Exploring the impact of phonological awareness, visual–spatial working memory, and preschool quantity–number competencies on mathematics achievement in elementary school: Findings from a 3-year longitudinal study

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Abstract

This longitudinal study explored the importance of kindergarten measures of phonological awareness, working memory, and quantity–number competencies (QNC) for predicting mathematical school achievement in third graders (mean age 8 years 8 months). It was found that the impact of phonological awareness and visual–spatial working memory, assessed at 5 years of age, was mediated by early QNC, which predicted math achievement in third grade. Importantly, and confirming our isolated number words hypothesis, phonological awareness had no impact on higher numerical competencies (i.e., when number words needed to be linked with quantities [QNC Level II and above]) but predicted basic numerical competencies (i.e., when number words were isolated from quantities [QNC Level I]), explaining the moderate relationship between early literacy development and the development of mathematical competencies.

Introduction

During the past three decades, numerous longitudinal studies on literacy development were carried out in various countries, and all of them demonstrated the predictive power of phonological information processing skills. In particular, individual differences in early phonological awareness—that is, the ability to analyze the sound structure of oral language—predicted differences in reading and spelling skills in elementary school (e.g., Bradley and Bryant, 1985, Schneider and Näslund, 1999, Wagner and Torgesen, 1987). Moreover, several studies focusing on the early training of phonological awareness were able to show that children trained during the last year of kindergarten outperformed untrained control children in reading and spelling at school and that training phonological awareness in kindergarten was also successful for children at risk for dyslexia, that is, children with very low levels of phonological awareness (e.g., Bradley and Bryant, 1985, Lundberg et al., 1988, Schneider et al., 1997, Schneider et al., 2000). Undoubtedly, phonological awareness plays an important role in literacy development regardless of the type of orthography under study (i.e., shallow vs. deep).

Although it is well known that relevant precursors of literacy development are already acquired during the preschool and kindergarten years, precursors of mathematical competencies such as quantity–number competencies (QNC) are not similarly well investigated. According to the available studies, it seems that linking imprecise nonverbal quantity concepts with the ability to count (which develops separately) forms the basis for understanding several major principles of the number system (cf. Gersten et al., 2005, Krajewski and Schneider, in press, Lemer et al., 2003, Okamoto and Case, 1996, Resnick, 1989). A recently proposed theoretical model (Krajewski, 2008; see also Krajewski & Schneider, in press) depicts how this linkage might occur and how early mathematical competencies are acquired via three developmental levels aiming at a deeper understanding of quantity to number word linkages (see Fig. 1). Given that the current article explores the impact of phonological awareness and working memory on the different developmental levels of math competency, relevant features of the theoretical model are described below.

Children are born with the capacity to discriminate quantities. As some researchers assume, this implies that infants can already differentiate between discrete quantities (see Antell and Keating, 1983, Bijeljac-Babic et al., 1993, Huntley-Fenner and Cannon, 2000, Starkey and Cooper, 1980, Wynn, 1992; for large numerosities, see Xu, Spelke, & Goddard, 2005). Others believe that infants differentiate between the spatial extent of quantities but not between discrete amounts (see Clearfield and Mix, 1999, Clearfield and Mix, 2001, Feigenson et al., 2002, Mix et al., 1996, Mix et al., 2002, Rousselle et al., 2004, Simon et al., 1995; for small numerosities, see Xu et al., 2005). As indicated by the first level of the theoretical model, we assume that children are at least able to differentiate between indiscrete amounts (quantity discrimination). With the acquisition of language, children also acquire the ability to discriminate quantities verbally. When comparing quantities, they use words such as more, less, and the same amount (see Resnick, 1989, protoquantitative comparison schema). Independently, they learn to count (recite number words) beginning at around 2 years of age and thereby already acquire precise number words and the exact number word sequence. However, they do not yet employ these number words to describe quantities (number words still isolated from quantities). As a consequence, verbal abilities such as phonological awareness should be more important for the acquisition of the number word sequence than visual–spatial abilities.

Somewhat later, young children become aware that number words are linked to quantities. At this point, they become able to understand the meaning of number words, enabling children to arrange numbers according to their size (see Gersten et al., 2005, Okamoto and Case, 1996). The understanding of the linkage between quantities and number words is typically acquired in two phases. First, children develop an imprecise vague conception of the attribution of number words to quantities and assign number words to rough quantity categories (i.e., imprecise quantity to number word linkage [Level IIa]). Based on this understanding, children at around 3 years of age can already distinguish between number words that are assigned to different rough quantity categories (e.g., two = a bit, twenty = much, hundred = very much) because these are discrete from each other. However, during this developmental phase, they are not yet able to distinguish between numbers that are assigned to the same quantity category because number words of the same quantity category represent the same amount (e.g., 20 = 22 = much). The ability to distinguish between adjacent numbers develops gradually when number words are also linked to exact quantities (i.e., precise quantity to number word linkage [Level IIb]), meaning when quantities (exactly enumerated) are precisely arranged to the exact number word sequence (for linking counting with quantity discrimination, see Gersten et al., 2005; Okamoto & Case, 1996). Only at this time are children able to judge which of two adjoining number words (e.g., 15 and 16) contains fewer or more. As Ansari and colleagues (2003) could show with normally developed 3-year-olds, individual differences in visual–spatial working memory become especially important when children must grasp the cardinality understanding at this developmental level.

Independent of the differentiation of the quantity to number word linkage, children gather experiences with relations between quantities (without reference to number words). They realize that a quantity can be divided into pieces that, taken together, equal the original quantity (see Resnick, 1989, protoquantitative part–whole schema from approximately 4 to 5 years of age) and that quantities change only if something is added or taken away (see Resnick, 1989, protoquantitative increase/decrease schema).

At the highest level of QNC, children understand that part–whole relations between quantities can also be represented with precise number words (decomposition of numbers). In addition, they discover that the difference between two numerical quantities yields a third numerical quantity (differences between numbers). At this point, thus, children arrive at the important insight that the difference between two numbers is another number. To understand (non)numerical relations between quantities (Levels II and III), preschoolers’ visual–spatial abilities should take on special importance because the nonverbal representation of quantities seems important to solve such tasks (cf. Rasmussen & Bisanz, 2005).

There is empirical evidence demonstrating that the described early QNC predict later mathematical school achievement (Aunola et al., 2004, Jordan et al., 2007, Koponen et al., 2007, Krajewski and Schneider, in press, Passolunghi et al., 2007, Stern, 1997, Stern, 1999, von Aster et al., 2007, Weißhaupt et al., 2006; for the special importance of Levels I and II, see also Gersten et al., 2005, and Okamoto & Case, 1996). Moreover, in studies focusing on children with mathematical learning disabilities, school children showed deficits in these quantity–number competencies (Gaupp et al., 2004, Geary et al., 2000, Geary et al., 2004, Landerl et al., 2004). Confirming the special importance of the quantity to number word linkage, for instance, the 7-year-olds with Williams syndrome in the study by Ansari and colleagues (2003) demonstrated an extremely delay in their understanding of the cardinality principle (Level II) despite the fact that they could recite the number word sequence (Level I) in the assessed number range almost without errors.

In several longitudinal studies on reading and math development, domain-general predictor variables were included in addition to the specific ones. For instance, indicators of working memory have been identified as potentially relevant unspecific predictors of both literacy and mathematics development in school (see Berg, 2008, Bull et al., 2008, Geary et al., 2007, Geary et al., 2000, Hecht et al., 2001, Landerl et al., 2004, Swan and Goswami, 1997, Wolf, 1984). Phonological aspects of working memory, particularly phonological recoding and phonological rehearsal processes as conceptualized in Baddeley’s (1986) working memory model, have been shown to affect subsequent literacy development during elementary school (see Daneman, 1987, Schneider and Näslund, 1999, Schuchardt et al., 2006, Swanson and Howell, 2001, Wagner and Torgesen, 1987). Findings of a recent study by Alloway and colleagues (2005) indicate that the central executive component of Baddeley’s model may also affect literacy development, particularly in the case of children at risk (see also Schuchardt et al., 2006).

Similar relationships with early individual differences in the phonological component of working memory have also been reported for math achievement (Berg, 2008, Hecht et al., 2001, Rasmussen and Bisanz, 2005; but see Lee et al., 2004, Swanson, 2006). Here effects of the central executive seem to be particularly strong (Bull et al., 2008, Geary et al., 1991, Geary and Hoard, 2001, Gersten et al., 2005, Leather and Henry, 1994, Lee et al., 2004, Lemaire et al., 1996, Logie et al., 1994; Passolunghi & Siegel, 2004; Passolunghi et al., 2007, Rammelaere et al., 2001, Swanson, 2006, Swanson and Beebe-Frankenberger, 2004, Thomas et al., 2006). In a recent study by Geary and colleagues (2007), it was shown that all components of working memory, including the visual–spatial sketchpad, are related to math performance in school. The impact of the visual–spatial sketchpad on math achievement was also confirmed in recent cross-sectional and longitudinal studies (Berg, 2008, Bull et al., 2008, Simmons et al., 2008; for inconsistent results, see Leather and Henry, 1994, Lee et al., 2004, Seitz and Schumann-Hengsteler, 2002, Swanson, 2006).

Several studies have found substantial intercorrelations between literacy and math competencies in school, ranging between r = .40 and r = .60 (e.g., Berg, 2008, Koponen et al., 2007, Lee et al., 2004, Schneider, 2009). This indicates that similar cognitive competencies influence performance development in these two areas of school achievement even though the two domains seem quite different from a substantive point of view. Meanwhile, several studies have addressed the issue that deficits in relevant precursor variables are related to problems in both literacy and math development (e.g., Geary, 1993). As a major finding, these studies not only highlighted the impact of working memory skills but also emphasized the role of phonological awareness for subsequent mathematical achievement. In a classic study, Bradley and Bryant (1985) reported that phonological awareness (sound categorization task) assessed in kindergarten not only correlated with reading and spelling (r = .50) but also correlated with math achievement assessed 3 years later (r = .33). Similarly, Alloway and colleagues (2005) found a substantial relationship (r = .49) between phonological awareness (rhyme detection and initial consonant detection) assessed at 4 and 5 years of age and mathematical competencies as judged by teachers at the beginning of first grade.

More recently, Koponen and colleagues (2007) reported a synchronic correlation of .58 between phonological awareness (initial sound matching and initial sound naming) and counting abilities in their longitudinal study and a diachronic correlation of .34 with calculation in Grade 4. Also, findings by Hecht and colleagues (2001) suggest substantial interrelationships among indicators of phonological awareness (phoneme elision, phoneme segmentation, sound categorization, and blending phonemes) and math performance in Grades 2 to 5, with correlations ranging between r = .47 and r = .56. In the cross-sectional study by Leather and Henry (1994), 31% of the variance in arithmetical achievement at 7 years of age was predicted by the composite phonological awareness score (initial and final consonant task, blending task, and phoneme tapping). Moreover, a longitudinal study by Simmons and colleagues (2008) showed that phonological awareness scores (rhyme task) assessed at 5 years of age predicted reading as well as arithmetic 1 year later.

Although there is a lot of support for the relationship between phonological awareness and math achievement, a few studies have come up with inconsistent findings. For instance, Durand, Hulme, Larkin, and Snowling (2005) could not confirm this relationship. In their study, phoneme deletion was a unique predictor of individual differences in reading but did not predict subsequent arithmetic skills. Also, Fuchs and colleagues (2005) reported that phonological processing (measured by rapid digit naming, first sound matching, and last sound matching) was a unique determinant of fact fluency but did not predict other aspects of math performance (e.g., story problems). In a subsequent study, Fuchs and colleagues (2006) reported similar results when phonological processing was measured by phonological decoding of pseudowords.

As Simmons and Singleton (2008) stated in their weak phonological representations hypothesis, phonological processing deficits will impair aspects of mathematics that involve the manipulation of verbal codes but will not impair areas of mathematics that are not verbally coded. When transferred to the assumption of different levels in Krajewski’s (2008) model, a plausible explanation for the correlation between phonological awareness and mathematical school achievement could be seen in an indirect influence of phonological awareness on mathematical school achievement via learning number words and the number word sequence (basic numerical skills [Level I]). Because phonological awareness reflects the ability to differentiate between meaningful segments of the language and to manipulate them, it should facilitate the differentiation and manipulation of single words in the number word sequence. Accordingly, the number word sequence should not be used as a string level anymore (“onetwothreefourfive” [see Fuson, 1988]); rather, it should be used as separated number words in a fixed order (“one two three four five” [see Fuson, 1988, unbreakable list]), an order that can even start without the first elements (“three four five” [see Fuson, 1988, breakable list]). One major goal of the current study was to explore this assumption in more detail.

Based on Krajewski’s (2008) model, we tried to understand the development of mathematical skill and its covariation with other skills pointing out a multicomponential approach, as claimed by Koponen and colleagues (2007). Two research questions were examined in this study. A first question concerned the role of working memory, especially the role of the visual–spatial working memory component, for the development of early QNC and mathematical school achievement. We assumed that visual–spatial working memory should, above all, predict higher level quantity–number competencies (QNC Levels II and III) as well as mathematical school achievement (see Bull et al., 2008, Simmons et al., 2008). According to the developmental model, it is only here that the ability to represent (and interpret) the quantitative information of numbers becomes important. Basic numerical skills (Level I), on the other side, should not be substantially predicted by visual–spatial working memory because they do not require the linking of numbers with quantity.

A second major question concerned the impact of early phonological awareness on later mathematical school achievement. More specifically, the question was whether the link is a direct one or only an indirect one. Our isolated number words hypothesis assumes that phonological awareness should influence the development of basic numerical skills (i.e., number words isolated from quantities [QNC Level I]). However, after controlling for its impact on these basic numerical skills, phonological awareness should not directly predict higher level competencies (i.e., when number words are linked with quantities [QNC Levels II and III]) and mathematical school achievement. Supporting evidence was found in the longitudinal study by Passolunghi and colleagues (2007), where only counting skills, but not phonological ability (assessed by first-sound repetition, last-sound repetition, and phonemic segmentation) predicted later mathematical achievement.

Section snippets

Sample and design of the study

The study began in September 2004 (T1) with 108 German preschoolers (55 girls and 53 boys) who attended their last year in kindergarten and lasted until the beginning of Grade 3 (T4). Informed consent was obtained from the parents. Sample size dropped to 91 by the last measurement point due to the fact that some children were not enrolled in school, moved away, or dropped out because of illness. Thus, the dropout rate was approximately 16%. At the beginning of the study, children’s average age

Results

Descriptive statistics are listed in the lower section of Table 1. Reliability scores of the tasks ranged between .67 and .94, indicating sufficient to good internal consistency. As can be seen from the Kolmogorov–Smirnov Z tests, the assumption of normal distribution was accepted for all tasks except digit span forward (Z = 1.8, p < .01), digit span backward (Z = 2.8, p < .01), and Arabic numbers (Z = 2.2, p < .01) and the spelling test (Z = 1.8, p < .01).

Discussion

The major aim of the current study was to investigate the respective roles of visual–spatial working memory (visual–spatial sketchpad) and phonological awareness for school achievement in mathematics. In particular, we were interested in the issue of whether influences of the visual–spatial sketchpad and phonological awareness on mathematical school achievement should be conceptualized as indirect (i.e., mediated by different levels of preschool QNC in Krajewski’s developmental model) or

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