What basic number processing measures in kindergarten explain ...

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Journal of Experimental Child Psychology 117 (2014) 12–28

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Journal of Experimental Child Psychology journal homepage: www.elsevier.com/locate/jecp

What basic number processing measures in kindergarten explain unique variability in ﬁrst-grade arithmetic proﬁciency? Dimona Bartelet a,⇑, Anniek Vaessen b, Leo Blomert b,1, Daniel Ansari c a

Top Institute for Evidence Based Education Research, Maastricht University, 6200 MD Maastricht, The Netherlands Faculty of Psychology and Neuroscience, Maastricht University, 6200 MD Maastricht, The Netherlands c Department of Psychology, University of Western Ontario, London, Ontario N6A 5C2, Canada b

a r t i c l e

i n f o

Article history: Received 13 February 2013 Revised 19 August 2013 Available online 12 October 2013 Keywords: Non-symbolic number processing skills Symbolic number processing skills Arithmetic proﬁciency Elementary school Task-speciﬁc effects Unique predictors

a b s t r a c t Relations between children’s mathematics achievement and their basic number processing skills have been reported in both crosssectional and longitudinal studies. Yet, some key questions are currently unresolved, including which kindergarten skills uniquely predict children’s arithmetic ﬂuency during the ﬁrst year of formal schooling and the degree to which predictors are contingent on children’s level of arithmetic proﬁciency. The current study assessed kindergarteners’ non-symbolic and symbolic number processing efﬁciency. In addition, the contribution of children’s underlying magnitude representations to differences in arithmetic achievement was assessed. Subsequently, in January of Grade 1, their arithmetic proﬁciency was assessed. Hierarchical regression analysis revealed that children’s efﬁciency to compare digits, count, and estimate numerosities uniquely predicted arithmetic differences above and beyond the non-numerical factors included. Moreover, quantile regression analysis indicated that symbolic number processing efﬁciency was consistently a signiﬁcant predictor of arithmetic achievement scores regardless of children’s level of arithmetic proﬁciency, whereas their non-symbolic number processing efﬁciency was not. Finally, none of the task-speciﬁc effects indexing children’s representational precision was signiﬁcantly associated with arithmetic ﬂuency. The implications of the results are 2-fold. First, the ﬁndings indicate that children’s efﬁciency to process symbols is important for the development of their arithmetic ﬂuency in Grade 1 above and beyond the inﬂuence of

⇑ Corresponding author. 1

E-mail address: [email protected] (D. Bartelet). In remembrance of coauthor Leo Blomert, who passed away after illness just before the ﬁnal revision of this article.

0022-0965/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jecp.2013.08.010

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non-numerical factors. Second, the impact of children’s non-symbolic number processing skills does not depend on their arithmetic achievement level given that they are selected from a nonclinical population. Ó 2013 Elsevier Inc. All rights reserved.

Introduction A key aim of primary education is teaching children arithmetic skills. Yet, some children in primary school experience severe difﬁculties in automatizing arithmetic facts (Swanson & Jerman, 2006). According to developmental theory, children’s mathematics capacity emerges from a cumulative process (Entwisle, Alexander, & Olson, 2005). Therefore, to understand the origins of arithmetic performance differences, it is necessary to acquire insights into which basic number processing skills in kindergarten uniquely predict individual differences in arithmetic skills at the onset of formal education. Empirical studies have increasingly addressed the issue to which extent basic number processing skills in kindergarten explain early mathematics achievement, but most were restricted to one time point in development (e.g., Holloway & Ansari, 2009; Inglis, Attridge, Batchelor, & Gilmore, 2011; Muldoon, Towse, Simms, Perra, & Menzies, 2013). Although Duncan and colleagues (2007) reported that knowledge of mathematics concepts in kindergarten was a more important predictor of later achievement in elementary and middle school than reading and attention skills, only a few studies have examined longitudinal data (Aunio & Niemivirta, 2010; Desoete, Ceulemans, de Weerdt, & Pieters, 2010; Kolkman, Kroesbergen, & Leseman, 2013; Libertus, Feigenson, & Halberda, 2013; Mazzocco, Feigenson, & Halberda, 2011b; Stock, Desoete, & Roeyers, 2009b, 2010). Furthermore, many studies have not assessed non-symbolic and symbolic number processing skills simultaneously to isolate unique predictors of arithmetic performance. However, this is essential to elucidate whether children’s (perhaps innate) ability to process non-symbolic magnitudes or their capacity to process culturally invented numerical symbols is a more important predictor of early arithmetic achievement. Therefore, in the current study, we examined one non-symbolic and three different symbolic number processing skills in kindergarten to determine which skills predict unique variance in ﬁrst-grade arithmetic performance. Another theoretical issue is whether individual differences in kindergarteners’ magnitude representations, children’s efﬁciency to process number information, or both account for signiﬁcant individual variation in arithmetic performance in Grade 1. To answer this question, studies need to incorporate different outcome measures. Task-speciﬁc effect measures, such as distance and ratio effects, are postulated to tap the representations underlying children’s performance on the number processing tasks (Maloney, Risko, Preston, Ansari, & Fugelsang, 2010). Other cognitive processes driving children’s attainment are captured by efﬁciency measures such as mean accuracy and mean reaction time (RT). The former quantiﬁes children’s ability to process magnitude information, and the latter quantiﬁes their speed of processing magnitude information. All of the measures have been related to mathematics achievement, but no measure was consistently found to be a signiﬁcant predictor of mathematics performance (see review by De Smedt, Noël, Gilmore, & Ansari, 2013). Hence, it remains unclear which measure is most sensitive to the prediction of early arithmetic achievement differences. In the current study, a third efﬁciency measure was introduced, namely a combined accuracy–speed outcome (acc–RT). It has long been established that individuals’ response behavior might be characterized by a decrease in response speed to ensure a correct answer or vice versa (Schouten & Bekker, 1967), but very few studies have included an outcome measure that takes a potential speed–accuracy trade-off into account (Sasanguie, De Smedt, Defever, & Reynvoet, 2012; Sasanguie, Van den Bussche, & Reynvoet, 2012). Thus, all of these measures generated by basic number processing tasks capture different performance features and, therefore, cannot be used interchangeably (Price, Palmer, Battista, & Ansari, 2012). Moreover, the inclusion of these measures allowed us to analyze how much of the variance in arithmetic achievement could be attributed to individual differences in

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underlying magnitude representations and how much could be attributed to processing efﬁciency of non-symbolic and symbolic magnitudes. Finally, researchers have hypothesized that the relationship between basic number processing skills and arithmetic performance is not only age dependent but also nonlinear (Bonny & Lourenco, 2013; Inglis et al., 2011). Yet, whether the explanatory power of children’s number processing efﬁciency skills is contingent on their arithmetic proﬁciency has rarely been studied in nonclinical samples. Recently, Bonny and Lourenco (2013) tested preschoolers’ non-symbolic comparison skills and mathematics abilities concurrently. Splitting their nonclinical sample into two groups based on children’s mathematics performance, they found non-symbolic comparison skills to signiﬁcantly explain the low mathematics achievers’ test scores but not the higher mathematics achievers’ test scores. This ﬁnding demonstrates the value of investigating whether the importance of basic number processing predictors varies for children with different levels of arithmetic proﬁciency. Basic number processing predictors and mathematics ability differences In the current study, four basic number processing measures were included, namely a non-symbolic and symbolic magnitude comparison paradigm, a counting paradigm, and an estimation paradigm. They were selected because they have been postulated to underlie early mathematics performance and have been used regularly in previous studies to measure basic number processing (e.g., Mazzocco, Feigenson, & Halberda, 2011a; Muldoon et al., 2013; Sasanguie, Göbel, Moll, Smets, & Reynvoet, 2013). Moreover, each task allowed for the indexing of task-speciﬁc effects. In the following, each of the tasks and the corresponding task-speciﬁc effects are discussed. Magnitude comparison skills In a non-symbolic magnitude comparison task, children are required to process and compare imprecise non-symbolic quantities (e.g., dots) (Mundy & Gilmore, 2009). This skill is thought to be supported by an inborn approximate number system (ANS) that provides a noisy, ratio-dependent representation of magnitudes (Dehaene, 2011). Speciﬁcally, the signal distributions of numerosities increasingly overlap with decreasing distance between numerosities (Feigenson, Dehaene, & Spelke, 2004). The preciseness of this representation is believed to be revealed by the task-speciﬁc numerical distance effect (NDE) and numerical ratio effect (NRE) (Moyer & Landauer, 1967; Price et al., 2012). The NDE refers to the observation that humans’ response accuracy and speed are superior for numerosities that are further apart (e.g., four and nine) as opposed to the ones that are close together (e.g., ﬁve and seven). The NRE, also referred to as the numerical size or magnitude effect, pertains to the observation that if distance is kept constant, it takes longer to compare high ratio magnitude pairs (e.g., nine and eight) than to compare lower ratio pairs (e.g., four and ﬁve) despite equal numerical distance. Because the NDE and NRE were found to be highly correlated, they have frequently been used interchangeably (Price et al., 2012). The NDE is already observed in infants (Izard, Sann, Spelke, & Streri, 2009) and animals (Cantlon, Platt, & Brannon, 2009). For example, newborn infants can differentiate two large numerosities if their ratio is approximately 1:3 (Izard et al., 2009). With increasing age, humans can successfully compare smaller ratios, suggesting that their representation of numerosities becomes more precise (Halberda & Feigenson, 2008). To compare exact symbolic quantities, researchers hypothesize that humans need to connect the culturally transmitted discrete symbols to their corresponding ANS representation (Dehaene, 2011). This proposition is based on the observation that the symbolic magnitude comparison task, a variant of a magnitude comparison task in which two symbolic stimuli (e.g., 4 and 6) are displayed, generates the NDE and NRE as well (Moyer & Bayer, 1976; Mundy & Gilmore, 2009). A number of previous studies have reported a smaller symbolic NDE, but not a smaller non-symbolic NDE, to be signiﬁcantly related to mathematics performance (Holloway & Ansari, 2009; Sasanguie, De Smedt, et al., 2012). This led researchers to hypothesize that differences in mathematics proﬁciency might be determined by people’s preciseness of mapping abstract symbols to the ANS representations rather than by ANS precision (Mundy & Gilmore, 2009; Sasanguie Göbel et al., 2013). However, in addressing this issue in older children, speciﬁcally 8- to 10-year-olds, Lonnemann, Linkersdörfer, Hasselhorn, and Lindberg

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(2011) found both their non-symbolic and symbolic NDE to be concurrently associated with mathematics achievement, which is not in line with this proposition. Recently, two studies investigated whether children’s non-symbolic and symbolic magnitude representation explains unique variance in mathematics achievement assessed at a later point in time. In addition, they addressed the predictive power of preschoolers’ efﬁciency to compare non-symbolic and symbolic magnitudes as well. Sasanguie, Van den Bussche, and colleagues (2012) administered non-symbolic and symbolic magnitude comparison tasks, number line estimation tasks, and number priming tasks to 72 children attending either Grade 1, 2, or 3. For both magnitude comparison tasks, they computed an efﬁciency measure, speciﬁcally adjusted RT scores that take into account a potential speed–accuracy trade-off, and the NDE slope. All number processing measures, in addition to children’s grade level and spelling achievement, were included in the regression model, but only children’s symbolic comparison speed, non-symbolic estimation accuracy, spelling ability, and grade level explained unique variance in mathematics achievement. Neither the non-symbolic NDE measure nor the symbolic NDE measure was found to be a signiﬁcant predictor of mathematics achievement 1 year later. Recently, Sasanguie Göbel (2013) related a similar regression model, not including the number priming tasks, to children’s timed arithmetic achievement. Again, children’s preciseness of their non-symbolic and symbolic magnitude representation, indexed by the Weber fraction and NDE, respectively, was not a unique predictor. Only children’s efﬁciency to compare Arabic digits, indexed by mean RT, and their spelling achievement uniquely predicted arithmetic performance 1 year later. Thus, both studies suggest that children’s efﬁciency to compare symbolic magnitudes, and not non-symbolic magnitudes, is the best predictor of elementary mathematics achievement. Yet, other studies addressing the role of children’s number processing efﬁciency measures have produced contradictory results. Desoete and colleagues (2010) administered three magnitude comparison tasks with different notations (dot, Arabic, and verbal number word) to kindergarteners, but only accuracy on the nonsymbolic comparison measure uniquely explained arithmetic fact retrieval 1 year later. In addition, Libertus and colleagues (2013) found preschool children’s non-symbolic processing efﬁciency to be predictive of early mathematics achievement after partialling out differences in expressive vocabulary and initial mathematics achievement. Finally, Mazzocco and colleagues (2011b) observed preschool children’s non-symbolic comparison efﬁciency to explain ﬁrst-grade differences in mathematics achievement, but not in subtests assessing children’s expressive vocabulary, perceptual organization, or numerical lexical retrieval. This suggests that non-symbolic comparison efﬁciency is a predictor speciﬁc to mathematics achievement. Note, however, that Libertus and colleagues (2013) and Mazzocco and colleagues (2011b) did not examine children’s symbolic representations simultaneously, thereby not allowing for the estimation of the unique variance in mathematics achievement explained by non-symbolic magnitude comparison abilities. Estimation ability Numerical estimation tasks assess children’s ability to approximate a quantity (Mazzocco et al., 2011a). Similar to the symbolic magnitude comparison task, children need to have an approximate understanding of the non-symbolic magnitude expressed by the symbolic magnitude in order to successfully complete estimation tasks. It is suggested that being more skillful in approximating magnitudes limits the range of possible answers from which children need to choose, which in turn might decrease the likelihood of making mistakes and increase arithmetic proﬁciency (Booth & Siegler, 2008). Estimation tasks asking participants to repeatedly estimate non-symbolic magnitudes enable researchers to record the precise difference between an estimate and target numerosity multiple times (Ansari, Donlan, & Karmiloff-Smith, 2007). This allows them to directly quantify an individual’s mapping precision between each discrete number word and its ANS representation rather than inferring it indirectly through metrics such as NDE (Ansari et al., 2007). It is hypothesized that mapping precision decreases in proportion to the size of the target numerosity (Whalen, Gallistel, & Gelman, 1999). To investigate this, previous studies used the coefﬁcient of variation (COV), a measure of dispersion that takes the magnitude of numerosities into account. The COV is deﬁned as the ratio of variability in estimates, namely standard deviation to mean (SD/M) estimate scores (Ansari et al., 2007).

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Therefore, the COV should remain constant across numerosities if an individual’s mapping variability increases proportional to the target quantity. Mazzocco and colleagues (2011a) examined the retrospective relation between the mapping precision, quantiﬁed by the COV, of children in Grade 9 and the arithmetic achievement scores in Grades 3 to 6. They administered an estimation paradigm requiring children to repeatedly estimate arrays of dots. Results revealed that higher COV scores, which are believed to reﬂect poorer mapping precision between the ANS and number words, correlates with weaker arithmetic proﬁciency. To the best of our knowledge, no previous study predictively related the estimation measure used in the present study, to in early primary education. Counting skills Dot enumeration tasks assess children’s mastery of the cardinality principle, which refers to their understanding that the last count word used in a sequence corresponds to the exact number of objects constituting the set. Through counting, children learn that each numerosity corresponds to one exact symbolic quantity (Cordes & Gelman, 2005). With respect to arithmetic, counting skills allow children to assign an exact meaning to disjoint numerosity sets and combine them step by step to derive a discrete answer (Butterworth, 2005; Le Corre & Carey, 2007). In line with this, children widely rely on counting skills, such as ﬁnger counting, when they ﬁrst learn to solve arithmetic problems (Butterworth, 2005). Previous studies argue for a relationship between children’s enumeration efﬁciency and their mathematics achievement. Muldoon and colleagues (2013) repeatedly analyzed preschool children’s enumeration ability and mathematics achievement. They found that at the beginning and end of the school year, children who successfully enumerated 20 dots scored signiﬁcantly better on the concurrently administered mathematics achievement test than children who failed to correctly count 20 dots. Reigosa-Crespo and colleagues (2012) extended these ﬁndings by observing that children’s dot enumeration efﬁciency is correlated with arithmetic ﬂuency in second- to ninth-grade children after controlling for Arabic comparison efﬁciency. In a study by Aunio and Niemivirta (2010), a composite measure of counting proﬁciency, which included enumeration items, was administered to an extensive kindergarten sample and found to be predictive of arithmetic ability in Grade 1. Note that this study did not address the unique explanatory power of children’s enumeration efﬁciency. Stock and colleagues (2009b) found kindergarteners’ counting accuracy to be signiﬁcantly correlated with arithmetic ﬂuency differences in Grade 1. However, when they took children’s intelligence, non-symbolic comparison, and logical ability skills into account, counting was not a signiﬁcant predictor. This result was conﬁrmed by Stock and colleagues (2010), who administered the same test battery, except for the intelligence measure, to a different sample. In addition to efﬁciency measures, some studies have computed linear regression slopes that summarize the positive relationship between cardinality (numerical size) and counting time. The steepness of the slopes generally differs for small and large numerosity ranges. It is postulated that this reﬂects a distinction between two underlying cognitive mechanisms, namely subitizing and counting (Schleifer & Landerl, 2011). Subitizing refers to the fast and accurate recognition of small numerosities (up to three or four objects) (Kaufmann, Lord, Reese, & Volkmann, 1949), whereas counting indexes the strength of the mapping between larger number words and their semantic representation (Schleifer & Landerl, 2011). A recent study found that ﬁrst-grade children’s subitizing slope contributed signiﬁcantly to calculation proﬁciency measured concurrently (Penner-Wilger et al., 2007). This is in line with studies reporting children with mathematics learning disabilities to have steeper subitizing slopes than controls (Landerl, Bevan, & Butterworth, 2004; Schleifer & Landerl, 2011). To the best of our knowledge, no predictive studies have been conducted in young children, nor have researchers addressed the role of the counting slope in explaining arithmetic differences. Non-numerical control variables Previous studies have also associated mathematics achievement with factors other than basic number processing measures, including school, gender, intelligence, and general processing speed (Geary,

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2011; Penner & Paret, 2008; Stock, Desoete, & Roeyers, 2009a). A general processing speed measure speciﬁes people’s ability and speed to serially encode and process low-level (perceptual) information without needing to rely on more advanced cognitive capacities (e.g., accessing letter meaning) or strategies (e.g., counting vs. retrieval) (Passolunghi & Lanfranchi, 2012). In a longitudinal study, Geary (2011) assessed children’s general processing speed, intelligence, and working memory, in addition to their magnitude comparison, conceptual counting, and number line estimation skills, in Grades 1 to 5. They found that processing speed, intelligence, and the visuospatial sketchpad (a working memory component) uniquely predicted achievement growth in mathematics above and beyond the basic number processing measures. The results of a study by Stock and colleagues (2009a) point toward an association between school and arithmetic ﬂuency. They assessed the explanatory power of kindergarteners’ conceptual counting skills for arithmetic ﬂuency in Grade 1, observing an intraclass correlation of .40. Several non-numerical factors, such as intelligence and general processing speed, were included in the current study to investigate which basic number processing skills affect arithmetic achievement above and beyond the non-numerical control variables. The current study The preceding review of the extant literature suggests that children’s ability to represent and process non-symbolic and symbolic magnitudes is related to mathematics achievement differences. Using four different basic number processing paradigms, we assessed kindergarten children’s processing efﬁciency and representation of non-symbolic and symbolic magnitudes and their relationship to ﬁrst-grade arithmetic proﬁciency after controlling for several non-numerical factors. The previous discussion points to three theoretical issues that were addressed in this study. First, we investigated which number processing efﬁciency measures assessed in kindergarten uniquely predict ﬁrst-grade arithmetic ﬂuency after partialling out the inﬂuence of non-numerical factors. We were primarily interested in whether non-symbolic number processing efﬁciency or symbolic number processing efﬁciency best explains individual differences in calculation skills. Second, we examined whether the relationship between children’s number processing efﬁciency and arithmetic achievement depends on the level of children’s calculation performance. Third, we sought to clarify whether children’s underlying magnitude representation or their number processing efﬁciency skills predominantly contribute to differences in early arithmetic achievement. Method Participants The initial sample consisted of 248 kindergarten children (134 boys and 114 girls). This sample was derived from an extensive test standardization study. For this purpose, kindergarten to sixth-grade children from six primary schools spread across The Netherlands were recruited. The children completed the cognitive measures during the second half of the school year. Subsequently, in January of Grade 1, the arithmetic ﬂuency test was administered to 209 children (113 boys and 96 girls); the other 39 children from the initial sample were not tested due to illness, grade retention, switching of schools, or parents’ refusal to give consent. Procedure After the contacted schools agreed to participate, parents were informed by letter. They could deny consent by returning an enclosed form. Each child was tested individually in a quiet room at school by trained project workers during one session lasting approximately 30 min. Only the arithmetic achievement test was centrally administered in January of Grade 1. Both the nonverbal IQ test (Raven’s Coloured Progressive Matrices [CPM]; Raven, Court, & Raven, 1995) and the arithmetic achievement test (Tempo Test Automatiseren [TTA]; de Vos, 2010) were paper-and-pencil tests. The remaining measures were computerized using a specialized response box with four buttons to precisely record

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children’s accuracy (% correct) and RT (ms/item) scores (Blomert, Vaessen, & Ansari, 2013). The reliabilities of the computerized outcome measures ranged from .83 to .94 except for the accuracy score of the baseline response task, which had a reliability of .66 (Blomert et al., 2013). Tasks Arithmetic achievement test Arithmetic fact retrieval was operationalized using the standardized TTA (de Vos, 2010). Its objective is to quantify the degree of automatization of basic arithmetic facts attained by children from Grade 1 to Grade 6. The entire test consists of four separate worksheets containing addition, subtraction, multiplication, and division problems of increasing difﬁculty. Because the multiplication and division subtests are not standardized for Grade 1, only the addition and subtraction worksheets were administered to the current sample. Then, a composite score was computed by summing the number of correct responses. For each worksheet, children were instructed to calculate as many of the 50 operations as possible within 2 min. Dot comparison In the dot comparison task, two arrays of dots, containing numerosities from 1 to 9 (Level 1) or from 10 to 39 (Level 2), were presented on a computer screen simultaneously and randomly. The side of the larger numerosity was counterbalanced to ensure that for each ratio the larger array was equally often presented on the right- and left-hand sides. Children were told that they would see two planes with dots. Subsequently, they were instructed to push the button corresponding to the numerosity that they thought was larger. It was explicitly stated that they were not supposed to count. The presentation time of the stimuli was not restricted. Hence, a new trial was initiated only after children pushed a button. The displays differed in numerosity based on one of four numerical ratios (.25–.33, .50, .66, and .75). Each ratio was presented 16 times, resulting in 64 test stimuli and 3 practice trials. After half of the trials, a break was included to prevent fatigue, and participants determined when to continue. Lastly, to ensure that total area and total perimeter were not predictive of numerosity, we included area-matched and perimeter-matched trials. Speciﬁcally, for half of the stimuli the total area of the dot arrays are the same, which causes the total perimeter to be greater on the side with more dots. For the other half of the stimuli, the total perimeter of the dots arrays are the same, which causes the total area to be greater on the side with less dots. These area-matched and perimeter-matched trials were randomly presented in each presentation condition in an effort to prevent participants from developing a strategy that relied on the relative size of the dot arrays. Arabic comparison The symbolic comparison task was composed of 4 practice and 32 test items that randomly displayed two one-digit Arabic numbers that differed based on one of four numerical ratios (.25–.33, .50, .66, and .75). Each ratio was presented eight times. Also in this task, the side of the larger number was counterbalanced for each ratio. It was explained to children that they would see two numbers and needed to determine which number was larger by pushing the corresponding button. As in the dot comparison task, a new item was presented only after children had responded. Dot enumeration The counting task consisted of 6 practice and 45 test items. On a black screen, children saw randomly one to nine white dots of varying sizes in a nonlinear order. Moreover, as in the dot comparison task, half of the trials were area matched and half of the trials were perimeter matched to control for the inﬂuence of perimeter and cumulative surface area. One third of the stimuli displayed quantities of three or less. The remaining items presented quantities ranging from four to nine. Each numerosity was presented ﬁve times, although no display contained the same number of dots as its immediate predecessor. Children were asked to count the dots and give their answers aloud. Children needed to push the green button of the response box simultaneously to ensure RT registration. The presentation time of the stimuli was not ﬁxed. Hence, a new item was displayed only after children had responded.

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Estimation In the estimation task, collections of 1, 2, 3, 4, 7, 11, or 16 white dots were presented for 750 ms on a black computer screen in random order. Children were told that they would see very brieﬂy a number of dots and needed to determine immediately how many dots were displayed. If they were hesitant to do so, the test administrator encouraged children to estimate or guess and reminded them that they were not expected to count or know the exact number (Ansari et al., 2007). The spatial arrangement of the dot clusters was random to avoid processing biases due to recognition of familiar patterns (Mandler & Shebo, 1982). Furthermore, as in the dot comparison and dot enumeration tasks, half of the trials had equivalent area and the other half had equivalent perimeter to ensure that total area and total perimeter were not predictive of numerosity. Each of 6 practice and 67 test items was followed by a mask after 750 ms. Three short breaks were included at ﬁxed time points to avoid fatigue.

Baseline response task The baseline response task (baseline RT) consisted of 20 trials in which children saw a row of four empty squares. Children were told that an animation ﬁgure would appear in one of these squares. When they saw the animation ﬁgure, they needed to push the button corresponding to its location as fast as possible.

Nonverbal IQ The CPM is a normed, untimed, visuospatial reasoning test for children in the age range from 5 to 11 years (Raven et al., 1995). Children saw a colored pattern and were asked to select the missing piece out of six choices. Van Bon (1986) reported reliabilities of .80 or higher for the Dutch version.

Results Children’s mean age in kindergarten and mean performance on the different measures are reported in Table 1. Except for the estimation task, trials on the computerized measures with a response faster than 200 ms were excluded. In addition to mean accuracy (% correct), mean RT (ms) and mean acc–RT (number of items correct/s) scores were calculated. The latter was derived by dividing the total number of correct items by the total response time in seconds. In the estimation task, extreme outliers, deﬁned as responses of 100 or higher, were excluded before calculating accuracy and COV (SD/M) scores. Finally, arithmetic achievement was operationalized as the composite score of the addition and subtraction fact retrieval subtests of the TTA.

Table 1 Age (in kindergarten) and task performance of the follow-up kindergarten sample.

Age (months) Raven IQ (ss 0–10) Arithmetic achievement (rs 0–55)a Number-speciﬁc cognitive tasks Dot comparison Arabic comparison Dot enumeration Estimation Baseline RT Note: ss, standard score; rs, raw score. a Administered in Grade 1.

Mean

SD

71.8 6.5 17.8

4.4 2.2 7.3

Accuracy (% correct)

Reaction time (ms)

Acc–RT (number of items correct/s)

Mean

SD

Mean

SD

Mean

SD

88.7 88.9 89.8 47.7 95.7

10.2 12.7 10.2 5.5 7.3

1893 2161 4391

519 774 1167

0.50 0.46 0.22

0.14 0.17 0.06

1105

243

0.90

0.17

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Arithmetic achievement and number processing efﬁciency At ﬁrst, a Pearson correlation analysis was conducted to analyze the relationship between the efﬁciency measures and arithmetic fact retrieval (see Table 2). To investigate whether the results are consistent across different efﬁciency measures, children’s accuracy, RT, and acc–RT scores were included if recorded. To correct for multiple pairwise comparisons, Holm’s correction method was used to compute an adjusted signiﬁcance level (Aickin & Gensler, 1996). Many of the measures were signiﬁcantly associated with each other (p 6 .001) and signiﬁcantly correlated with arithmetic achievement (p 6 .001). However, children’s dot comparison speed (r = –.14, p > .001), their Arabic comparison ability (r = .15, p > .001), and their counting ability (r = .21, p > .001) were not signiﬁcantly associated with children’s arithmetic performance. In addition, two hierarchical multiple regression analyses were conducted to investigate the significant simple correlations between children’s number efﬁciency measures and arithmetic ﬂuency in more detail. In both regressions, the control variables time elapsed between kindergarten assessment and follow-up testing (DL difference T1 and T2), gender, baseline RT, and school were entered in Step 1. Because school is a categorical variable, we operationalized it using ﬁve school dummies. In addition, in Step 2, the efﬁciency measures revealing signiﬁcant simple correlations were included. The process of entering all control variables in a ﬁrst block, followed by the addition of all remaining explanatory variables of interest in a second block, allows us to distinguish the variance in arithmetic achievement attributable to the basic number processing measures from the variance captured by the control variables (Pedhazur, 1997). In the ﬁrst regression analysis, efﬁciency on the timed number processing tasks was indexed by acc–RT score. Because the estimation task was untimed, children’s efﬁciency on this measure was indexed by accuracy scores. The results are reported in Table 3. A total of 45% of the variance was explained by the complete model, F(14, 194) = 11.49, p < .001. The number processing measures accounted for an additional 25% of the variance, F(4, 194) = 21.91, p < .01. To further investigate the importance of the second predictor block, the effect size was estimated using Cohen’s f2. Cohen advanced the following guidelines for effect size interpretation: small = .02, medium = .15, and large = .35 (Cohen, Cohen, West, & Aiken, 2003). Based on these references, we found a large effect size of .46. In addition, the regression results revealed that except for dot comparison (p > .05), the number processing variables included in the model (p < .01) as well as the control variables gender (p < .05) and School Dummy 4 (p < .01) were signiﬁcant individual predictors. In the second regression analysis, children’s efﬁciency on the timed number processing tasks was indexed by accuracy and RT scores separately. Again, only the measures signiﬁcantly correlated with arithmetic ﬂuency were included. The results are reported in Table 4. The complete model explained 42% of the variance in arithmetic fact retrieval, F(14, 194) = 10.23, p < .01, and the number processing

Table 2 Pearson correlations between arithmetic achievement test and basic number processing skills (after applying Holm’s correction method). Measure 1. Arithmetic achievement 2. Dot comparison (acc) 3. Dot comparison (RT) 4. Dot comparison (acc–RT) 5. Arabic comparison (acc) 6. Arabic comparison (RT) 7. Arabic comparison (acc–RT) 8. Counting (acc) 9. Counting (RT) 10. Counting (acc–RT) 11. Estimation (acc) *

p < .05 (after Holm’s correction).

1

2 .24*

3

4 .14 .09

5 .22* .32* .85*

6 .15 .10 .06 .08

7 .38* .07 .39* .41* .20

8 .45* .15 .38* .45* .45* .85*

9 .21 .25* .05 .15 .13 .08 .16

10 .22* .20 .22* .27* .06 .22 .19 .21

.37* .23* .28* .36* .04 .28* .31* .18 .84*

11 .38* .35* .02 .15 .16 .17 .26* .25* .23* .34*

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Table 3 Results of hierarchical multiple regression analysis with dependent arithmetic achievement test and adding basic number processing skills in Step 2. B Step 1 Constant Age in months DL difference T1 and T2 Gender Nonverbal IQ Baseline RT School Dummy 2 School Dummy 3 School Dummy 4 School Dummy 5 School Dummy 6 Step 2 Dot comparison (acc–RT) Arabic comparison (acc–RT) Counting (acc–RT) Estimation (acc)

SE B

Signiﬁcance (p)

b

8.17 0.04 0.22 1.71 0.05 0.00 1.84 0.84 6.36 1.15 1.20

10.08 0.10 0.44 0.82 0.09 0.00 2.76 1.34 1.28 1.32 1.37

.02 .03 .12 .04 .02 .04 .05 .35 .06 .06

.42 .69 .62 .04* .57 .69 .51 .53 .00** .39 .38

1.80 12.11 23.06 0.40

3.38 2.83 8.21 0.08

.04 .28 .17 .30

.59 .00** .01* .00**

Note: R2 = .20 for Step 1; DR2 = .25 for Step 2. * p < .05. ** p < .01.

Table 4 Results of hierarchical multiple regression analysis with dependent arithmetic achievement test and adding basic number processing skills (accuracy and reaction time separately) in Step 2. B

SE B

Signiﬁcance (p)

b

Step 1 Constant Age in months DL difference T1 and T2 Gender Nonverbal IQ Baseline RT School Dummy 2 School Dummy 3 School Dummy 4 School Dummy 5 School Dummy 6

1.19 0.07 0.28 2.33 0.07 0.00 1.50 0.52 6.64 1.09 1.33

10.75 0.10 0.45 0.83 0.09 0.00 2.78 1.36 1.29 1.34 1.40

.04 .04 .16 .05 .00 .03 .03 .37 .06 .07

.91 .52 .54 .01** .44 1.00 .59 .70 .00** .42 .34

Step 2 Dot comparison (acc) Arabic comparison (RT) Counting (RT) Estimation (acc)

0.08 0.00 0.00 0.44

0.04 0.00 0.00 0.08

.12 .23 .05 .33

.05 .00** .37 .00**

Note: R2 = .20 for Step 1; DR2 = .22 for Step 2. ** p < .01.

measures accounted for an additional 22% of the variance, F(4, 194) = 18.40, p < .01. Moreover, Cohens f2 was found to be large at .38. Children’s ability to compare non-symbolic magnitudes and their counting speed did not contribute additional individual explanatory power to the model (p > .05), whereas their estimation ability and Arabic comparison speed did (p < .01).

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A contingent relationship? To address whether the relationship between children’s number processing efﬁciency and arithmetic fact retrieval is contingent on their arithmetic proﬁciency, we ﬁrst checked for heteroskedasticity. The signiﬁcant Breusch–Pagan test, P(v2(14) = 36.25) = .001, points toward a nonlinear relationship. Subsequently, we ran a quantile regression analysis in Stata. This statistical method models the relationship between an outcome variable and its predictor variables at different points of the conditional distribution of the outcome measure (Cameron & Trivedi, 2009). We estimated the rate of change in the 16th, 50th, and 84th percentiles of arithmetic achievement brought about by a 1-unit change in the predictor variables. The two extreme percentile ranks correspond to a performance score of 1 SD above or below the mean under a normal curve. They were chosen because we were primarily interested in the explanatory regression models for children who exhibited arithmetic performance substantially higher or lower than the mean. Children’s number processing efﬁciency measures, indexed by acc–RT scores, and the non-numerical control factors were inserted as predictors into the quantile regression analyses. The results of the quantile regression analysis are presented in Table 5. The basic number processing tasks Arabic comparison (p < .05) and estimation (p < .01) were signiﬁcant individual predictors across all quantile levels, whereas the counting measure (p < .05) explained unique variance only at the highest quantile level of arithmetic performance. Regarding the control variables, only School Dummy 4 (p < .05) was a unique predictor, although this variable no longer signiﬁcantly explained variance at the highest quantile level. Additional analysis in Stata revealed that the variable School Dummy 4 had a signiﬁcantly higher impact at the lower quantile of arithmetic performance. However, the regression coefﬁcients of the basic number processing measures, including counting, did not differ signiﬁcantly across the quantile levels. A visual comparison of the quantile and linear regression coefﬁcients showed that the latter sometimes underestimates or overestimates the rate of change explained by variables, depending on the quantile of arithmetic achievement analyzed. Finally, the unique

Table 5 Results of quantile regression analysis with dependent arithmetic achievement test for the 16th, 50th, and 84th percentiles. Quantile Regression 16th percentile B Constant Age in months DL difference T1 and T2 Gender Nonverbal IQ Baseline RT School Dummy 2 School Dummy 3 School Dummy 4 School Dummy 5 School Dummy 6 Dot comparison (acc–RT) Arabic comparison (acc–RT) Counting (acc–RT) Estimation (acc)

50th percentile

SE B

Signiﬁcance (p)

13.54 0.02 0.70 1.99 0.07 0.00 0.36 1.23 10.07 0.07 0.31 1.09

12.40 0.11 0.69 1.09 0.12 0.00 3.09 1.22 1.65 1.79 1.43 3.62

.28 .88 .31 .07 .59 .64 .91 .32 .00** .97 .83 .76

7.78

3.90

10.02 0.42

10.66 0.12

B

84th percentile

SE B

Signiﬁcance (p)

6.77 0.00 0.07 1.50 0.01 0.00 1.51 1.02 4.79 1.22 0.26 0.62

12.09 0.14 0.52 1.09 0.13 0.00 3.52 1.59 2.35 1.64 1.87 4.86

.58 .98 .89 .17 .95 .43 .67 .52 .04* .46 .89 .90

.05*

11.78

4.36

.35 .00**

25.86 0.32

14.23 0.11

B

SE B

Signiﬁcance (p)

21.34 0.03 0.20 1.67 0.05 0.00 4.50 2.05 3.21 2.24 1.76 0.61

16.01 0.16 0.56 1.25 0.15 0.00 3.85 2.28 1.79 2.75 2.02 6.77

.18 .83 .73 .19 .73 .59 .24 .37 .08 .42 .38 .93

.01**

16.89

5.97

.00**

.07 .01**

34.99 0.51

14.17 0.17

.01* .00**

Note: Pseudo R2 = .37 for 16th percentile model; pseudo R2 = .23 for 50th percentile model; pseudo R2 = .29 for 84th percentile model. * p < .05. ** p < .01.

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explanatory power of gender observed in the linear regression model was absent in all three quantile regression models. Arithmetic achievement and magnitude representations We examined whether the task-speciﬁc measures of numerical magnitude processing were significantly related to individual differences in children’s arithmetic achievement. First, we needed to verify that the postulated task-speciﬁc effects were observed in the current sample. Because mean accuracy scores were high on the timed number processing measures, which considerably reduced the discernible variability, RT scores were used to compute task-speciﬁc effects. Given that no RT scores were available for the estimation task, the COV was calculated using accuracy scores. Symbolic and non-symbolic NRE First, a 2 (Task) 4 (Ratio) repeated-measures analysis of variance (ANOVA) was executed. Because the assumption of sphericity was violated, the Greenhouse–Geisser adjustments are reported for ratio and task ratio. A signiﬁcant ratio effect (NRE), F(2.64, 548.77) = 104.57, p < .001, and a signiﬁcant task effect, F(1, 208) = 23.91, p < .001, were found. To further examine the relationship between the nonsymbolic and symbolic comparison NRE, numerical ratio slopes were calculated. The size of the numerical ratio slope (M = 1831, SD = 1566) was larger for the non-symbolic measure than for the symbolic measure (M = 903, SD = 1585). A dependent t test showed that the non-symbolic slope was signiﬁcantly steeper than the symbolic one, t(208) = 6.08, p < .001. Counting-speciﬁc effects Behavioral performance on an enumeration task is frequently characterized by a subitizing and counting effect. Because the quantity of objects falling within the subitizing range increases across development, accuracy and RT scores were explored to decide on the appropriate cutoff point for the current sample (Fischer, Gebhardt, & Hartnegg, 2008). A sharp and continuous increase in RT and a drop in accuracy scores below 99% after numerosity 3 were observed. This indicated that the enumeration of 1 to 3 objects constitutes the subitizing range, whereas the items with more dots reﬂect the counting range. A repeated-measures ANOVA with number as the within-participant variable conﬁrmed the number effect for accuracy, F(3.53, 734.64) = 97.13, p < .001, and RT scores, F(4.25, 819.95) = 591.07, p < .001. Note that the Greenhouse–Geisser adjustment was used because the data violated the sphericity assumption. For the purpose of analyzing the relationship between arithmetic achievement and both the subitizing and counting effects, we computed a subitizing slope and a counting slope. Estimation effect The variability in children’s verbal responses on an estimation task was captured by the COV, which was computed as SD/M per numerosity. Because it has been suggested that the processing of small and large numerosities might be supported by different cognitive processes (Schleifer & Landerl, 2011), two separate repeated measures were conducted, including numerosities falling either within the subitizing range (1, 2, and 3) or within the counting range (5, 7, 11, and 16). The COV was constant for the subitizing range, F(1, 207) = 0.01, p = .916, but not for the counting range, F(1, 207) = 15.56, p < .001. A Bonferroni post hoc comparison including the counting range numerosities revealed that only mean COVs for 11 and 16 dots were constant. Unexpectedly, both COVs were smaller than the COV of numerosity 7. If this decrease in variability was caused by a deﬁcient understanding of the meaning of twodigit numbers inducing children to repeatedly respond with the same number word when viewing displays containing two-digit numerosities, then the SD for numerosities 11 and 16 should be smaller than that for numerosity 7. However, the SD for numerosities, 7, 11, and 16 were 2.59, 3.7, and 6.4, respectively. Task-speciﬁc effects and arithmetic achievement To examine the relationship between task-speciﬁc effects and arithmetic achievement, a Pearson’s correlations analysis was conducted (see Table 6). However, after Holm’s correction, none of the

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Table 6 Pearson correlations between arithmetic achievement test and task-speciﬁc effects. Measure 1. 2. 3. 4. 5. 6. 7. *

Arithmetic achievement Non-symbolic NRE Symbolic NRE Subitizing effect (slope) Counting effect (slope) Estimation COV subitizing range Estimation COV counting range

1

2 .10

3

4 .13 .02

5 .16 .03 .01

6 .03 .10 .09 .04

7 .10 .05 .02 .04 .13

.01 .05 .02 .05 .02 .64*

p < .05.

measures was signiﬁcantly associated with arithmetic ﬂuency (p > .05). Hence, the relationship between the task-speciﬁc effects and arithmetic performance was not investigated further. Discussion The ﬁrst aim of this study was to clarify which preschool number processing efﬁciency measures uniquely predict ﬁrst-grade arithmetic achievement while controlling for non-numerical factors. Second, we wanted to create a more reﬁned picture of this relationship. Therefore, we investigated whether the relationship between children’s number processing efﬁciency and arithmetic performance varied depending on children’s arithmetic proﬁciency. Moreover, we examined whether the preciseness of children’s underlying magnitude representations or their number processing efﬁciency accounts for most variation in arithmetic proﬁciency. Kindergarteners’ efﬁciency to process non-symbolic and symbolic magnitudes was signiﬁcantly correlated with calculation ﬂuency at the onset of formal education. However, after controlling for non-numerical factors, the tasks assessing children’s efﬁciency to process symbolic magnitudes and the control variables gender and school were the only unique predictors. Quantile regression analysis revealed that children’s Arabic comparison and estimation efﬁciency, but not their dot comparison efﬁciency, were signiﬁcant individual predictors across all three quantile regression models. Children’s counting efﬁciency explained unique variance only in the 84th percentile model. Yet, the regression coefﬁcients of the counting measure did not differ signiﬁcantly across the quantile regression models. The non-numerical factor school no longer contributed individual explanatory power in the highest percentile of arithmetic achievement. Additional analysis showed that the regression coefﬁcient of the variable school in the 84th quantile model was signiﬁcantly different from the coefﬁcient found in the 16th and 50th quantile models. Finally, no signiﬁcant simple correlations were observed between task-speciﬁc effect measures, which are thought to index individual differences in magnitude representations, and arithmetic performance. Contribution of speciﬁc cognitive skills In accordance with previous studies, we found kindergarteners’ efﬁciency to compare symbolic magnitudes, but not non-symbolic magnitudes, to be uniquely predictive of ﬁrst-grade arithmetic ﬂuency above and beyond several control factors (De Smedt & Gilmore, 2011; Kolkman et al., 2013; Sasanguie Göbel et al., 2013; Sasanguie, Van den Bussche, et al., 2012; Stock et al., 2010). Children’s procedural counting efﬁciency and their ability to verbally estimate numerosities have been studied less extensively as individual predictors of arithmetic ﬂuency. The current ﬁnding that children’s counting and estimation efﬁciency explained a unique part of the variance in arithmetic ﬂuency as well further supports the hypothesis that children’s efﬁciency to process symbolic magnitudes is an important kindergarten predictor of early arithmetic performance. However, Stock and colleagues (2010) did not ﬁnd a signiﬁcant relationship between children’s counting skills and arithmetic achievement. Moreover, some researchers observed non-symbolic comparison skills to be related to arithmetic skills (Desoete et al., 2010; Libertus et al., 2013; Mazzocco et al., 2011b). Some of the

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reasons that could account for these divergent ﬁndings include the use of accuracy scores as an index of number processing efﬁciency, different operationalization of the tasks and the arithmetic achievement test, and differences in the age and size of the samples (Price et al., 2012). Furthermore, those studies that found an association between non-symbolic comparison and arithmetic did not use symbolic comparison tasks concurrently and therefore could not, unlike the current study, estimate the amount of unique variance explained by non-symbolic magnitude comparison performance. These contradictory ﬁndings suggest that the role of children’s basic number processing skills in the development of mathematics achievement is not straightforward. Researchers have emphasized that advanced mathematics abilities build on lower ones (Bonny & Lourenco, 2013; Entwisle et al., 2005). It might be that the initial learning of arithmetic requires the support of not only a broad network of basic number processing skills but also more basic ones, whereas automatizing one’s arithmetic abilities relies more on number skills closer in content and developmental time. In line with such an argument is the hypothesis that children’s efﬁciency to compare non-symbolic magnitudes might be a signiﬁcant individual predictor of arithmetic ﬂuency only for lower achieving children. But the quantile regression results in the current study do not disclose a differential effect of the non-symbolic number processing measure. Children’s non-symbolic number processing efﬁciency did not explain unique variance in arithmetic achievement, whereas their symbolic number processing efﬁciency was a consistent predictor. This indicates that early arithmetic development is primarily explained by basic number processing skills that require an understanding of symbolic notations. In addition, the results show that in a nonclinical sample, the relationship between number processing efﬁciency and arithmetic performance does not vary for different points in the conditional distribution of arithmetic scores. Both ﬁndings are contrary to Bonny and Lourenco (2013), who observed a signiﬁcant correlation between children’s non-symbolic magnitude processing efﬁciency for children with lower mathematics scores but not for children with higher scores. However, the researchers measured children’s non-symbolic number processing efﬁciency and mathematics achievement concurrently and did not include multiple number skills simultaneously in their explanatory model. The current study also shed light on the individual contribution of non-numerical factors to calculation ﬂuency. Children attending School 4 performed signiﬁcantly worse than the reference group despite the fact that they used the same mathematics curriculum. However, other characteristics could possibly account for the observed difference in performance. For example, the reference school was located in a more rural area, the number of enrolled students was substantially lower, and it had a Catholic denomination as opposed to a public one. Together with the previous study by Stock and colleagues (2009a) in which a substantial intraclass correlation was reported, the current ﬁnding implies that school characteristics should be included as control variables in future research. Furthermore, in line with past studies, gender (Geary, Saults, Liu, & Hoard, 2000; Penner & Paret, 2008) was found to be uniquely predictive of average arithmetic differences, with girls scoring lower than boys. Finally, neither nonverbal IQ nor general response speed uniquely predicted arithmetic achievement differences, indicating that these variables are of little additional informative value for the prediction of arithmetic ﬂuency at the beginning of formal schooling. Although this is in line with a number of studies (Jordan, Kaplan, Nabors Oláh, & Locuniak, 2006; Krajewski & Schneider, 2009; Passolunghi, Vercelloni, & Schadee, 2007), it contradicts others (Geary, 2011; LeFevre et al., 2006; Penner & Paret, 2008). A possible explanation for these conﬂicting ﬁndings could be the difference in operationalization of IQ and general response speed. Note that Passolunghi and Lanfranchi (2012) found verbal IQ to be related to mathematics achievement, whereas performance IQ was not. Furthermore, the variance in arithmetic ﬂuency explained by children’s general response speed might have been captured by the timed number processing measures. In support of this argument, we found the general response speed measure to be strongly correlated with the timed number processing tasks. Magnitude representations and arithmetic ﬂuency Some researchers have postulated that differences in mathematics achievement can be attributed to the preciseness of children’s ANS and the analogue representation of symbolic magnitudes (Dehaene, 2011; Mundy & Gilmore, 2009). However, the absence of signiﬁcant associations between arithmetic ﬂuency and multiple task-speciﬁc effects in the current study speak against this view.

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Contrary to our results, Bugden and Ansari (2011), who studied ﬁrst- and second-grade children, found the symbolic NRE to uniquely explain arithmetic ﬂuency measured concurrently. To the best of our knowledge, previous studies with younger children have examined only the predictive relation between NDE, a metric strongly correlated with the NRE, and calculation ﬂuency. For example, De Smedt, Verschaffel, and Ghesquière (2009) reported symbolic NDE in Grade 1 to uniquely predict mathematics achievement in Grade 2. Using the COV as a metric indexing the mapping precision between the ANS and symbolic number words, Mazzocco and colleagues (2011a) found children’s mapping precision in Grade 9 to be retrospectively correlated with their arithmetic achievement scores in Grades 3 to 6. Yet, our study differs from these previous studies in certain aspects, including the sample size and the age of the children. In addition, the subitizing and counting effect was not signiﬁcantly correlated with early arithmetic ﬂuency in the current study. This ﬁnding is not in line with a previous study by Penner-Wilger and colleagues (2007) in which the subitizing slope was linearly related to calculation skills in nonclinical ﬁrst-grade children. However, their measures were administered concurrently and analyzed a different age group. In addition, they computed the subitizing slope slightly different, ﬁrst calculating median response times for the subitizing range values (1–3) for each child and subsequently drawing a regression line through these medians. In our study, we computed a slope through the observed reaction times for all subitizing range items (1–3) for each child. The current ﬁndings lead to the conclusion that children’s efﬁciency to process symbolic magnitudes and not the precision or acuity of their analog representation of non-symbolic and symbolic magnitudes is a central, predictive preschool competence for arithmetic ﬂuency development. Nevertheless, the absence of signiﬁcant ﬁndings for the task-speciﬁc effects might ensue from a low reliability of the task-speciﬁc effect measures. Regarding the NDE, researchers recently found this measure to have low, albeit signiﬁcant, split-half reliabilities and, therefore, proposed that one should interpret results based on an NDE measure carefully (Maloney et al., 2010). Furthermore, consistent with the notion of a strong difference between symbolic and non-symbolic magnitude processing, it was recently proposed that the symbolic magnitude representation system might not be mapped onto the ANS but rather is a distinct exact system (Lyons, Ansari, & Beilock, 2012). The ﬁndings of a recent study by Sasanguie, Defever, Maertens, and Reynvoet (2013) support this view. Their analysis revealed that kindergarteners’ non-symbolic comparison accuracy was not signiﬁcantly correlated with their symbolic comparison ability 6 months later. The current ﬁnding that children’s symbolic NDE and estimation COV were both not signiﬁcantly associated with arithmetic performance suggests that children’s analog representation of numerical symbols in kindergarten is not important to early arithmetic development. However, we cannot exclude the possibility that individual differences in arithmetic performance are driven by a deﬁcient, qualitatively different exact symbolic representational system. Conclusions The current study suggests that children’s efﬁciency to process discrete symbols is important to the development of adequate calculation ﬂuency above and beyond non-numerical factors. Children’s non-symbolic magnitude comparison efﬁciency, on the other hand, did not uniquely explain arithmetic performance regardless of children’s level of arithmetic proﬁciency. This suggests that non-symbolic magnitude comparison efﬁciency is not a critical predictor of early arithmetic proﬁciency. Finally, children’s magnitude representations indexed by task-speciﬁc effects were not signiﬁcantly correlated with arithmetic achievement. To conclude, children’s efﬁciency to process symbolic magnitudes, rather than underlying magnitude representations, should be considered as potential candidates to be addressed in successful teaching or remediation programs (Gersten, Jordan, & Flojo, 2005) aimed at young children. References Aickin, M., & Gensler, H. (1996). Adjusting for multiple testing when reporting research results: The Bonferroni vs. Holm methods. American Journal of Public Health, 86, 726–728.

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