which the process question was answered correctly. Cronbach’s coeYcient  was .79.

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Digit/Sentence Span This task assesses children’s ability to remember numerical information embedded in a short sentence (Swanson, 1992, 1995). Before stimulus presentation, children were shown a card depicting four strategies for encoding numerical information to be recalled. The pictures portrayed the strategies of rehearsal, chunking, association, and elaboration. The experimenter described each strategy to the children before administration of targeted items. After all strategies had been explained, children were presented numbers in a sentence context. For example, Item 3 stated, “Now suppose somebody wanted to have you take them to the supermarket at 8 6 5 1 Elm Street.” The numbers were presented at 2-s intervals, followed by a process question (i.e., “What was the name of the street?”). Then children were asked to select a strategy from the array of four strategies that represented the best approximation of how they practiced the information for recall. Finally, the examiner prompted children to recall the numbers from the sentence in order. No further information about the strategies was provided. Students were allowed 30 s to remember the information. Recall diYculty for this task ranged from 3 to 14 digits; the dependent measure was the highest set recalled correctly (range D 0–9) in which the process question was answered correctly. Cronbach’s coeYcient  was .79. Backward Digit Span We also administered the Backward Digit Span task to assess WM. This task was taken from the WISC-III and required participants to recall sets of digits in reverse order. The dependent measure was the highest set of items recalled in order (range D 0–7). Cronbach’s  for the Backward Digit Span task was .84. Updating In contrast to the aforementioned WM measures that involved a dual-task situation where participants answered questions about the tasks while retaining information (words or spatial location of dots), the Updating task involved the active manipulation of information such that the order of new information was added to or replaced the order of old information. The experimental Updating task was adapted from Morris and Jones (1990). For this task, a series of one-digit numbers that varied in set lengths of 9, 7, 5, and 3 digits was presented. No digit appeared twice in the same set. The examiner stated that the list may be either long or short and that children should remember only the last three numbers in the same order as presented. Each digit was presented at approximately 1-s intervals. The four practice trials used list lengths of 3, 5, 7, and 9 digits each in random order. It was stressed that some of the lists of digits would be short and that children should not ignore any items. That is, to recall the last three digits in an unknown (ns D 3, 5, 7, 9) series of digits, the order of old information must be kept available (previously presented digits) along with the order of newly presented digits. The dependent measure was the total number of digits repeated correctly (range D 0–16). Cronbach’s  for the current sample was .94. Visual–spatial sketchpad Visual Matrix The purpose of this task was to assess the ability of participants to remember visual sequences within a matrix (Swanson, 1992, 1995). In contrast to the standardization procedures (Swanson, 1995), the Visual Matrix task was administered in small groups.

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An overhead projector was used to display stimuli to groups of children instead of individually by use of the examiner’s manual. This change in format required students to circle their answers to the process questions rather than to respond verbally. Otherwise, the task was administered as per the manual instructions. Participants were presented a series of dots in a matrix and were allowed 5 s to study the matrix. The matrix was then removed and participants were asked, “Are there any dots in the Wrst column?” To ensure students’ understanding of columns prior to testing, the experimenter pointed to the Wrst column on the blank matrix (a grid with no dots) as a reminder of the Wrst column. After answering the discrimination question (by circling “yes” or “no”), students were asked to draw the dots they remembered seeing in the corresponding boxes of their blank matrix response booklets. The task diYculty ranged from a matrix of 4 squares and 2 dots to a matrix of 45 squares and 12 dots. The dependent measure was the highest set recalled correctly (range D 0–11) in which the process question was answered correctly. Cronbach’s  was .42. Mapping This task required children to remember a sequence of directions on a map (Swanson, 1992, 1995). Before stimulus presentation, children were shown a card depicting four strategies. The pictures portrayed the strategies for processing patterns. The experimenter described each strategy to the children before administration of targeted items. After all strategies had been explained, children were presented with a street map with dots connected by lines and with arrows illustrating the direction a bicycle would go to follow this route through the city. The dots represented stoplights, whereas lines and arrows mapped the route through the city. Children were allowed 5 s to study the map. After the map was removed, children were asked a process question (i.e., “Were there any stoplights on the Wrst street [column]?”) Children were then presented a blank matrix on which to draw the street directions (lines and arrows) and stoplights (dots). DiYculty on this subtest ranged from 2 to 19 dots. The dependent measure was the highest set of a correctly drawn map (range D 0–9) in which the process question was answered correctly. Cronbach’s  was .94. Inhibition measures (random generation and Xuency) Random Number/Letter Generation The task was assumed to measure inhibition because participants are required to actively monitor candidate responses and suppress responses that would lead to welllearned sequences such as “1-2-3-4” and “a-b-c-d” (Baddeley, 1996). Because this task has been used primarily with adult samples that have quicker access to letters and numbers, it was modiWed for the age groups in this study. Instead of being given orally as is done in adult studies, children were asked to write as quickly as possible numbers (or letters) Wrst in sequential order to establish a baseline. Children were then asked to quickly write numbers (or letters) in a random nonsystematic order (e.g., no telephone numbers, patterned responses, or real words). For example, for the number section, students were Wrst asked to write numbers from 0 to 9 in order (i.e., 1, 2, 3, 4) as quickly as possible within a 30-s period. They were then asked to write numbers “out of order” (including number repetition) as quickly as possible within a 30-s period. Scoring included an index for randomness, information redundancy, and percentage of paired responses to assess the tendency of participants to suppress response repetitions.

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The measure of inhibition was calculated as the number of sequential letters or numbers minus the number of correctly unordered numbers or letters. Cronbach’s  for the Random Number Generation task in the current sample was .89 and for the Random Letter Generation task was .91. Categorical Fluency The experimental measure was adapted from Harrison, Buxton, Husain, and Wise (2000). Children were given 60 s to generate as many names of animals as possible. The experimenter told children, “I want to see how many items you can name that go together in a category. For example, if the category was ‘fruit,’ you could say ‘apple, banana, orange ƒ’ Do you understand? Okay, try not to repeat yourself and keep going until I tell you to stop. Speak clearly and loud enough so that I can hear the words you are saying. The category is ‘animals.’ Ready, begin.” If children paused for more than 10 s, the task was stopped. Repetitions were deleted from the analysis. The dependent measure was the number of words stated correctly within 60 s. The coeYcient  for the experimental categorical Xuency task was .91. Letter Fluency This experimental measure was adapted from Harrison and colleagues (2000). Children were given 60 s to generate as many words as possible beginning with the letter B. The experimenter told children, “I want to see how many words you can say that begin with a certain letter. For example, if the letter was F, you could say ‘fun, friend, fantastic, Xying ƒ’ Do you understand? Okay, don’t say the names of people or places or numbers or the same word with diVerent endings, and try not to repeat yourself. Keep naming words that start with the letter until I say ‘stop.’ Speak clearly and loud enough so that I can hear the word you are saying. The letter is B. Ready, begin.” Repetitions, proper name errors, and contravention of the stem repetition were deleted from the analysis. The dependent measure was the number of words stated correctly within 60 s. Cronbach’s coeYcient  was .88. Procedures Three doctoral-level graduate students trained in test administration tested all participants in their schools. One session of approximately 45–60 min was required for small group test administration, and one session of 45–60 min was required for individual administration. Small group testing occurred in the children’s classrooms, and individual testing took place in a separate room. During the group testing session, data were obtained from the Raven’s Colored Progressive Matrices, WIAT, WRAT-III, Visual Matrix task, and Random Number/Letter Generation task. The remaining tasks were administered individually. Test administration was counterbalanced to control for order eVects. Task order was random across participants within each test administrator.

Results The means and standard deviations for mathematical computation, intelligence, reading, naming speed, STM, WM, and inhibition are shown in Table 1. The analyses and results were divided into two sections. The Wrst section focused on ability group diVerences. Children precocious in math were compared with children average in math. This approach

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lends itself to comparing ability groups but has the disadvantage of ignoring information about the variability of participants in each group. The second section focused on correlations between WM and mathematical performance in the complete sample. This approach allowed us to study the entire range of scores in WM and mathematical performance as well as to focus on common and unique variance. Regression models were computed to isolate unique processes that underlie mathematical performance. Ability group comparisons Because signiWcant diVerences were found between groups on word recognition (WRAT-III) and Xuid intelligence (Raven’s matrices), the measures were used as covariates in the subsequent analyses (described subsequently). Thus, a series of one-way (precocious vs. average math achievers) factorial multivariate analyses of variance (MANCOVAs) were conducted to examine diVerences in means for the following processes: (a) naming speed, (b) STM (phonological loop), (c) WM (executive), (d) visual– spatial WM (visual–spatial sketchpad), and (e) inhibition. The  level was set to .05 unless otherwise speciWed. All univariate F ratios and eVect sizes are reported in Table 1. 2 values of .13, .05, and .02 corresponded to Cohen’s d values of .80, .50, and .20, respectively. Fluid intelligence and reading As shown in Table 1, the univariates for the standard scores from the Raven’s matrices and the WRAT-III reading were signiWcant. Because precocious children outperformed average achievers on both measures, these are used as covariates in the analyses. Cognitive measures Speed A MANCOVA was conducted examining processing speed (CTOPP Rapid Letter Naming and CTOPP Rapid Digit Naming). A signiWcant multivariate main eVect was found for ability, Wilks’s  D .93, F(2, 123) D 4.53, p < .05, 2 D .07, indicating that precocious students demonstrate faster processing speed than do average students. Short-term memory (phonological loop) A MANCOVA was conducted examining short-term memory (WISC-III Forward Digit Span, Pseudoword Span, and Word Span). The multivariate main eVect was not signiWcant, Wilks’s  D .97, F(3, 122) D 1.14, p D .34, 2 D .03. As shown in Table 1, precocious and average achievers were statistically comparable on the STM measures. Working memory (executive system processing) A MANCOVA was conducted examining WM (WISC-III Digit Span Backward, Listening/Sentence Span, Digit/Sentence Span, Updating). The group multivariate main eVect was signiWcant, Wilks’s  D .92, F(4, 120) D 2.78, p < .05, 2 D .08, indicating that signiWcant diVerences emerged between math-precocious and average-achieving students in WM processes. Stepdown univariate analyses yielded signiWcant between-group eVects for all measures except Digit/Sentence Span.

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Working memory (visual–spatial sketchpad) A MANCOVA was conducted on the visual–spatial WM tasks (Visual Matrix and Mapping). The group multivariate main eVect was not signiWcant, p > .05, Wilks’s  D .99, F(2, 122) D .26, p D .77, 2 D .01. Inhibition A MANCOVA was conducted examining measures of inhibition (Random Number/ Letter Generation, Categorical Fluency, and Letter Fluency). The group eVects were signiWcant, Wilks’s  D .81, F(4, 121) D 6.90, p < .0001, 2 D .19. These results showed that inhibition scores were higher for precocious students than for average students. Between-group eVect sizes were in the high range for both measures. Stepdown univariate analyses yielded signiWcant between-group eVects for all measures. In summary, the results showed clearly that all MANCOVAs were signiWcant except STM and visual–spatial WM. Correlations The next analyses examined the relation between WM and mathematical computation in the total sample. We predicted that if the executive system of WM played an important role in accounting for individual diVerences in math independent of the phonological loop, Table 2 Intercorrelations among achievement and cognitive measures 1

2

3

4

5

6

7

8

9

10

11 12 13 14

15 16

17 18 19 20

1. Age 1.00 2. Gender .01 1.00 3. WRAT-M .71 .07 1.00 4. WIAT-M .75 .01 .88 1.00 5. Reading .76 .06 .71 .77 1.00 6. Speed-L ¡.49 .09 ¡.50 ¡.55 ¡.51 1.00 7. Speed-N ¡.54 ¡.01 ¡.56 ¡.61 ¡.59 .86 1.00 8. Fluency-C .24 .16 .35 .38 .28 ¡.25 ¡.30 1.00 9. Fluency-L .21 ¡.02 .33 .29 .25 ¡.28 ¡.24 .27 1.00 10. Gener-L .41 .17 .52 .48 .43 ¡.43 ¡.48 .27 .23 1.00 11. Gener-N .55 .02 .58 .62 .54 ¡.50 ¡.58 .22 .29 .49 1.00 12. Digit-F .10 .00 .13 .09 .11 ¡.08 ¡.03 .09 .04 .13 .10 1.00 13. Phon-Span .20 ¡.02 .15 .15 .19 ¡.16 ¡.15 .13 .07 .22 .23 .32 1.00 14. Word-Span .08 .05 .21 .20 .16 ¡.13 ¡.16 .18 .10 .26 .14 .30 .31 1.00 15. Update .22 ¡.07 .32 .34 .27 ¡.18 ¡.16 .19 .14 .40 .14 .25 .29 .31 1.00 16. Digit-B .10 ¡.09 .25 .23 .22 ¡.11 ¡.11 .22 .13 .01 .02 .10 .10 .17 .23 1.00 17. Lis/Sent. .24 ¡.03 .41 .41 .33 ¡.26 ¡.25 .36 .24 .33 .26 .12 .20 .28 .18 .21 1.00 18. Digit/Sent. .32 ¡.01 .38 .37 .41 ¡.21 ¡.27 .23 .25 .32 .25 .10 .20 .14 .18 ¡.11 .33 1.00 19. Vis-Matrix .20 ¡.07 .21 .25 .22 ¡.19 ¡.20 .16 .05 ¡.01 .23 .02 .04 ¡.08 .07 .05 .15 .18 1.00 20. Mapping .36 .07 .35 .34 .34 ¡.22 ¡.23 .21 .16 .21 .20 .05 .10 .06 .06 .09 .27 .33 .22 1.00 Note. WRAT-M D WRAT-III math subtest, WIAT-M D WIAT math subtest, Word Prob D WISC-III word problems, Word-Sem D word problems semantic variations, Reading D WRAT-III reading subtest, Speed-L D letter naming speed, Speed-N D number naming speed, Fluency-C D categorical Xuency task, Fluency-L D letter Xuency task, Gener-L D random letter generation, Gener-N D random number generation, Digit-F D WISC-III digit forward, Phon Span D psuedoword span task, Word Span D real word span task, Digit-B D WISC-III digit backward, List/Sentence D listening sentence span, Dig/Sentence D digit sentence span, Mapping D Mapping task.

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then WM measures would predict math performance after various measures of the phonological system had been partialed out of the analysis. We examined this hypothesis through a series of regression analyses in which component processes of WM (e.g., phonological loop, executive processing) were the independent variables and mathematical and word problem-solving accuracy were the criterion measures. Prior to our regression analysis, however, the intercorrelations between WM and math measures were examined. The intercorrelations among all variables used in the analysis are reported in Table 2.  was set to .001 (rs > .31) due to the number of comparisons. Although correlations among the memory tasks shown in Table 2 were signiWcant, two Wndings are worth noting. First, a large number of weak to moderate correlations emerged for measures that we assumed were tapping the executive system. This Wnding is consistent with Miyake, Friedman, Rettinger, Shah, and Hegarty’s (2001) analysis showing that correlations among WM tasks are frequently lower than other within-construct correlations “partly because these complex tasks often involve a good deal of variance related to non-executive processes as well as measurement error” (p. 630). Fortunately, latent variable analysis, which was used in the subsequent analysis, is particularly useful in these circumstances because the analysis extracts the common and perfectly reliable variance between the tasks chosen to tap WM. Number naming speed was highly intercorrelated with letter naming speed, suggesting that proWciency in processing numbers is also related to processing letter information. Component structure of WM Prior to the regression analysis, we tested whether the component structure of the WM structure was consistent with Baddeley’s multicomponent model. We assumed that structure of the memory measures reXected three factors: phonological loop (STM), executive processing, and visual–spatial sketchpad. To address this issue, we ran a conWrmatory factor analysis using the CALIS (Covariance Analysis and Linear Structural Equation) software program (SAS Institute, 1992) with measures of the phonological loop (STM) loading onto one factor (Forward Digit Span, Pseudoword Span, and Word Span), measures of the central executive (Backward Digit Span, Listening/Sentence Span, Digit/Sentence Span, and Updating) loading onto one factor, and visual–spatial measures (Visual Matrix and Mapping) loading onto a third factor. The three-factor solution conWrmed that WM can be separated into three components for children of this age. The Wt statistics were .91 for the comparative Wt index (CFI) (Bentler & Wu, 1995) and .074 for the root mean square residual error of approximation (RMSEA) (Joreskog & Sorbom, 1984). Parameter estimates for the three-factor model are shown in Table 3. All standardized parameters were signiWcant at the .01  level. The unique variance related to each factor and its contribution to math calculation are assessed in the next analysis. Correlations among latent measures For the subsequent analysis, measures (manifest variables) were aggregated into latent measures. The CALIS program (SAS Institute, 1992) was used to create latent measures. The latent measure of mathematical computation (WRAT-III math and WIAT math raw scores) served as the criterion measure. In addition to the memory factors, three additional latent measures served as predictor variables: speed (rapid naming of letters and numbers),

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Table 3 Standardized parameter estimates and test statistics (t values) for conWrmatory factor analysis Path

t value

Phonological loop ! measure STM ! Pseudoword Span STM ! Word Span

.59 .56

6.88¤¤¤ 6.55¤¤¤

Executive system ! measure WM ! Listening/Sentence Span WM ! Digit/Sentence Span WM ! Backward Digit Span STM ! Updating

.59 .52 .35 .46

8.87¤¤¤ 6.14¤¤¤ 4.17¤¤¤ 5.43¤¤¤

Visual sketchpad ! measure WM ! Visual Matrix WM ! Mapping

.29 .50

2.92¤¤ 4.40¤¤¤

¤¤ ¤¤¤

p < .01. p < .001.

random generation (number and letter random generation), and Xuency (letters and categories). Table 4 shows the intercorrelations among the latent measures, reading from the WRAT-III (raw scores), and chronological age. Because of the number of comparisons,  was set to .001 (N D 127, rs >.31, p < .001). As shown in this table, all correlations were signiWcant, and close inspection reveals three important Wndings (to interpret the results, we considered rs > .50 as substantial correlations). First, age was signiWcantly correlated with all latent measures and reading. Second, reading performance was substantially correlated with math calculation, naming speed, random generation, Xuency, and all components of WM. Finally, components of WM were correlated with measures of mathematical computation. To examine this relation further, we partialed age out of the correlation analysis. Table 5 shows these partial correlations. There are three important Wndings when comparing Tables 4 and 5. First, in contrast to Table 4, Table 5 shows that only four partial correla-

Table 4 Intercorrelations among mathematics, reading, and cognitive processing variables

1. Age 2. Gender 3. Mathematical computation 4. Random Number/Letter Generation 5. Digit/Letter Naming Speed 6. Categorical/Letter Fluency 7. STM (phonological loop) 8. WM (executive system) 9. WM (visual–spatial sketchpad) 10. Reading

1

2

3

4

5

6

7

8

9

1.00 .01 .74 .63 ¡.49 .52 .44 .53 .62 .76

1.00 .01 .05 .08 .12 .02 ¡.02 .04 .06

1.00 .71 ¡.55 .66 .49 .68 .50 .75

1.00 ¡.57 .61 .52 .58 .43 .64

1.00 ¡.47 ¡.36 ¡.43 ¡.32 ¡.51

1.00 .55 .73 .51 .59

1.00 .66 .50 .51

1.00 .70 .66

1.00 .67

Note. Age: chronological age in months; phonological loop: Forward Digit Span, Word Span, and Pseudoword Span; executive system: Digit/Sentence Span, Backward Digit Span, Listening/Sentence Span, and Updating; visual–spatial sketchpad: Mapping and Visual Matrix; reading: word recognition raw scores. rs > .31. p < .001.

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Table 5 Intercorrelations among mathematics, reading, and cognitive processing variables with age partialed out

1. Gender 2. Mathematical computation 3. Random Number/Letter Generation 4. Digit/Letter Naming Speed 5. Categorical/Letter Fluency 6. STM (phonological loop) 7. WM (executive system) 8. WM (visual–spatial sketchpad) 9. Reading

1

2

3

4

5

6

7

8

1.00 ¡.01 .06 .10 .13 .02 ¡.04 .04 .09

1.00 .45 ¡.32 .48 .28 .50 .40 .44

1.0 ¡.39 .42 .36 .38 .33 .31

1.00 ¡.29 ¡.18 ¡.23 ¡.21 ¡.24

1.00 .42 .62 .48 .36

1.00 .50 .32 .30

1.00 .56 .46

1.00 .37

Note. Phonological loop: Forward Digit Span, Word Span, and Pseudoword Span; executive system: Digit/Sentence Span, Backward Digit Span, Listening/Sentence Span, and Updating; visual–spatial sketchpad: Mapping and Visual Matrix; reading: word recognition raw scores. rs > .31. p < .0001.

tion coeYcients were at or above a magnitude of .50. The executive component of WM was correlated with mathematical computation, Xuency, phonological loop and visual–spatial sketchpad. Second, Table 5 shows that calculation performance was more highly correlated with the executive system and visual–spatial sketchpad than with the phonological loop. Finally, Table 5 shows that the correlation between speed and the phonological loop was not signiWcant (p > .001). Thus, contrary to our expectations, naming speed was unrelated to performance on STM measures. Predictions of mathematical performance The next analysis determined those variables that may mediate the relation between WM and mathematical performance. We investigated whether the relation between math calculation and WM was maintained when blocks of variables related to age, speed, and inhibition were entered into the analysis simultaneously. The criterion and predictor variables were the same as those shown in Table 4. The criterion measure was the latent score for math calculation. Predictor variables were gender, age, reading, and latent measures related to components of WM, random generation, Xuency, and speed. Chronological age was calculated in months, and reading was calculated in raw scores. For our Wrst set of analyses (Table 6), we determined the amount of variance in mathematical performance that was accounted for by age and gender alone (Model 1). Gender was included because several studies of elementary school children have reported gender diVerences in mathematical performance (for a review of the literature, see Robinson et al., 1996). As shown in Table 6, Model 1 contributed approximately 55% of the variance to mathematical computation. For each subsequent model, variables were entered simultaneously such that the  values reXected unique variance (with the inXuence of all other variables partialed out). In Model 2, we determined the contribution of the latent scores related to the components of WM when entered into the model. Age was retained in the model, but gender was dropped because it contributed no unique variance in Model 1. As shown in Model 2, the executive processing component of WM contributed signiWcant variance to math calculation even when the inXuence of the phonological loop, visual–spatial sketchpad, and age were partialed out of the

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Table 6 Hierarchical analysis of math calculation and word problem solving Mathematical computation B

SE



Model 1 Gender F(2, 124) D 74.22¤¤¤, R2 D .55

¡.01

.08

¡.04

¡.14

Model 2 Phonological loop Executive system Visual–spatial sketchpad Age F(4, 122) D 62.12¤¤¤, R2 D .67

¡.01 .44 .20 .67

.09 .11 .11 .09

¡.01 .34 .14 .47

¡.18 4.00¤¤¤ 1.75 6.96¤¤¤

Model 3 Phonological loop Executive system Visual–spatial sketchpad Age Reading F(5, 121) D 54.50¤¤¤, R2 D .69

¡.03 .35 .15 .47 .37

.09 .11 .11 .11 .12

¡.02 .27 .10 .33 .26

¡.33 3.17¤¤¤ 1.35 4.44¤¤¤ 2.92¤¤¤

Model 4 Phonological loop Executive system Visual–spatial sketchpad Age Reading F(6, 120) D 48.82¤¤¤, R2 D .71

¡.04 .34 .13 .43 .32

.09 .10 .11 .11 .12

¡.03 .26 .09 .30 .23

¡.48 3.13¤¤¤ 1.18 3.84¤¤¤ 2.61¤¤

Model 5 Random Number/Letter Generation Categorical/Letter Fluency Phonological loop Executive system Visual–spatial sketchpad Age Reading F(7, 119) D 47.87¤¤¤, R2 D .73

.24 .18 ¡.10 .24 .07 .38 .30

.08 .10 .09 .11 .11 .11 .11

.20 .14 ¡.07 .18 .05 .27 .21

2.92¤¤ 1.78 ¡1.09 2.15¤ .64 3.45¤¤¤ 2.49¤¤

Model 6 Random Number/LetterGeneration Categorical/Letter Fluency Digit/Letter Naming Speed Phonological loop Executive system Visual–spatial sketchpad Age Reading F(8, 118) D 44.97¤¤¤, R2 D .73

.21 .17 ¡.12 ¡.10 .25 .06 .37 .28

.08 .10 .10 .09 .11 .11 .11 .12

.18 .12 ¡.07 ¡.07 .19 .04 .27 .20

2.48¤ 1.64 ¡1.23 ¡1.09 2.18¤ .61 3.35¤¤¤ 2.35¤

t

Note. Phonological loop: Forward Digit Span, Word Span, and Pseudoword Span; executive system: Digit/Sentence Span, Backward Digit Span, Listening/Sentence Span, and Updating; visual–spatial sketchpad: Mapping and Visual Matrix, age: chronological age in months; reading: word recognition raw scores. ¤ p < .05. ¤¤ p < .01. ¤¤¤ p < .001.

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analysis. As shown, neither the phonological loop nor the visual–spatial system contributed signiWcant variance. The predictor variables in Model 2 contributed approximately 67% of the variance to math calculation. When compared with Model 1, Model 2 signiWcantly improved (R2 D .12) the predictions for math calculation, Finc(3, 123) D 15.00, p < .001. In Model 3, we tested whether the executive component would continue to be a signiWcant predictor when reading scores were added to the model. As shown, Model 3 accounted for 69% of the variance in math calculation. The executive component of WM, chronological age, and reading contributed unique variance in predicting mathematical computation performance. When compared with Model 2, Model 3 improved (R2 D .02) the predictions of math calculation, Finc(1, 121) D 7.81, p < .001. In Model 4, we assessed the contribution of WM (executive system) when latent measures related to speed were added to Model 3. Although naming speed was found to be unrelated to the phonological loop in Table 5, previous studies have reported an association between articulation speed (naming speed in this case) and STM (for reviews, see Gathercole, 1998; Henry & Millar, 1993). When compared with Model 3, Model 4 accounted for a 2% increase in predictions of math calculation, Finc(1, 120) D 8.27, p < .05. However, naming speed, executive processing, age, and reading contributed signiWcant variance to math calculation. The important Wnding was that the relation between WM and math calculation was not directly mediated by the phonological loop. If this had been the case, then the relation between calculation and WM should have been eliminated when measures of the phonological system (e.g., STM, naming speed) were partialed out of the analysis. Such was not the case in this analysis. In Model 5, we assessed the contribution of WM to calculation when latent measures related to inhibition (random generation and Xuency) were added to Model 3. These measures were assessed due to their association with executive processing (e.g., Miyake et al., 2000). As shown, Model 5 accounted for 73% of the variance in math calculation. Only latent measures of random generation and the executive processing component of WM predicted mathematical computation performance. When compared with Model 3, Model 5 improved the prediction (R2 D .04) for calculation, Finc(2, 119) D 17.62, p < .001. Importantly, entering random generation and Xuency into the regression model did not remove the signiWcant inXuence of the executive component of WM in predicting mathematical computation. In particular, we argued that if individual diVerences in WM and mathematical computations were mediated by inhibition, then the relationship between calculation and WM should be eliminated when activities related to controlled attention (e.g., inhibition) were partialed out of the analysis. Such was not the case here. However, it could be argued that both random generation and Xuency reXect speed measures rather than inhibition of responses; therefore, speed was entered into Model 6. Model 6 captured 73% of the variance in math calculation. The important Wnding related to the Wnal model was that the signiWcant inXuence of inhibition as measured by random generation was maintained when naming speed was entered into the regression model. As was the case in all previous models, reading performance and chronological age also contributed signiWcant variance to calculation. In summary, these Wndings in the full model indicated that random generation, reading, and age contributed unique variance to math calculation. However, the results showed that the executive component of WM contributed unique variance to math calculation. These results suggested that processes related to inhibition are not the only processes that underlie the relation between the executive components of WM and mathematical computation.

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Discussion The purpose of this study was to determine the mechanisms that underlie young children’s precociousness in mathematics. Before discussing the results related to the WM model, however, we brieXy summarize ability group diVerences across achievement and cognitive measures. Of particular interest in our study was isolating those components of WM most related to math ability. Performance of average-achieving children was below that of precocious children on aggregate measures related to naming speed, random generation, and Xuency as well as on measures of WM related to the executive system. No signiWcant diVerences emerged between the groups on measures related to the phonological loop or the visual–spatial sketchpad. Thus, the component of WM that played a major role in predicting precociousness was the central executive. Of particular interest, however, was identifying those cognitive processes that, when partialed out of the analysis, would mitigate the relation between individual diVerences in WM and mathematical computation. The results related to the regression showed a signiWcant relation between the executive component of WM and math calculation when the inXuence of inhibition and naming speed was partialed out of the analysis. The implications of these Wndings are discussed below. The important Wndings, however, relate to testing two models on the inXuence of WM on math performance. The Wrst model tests whether processes related to the phonological loop (e.g., STM) played a major role in predicting math performance in young children. The model follows logically from the arithmetic literature that links phonological skills and mental calculation (e.g., Logie et al., 1994). The model assumes that low-order processing, such as phonological coding, provides a more parsimonious explanation for ability group diVerences in math performance than do measures related to the executive system. The model suggests that precocious children have an advantage in the processing of phonological information (e.g., digits), thereby increasing the storage of information for higher levels of processing. The second model suggests that mathematical performance relates to executive processing independent of the inXuence of the phonological system. This assumption follows logically from the problem-solving literature suggesting that abstract thinking, such as comprehension and reasoning, requires the coordination of several basic processes (e.g., Engle et al., 1999; Just, Carpenter, & Keller, 1996; Kyllonen & Christal, 1990). Measures of executive processing in this study were related to the residual variance of WM (latent measures with the inXuence of STM partialed out) and measures assumed to reXect activities on the executive system such as inhibition (random generation of numbers). The Wndings for these two models are as follows. First, the WM executive system contributes unique variance to mathematic performance beyond that contributed by the phonological loop, the visual–spatial sketchpad, naming speed, random generation, and Xuency. The results show that WM contributes approximately 12% of the variance to math calculation after age has been partialed out of the analysis (Model 2). Furthermore, although age-related and individual diVerences emerge on measures of naming speed, entering naming speed into the regression analysis did not eliminate the signiWcant inXuence of WM on mathematical performance. There is clear evidence that the executive system of WM does contribute important variance to mathematical performance beyond processes that relate to the phonological system. Thus, the results do not support a conservative form of the Wrst model. There is weak support for the assumption that the phonological loop mediates individual diVerences in WM performance and its inXuence on

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math. A reWnement of this model might consider the phonological loop as important to mathematical accuracy but no more important than other processes. Second, the executive system of WM captures unique variance in predicting calculation skill beyond measures of inhibition. We assumed in our regression modeling that the residual variance related to WM measures (with the inXuence of the phonological loop and visual–spatial sketchpad partialed out) tapped a function of executive processing referred to as controlled attention (e.g., Engle et al., 1999). We investigated whether this residual variance was mediated (i.e., signiWcant inXuence of WM on math measures eliminated) when measures of inhibition (e.g., random generation, Xuency) were entered into the regression analysis. The results suggested that the executive system of WM were not directly mediated by measures of inhibition. Overall, these Wndings give partial support to Model 2—“partial” because the role of the executive system (i.e., controlled attention) has not been clearly delineated. However, the study provides information on two hypotheses discussed in the literature as playing a major role in accounting for individual diVerences in math ability and WM. One hypothesis relates to the speed of processing. A simple version of this hypothesis states that individuals precocious in math are faster at processing language information than are average-achieving children and that this processing advantage underlies precocious students’ advantage in WM performance. Several models of WM assume that operations related to language are time-consuming (e.g., Salthouse, 1996). Therefore, speed of processing might underlie the general pattern of WM advantages noted in the current study. Furthermore, Kail (1993) argued that a common pool of cognitive resources related to processing speed is used to perform a variety of tasks, with the pool increasing across age in children. Clearly, our Wndings (Table 5) show a signiWcant relation between processing speed and latent measures of mathematical computation, inhibition, Xuency, and components of WM (rs range from ¡.36 to ¡.43 [Table 4]). In addition, math-precocious children have faster naming scores than do average achievers, even when reading and Xuid intelligence are partialed out of the analysis. Although speed does play an important role in the earlier regression models (Model 4), it did not contribute unique variance in the complete model. However, it is important to note that we focused on naming speed. Other measures of processing speed (reaction time and decision-making speed) may yield diVerent results. A second hypothesis considers whether precocious children are better at inhibition than are average-achieving children (for further discussion of this model, see Rosen & Engle, 1997; Towse, 1998). Such a hypothesis assumes that an inhibition deWcit limits some participants’ ability to prevent irrelevant information from entering WM during the processing of targeted information (for a discussion of this model, see Passolunghi & Siegel, 2001). An activity related to the central executive that has been implicated in children less proWcient in math is their ability to suppress irrelevant information under high processing demand conditions (e.g., De Beni, Palladino, Pazzaglia, & Cornoldi, 1998). Earlier studies showed that lower performing children in math vary from controls in their ability to recall targeted (relevant) and nontargeted (incidental) information (e.g.,Passolunghi & Siegel, 2001). Furthermore, there is evidence of interference eVects with children who have learning disabilities in math calculation (Barrouillet, Fayol, & Lathulière, 1997). In contrast, individuals with higher math ability have less diYculty in preventing unnecessary information from entering WM and, therefore, are less likely than average achievers to consider alternative interpretations of material that are not central to the task. This interpretation Wts within

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several recent models that explain individual diVerences in memory performance as related to inhibitory mechanisms (e.g., Conway & Engle, 1994). Our results show that precocious children experience a clear advantage on the random generation and Xuency tasks when compared with average achievers. Our Wndings are consistent with the hypothesis that average-achieving children may use more WM capacity than do mathematically precocious children to inhibit or resist potential interference from irrelevant items. However, partialing out performance on the inhibition measure in the regression analysis did not eliminate the signiWcant relation between the executive component of WM and math calculation. Although the WM tasks we administered appeared to have put demands on participants’ ability to suppress competing information, individual diVerences in inhibition were not the only processes that mediate WM performance. Implications There are two implications of our Wndings for current literature. First, the executive system, and not the phonological loop, is the primary component of WM that predicts math precociousness in the early grades. Our Wndings indicate that although math skills are associated with the phonological loop (i.e., naming speed and STM), they are no more important than the executive system. Such a Wnding qualiWes bottom-up models of problem solving in children by suggesting that if low-order processes such as phonological processes (e.g., STM) mediate the inXuence of executive processing (WM) on mathematical and problem-solving performance, then their eVects may be indirect or minimal for children who have perhaps met a minimum threshold in mathematics. Of course, these results apply only to the age and ability groups represented in this sample. Furthermore, we have not adequately assessed the role of visual–spatial processes in this study; therefore visual–spatial WM may be relevant to other types of mathematic performance (Zorzi, Priftis, & Umiltá, 2002). In addition, we have not investigated various components of number sense (for reviews, see Berch, 2005; Dehaene, 1997) such as mental representations for number comparison and approximate calculation. The second implication is that we Wnd that processes related to Baddeley’s three components can be separated out and their unique contribution to mathematical performance can be analyzed. This is an important Wnding because WM and STM tasks have been found to load onto the same factor in young children. For example, Hutton and Towse (2001) found, via a principal component analysis, that both WM and STM tasks loaded onto the same factor for 8- to 11-year-olds. Their results also showed that correlations related to WM and STM on measures of number skills (.33 vs. 38) were of the same magnitude, suggesting that WM and STM share the same construct (for a similar Wnding, see also Cowan et al., 2003). As stated by Hutton and Towse (2001), “It appears that what holds for WM in adults may not be equally true for children, and vice versa. The study highlights the value of taking account of children’s online processing during WM tasks, and in so doing suggests that WM and STM may, at least in some circumstances, be rather equivalent” (p. 392). These Wndings, however, contrast with those of Gathercole, Pickering, Ambridge, and Wearing (2004), who investigated the structure of WM in 4- to 15-year-olds. Their results supported a basic modular structure of WM that was consistent with a tripartite adultbased model that includes phonological STM, executive processing, and the visual–spatial sketchpad. However, it was unclear from their Wndings whether the processes that contributed to these structures were distinct. For example, the average factor correlation between

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measures of STM (phonological loop) and executive processing was .80 (p. 186). These authors explained the close association between the factors associated with the phonological STM and the executive component of WM as follows: “The central executive’s identiWcation was based on tasks that are constrained by phonological loop capacity” (p. 188). Given these Wndings, our Wnding with young children was important. We found that a three-factor model Wts the data fairly well. We also found that the correlations among the three components of WM were signiWcant (rs ranged from .50 to .71 [Table 5]). However, we found that only the executive component contributed unique variance to math calculation and problem solving and that this component operated independent of any inXuence of tasks related to the phonological loop or visual–spatial sketchpad.

Conclusion Our Wndings converge with those of studies Wnding that individual diVerences in the WM executive system play a critical role in math precociousness. We argue that this system plays a major role because (a) it holds recently processed information to make connections with the latest input and (b) it maintains the gist of information for the construction of an overall representation of the problem. Yet WM is not the exclusive contributor to variance in math precociousness. The study also supports previous research regarding the importance of processing speed and inhibition in mathematical reasoning in children. However, we believe that one of the core advantages that precocious children have in mathematics relates to operations ascribed to a central executive.

Acknowledgments This study is part of a longitudinal project funded by the U.S Department of Education, Cognition, and Student Learning (USDE R305H020055), U.S. Department of Education, Institute of Education Sciences. The author is indebted to Georgia Doukas, Diana Dowds, and Rebecca Gregg for data collection and to Margaret Beebe-Frankenberger for serving as project manager. Special appreciation is given to the Colton School District and Tri City Christian Schools. The author also appreciates the incisive comments of three anonymous reviewers on an earlier version of this manuscript. This article does not necessarily reXect the views of the U.S. Department of Education or the school districts.

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