Task-Focused Behavior Mediates the Associations Between Supportive Interpersonal Environments and Students' Academic Performance

Linguistic and Spatial Skills Predict Early Arithmetic Development viaCounting Sequence Knowledge

Xiao ZhangHong Kong Institute of Education

Tuire Koponen and Pekka R€as€anenNiilo M€aki Institute

Kaisa Aunola, Marja-Kristiina Lerkkanen, and Jari-Erik NurmiUniversity of Jyv€askyl€a

Utilizing a longitudinal sample of Finnish children (ages 6–10), two studies examined how early linguistic(spoken vs. written) and spatial skills predict later development of arithmetic, and whether counting sequenceknowledge mediates these associations. In Study 1 (N = 1,880), letter knowledge and spatial visualization,measured in kindergarten, predicted the level of arithmetic in first grade, and later growth through thirdgrade. Study 2 (n = 378) further showed that these associations were mediated by counting sequence knowl-edge measured in first grade. These studies add to the literature by demonstrating the importance of writtenlanguage for arithmetic development. The findings are consistent with the hypothesis that linguistic and spa-tial skills can improve arithmetic development by enhancing children’s number-related knowledge.

Mathematics competence during formal educationis severely compromised if children have difficul-ties in learning arithmetic facts and achieving flu-ency in arithmetic calculations (Geary, 1993). Overthe past decade, an increasing number of studieshave investigated the cognitive foundation of arith-metic development. For example, there is growingconsensus that arithmetic achievement in elemen-tary school can be traced to early number compe-tence, such as counting sequences of numbers (i.e.,counting sequence knowledge), number recogni-tion, and number comparisons (Aunola, Leskinen,Lerkkanen, & Nurmi, 2004; Jordan, Kaplan, Ramin-eni, & Locuniak, 2009). In addition to such numberskill precursors, domain-general factors, such aslinguistic (e.g., Durand, Hulme, Larkin, & Snow-ling, 2005) and spatial (e.g., LeFevre et al., 2010)skills, have also been found to contribute to arith-metic competence. However, existing research inthe field has at least three limitations. First, fewattempts have been made to examine the precur-sors of arithmetic development by using a multi-wave longitudinal design (for an exception, see

Aunola et al., 2004). Second, little research hasexplored how different types of precursors areinterrelated in the prediction of arithmetic compe-tence (for an exception, see Cirino, 2011). Third, lit-tle work has been done to delineate the role of theability to understand written versus spoken lan-guage in the development of arithmetic. The aimof this research was to investigate the extent towhich children’s linguistic (written vs. spoken) andspatial skills, measured before formal education,are associated with subsequent development ofarithmetic from first to third grades, and whetherthese associations are mediated by countingsequence knowledge.

Linguistic Skills and Arithmetic Development

During the elementary school years, arithmeticcompetence is often associated with reading compe-tence (Duncan et al., 2007), and many children withdifficulties in arithmetic also have difficulties inreading (Landerl & Moll, 2010). Moreover, achiev-ing fluency in arithmetic has been found to be acommon obstacle in children with languageimpairment (Donlan, Cowan, Newton, & Lloyd,2007; Koponen, Mononen, R€as€anen, & Ahonen,

This research was funded by a grant from the Academy of Fin-land to the Finnish Center of Excellence in Learning and Motiva-tion Research (213486).

Correspondence concerning this article should be addressed toXiao Zhang, Department of Early Childhood Education, TheHong Kong Institute of Education, 10 Lo Ping Road, Tai Po,New Territories, Hong Kong, China. Electronic mail may be sentto [email protected].

© 2013 The AuthorsChild Development © 2013 Society for Research in Child Development, Inc.All rights reserved. 0009-3920/2014/8503-0019DOI: 10.1111/cdev.12173

Child Development, May/June 2014, Volume 85, Number 3, Pages 1091–1107

2006). These findings suggest that linguistic pro-cesses are involved in the acquisition of arithmetic.In his neuropsychological model of numerical pro-cessing, Dehaene (2011; Dehaene, Molko, Cohen, &Wilson, 2004) also discussed the role of language inarithmetic. According to this model, arithmetictasks, such as retrieving answers to arithmetic com-binations (e.g., 2 + 4), activate a region of the leftangular gyrus in the brain that is linked to lan-guage processing. Recently, researchers have furtherpostulated that linguistic skills predict mathematicsor numeracy measures that involve manipulation ofthe symbolic number system (i.e., Arabic digits orwords) but not those that are not symbolicallycoded (Simmons & Singleton, 2008).

Several linguistic skills have been found to beassociated with arithmetic development. First, pho-nological processing, especially phonological aware-ness, the ability to differentiate and manipulatemeaningful segments of a spoken language (e.g.,phoneme, syllable, and rhyme), has been linked toarithmetic competence (see Simmons & Singleton,2008, for a review; see Durand et al., 2005, forinconsistent results). Geary (1993) argued that rep-resenting and retrieving phonological informationfrom the long-term memory may underlie problemsin learning arithmetic facts. Second, many aspectsof verbal abilities, including listening comprehen-sion, verbal reasoning, and vocabulary, have alsobeen found to predict arithmetic competence (Aun-ola et al., 2004; Durand et al., 2005; see Fuchs et al.,2010, for inconsistent results). LeFevre et al. (2010)argued recently that receptive vocabulary mayreflect children’s abilities to acquire vocabulary inthe number system.

In this growing body of research, linguistic skillshave been conceptualized in different ways buthave nonetheless been used to refer to abilities todecode or understand spoken language. However,conventional arithmetic tasks often require thatchildren understand written symbols involvingnumbers and operators. To understand a writtenlanguage, including the Arabic number system,one must learn to use written symbols to representthe spoken language. It is possible that early expo-sure to print, in which children are provided withexperience of mapping sounds to written letters inan alphabetic writing system, improves their abili-ties to use and manipulate written symbols fornumbers and operators. Such abilities have beensuggested to aid later learning of symbolic process-ing in mathematics (Hale & Fiorello, 2004). Vygot-sky (1978) also argued that the acquisition ofwritten symbols, which he refers to as a specific

cultural tool, extends children’s mental capacities.As a result of using this tool, children develophigher mental functions (focused attention, deliber-ate memory, etc.) that are critical for concept for-mation across domains, including arithmetic. Insupport of these arguments, two recent studieshave shown that letter knowledge or awareness,the ability to decode written linguistic codesinvolving letters, is associated with arithmetic com-petence (Koponen, Aunola, Ahonen, & Nurmi,2007; Purpura, Hume, Sims, & Lonigan, 2011). Onegoal of the present research was to examine theextent to which children’s early abilities to decodeand understand written (i.e., letters) versus spoken(i.e., phoneme and vocabulary) language contributeto their arithmetic development in the early schoolyears.

Spatial Skills and Arithmetic Development

In addition to linguistic processes, spatial pro-cesses may also underlie arithmetic development.Evidence from psychophysical and neuropsycholog-ical studies has suggested that number magnitudeis coded spatially from left to right in the mindalong an internal number line (Dehaene, Dupoux, &Mehler, 1990; Gallistel & Gelman, 1992). Morerecently, Siegler and Booth (2004) showed that chil-dren shift from logarithmic to linear in their numbermagnitude representations. Moreover, they furtherfound that the use of a linear representation (i.e.,the linear number line) was related to children’shigher arithmetic competence. Dehaene (2011; Deh-aene et al., 2004) has also suggested that the spatialprocess is involved in arithmetic: The posteriorsuperior parietal lobule, which is linked with spatialprocessing, is activated in arithmetic operations.Consequently, the second goal of the presentresearch was to examine the role of spatial skills inearly arithmetic development.

Linn and Petersen (1985) distinguished threetypes of spatial skills, namely, spatial perception(identifying spatial relations with respect to one’sown orientation), mental rotation (mentally rotatinga 2-D or 3-D object), and spatial visualization (pro-cessing complicated, multistep manipulations ofspatial information). According to these authors,tests of spatial perception (e.g., draw a horizontalline in a tilted bottle) and mental rotation (e.g.,identify a response that shows the standard in adifferent orientation) both require only one solutionstrategy. In contrast, tests of spatial visualization(e.g., indicate how a piece of holed, folded paperwould look when unfolded) require multiple solu-

1092 Zhang et al.

tion strategies. Although it is theoretically impor-tant to examine whether certain components ofspatial skills relate differentially to arithmetic com-petence (Gunderson, Ramirez, Beilock, & Levine,2012), in this research our focus is on spatial visual-ization.

Prior studies with elementary and high schoolstudents have shown that spatial visualizationability is associated strongly with success in solv-ing arithmetic problems (Hegarty & Kozhevnikov,1999; van Garderen, 2006). In a study among 5-year-olds, Barnes et al. (2011) found that spatialvisualization predicted oral counting (the highestnumber counted from 1), quantitative concepts,and object-related arithmetic competence, evenafter controlling for the effects of linguistic (i.e.,vocabulary and phonological awareness) andmotor skills. Although these studies provide sup-port for the view that spatial visualization andarithmetic competence are related, they are basedmainly on cross-sectional data, leaving the direc-tion of causality unclear. Consequently, the pres-ent research employed a longitudinal design toassess whether early spatial visualization abilitywould predict later growth of arithmeticcompetence.

Counting Sequence Knowledge and ArithmeticDevelopment

The third goal of this research was to examinehow counting sequence knowledge is interrelatedto linguistic and spatial skills in the prediction ofarithmetic development. We define countingsequence knowledge as the capacity to verballycount numbers in their correct forward or back-ward sequence (Aunola et al., 2004). According toFuson (1982), counting sequence knowledge evolvesfrom the reproduction of a number sequence to thecreation of a “breakable chain” of numbers.

Learning to solve arithmetic problems developsgradually, at least partially via the development ofcounting sequences of numbers. When learning tocalculate, children use strategies based on countingforward or backward either verbally or on theirfingers (Siegler & Shrager, 1984). In this sense,counting sequence knowledge is inherent to thestrategies that children typically first use to calcu-late. For example, in addition tasks, children firstcount both numbers presented—the counting-allprocedure—and later shift to counting on from thecardinal value of the first (counting-on first or max)or larger number (counting-on min) presented(Fuson, 1982). It is obvious that children’s ability to

accurately recite the numbers from 1 or after agiven number (i.e., forward counting knowledge) isessential when using the counting-all or counting-on strategies. Similarly, in subtraction the most fre-quently used counting-based strategies amongprimary school children are counting all, countingdown (e.g., 5-2, “four, three”), and counting up(e.g., 5-2, “three, four, five”; Siegler & Shrager,1984). Whereas forward counting knowledge isinvolved in counting all and counting up, backwardcounting knowledge, or the ability to produce thenumbers preceding a given number, is crucial whena child starts to use counting down to subtract.Finally, the frequent and successful use of countingsequence knowledge is assumed to increase mem-ory representations of arithmetic facts and lead to amore mature strategy of retrieving these facts fromlong-term memory (i.e., fact retrieval; Siegler &Shrager, 1984). Consequently, it has been foundrepeatedly that counting sequence knowledge is astrong predictor of arithmetic competence duringthe elementary school years (Aunola et al., 2004;Krajewski & Schneider, 2009).

Recent theoretical models suggest that linguisticand spatial factors contribute to arithmetic develop-ment indirectly via number skills, including count-ing sequence knowledge (Krajewski & Schneider,2009; LeFevre et al., 2010). The models essentiallylocate the development of arithmetic within abroader context, as linguistic and spatial skills havebeen directly connected to arithmetic competence. Itshould be emphasized that number skills may be astepping stone from more fundamental linguisticand spatial skills to the acquisition of arithmeticknowledge.

The literature has suggested that countingsequence knowledge mediates the link between lin-guistic skills and arithmetic competence. The abilityto understand and express symbolic number sys-tems relies on language skills. Counting sequencesof numbers requires mastering the number systemvocabulary (LeFevre et al., 2010) and retrieving thecorrect phonological codes for number words(Logie & Baddeley, 1987). For young children,fluent and accurate counting, especially backwardcounting, demands higher mental functions, such asfocused attention and memory skills (No€el, 2009).These capacities, according to Vygotsky (1978),can be acquired gradually through the use ofwritten language. Recently, counting sequenceknowledge has been found to mediate the linkbetween phonological awareness and arithmeticcompetence (Cirino, 2011; Krajewski & Schneider,2009). In this research, we examined whether the

Early Arithmetic Development 1093

relations between a broader range of linguistic skills(phonological awareness, vocabulary, and letterknowledge) and arithmetic competence would alsobe mediated by counting sequence knowledge. Theinclusion of a broader set of linguistic skills is help-ful in determining whether the mediator effects per-tain to skills beyond phonological awareness.

Spatial skills may also act indirectly on arithme-tic outcomes via counting sequence knowledge.Numerical sequences might be organized spatiallyin the mind (Dehaene et al., 1990). Prior studieshave shown that counting sequences, especiallybackward counting, involve not only linguistic butalso spatial processes (Hoshi et al., 2000; Zhouet al., 2006). Moreover, the relation between visuo-spatial working memory (memory skills that areused to store spatial information) and arithmeticcompetence has been found to be mediated bycounting sequence knowledge (Cirino, 2011; Kra-jewski & Schneider, 2009). In this research, weinvestigated whether such knowledge would alsomediate the relation between spatial visualizationability and arithmetic competence.

The Present Research

Recent research on the cognitive precursors ofarithmetic development has suggested a compre-hensive model in which linguistic and spatial pre-cursors independently contribute to arithmeticcompetence via number skills (Cirino, 2011; Krajew-ski & Schneider, 2009; LeFevre et al., 2010). Thisresearch, however, has several limitations. First,many studies have used small convenience samples,which limits the generalizability of the results (foran exception, see LeFevre et al., 2010). Second, moststudies have assessed arithmetic competence at aparticular time point (for an exception, see Aunolaet al., 2004), which is not an ideal setting for study-ing the precursors of the level of arithmetic compe-tence and its growth over time. Third, the existingliterature linking linguistic skills with arithmeticcompetence has focused extensively on the abilityto understand spoken language rather than writtenlanguage, leaving it unclear whether the ability todeal with written linguistic symbols, such as letterknowledge, is involved in arithmetic developmentor not.

This research adopted a multiwave longitudinaldesign and consisted of two studies. In Study 1, wefollowed up 1,880 Finnish children from kindergar-ten to third grade and examined the extent towhich linguistic (spoken and written; i.e., phonemicawareness, vocabulary, and letter knowledge) and

spatial (i.e., spatial visualization) skills, measured inkindergarten, would predict the development ofarithmetic from first to third grades. In Study 2, weselected 378 children from Study 1, measured theircounting sequence knowledge in first grade, andfurther examined whether such knowledge wouldmediate the relations between linguistic and spatialskills and the development of arithmetic. On thebasis of the available literature, we hypothesizedthat early individual differences in phonemicawareness, vocabulary, letter knowledge, and spa-tial visualization would predict both the subsequentlevel and growth rate of arithmetic. We alsohypothesized that counting sequence knowledgewould mediate the relations between linguistic andspatial skills and arithmetic development.

For exploratory purposes, we examined the spe-cific mediator effects of forward versus backwardcounting knowledge. Although forward and back-ward counting employs a common linguistic repre-sentation (Richardson, 1977), it has been found thatthe spatial process underlies backward but not for-ward counting (Hoshi et al., 2000; Zhou et al.,2006). We thus speculated that the relation betweenlinguistic skills and arithmetic development wouldbe mediated by both forward and backward count-ing knowledge, whereas the relation between spa-tial skills and arithmetic development would bemediated by backward but not forward counting.Because gender (boys perform better than girls) andparental education (children whose parents are welleducated perform better than those whose parentsare poorly educated) have a well-established historyof being predictive of arithmetic competence (e.g.,Jordan et al., 2009), we included these backgroundvariables as covariates in our research.

Study 1

Method

Participants and Procedure

This study is part of an ongoing longitudinalstudy (Lerkkanen et al., 2006) in which 1,880 chil-dren (896 girls; age at kindergarten entry: M = 74.0� 3.6 months) were recruited from four municipali-ties in Finland (two in Central, one in Western, andone in Eastern Finland) and followed up from kin-dergarten to third grade (age of 10). The childrencomprised the whole age cohort from three munici-palities and about a half of the age cohort from thefourth one. The vast majority (80%) of the childrencame from nuclear families, 10% from single-parent

1094 Zhang et al.

families, 8% from blended families, and 2% fromfamilies where the parents were divorced and thechild lived in two homes. Parents were asked fortheir written consent for their child’s participation.

The children were tested longitudinally for atotal of six times between kindergarten and thirdgrade: in the fall (September, Time 1 [T1];n = 1,867) and spring (April, Time 2 [T2]; n = 1,839)of kindergarten, in the fall (September, Time 3 [T3];n = 1,837) and spring (April, Time 4 [T4]; n = 1,711)of first grade, in the spring of second grade (April,Time 5 [T5]; n = 1,659), and in the spring of thirdgrade (April, Time 6 [T6]; n =1,639). All the testswere carried out by trained testers (researchers orstudents of psychology and education).

To assess the problem of longitudinal attrition,we conducted a logistic regression and examinedthe extent to which absence at the final time pointwas related to age, gender (0 = female, 1 = male),family structure (0 = nuclear family, 1 = nonnuclearfamily), and parental education. Because paternaleducation was highly correlated with maternal edu-cation (r = .68, p < .001), we used the highest educa-tion in the household as the indicator of education.This decision also allowed us to use data on educa-tion for almost all family structures, thus reducingthe amount of missing data. Education was codedas follows: 1 = no education beyond comprehensiveschool (3.0% of the families), 2 = vocational courses(2.2%), 3 = vocational school degree (26.8%), 4 = voca-tional college degree (22.8%), 5 = polytechnic degree orbachelor’s (BA) degree (12.4%), 6 = master’s degree(MA; 25.4%), and 7 = licentiate or doctoral degree(7.4%). The results indicated that family structuresignificantly predicted the odds of being absent fromthe study at the last time point (B = 1.02, p < .001,OR = 2.78). Children from nonnuclear families wereapproximately 2.78 times more likely to be absentcompared to their peers from nuclear families. Gen-der, age, and education did not predict the odds ofbeing absent. The differential attrition in this studymay limit the generalizability of our findings andshould be kept in mind when interpreting theresults.

Measures

A total of five tests were used at different timepoints. Linguistic and spatial precursors of arithme-tic competence were tested on an individual basis:Phonemic awareness and letter knowledge weretested at both T1 and T2, and vocabulary and spa-tial visualization were tested at T2. Arithmetic com-petence was tested in a group situation at each time

point from T3 to T6. Raw sum scores of correctitems were used for all variables in all analyses.

Phonemic awareness. Phonemic awareness wasassessed using an initial phoneme identificationtask that contains 10 items (maximum score: 10;see Lerkkanen, Poikkeus, & Ketonen, 2006). Ineach item, the child was shown four pictures ofobjects with simultaneous presentation of theirnames and asked to indicate which of the picturesshows the object whose name starts with therequested phoneme (e.g., “At the beginning ofwhich word do you hear ____?”). Cronbach’salpha coefficients were .78 and .76 at T1 and T2,respectively.

Letter knowledge. To assess the ability to decodewritten linguistic symbols involving letters, thechild was tested on a task containing all 29 lettersin the Finnish language (maximum score: 29; seeLerkkanen, Niemi, et al., 2006). The 29 uppercaseletters were in random order and arranged in threerows. The child was asked to name the letters, onerow at a time, while the other rows were covered.Cronbach’s alpha coefficients were .96 and .94 at T1and T2, respectively.

Vocabulary. A 30-item shortened version of thePeabody Picture Vocabulary Test–Revised (PPVT–R,Form L; Dunn & Dunn, 1981) was used to measurereceptive vocabulary (maximum score: 30). ThePPVT requires the child to select from four alterna-tives the picture that correctly depicts a spokenword. The items for the shortened version wereselected based on the data from the full-scale admin-istration of the PPVT–R to the control group inanother Finnish study called the Jyv€askyl€a Longitu-dinal Study of Dyslexia (see Lyytinen et al., 2004).Cronbach’s alpha coefficient was .61.

Spatial visualization. Spatial visualization wasassessed using a subtest of spatial relations from theWoodcock and Johnson (1977) test battery. The testrequires the child to identify the subset of piecesneeded to form a complete shape with multiple-point scored items (i.e., “Two of these pieces () gotogether to make this (). Tell me which two pieces.”).It involves complicated, multistep manipulations ofspatial information (i.e., detecting multiple spatialforms or shapes, rotating or manipulating them inthe imagination, and matching). According to Linnand Petersen (1985), the test can be seen as a mea-sure of spatial visualization ability. A total of 31tasks can be attempted within a 3-min time limit(maximum score: 31). Cronbach’s alpha coefficientwas .71.

Arithmetic competence. Arithmetic competencewas assessed using the Basic Arithmetic Test

Early Arithmetic Development 1095

(Aunola & R€as€anen, 2007). In the test, a total of 28items containing 14 addition (e.g., 2 + 1 =; 3 + 4 +6 =) and 14 subtraction (e.g., 4 – 1 =; 20 – 2 – 4 =)items can be attempted within a 3-min time limit(maximum score: 28). Task difficulty increases grad-ually across the test. The test indexed a combina-tion of speed and accuracy of performance ofarithmetic in this study. The accuracy of the chil-dren’s performance increased largely from first tothird grades (M � SD = 58 � 28%, 85 � 15%,91 � 9%, and 92 � 8% at T3, T4, T5, and T6,respectively). Cronbach’s alpha coefficients were0.77, 0.85, 0.89, and 0.87 at T3, T4, T5, and T6,respectively.

Statistical Analyses

The analyses were performed using Mplus 6(Muth�en & Muth�en, 1998–2010). Attrition at anytime point in this study was found to be duemainly to children moving away from the schooldistrict (rather than withdrawing from the study orbeing absent on the day of testing), and there is noevidence that it resulted from the actual perfor-mance of the children at an earlier time point. Thus,the standard “missing at random” (MAR) approachwas applied (Muth�en & Muth�en, 1998–2010). Theparameters of the models were estimated using thefull information maximum likelihood estimationwith non-normality robust standard errors (MLR;Muth�en & Muth�en, 1998–2010). The goodness of fitof the estimated models was evaluated by five indi-cators: chi-square test, comparative fit index (CFI),Tucker–Lewis index (TLI), root mean square errorof approximation (RMSEA), and standardized rootmean square residual (SRMR). Growth-curve analy-ses were conducted to estimate the level (i.e., inter-cept) and growth (i.e., slope) parameters ofarithmetic competence and to examine how theseparameters were predicted by various linguisticand spatial precursors after accounting for demo-graphic covariates. A significance level of .01 wasused due to the large sample size.

Results

Descriptive Statistics and Correlations Among Variables

Table 1 presents descriptive statistics for and cor-relations among the linguistic and spatial precursors.Table 2 presents descriptive statistics of arithmeticoutcomes and correlations between the outcomesand precursors. The measures of phonemic aware-ness and letter knowledge had a moderate to high

ceiling at T2 and were therefore not further analyzed.Arithmetic competence, measured at each time point,correlated with all the linguistic and spatial precur-sors at the significance level of .001.

Growth-Curve Analyses

First, unconditional growth-curve analyses werecarried out to estimate the level (i.e., intercept) andgrowth (i.e., slope) of arithmetic. Both linear andquadratic models were tested. We fixed the loadingsof the observed arithmetic variables across T3–T6 to1 on the level factor; to 0, 1, 3, and 5 on the linearslope factor; and to 0, 1, 9, and 25 on the quadraticslope factor. Both models showed poor fit,v2(df = 7) = 2934.98, p < .001, CFI = .01, TLI = .15,RMSEA = .48, SRMR = .40 for the linear model, andv2(df = 5) = 646.95, p < .001, CFI = .78, TLI = .74,RMSEA = .26, SRMR = .13 for the quadratic model,indicating that the growth curve was neither linearnor quadratic. The modification indices further sug-gested that freeing the slope factor loadings wouldimprove the model fit. These findings corroborateprevious research on early arithmetic developmentamong Finnish children (Aunola et al., 2004). Wethus consider it theoretically appropriate to free theslope factor loadings (see also Aunola et al., 2004).

We then constructed a growth model with esti-mated time scores, in which the first loading on theslope factor was set at 0 and the last at 1, and theloadings between the two time points were allowedto be estimated freely. By doing so, the level can beinterpreted as the level at T3 (i.e., fall of first grade),

Table 1Means and Standard Deviations of and Correlations Among CognitivePrecursors in Study 1

Cognitiveprecursors M SD 1 2 3 4 5 6

1. T1_Phonemicawareness

7.47 2.45 —

2. T2_Phonemicawareness

8.94 1.68 .56 —

3. T1_Letterknowledge

16.93 9.01 .58 .53 —

4. T2_Letterknowledge

23.30 6.61 .52 .64 .79 —

5. T2_Receptivevocabulary

19.83 3.37 .32 .29 .34 .29 —

6. T2_Spatialvisualization

14.23 2.38 .27 .29 .28 .29 .30 —

Note. N = 1,880. T1 = fall of kindergarten; T2 = spring of kinder-garten. All correlations were significant at p < .001, two-tailed.

1096 Zhang et al.

and the slope can be interpreted as the amount ofchange from T3 to T6 (i.e., spring of third grade).This point is of particular importance for ourresearch, as we are interested in the cognitive skillsthat predict the overall amount of growth overyears. The results showed that the model had abetter fit than the linear and quadratic models, butstill did not show a good fit, v2(df = 3) = 125.15,p < .001, CFI = .96, TLI = .92, RMSEA = .15,SRMR = .10. Based on the modification indices, theestimated residual terms between the measure-ments at T4 and T5 were allowed to correlate, indi-cating that the two measurements shared someunique variance that was not included in theoverall model. We consider this modification theo-retically appropriate, as it is consistent with previ-ous research on Finnish children’s arithmeticdevelopment (Aunola et al., 2004). After the modifi-cation, the model showed an acceptable fit,v2(df = 2) = 24.33, p < .001, CFI = .99, TLI = .98,RMSEA = .08, SRMR = .04, although the chi-squarestatistic was significant, presumably due to thelarge sample size (Bentler & Bonnet, 1980). Thus,the modified model was chosen as our final uncon-ditional model. The results showed a relatively lowinitial level (i.e., intercept) of arithmetic at T3(M = 3.56, SE = .06, p < .001), followed by positivegrowth (i.e., slope) from T3 to T6 (M = 15.97,SE = .10, p < .001). There were significant individ-ual differences in the intercepts (r2

i = 6.01,p < .001) and slopes (r2

i = 11.68, p < .001), indicat-ing that children differed in their initial level andgrowth rate in arithmetic. The correlation betweenthe intercept and slope was not significant (r = .07,p = .25), suggesting that initial arithmetic compe-tence was not related to later growth.

Next, we added the effects of the linguistic (pho-nemic awareness, vocabulary, and letter knowl-edge) and spatial (spatial visualization) precursors

and demographic covariates (gender and education)to the unconditional model and tested to whatextent the precursors predicted the level andgrowth of arithmetic. Figure 1 presents this modelwith a completely standardized solution. The modelshowed a good fit, v2(df = 14) = 37.32, p < .001,CFI = .99, TLI = .99, RMSEA = .03, SRMR = .02.Letter knowledge and spatial visualization posi-tively predicted the level and growth of arithmeticafter controlling for the effects of gender and educa-tion (the results did not change when gender andeducation were excluded from the model). Childrenwith stronger letter knowledge and spatial visuali-zation in kindergarten had higher arithmetic com-petence in the fall of first grade and steeper rates ofgrowth in arithmetic through third grade. The othertwo linguistic measures, phonemic awareness andvocabulary, did not significantly predict the level orgrowth of arithmetic. In total, the model explained28% of the variance of level and 6% of the varianceof growth.

Finally, because the effect of letter knowledge hasrarely been considered in prior studies, as supple-mentary analyses we reran the model and examinedthe effect of phonemic awareness and vocabularywith letter knowledge excluded from the model.The fit of the model was good, v2(df = 12) = 34.43,p < .001, CFI = .99, TLI = .99, RMSEA = .03, SRMR= .02. When controlled for gender and education(the results did not change when uncontrolled forthese variables), spatial visualization significantlypredicted the level (standardized b = .26, p < .001)and growth (standardized b = .19, p < .001) of arith-metic, whereas phonemic awareness and vocabularysignificantly predicted level (phonemic awareness:standardized b = .19, p < .001; vocabulary: stan-dardized b = .10, p < .001), but not growth (phone-mic awareness: standardized b = .06, ns; vocabulary:standardized b = .00, ns).

Table 2Means and Standard Deviations of Arithmetic Competence and Its Correlations With Cognitive Precursors in Study 1

Outcome variables M SD

Phonemicawareness

Letterknowledge

T2_Receptivevocabulary

T2_SpatialvisualizationT1 T2 T1 T2

T3_Arithmetic competence 3.57 2.72 .27 .27 .39 .38 .24 .31T4_Arithmetic competence 10.60 4.09 .24 .25 .35 .35 .17 .32T5_Arithmetic competence 16.27 4.88 .24 .25 .35 .38 .20 .33T6_Arithmetic competence 19.76 4.52 .24 .25 .35 .37 .20 .33

Note. N = 1,880. T1 = fall of kindergarten; T2 = spring of kindergarten; T3 = fall of first grade; T4 = spring of first grade; T5 = spring ofsecond grade; T6 = spring of third grade. All correlations were significant at p < .001, two-tailed.

Early Arithmetic Development 1097

Discussion

Study 1 examined the extent to which early lin-guistic (written vs. spoken) and spatial skillswould predict later development of arithmeticfrom first to third grades. In line with findingsfrom recent longitudinal studies of arithmeticdevelopment in the elementary school years (Aun-ola et al., 2004; Jordan et al., 2009), the resultsshowed significant variations in both the initiallevel and rate of growth in arithmetic. Moreimportant, letter knowledge and spatial visualiza-tion, tested in kindergarten, independently pre-dicted both the level and growth of arithmeticfrom first to third grades, even after controlling forthe effects of child gender and parental education.This finding, in conjunction with a growing bodyof research linking diverse linguistic and spatialprecursor measures to arithmetic outcomes (e.g.,Aunola et al., 2004; Fuchs et al., 2010), underscoresthe importance of these fundamental skill precur-sors in the acquisition of arithmetic during theearly years of schooling.

It is interesting to note the links from phonemicawareness and vocabulary—the two linguistic skillsinvolved in understanding spoken language—toarithmetic development. When letter knowledge—the linguistic skill involved in decoding writtenlanguage—was excluded from the model, phonemic

awareness and vocabulary significantly predictedthe level of arithmetic. However, when the measure-ments of spoken and written linguistic skills wereincluded simultaneously in the model, the linksfrom phonemic awareness and vocabulary to thearithmetic level lost significance. Instead, letterknowledge significantly predicted both the leveland growth of arithmetic. These findings suggestthat letter knowledge, an important skill relevant todecoding the written language system, not onlyoverrode the prediction of arithmetic level from thespoken language-related skills but also demon-strated unique utility in the prediction of arithmeticgrowth. Notably, however, the reliabilities wererelatively low for phonemic awareness and vocabu-lary. Hence, this findings need to be replicated withmore reliable skill measures.

In sum, the results of Study 1 lend support tothe notion that linguistic and spatial skills areimportant precursors of arithmetic development.Moreover, the results highlight the unique role ofletter system knowledge in engendering positivegrowth in arithmetic during the early school years.These results provide a prerequisite condition fortesting the mediator effect of counting sequenceknowledge in the associations between theseprecursor skills and arithmetic development. Con-sequently, the mediator effect was examined inStudy 2.

1 .78.430 11 11

L-Arithmetic (R2 = .28***)

S-Arithmetic (R2 = .06***)

T3_Arithmetic T4_Arithmetic T5_Arithmetic T6_Arithmetic

.28***

Parental Education

T2_Spatial Visualization

T1_Letter Knowledge

T1_Phonemic Awareness

T2_Receptive Vocabulary

Gender .23***

.35***

.18***.09***

.12***

Figure 1. The final model with a completely standardized solution in Study 1. N = 1,880. T1 = fall of kindergarten; T2 = spring of kin-dergarten; T3 = fall of first grade; T4 = spring of first grade; T5 = spring of second grade; T6 = spring of third grade. For ease of pre-sentation, predictive paths that were not significant at p < .01 (two-tailed) are not shown.***p < .001, two-tailed.

1098 Zhang et al.

Study 2

Method

Participants and Procedure

Participants were 378 children selected randomlyfrom the 1,880 children of Study 1. The reason forcreating the subsample was that we wanted to testmathematics competence in more detail than wasdone in the tests conducted for the entire sample.We created the subsample by selecting 1–4(M = 2.5 � 0.7) children from each first-grade class-room (number dependent on class size). The major-ity (77.6%) of them came from nuclear families,11.5% from single-parent families, 9.6% fromblended families, and 1.3% from families where theparents were divorced and the child had twohomes.

The children underwent the six test occasionsdescribed in Study 1. A total of 376, 377, 374, 377,364, and 362 children participated in T1, T2, T3, T4,T5, and T6, respectively. We conducted a logisticregression similar to that in Study 1 to examinewhether absence at the last time point was relatedto age, gender, education, or family structure. Theresults suggested that age significantly predictedthe odds of being absent from the study at the lasttime point (B = 0.40, p = .03, OR = 1.50). Older chil-dren were more likely to be absent than youngerchildren. Gender, education, and family structuredid not predict the odds of being absent.

Measures

Alongside the tests described in Study 1, weadded a test on counting sequence knowledge at T3(i.e., fall of first grade). The test contained sevenitems in which children were asked to count aloudforward (from 1 to 5, 6 to 13, and 18 to 25) andbackward (from 12 to 7, 23 to 18, 33 to 17, and 23,five items backward). The items were scored usinga 3-point scale: 2 = no errors, 1 = one small error(e.g., the child stopped counting one number tooearly), and 0 = two or more errors. We calculated theraw sum scores for the seven tasks (maximumscore: 14; a = .84), the three forward counting tasks(maximum score: 6; a = .62) and the four backwardtasks (maximum score: 8; a = .72).

Statistical Analyses

As in Study 1, missing data were handled byusing MLR under the assumption of MAR. Growth-

curve analyses were conducted to estimate the leveland growth parameters of arithmetic competenceand to examine whether counting sequence knowl-edge mediated the links between linguistic andspatial skills and arithmetic competence afteraccounting for demographic covariates. A signifi-cance level of .05 was used.

Results

Descriptive Statistics and Correlations Among Variables

Table 3 presents the descriptive statistics for andcorrelations among the linguistic, spatial, and num-ber skill precursors. Table 4 presents the descriptivestatistics for the arithmetic outcomes and the corre-lations between the outcomes and the precursors.Counting sequence knowledge correlated witharithmetic competence at each time point and withall the linguistic and spatial precursors at the signif-icance level of .001.

Growth-Curve Analyses

Three growth-curve models were computed: (a)an unconditional model, which provided estimatesof the level (i.e., intercept) and growth (i.e., slope)parameters of arithmetic competence; (b) a direct-effect model, which added the direct effects of thelinguistic (phonemic awareness, vocabulary, andletter knowledge) and spatial (spatial visualization)skills while controlling for the effects of gender andeducation; and (c) a mediation model, which fur-ther added the mediator effect of counting sequenceknowledge.

First, we employed unconditional growth-curveanalyses similar to those in Study 1. The finalunconditional model was a growth model with esti-mated time scores, in which the first loading on theslope factor was set at 0 and the last at 1, and theloadings between these two time points wereallowed to be estimated freely. As in Study 1, theresidual terms between the measurements at T4and T5 were allowed to correlate. The modelprovided an acceptable fit to the data, v2(df = 2)= 9.67, p = .01, CFI = .99, TLI = .97, RMSEA = .08,SRMR = .05. On average, the initial level in arith-metic competence was relatively low (M = 3.72,SE = .13, p < .001). This was followed by positivegrowth in arithmetic from T3 to T6 (M = 15.94,SE = .21, p < .001). There were significant variationsin the intercepts (r2

i = 4.88, p < .001) and slopes(r2

i = 10.75, p < .001), indicating that children dif-fered in their initial arithmetic competence and in

Early Arithmetic Development 1099

their rates of growth over time. The correlationbetween the intercept and the slope was not signifi-cant (r = .26, p = .06), suggesting that initial arith-metic competence was not associated with latergrowth.

Next, we estimated the direct-effect modeldescribed earlier. The model had a good fit,v2(df = 14) = 26.26, p = .02, CFI = .99, TLI = .97,RMSEA = .05, SRMR = .03. The results showed thatboth the level and growth of arithmetic were pre-dicted by letter knowledge (unstandardized Β = .07,standardized b = .29, p < .001 for level; unstandard-ized Β = .05, standardized b = .15, p < .05 forgrowth) and spatial visualization (unstandardizedΒ = .24, standardized b = .25, p < .001 for level;unstandardized Β = .29, standardized b = .21,p < .01 for growth) after controlling for the effectsof gender and education. Children with strongerletter knowledge and spatial visualization showedhigher arithmetic competence in the fall of firstgrade and steeper rates of growth through thirdgrade. Phonemic awareness and vocabulary pre-dicted neither the level nor growth of arithmetic. In

total, the model explained 25% of the variance oflevel and 7% of the variance of growth.

We then estimated the mediation model. Themodel showed a good fit, v2(df = 16) = 29.54,p = .02, CFI = .99, TLI = .97, RMSEA = .05,SRMR = .03. Figure 2 presents the model with acompletely standardized solution. Countingsequence knowledge predicted both the level andgrowth of arithmetic and was predicted by letterknowledge and spatial visualization. Moreover,when the mediator effect of counting sequenceknowledge was taken into account, the direct effectsof letter knowledge and spatial visualization on thelevel of arithmetic were reduced to unstandardizedΒ = .04 (p = .05; percent variance reduction = 48.6%)and Β = .18 (p < .001; percent variance reduc-tion = 26.9%), respectively, and on growth to unstan-dardized Β = �.01 (ns; percent variancereduction = 100%) and Β = .20 (ns; percent variancereduction = 36.1%), respectively. These reductionswere all statistically significant, as shown in Table 5by the positive indirect effects with the 99% confi-dence interval (CI) not including zero (the 99% CI

Table 3Means and Standard Deviations of and Correlations Among Cognitive Precursors in Study 2

Cognitive precursors M SD 1 2 3 4 5 6 7

1. T1_Phonemic awareness 7.63 2.34 —

2. T2_Receptive vocabulary 20.08 3.28 .36 —

3. T1_Letter knowledge 16.90 8.99 .52 .34 —

4. T2_Spatial visualization 14.33 2.31 .21 .22 .24 —

5. T3_Counting sequence knowledge 9.20 3.45 .30 .22 .44 .28 —

6. T3_Forward counting sequence knowledge 4.67 1.55 .34 .18 .39 .24 .85 —

7. T3_Backward counting sequence knowledge 4.53 2.29 .23 .23 .41 .27 .93 .60 —

Note. N = 378. T1 = fall of kindergarten; T2 = spring of kindergarten; T3 = fall of first grade. All correlations were significant atp < .001, two-tailed.

Table 4Means and Standard Deviations of Arithmetic Competence and Its Correlations With Cognitive Precursors in Study 2

Outcome variables M SDT1_Phonemicawareness

T2_Receptivevocabulary

T1_Letterknowledge

T2_Spatialvisualization

T3_Counting sequenceknowledge

Total Forward Backward

T3_Arithmetic competence 3.69 2.50 .25*** .19*** .33*** .30*** .45*** .32*** .48***T4_Arithmetic competence 10.51 4.14 .15*** .06 .28*** .30*** .47*** .36*** .47***T5_Arithmetic competence 16.29 4.92 .15*** .11* .26*** .34*** .53*** .42*** .51***T6_Arithmetic competence 19.69 4.76 .20*** .11 .29*** .29*** .51*** .39*** .51***

Note. N = 378. T1 = fall of kindergarten; T2 = spring of kindergarten; T3 = fall of first grade; T4 = spring of first grade; T5 = spring ofsecond grade; T6 = spring of third grade.*p < .05. ***p < .001, two-tailed.

1100 Zhang et al.

was used because multiple tests were applied).Taken together, these results indicated that countingsequence knowledge mediated the associationsbetween linguistic (i.e., letter knowledge) and spatial(i.e., spatial visualization) skills and the level andgrowth of arithmetic. In total, the model explained37% of the variance of level and 23% of the variance

of growth. It also explained 26% of the variances ofcounting sequence knowledge.

We also tested the specific mediator effects offorward versus backward counting knowledge.Because the two potential mediators were highlycorrelated, they were tested separately. The resultsshowed that forward counting mediated the linksto the level and growth of arithmetic from letterknowledge (percent variance reduction = 18.8% and61.2% for level and growth, respectively; ps < .01),but not those from spatial visualization (percentvariance reduction = 11.1% and 22.6% for level andgrowth, respectively; ns). In contrast, backwardcounting mediated the links to the level and growthof arithmetic from both letter knowledge (percentvariance reduction = 53.1% and 91.3% for level andgrowth, respectively; ps < .01) and spatial visualiza-tion (percent variance reduction = 27.7% and 30.7%for level and growth, respectively; ps < .01). Bothforward and backward counting predicted the level(forward counting: standardized b = .20, p < .001;backward counting: standardized b = .42, p < .001)and growth (forward counting: standardizedb = .34, p < .001; backward counting: standardizedb = .39, p < .001) of arithmetic.

Finally, we reran the above direct-effect andmediation models with the exclusion of letterknowledge. The results for spatial visualization

Table 5Test of Indirect Effects in Study 2

Potential paths mediated bycounting sequence knowledge

Test of indirect effects

99% CI

Coef. Lower Upper Stand.

Letter knowledge ?L_Arithmetic

.035*** .017 .054 .144

Spatial visualization ?L_Arithmetic

.065*** .013 .117 .069

Letter knowledge ?S_Arithmetic

.059*** .025 .094 .166

Spatial visualization ?S_Arithmetic

.110** .017 .204 .079

Note. N = 378. L_Arithmetic = the level factor of arithmetic com-petence; S_Arithmetic = the slope factor of arithmetic competence;CI = confidence interval; Coef. = estimate of unstandardizedindirect effect; Stand. = estimate of standardized indirect effect.**p < .01. ***p < .001, two-tailed.

T3_Counting SequenceKnowledge

(R2 = .26***)

1 .79.430 11 11

L-Arithmetic (R2 = .37***)

S-Arithmetic (R2 = .23***)

T3_Arithmetic T4_Arithmetic T5_Arithmetic T6_Arithmetic

.33***

Parental Education

T2_Spatial Visualization

T1_Letter Knowledge

T1_Phonemic Awareness

T2_Receptive Vocabulary

Gender

.12*

.19***

.15*

.37***

.18***

.18***

.44***.39***

Figure 2. The final model with a completely standardized solution in Study 2. N = 378. T1 = fall of kindergarten; T2 = spring of kinder-garten; T3 = fall of first grade; T4 = spring of first grade; T5 = spring of second grade; T6 = spring of third grade. For ease of presenta-tion, predictive paths that were not significant at p < .05 (two-tailed) are not shown.*p < .05. ***p < .001, two-tailed.

Early Arithmetic Development 1101

were similar to those presented earlier. Phonemicawareness predicted the level (standardized b = .20,p < .001) but not the growth (standardized b = .06,p = .39) of arithmetic in the direct-effect model. Therelation between phonemic awareness and levelwas mediated by counting sequence knowledge(percent variance reduction = 53.9%; p < .01). Thismediator effect applied to both forward and back-ward counting (percent variance reduction = 21.3%and 56.5% for forward and backward counting,respectively; ps < .01). In contrast, vocabularypredicted neither the level (standardized b = .05,p = .40) nor the growth (standardized b = �.07,p = .40) of arithmetic.

Discussion

Study 2 investigated whether counting sequenceknowledge mediated the relations between linguis-tic and spatial skills and arithmetic development.The results showed support for the mediation pro-cess: Counting sequence knowledge, measured inthe fall of first grade, mediated the links from letterknowledge and spatial visualization in kindergartento both the level and growth of arithmetic from firstto third grades. This finding, in conjunction withthe findings of two recent longitudinal studies (Ciri-no, 2011; Krajewski & Schneider, 2009), highlightsthe mediating role of number skills between earlylinguistic and spatial precursors and later arithmeticcompetence.

We further examined the specific mediator effectof forward versus backward counting knowledgeon the link between linguistic and spatial skills andarithmetic development. The results showed thatthe relation between linguistic skills and arithmeticwas mediated by both forward and backwardcounting knowledge. By contrast, the relationbetween spatial skills and arithmetic was mediatedby backward but not forward counting. These find-ings are consistent with the notion that the linguis-tic process is involved in both forward andbackward counting, whereas the spatial processunderlies backward but not forward counting (Ho-shi et al., 2000; Zhou et al., 2006). The findings,however, need further replication, as the reliabilitywas relatively low for the forward counting tasks.

It is noteworthy that counting sequence knowl-edge partially mediated the relations of letterknowledge and spatial visualization with the levelof arithmetic. In contrast, it fully mediated theirrelations with the growth of arithmetic. Presum-ably, other number skill mediators, such as numberrecognition (Cirino, 2011) and knowledge of numer-

ical magnitude (LeFevre et al., 2010), are alsoinvolved in the mechanism that underlies the devel-opmental level of arithmetic. Future research shouldexplore this possibility.

General Discussion

Although a growing body of research has been con-ducted to examine various types of cognitive (e.g.,linguistic, spatial, and numeracy) precursors ofarithmetic development, few attempts have beenmade to examine these precursors simultaneouslyby using multiwave longitudinal data. This researchwas designed to assess the relations between a vari-ety of linguistic, spatial, and number skill precur-sors in the prediction of the level and growth ofarithmetic competence in the early years of school-ing. The results showed that linguistic and spatialskills, measured in kindergarten, had unique andindependent impacts on both the level and growthof arithmetic from first to third grades. Moreover,the results further showed that these effects weremediated by counting sequence knowledge.

Among the three linguistic measures (i.e., phone-mic awareness, receptive vocabulary, and letterknowledge) examined in our studies, letter knowl-edge was found to be of particular importance forthe development of arithmetic. When letter knowl-edge was excluded from the model, phonemicawareness predicted the initial level of arithmetic inboth Study 1 and Study 2, and receptive vocabularypredicted the initial level in Study 1. This finding isgenerally consistent with the extensive literaturelinking phonological awareness (for a review, seeSimmons & Singleton, 2008) and vocabulary (e.g.,LeFevre et al., 2010) with the level of arithmeticcompetence. Notably, however, neither of the spo-ken linguistic skills predicted arithmetic develop-ment when letter knowledge was taken intoaccount. Instead, letter knowledge significantly pre-dicted both the level of arithmetic in the fall of firstgrade and the growth of arithmetic from first tothird grades. Taken together, these results under-score the importance of written (rather than spo-ken) linguistic skills for arithmetic development.

There might be several explanations for the asso-ciation between letter knowledge and the develop-ment of arithmetic. First, early exposure to writtenletters may provide children with the ability todecode, comprehend, and manipulate other writtensymbols, such as written numbers and arithmeticoperators. For example, the ability to map soundsto print may be improved by early experience with

1102 Zhang et al.

the written language system. Hale and Fiorello(2004) have suggested that such ability is critical forarithmetic learning. Second, a more general sym-bolic capacity may underlie the acquisition of bothlinguistic and numerical written symbols. In otherwords, such symbolic capacity may aid learning ofvarious written symbols, including letters, numbers,and operators. Nevertheless, this capacity may alsobe malleable and be influenced by early experiencewith letters and numbers. Third, children mayacquire higher mental capacities (e.g., focused atten-tion, deliberate memory) through using a writtenlanguage as a specific cultural tool. According toVygotsky (1978), the higher mental capacities arecrucial for cognitive development across domains.A number of recent studies have also demonstratedthat such capacities predict arithmetic competencein the early years of schooling (e.g., No€el, 2009).Further studies are needed to clarify how a child’swritten system knowledge forms an important cog-nitive foundation for his or her arithmetic develop-ment.

Our studies also provide evidence for the impor-tance of spatial skills for arithmetic development.Spatial visualization in kindergarten positively pre-dicted the level and growth of arithmetic from firstto third grades. That is, children with stronger spa-tial visualization ability in kindergarten had higherarithmetic competence upon entry into elementaryschool, and also later showed a faster rate ofgrowth in arithmetic. This finding is consistent withan increasing body of research linking arithmeticcompetence with spatial skills in general (e.g., LeFe-vre et al., 2010) and spatial visualization in particu-lar (e.g., Barnes et al., 2011). It also extendsprevious work by demonstrating the predictivepower of spatial visualization ability for the growthof arithmetic in the early elementary years.

In our studies, letter knowledge and spatial visu-alization both showed a unique contribution to thelevel and growth of arithmetic. In addition, the cor-relations between spatial visualization and linguis-tic precursors were low (rs ≤ .30). These findingsemphasize the separation of the influences of earlylinguistic and spatial skills on later arithmeticdevelopment. Consequently, they are consistentwith recent neuropsychological perspectives onnumerical processing (Dehaene, 2011; Dehaeneet al., 2004) and recent findings linking linguisticand spatial skills independently to arithmetic com-petence (Cirino, 2011; Krajewski & Schneider, 2009;LeFevre et al., 2010). It seems that linguistic andspatial abilities are two separate domain-general,fundamental underlying skills that have critical

importance for the growth of arithmetic during chil-dren’s early years of schooling.

Notably, in our studies, linguistic and spatialskills explained a meaningful, but small, proportionof the variance of the growth of arithmetic.Although the inclusion of counting sequenceknowledge as a mediator largely increased the pro-portion of the variance explained, future studies areneeded to examine the role of more fundamentalskills (e.g., quantitative skill) that may contribute toarithmetic development. Indeed, Dehaene’s (Dehae-ne, 2011; Dehaene et al., 2004) neuropsychologicalmodel of numerical processing involves three sepa-rate neural circuits that are linked to three forms ofcognitive processing, namely, linguistic, spatial, andquantitative knowledge. It would be important infuture research on the development of arithmetic totest this model by using multiwave longitudinaldata and incorporating all three forms of cognitiveprecursors (see LeFevre et al., 2010).

It is also important to note that the initial levelof arithmetic, measured in the fall of first grade,was weakly related to later growth through thirdgrade, which was against our expectation. This isprobably due to the fact that a time-limited mea-sure was used to assess arithmetic competence. Inthis research the accuracy of children’s perfor-mance in arithmetic increased largely from first tothird grades. It thus seems that the arithmetic mea-sure used here indexed accuracy of performancemore at the earlier time points and speed of per-formance more at the later time points. It wouldtherefore be valuable in future studies to replicatethis research using time-unlimited measures ofarithmetic.

It was assumed in this research that countingsequence knowledge would capture children’sknowledge about the symbolic number systemwhen they have not yet received much formalinstruction. Consistent with findings from recentlongitudinal studies (e.g., Aunola et al., 2004),counting sequence knowledge was strongly predic-tive of the level and growth of arithmetic. In addi-tion, in accordance with recent research linkingnumber skills with linguistic (e.g., Koponen et al.,2007) and spatial (e.g., Cirino, 2011) precursors,counting sequence knowledge was predicted signif-icantly by both linguistic and spatial skills. Moreimportant, this research made a meaningful connec-tion between these two lines of research, suggestingthat counting sequence knowledge mediated thelinks between linguistic and spatial skills and arith-metic development. Specifically, counting sequenceknowledge mediated the linkages from letter

Early Arithmetic Development 1103

knowledge and spatial visualization to both thelevel and growth of arithmetic.

These findings underscore previous indicationsthat the effect of early fundamental skill precursorson basic arithmetic competence operates viaenhancing children’s number system knowledge. InStudy 2, the direct effects of the skill precursorswere reduced significantly when counting sequenceknowledge was added to the model. This pattern ofresults is in accordance with prior findings in whichlinguistic and spatial precursors have explained avery small part of the variance of arithmeticoutcomes when used as predictors along with num-ber skills (Aunola et al., 2004; Koponen et al., 2007).Fuchs et al. (2010) found that as much as over 60%of the explained variance was shared by numerosi-ty measures and domain-general factors, includinglanguage and spatial skills, although they did notconceptualize the numerosity measures as anunderlying mediation process. Two recent studies(Cirino, 2011; Krajewski & Schneider, 2009) furtherdemonstrated that linguistic (i.e., phonologicalawareness) and spatial (i.e., visuospatial workingmemory) skills showed significant associations witharithmetic outcomes, and that these associationswere no longer significant when early number skillswere added to the model. Hence, this research,along with these earlier studies, highlights the needto understand these domain-general and domain-specific precursors of arithmetic competence in acomprehensive mediation model (Krajewski &Schneider, 2009; LeFevre et al., 2010).

The finding that letter knowledge predictedarithmetic development via counting sequenceknowledge could be explained in light of the theorythat higher mental capacities (e.g., focused attentionand deliberate memory) are a stepping stone fromknowledge of written symbols to knowledge ofcounting sequences. As stated previously, thehigher mental capacities can be acquired throughearly experience with a written language (Vygotsky,1978). These capacities have been demonstrated tobe crucial for fluent and accurate counting and cal-culations among young children (e.g., No€el, 2009).Nevertheless, the correlation between letter knowl-edge and counting sequence knowledge is alsoprobably due to the fact that both tasks require theproduction of verbal codes. The finding that spatialvisualization predicted arithmetic development,mediated also by counting sequence knowledge,can be explained in accordance with the theory thatnumbers are organized spatially in the mind (Deh-aene, 2011; Dehaene et al., 2004). Interestingly, therelation between spatial visualization and arithme-

tic was mediated by backward but not forwardcounting knowledge. This finding is consistent withthe literature suggesting that the spatial processunderlies backward but not forward counting (Ho-shi et al., 2000; Zhou et al., 2006). Possibly, forwardcounting involves only the process of producingverbal numbers, at least for first graders and olderchildren. By contrast, backward counting mayinvolve a multistep process that requires verbalproduction and updating of working memory(especially visuospatial working memory), the latterof which is also needed in the spatial visualizationtasks. Thus, children with higher spatial visualiza-tion ability can use their visuospatial workingmemory resources to support their memorizationand counting of a backward number sequence.

Certain limitations of our studies should be notedbefore making any generalizations based on theresults. First, counting sequence knowledge, whichrepresents only one aspect of children’s knowledgeof the symbolic number system, was the only num-ber skill mediator that was tested in the studies. Ithas been suggested previously that other aspects ofsymbolic number knowledge, such as number recog-nition (Cirino, 2011) and comparisons (Krajewski &Schneider, 2009), and aspects of nonsymbolic num-ber knowledge, such as numerical magnitude pro-cessing (Gunderson et al., 2012; LeFevre et al., 2010),may also be involved in the mediation process.Second, arithmetic competence was measured solelyby a paper-and-pencil factual and procedural com-putational task. Future research could include otherarithmetic domains, such as word problems, andinvestigate whether growth of competence in thosedomains can also be predicted by linguistic andspatial precursors and mediated by number skills.Finally, it remains to be seen whether our results forFinnish children can be generalized to children else-where, although there is no evidence to suggest thatthere are any features specific to the Finnish lan-guage that would confine the results to the Finnishcontext. For example, the Finnish number–word sys-tem has a clear base-ten structure from 20 onward,which is similar to the number word systems inmany other languages, such as English. AlthoughFinnish is transparent with a high grapheme–pho-neme correspondence, growing evidence suggeststhat similar cognitive mechanisms underlie theacquisition of writing systems of varying ortho-graphic complexity (Landerl et al., 2013). Hence, weexpect that the relations revealed here can be gener-alized to other contexts. Nevertheless, future studiesthat include samples from other language and writ-ing systems are needed to broaden our understand-

1104 Zhang et al.

ing of the relations between linguistic and numericalsymbolic systems.

In conclusion, our studies provided support forthe view that linguistic and spatial skills are impor-tant for early arithmetic development. Moreover,these fundamental precursors of arithmetic oper-ated via children’s counting sequence knowledge,which further predicted the development of arith-metic. These findings carry important theoreticaland applied implications. First, our studies addresstheorizing about the role of language in arithmetic.Drawing on neuropsychological research, Dehaene(2011; Dehaene et al., 2004) proposes that linguisticprocesses are involved in the acquisition of arithme-tic. In prior studies, however, “linguistic processes”have nearly always been operationalized usingmeasures of spoken language and thus have beenunable to assess processes related to written lan-guage. In an effort to extend this earlier work, ourstudies examined the role of both spoken and writ-ten language in the development of arithmetic. Theresults indicate that letter knowledge, or the abilityto deal with components of the written languagesystem, such as letters in an alphabetic system ofwriting, has a substantial impact on the level andgrowth of arithmetic competence in the early schoolyears. More important, this impact is above andbeyond the effect of the linguistic skills relevant tounderstanding the oral language system, includingphonological awareness and receptive vocabulary.

Second, in this research, different types of earlyskill precursors, that is, linguistic, spatial, and num-ber skills, were evaluated simultaneously in the con-text of a comprehensive mediation model ofarithmetic development (Krajewski & Schneider,2009; LeFevre et al., 2010). To our knowledge, thisresearch is the first endeavor to test this model in amultiwave longitudinal sample using a growth-curve approach, thereby allowing the level andgrowth of arithmetic to be studied separately. Ourfindings showed that linguistic and spatial skillsindependently predicted both the level and growthof arithmetic via the enhancement of number skills.This is crucial, as linguistic, spatial, and numberskills are all malleable and can be improved consid-erably by appropriate early experiences (Elbro &Petersen, 2004; Siegler & Ramani, 2008; Uttal et al.,2013). An important question for future research iswhether training in spatial and linguistic skills—forexample, spatial visualization and knowledge ofwritten symbols—would improve subsequent num-ber-related knowledge and, in turn, the develop-ment of arithmetic. It is especially important todevelop and test such interventions among young

children who have not yet received formal instruc-tion, and thereby obtain information of value inplanning early childhood education.

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