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The Cognitive and Numerical Predictors of Early Mathematical Achievement: A Latent Growth Curve Analysis

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Title: The Cognitive and Numerical Predictors of Early Mathematical Achievement: A Latent Growth Curve Analysis
Language: English
Authors: Abbie Cahoon (ORCID 0000-0001-7587-6670), Emine Simsek (ORCID 0000-0003-1679-1276), Camilla Gilmore (ORCID 0000-0002-5879-2683), Victoria Simms (ORCID 0000-0001-5664-6810)
Source: Journal of Cognition and Development. 2025 26(3):443-463.
Availability: Routledge. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 21
Publication Date: 2025
Document Type: Journal Articles
Reports - Research
Education Level: Elementary Education
Preschool Education
Early Childhood Education
Descriptors: Mathematics Achievement, Preschool Children, Elementary School Students, Individual Differences, Achievement Gains, Child Development, Foreign Countries, Mathematics Skills, Predictor Variables
Geographic Terms: United Kingdom
Assessment and Survey Identifiers: British Ability Scales
DOI: 10.1080/15248372.2024.2434036
ISSN: 1524-8372
1532-7647
Abstract: Longitudinal studies are essential for understanding causes of developmental change and growth rates of mathematical achievement. One hundred and twenty-eight UK-based children (M[subscript age] = 4 years; SD[subscript age] = 3.3 months; age range 43-54 months; 70 female) were tracked for 15 months, from the beginning of preschool until the end of the first year of primary school (i.e., across 7 preschools to 18 primary schools) and were assessed at three time points. At the beginning of preschool, data were collected from parents and children, including background demographics, domain-specific mathematical skills, domain-general cognitive skills, and language skills. Mathematical achievement was assessed once during preschool and at two time points during the first year of primary school. Using a latent growth model, we examined the contribution of the predictors to the growth patterns in mathematical achievement and the stability of initial individual differences during preschool to school transitions. Results showed that over a period of 15-months, children displayed substantial growth in mathematical achievement. This growth in mathematical achievement was linear and there was little variability in children's rate of development. In contrast, there was substantial variance in initial mathematical achievement, and this variance was explained by children's cardinality understanding and receptive vocabulary. These early variations highlight the importance of exposure to mathematical language and concepts in early childhood to ensure the development of broader mathematical skills.
Abstractor: As Provided
Entry Date: 2026
Accession Number: EJ1494947
Database: ERIC
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  Value: <anid>AN0186129863;7m701may.25;2025Jun26.02:29;v2.2.500</anid> <title id="AN0186129863-1">The Cognitive and Numerical Predictors of Early Mathematical Achievement: A Latent Growth Curve Analysis </title> <sbt id="AN0186129863-2">Introduction</sbt> <p>Longitudinal studies are essential for understanding causes of developmental change and growth rates of mathematical achievement. One hundred and twenty-eight UK-based children (M<sub>age</sub> = 4 years; SD<sub>age</sub> = 3.3 months; age range 43–54 months; 70 female) were tracked for 15 months, from the beginning of preschool until the end of the first year of primary school (i.e., across 7 preschools to 18 primary schools) and were assessed at three time points. At the beginning of preschool, data were collected from parents and children, including background demographics, domain-specific mathematical skills, domain-general cognitive skills, and language skills. Mathematical achievement was assessed once during preschool and at two time points during the first year of primary school. Using a latent growth model, we examined the contribution of the predictors to the growth patterns in mathematical achievement and the stability of initial individual differences during preschool to school transitions. Results showed that over a period of 15-months, children displayed substantial growth in mathematical achievement. This growth in mathematical achievement was linear and there was little variability in children's rate of development. In contrast, there was substantial variance in initial mathematical achievement, and this variance was explained by children's cardinality understanding and receptive vocabulary. These early variations highlight the importance of exposure to mathematical language and concepts in early childhood to ensure the development of broader mathematical skills.</p> <p></p> <hd id="AN0186129863-3">Importance of mathematics and understanding developmental processes in early childhood</hd> <p>Childhood experiences during the first 5 years of life influence many aspects of a child's mathematical development (Davis‐Kean et al., [<reflink idref="bib14" id="ref1">14</reflink>]). These early learning experiences at home and school can have considerable consequences and long-lasting effects throughout childhood, adolescence, and adulthood (Davis‐Kean et al., [<reflink idref="bib14" id="ref2">14</reflink>]; Hoff, [<reflink idref="bib33" id="ref3">33</reflink>]; World Bank, [<reflink idref="bib79" id="ref4">79</reflink>]). For instance, low mathematical performers are more at risk of poverty in later life and tend to have poorer health (Murray, [<reflink idref="bib53" id="ref5">53</reflink>]; OECD, [<reflink idref="bib56" id="ref6">56</reflink>]; World Bank, [<reflink idref="bib78" id="ref7">78</reflink>]). Furthermore, children with low levels of mathematics skills at school entry display low growth in mathematical achievement during school transitions and throughout schooling (Davis‐Kean et al., [<reflink idref="bib14" id="ref8">14</reflink>]), whereas children who entered preschool with foundational mathematics skills were more likely to succeed in school (Dowker, [<reflink idref="bib18" id="ref9">18</reflink>]; Duncan et al., [<reflink idref="bib21" id="ref10">21</reflink>]; W. Holmes & Dowker, [<reflink idref="bib35" id="ref11">35</reflink>]). Hence, there are clear differences in children's mathematical skills at school entry, prior to the beginning of formal schooling (Davis‐Kean et al., [<reflink idref="bib14" id="ref12">14</reflink>]). However, these studies typically focus on children's achievement at the start of formal schooling and how it predicts later development (e.g., Davis-Kean & Jager, [<reflink idref="bib15" id="ref13">15</reflink>]; Davis‐Kean et al., [<reflink idref="bib14" id="ref14">14</reflink>]; Kuhfeld et al., [<reflink idref="bib42" id="ref15">42</reflink>]; Paschall et al., [<reflink idref="bib57" id="ref16">57</reflink>]). Less is known about developmental processes in the early years (i.e., 0 to 5 years old) that occur independently of school and how this may lead to the individual differences observed at the start of schooling using longitudinal research. herefore, this research is novel in this empirical approach with regards to this early age group (i.e., 3–5 years old).</p> <hd id="AN0186129863-4">Stability of initial individual differences and lack of longitudinal studies in early years</hd> <p>A common feature of these predictive longitudinal studies (e.g., Davis-Kean & Jager, [<reflink idref="bib15" id="ref17">15</reflink>]; Davis‐Kean et al., [<reflink idref="bib14" id="ref18">14</reflink>]) is the stability of initial individual differences over time. Davis‐Kean et al. ([<reflink idref="bib14" id="ref19">14</reflink>]) found that after controlling for key demographic characteristics (i.e., background and home environment factors) the numeric competency (i.e., counting, concrete representational arithmetic, and abstract arithmetic operations) that children master prior to school entry (i.e., 54 months of age in the USA) related to educational transitions (i.e., course selection and applying to college) in secondary and post-secondary education. Therefore, there is stability of initial mathematical competencies on later academic development for individuals prior to school entry. Most studies focus on children who are receiving educational input in kindergarten or school and later trajectory in academic achievement (i.e., 5-year-olds and above: Davis-Kean & Jager, [<reflink idref="bib14" id="ref20">14</reflink>]; Kuhfeld et al., [<reflink idref="bib42" id="ref21">42</reflink>]; Paschall et al., [<reflink idref="bib57" id="ref22">57</reflink>]). For instance, Davis-Kean and Jager ([<reflink idref="bib15" id="ref23">15</reflink>]) investigated achievement trajectories for children in kindergarten through to Grade 5 within different race/ethnicity groups and found that some groups of children (classified by high and low achieving groups within different race/ethnicity groups) caught up to the highest achieving groups by Grade 5. This highlights the positive impact of schooling on subgroups of children. Thus, achievement gaps can either grow or shrink as children progress through school due to various reasons (Davis-Kean & Jager, [<reflink idref="bib15" id="ref24">15</reflink>]; Kuhfeld et al., [<reflink idref="bib42" id="ref25">42</reflink>],; Reardon, [<reflink idref="bib62" id="ref26">62</reflink>]). However, to understand the etiology of these individual differences in mathematical achievement we need to examine development at an earlier age (i.e., broadly 3–5-year-olds).</p> <hd id="AN0186129863-5">Growth during early school transitions</hd> <p>A key aspect of development during this early period is the transition from early years settings to more formal educational settings. There are few longitudinal studies tracking children's development during this crucial transition period (i.e., preschool to primary school; 3–5 years) with regards to different mathematical component skills and mathematical development (Cahoon et al., [<reflink idref="bib11" id="ref27">11</reflink>]), and the relation between cognitive skills and mathematical development (Xenidou‐Dervou et al., [<reflink idref="bib81" id="ref28">81</reflink>]). Given age differences across different educational contexts, much research with children at the preschool to school transition involve children of 5 years or older. Geary et al. ([<reflink idref="bib28" id="ref29">28</reflink>]) investigated the quantitative competencies of 141 children (mean age 3 years, 10 months at first assessment) over 2 years and found that cardinality and age of acquisition of cardinality were key to success in later mathematical development. Yet, there is still a relative lack of longitudinal research tracking children's development during this time period. Therefore, a dynamic framework of longitudinal research is necessary for a more complete understanding of early developmental processes and individual differences in mathematical achievement (Cahoon et al., [<reflink idref="bib11" id="ref30">11</reflink>]; Molenaar, [<reflink idref="bib51" id="ref31">51</reflink>]; Nesselroade & Schmidt McCollam, [<reflink idref="bib54" id="ref32">54</reflink>]). An approach which enables the exploration of developmental processes is growth analyses, in which the initial status, rates of growth and the shape of the growth trajectory are of great importance (McCoach et al., [<reflink idref="bib49" id="ref33">49</reflink>]).</p> <p>Most evidence into the development of mathematical skills are with older children in school and are cross-sectional in design (e.g., Sowinski et al., [<reflink idref="bib69" id="ref34">69</reflink>]). In response to this, our previous longitudinal research (i.e., Cahoon et al., [<reflink idref="bib11" id="ref35">11</reflink>]) investigated multiple component numeric skills (i.e., cardinal principle knowledge, mapping between symbols and quantities, digit recognition, and order processing) within a preschool population (i.e., age range 43–54 months at Time 1), examining developmental pathways of mathematical learning during the transition from preschool to formal schooling. Five developmental pathways of mathematical learning were found with children in the low number skill pathway showing a lower rate of growth than more advanced pathways with some groups of children displaying a more rapid shift in pathway membership upon school-entry possible due to higher receptive vocabulary scores. Further, children in the advanced number skills pathway had higher working memory than children in the other four lower number skill pathways. Additionally, there was an association between working memory scores and the school-entry transition pathways where those that made gains at school entry had higher working memory scores. This study (i.e., Cahoon et al., [<reflink idref="bib11" id="ref36">11</reflink>]) is unique as it focuses on multiple component numeric skills development over early transitions using a longitudinal design, however, it did not consider growth in children's overall mathematics achievement or identify predictors of this.</p> <hd id="AN0186129863-6">Predictors of mathematical achievement</hd> <p>Evidence from cross-sectional studies have identified a set of mathematics component skills (Bisanz et al., [<reflink idref="bib7" id="ref37">7</reflink>]; Dowker, [<reflink idref="bib19" id="ref38">19</reflink>]), cognitive skills (Krajewski & Schneider, [<reflink idref="bib41" id="ref39">41</reflink>]; LeFevre et al., [<reflink idref="bib43" id="ref40">43</reflink>]), characteristics of the parents (e.g., highest education, socio-economic status, income, family size, marital status) and/or home environment (e.g., reading, number of books, visits to the library) that are fundamental factors for early mathematical achievement (Duncan et al., [<reflink idref="bib20" id="ref41">20</reflink>]; Guo & Harris, [<reflink idref="bib30" id="ref42">30</reflink>]). However, there is a lack of studies that bring together groups of domain-specific skills and domain-general skills that predict mathematical achievement longitudinally (Cahoon et al., [<reflink idref="bib11" id="ref43">11</reflink>]; Xenidou‐Dervou et al., [<reflink idref="bib81" id="ref44">81</reflink>]).</p> <p>Some basic numerical processing abilities may be more important in the development of early mathematics skills during preschool (i.e., 3–5-years-old). Hodgen et al. ([<reflink idref="bib32" id="ref45">32</reflink>]) offers a theoretical framework by summarizing the current literature on the role of different component numeric skills in the development of early mathematics skills. Four component numeric skills capture distinct elements and offer a window of time in mathematical skill development (i.e., cardinal principle knowledge, mapping between symbols and quantities, digit recognition, and order processing; Cahoon et al., [<reflink idref="bib11" id="ref46">11</reflink>]; Hodgen et al., [<reflink idref="bib32" id="ref47">32</reflink>]). These four skills have been shown to develop in close proximity, in a hierarchy of steps and in overlapping ways; reinforcing and influencing one another across development (Cahoon et al., [<reflink idref="bib11" id="ref48">11</reflink>]; Dowker, [<reflink idref="bib19" id="ref49">19</reflink>]; Hodgen et al., [<reflink idref="bib32" id="ref50">32</reflink>]; Sarnecka & Carey, [<reflink idref="bib64" id="ref51">64</reflink>]). They have also been shown to be predictors of early mathematical skills (e.g., mapping magnitude: Attout et al., [<reflink idref="bib3" id="ref52">3</reflink>]; cardinality: Geary et al., [<reflink idref="bib28" id="ref53">28</reflink>]; numerical ordering: Lyons et al., [<reflink idref="bib45" id="ref54">45</reflink>]) and will be used within this study as predictors of mathematical achievement. To represent children's understanding of the counting system, cardinal principle knowledge was the chosen factor. To capture children's proficiency with semantic bases of number processing, mapping and order processing tasks were chosen (Batchelor et al., [<reflink idref="bib6" id="ref55">6</reflink>]; Lyons et al., [<reflink idref="bib45" id="ref56">45</reflink>]). Digit recognition was also a chosen measure, as it is crucial for children to apply their understanding of number knowledge to Arabic digits and abstract arithmetic (Condry & Spelke, [<reflink idref="bib13" id="ref57">13</reflink>]; Wright et al., [<reflink idref="bib80" id="ref58">80</reflink>]).</p> <p>Additionally, two cognitive skills (i.e., verbal working memory and auditory sustained attention) and receptive vocabulary will also be used as predictors of mathematical achievement. Younger children's executive functions (i.e., verbal, and visual working memory) have been best explained as a unitary construct (Hughes et al., [<reflink idref="bib37" id="ref59">37</reflink>]; Schmitt et al., [<reflink idref="bib67" id="ref60">67</reflink>]) that becomes differentiated as children get older (Huizinga et al., [<reflink idref="bib38" id="ref61">38</reflink>]; Lehto et al., [<reflink idref="bib44" id="ref62">44</reflink>]) hence, verbal working memory was chosen. Verbal working memory plays an important role in explaining individual differences of mathematical learning both for children with mathematical disabilities (e.g., Passolunghi & Siegel, [<reflink idref="bib58" id="ref63">58</reflink>]; H. L. Swanson & Beebe-Frankenberger, [<reflink idref="bib71" id="ref64">71</reflink>]; Van Der Sluis et al., [<reflink idref="bib75" id="ref65">75</reflink>]) and typically developing children (i.e., J. Holmes & Adams, [<reflink idref="bib34" id="ref66">34</reflink>]; L. Swanson & Kim, [<reflink idref="bib72" id="ref67">72</reflink>]). While research relating attention with later academic achievement are uncommon, there is consistent evidence that sustained attention and participation in classroom activities predicts achievement during pre-school and early school years (Alexander et al., [<reflink idref="bib1" id="ref68">1</reflink>]; Raver et al., [<reflink idref="bib61" id="ref69">61</reflink>]). These correlational studies on working memory and mathematical achievement have been cross-sectional in design, with longitudinal studies lacking, which does not allow examination into the nature of the relationship (DeSmedt et al., [<reflink idref="bib17" id="ref70">17</reflink>]). Therefore, verbal working memory and sustained attention will be used as a longitudinal predictor of mathematical achievement within this study.</p> <hd id="AN0186129863-7">The current study</hd> <p>Longitudinal studies are essential for understanding causes of developmental change and growth rates in predicting mathematical outcomes. This longitudinal research study aimed to explore how young children's mathematical achievement evolves during the transition from preschool to primary school. Specifically, this study examined changes in children's mathematical skills over 15 months, focusing on how their initial mathematical knowledge contributes to their subsequent development. By identifying which numerical and cognitive skills – such as cardinality understanding, digit recognition, and receptive vocabulary – predict growth in mathematical achievement, this study sought to provide a deeper understanding of the factors that drive early mathematical development and the impact of schooling on this growth. The research questions of the study were as follows:</p> <hd id="AN0186129863-8">RQ1:</hd> <p>How does children's mathematical achievement change across time?</p> <hd id="AN0186129863-9">RQ2:</hd> <p>Does children's initial mathematical knowledge relate to development of their mathematical achievement across time?</p> <hd id="AN0186129863-10">RQ3:</hd> <p>Which cognitive skills predict children's initial mathematical knowledge and their growth in mathematical achievement over time?</p> <p>To extend the body of literature on early mathematical development, previously identified predictors (i.e., age, cardinal principle knowledge, digit recognition, numeric ordering, symbolic-quantity mapping, verbal working memory, auditory sustained attention, and receptive vocabulary) of mathematical achievement were investigated to identify predictors of growth patterns in mathematical achievement. At the beginning of preschool, data were collected on children's domain-specific mathematical skills, domain-general cognitive skills, and language skills, as well as background demographics from parents/guardians. Preschool children were tracked for 15 months until the end of primary one (the first year of primary school), and their mathematical achievement was assessed at three-time points: at the beginning of preschool (T1), at the beginning of primary one (T2), and at the end of primary one (T3). Using a latent growth model, children's mathematical achievement was modeled to examine the contribution of the predictors to the growth patterns in mathematical achievement.</p> <p>It was initially hypothesized that there would be a linear growth (slope) in mathematical achievement across time and that there would be a relationship between a child's initial status (intercept) and the growth of mathematical achievement over time. This study allowed comparison of rates of mathematical achievement growth during the transition from preschool to the beginning of school (i.e., from T1 to T2) and during the school year (i.e., from T2 to T3). The first time-period (i.e., from T1 to T2) included preschool which involves no direct mathematics instruction in the UK, and the summer period when children do not usually attend school. The second time-period (i.e., from T2 to T3) was when direct mathematics instruction began. Comparing the rates of mathematical achievement growth will aid understanding if there was a "schooling effect." If rates of growth were steeper during the school year, we could conclude that schooling facilitates the development of certain mathematical skills.</p> <hd id="AN0186129863-11">Method</hd> <p>This research is part of a larger study (i.e., Cahoon et al., [<reflink idref="bib11" id="ref71">11</reflink>]) which involved tracking children at four time points over 15-months encompassing the transition from preschool through to their first year of primary education. Sufficient information is provided within the method for the aims of this study however, for more detail on the methods used see Cahoon et al. ([<reflink idref="bib11" id="ref72">11</reflink>]).</p> <hd id="AN0186129863-12">Ethical considerations</hd> <p>This research project was approved by Ulster BLINDEDUniversity Research Ethics Committee before the study commenced, an additional data collection point (i.e., included in this study called T3) was approved during the project but obtained before T3 data collection commenced. Signed consent was obtained from all parents/guardians for their children to participate in the original study, and additional consent was necessary and obtained from all parents/guardians before T3 data collection took place. All children provided assent before the researcher began to administer the tasks at all time points. Our ethics approval did not include consent for public sharing of children's data. Therefore, the data are not publicly available but further information is available on request.</p> <hd id="AN0186129863-13">Participants</hd> <p>One hundred and fifty-two parents consented for their child's participation in the study and a total of 136 parents (124 mothers and 12 fathers) completed a demographic questionnaire at T1. Children were assessed at three time points over a 15-month period from preschool to primary school, comprising one time point in preschool and two in primary school at the beginning and end of the school year. A total of 128 children (<emph>M</emph><subs>age</subs> = 4 years; <emph>SD</emph><subs>age</subs> = 3.3 months; age range 43–54 months; 70 female, 58 male) completed the tasks at T1. 118 children (65 female) completed the tasks at T2, and 106 children (59 female) completed the tasks at T3. There was a 17.2% attrition of children from T1 to T3.</p> <hd id="AN0186129863-14">Procedure</hd> <p>Ten primary schools in the UK that had preschool provision were contacted and invited to take part. Seven preschools (Urban = 5; Rural = 2; Controlled (i.e., managed and funded by the Education Authority through school Boards of Governors) = 7) accepted the invitation, giving a potential recruitment pool of approximately 341 dyads. At T1, 7 preschools were involved, and during the transition 11 primary schools were recruited to continue tracking children who were moving to those schools to give a total of 18 primary schools at T2 and T3.</p> <p>At T1, parents/guardians completed and returned a demographics questionnaire to the preschool teacher to be collected by the researcher. This study is part of a larger study on mathematical development which involved 4 time points. In this paper we call the time points T1, T2 and T3; however, there was an additional time point between T1 and T2 not included in this analysis, as mathematical achievement was not measured at this time point. The tasks measured included four domain-specific skills (i.e., cardinal principle knowledge, digit recognition, numeric ordering, and symbolic-quantity mapping), two domain-general skills (i.e., verbal working memory and auditory sustained attention), receptive vocabulary and general mathematical achievement (measured by the British Ability Scale (BAS)). At T1 and T2, children completed two 20-min sessions and were administered all measures. At T3, the children only completed the BAS measure.</p> <hd id="AN0186129863-15">Measures</hd> <p>For more details of each measure see Cahoon et al. ([<reflink idref="bib11" id="ref73">11</reflink>]). Proportion correct scores were used for the four component numeric skills as proportion scores offer unidimensional scores and demonstrate convergent validity (Barchard & Russell, [<reflink idref="bib4" id="ref74">4</reflink>]). For both the receptive vocabulary and mathematical achievement measures, the correct responses were totaled to give a raw score for each participant as opposed to age-equivalent scores or standardized scores as age will be used as a predictor variable.</p> <hd id="AN0186129863-16">Digit recognition</hd> <p>Children were presented with 2 blocks of 6 Arabic digits (i.e., single digit numbers; order 3, 2, 5, 8, 7, 9 and double digits; order 12, 14, 11, 16, 20, 18) and asked, "What number is this?." When children made a mistake with four or more numbers in one block, the task was stopped, and a proportion correct score out of 12 was calculated for each participant. The Cronbach's alpha for this task was 0.91 in Batchelor et al. ([<reflink idref="bib6" id="ref75">6</reflink>]).</p> <hd id="AN0186129863-17">Cardinal principle knowledge</hd> <p>The Give-N task was used to measure cardinal principle knowledge and involved 18 plastic counters placed in front of the child (see Cahoon et al., [<reflink idref="bib11" id="ref76">11</reflink>] for diagram). The child was asked to give a puppet a given number (i.e., 3, 4, 6, 11 and 15, with each number being requested up to three times) of counters by placing the counters on a plate. For each correct response children were awarded one point. A proportion correct score out of 15 was calculated for each participant. The Cronbach's alpha was 0.76 in Batchelor et al. ([<reflink idref="bib6" id="ref77">6</reflink>]) which used a similar measure.</p> <hd id="AN0186129863-18">Numeric ordering</hd> <p>Three cards with Arabic digits were presented out of order (i.e., four consecutives (e.g., 8, 7, 9), gaps of 2 (e.g., 5, 9, 7), and gaps of 3 (e.g., 1, 7, 4); numbers counterbalanced) to children who were then asked to put the numbers in the right order. Four practice trials were given, followed by 12 experimental trials. Corrections were given only during the four practice trials and there was no time limit in place. A proportion correct score out of 12 was calculated for each participant. The Cronbach's alpha for this task was 0.48, as reported in Batchelor et al. ([<reflink idref="bib6" id="ref78">6</reflink>]).</p> <hd id="AN0186129863-19">Symbolic-quantity mapping</hd> <p>Symbolic-quantity mapping was measured by the cross-notation comparison task. Children were instructed to choose between two characters (i.e., duck and frog) who had the most balls. One of the characters' balls were presented as a non-symbolical dot array and the other characters' balls were presented as a verbal number word with the 12 trials, numerosities used ranged between 1 and 15, being counterbalanced. The child was required to directly assess mapping between magnitude representations without counting and children responded by pointing and/or naming the character. A proportion correct score was calculated for each participant out of 12. The Cronbach's alpha for this task was 0.64, as reported in Batchelor et al. ([<reflink idref="bib6" id="ref79">6</reflink>]).</p> <hd id="AN0186129863-20">Verbal working memory</hd> <p>Verbal working memory was measured using an animal recall task (McCormack et al., [<reflink idref="bib50" id="ref80">50</reflink>]) which involved alternating pictures of animal cards and smiley faces. The children were asked to name the colors of the smiley faces as the cards were presented. Twenty-two animals were used, and at the end of each set shown the children were asked to recall the animals in the correct order. The trials involved five levels of increasing difficulty with four sets of one-through five-animals. The task was stopped when the child failed to recall all the animals at a level for the four trials. A correct response was classified as an animal recalled in the correct position. Children were awarded one point and the accuracy score was used in the analysis. The maximum score was 60 however, no child scored above 20 therefore, a total accuracy score was used instead of a proportion score as a proportion score would be heavily skewed if used.</p> <hd id="AN0186129863-21">Auditory sustained attention</hd> <p>An adapted computerized version of the Auditory Continuous Performance Test-Preschool (ACPT-P; Mahone et al., [<reflink idref="bib46" id="ref81">46</reflink>]) was used in this study. The task included 44 trials consisting of 22 non-targets (i.e., cat meowing) and 22 targets (i.e., dog barking), where participants were required to press the spacebar for the target sound and to inhibit their response for the non-target. The stimuli were presented for 1 seconds, with 4 seconds to respond. The children were seated in front of a laptop and at different intervals heard a dog barking and cat meowing. Children were given a familiarization and practice phase, to confirm they understood the instructions. All participants successfully completed the practice trials before continuing to the test phase. The accuracy score, total omission, total commission (first three scores had a maximum score of 22) and response rate (ranged from 0 to 4 s) were calculated. An inverse efficiency score was used to enable the combination of speed and error metrics (Townsend & Ashby, [<reflink idref="bib73" id="ref82">73</reflink>]; [<reflink idref="bib74" id="ref83">74</reflink>]).</p> <hd id="AN0186129863-22">Receptive vocabulary</hd> <p>Receptive vocabulary was measured using the British Picture Vocabulary Scale – Third Edition (BPVS-III; Dunn et al., [<reflink idref="bib22" id="ref84">22</reflink>]). The child responds by selecting a picture from four options that best illustrates the word meaning the researcher states. There is a maximum score of 168 with 14 sets and raw scores were used. The BPVS-III was normed on 3278 children aged 3–16 years with and without disabilities and has an internal reliability of <emph>r</emph> = 0.91 (Dunn et al., [<reflink idref="bib22" id="ref85">22</reflink>]).</p> <hd id="AN0186129863-23">Mathematical achievement (measured by the British Ability Scale)</hd> <p>The British Ability Scale (BAS-II) Early Number Concepts (Elliot et al., [<reflink idref="bib23" id="ref86">23</reflink>]) was administered as an outcome measure of mathematical achievement. For this task, the child answered questions about size, number and other numerical concepts. The task stimuli include 10 green plastic squares and an easel used to present a series of pictures (Diagnostic Scales Stimulus Booklet 1). There were 30 questions in total with a maximum score of 35. The assessment was stopped when the child made five consecutive errors. There are three suggested starting points based on the child's age. At T1, the researcher started at item 1 but at T3 the researcher started at item 4. If the child got fewer than 3 correct within that decision point, the researcher went back to the previous starting point, if applicable. The researcher coded the child's answers with a score of 1 if correct and a score of 0 if incorrect. Item 3 was an exception to this and was scored between 0 and 3. The researcher was to provide only neutral encouragement to the child during the task, except for the designated teaching items (items 4 and 5). For these items, the researcher provided specific feedback, e.g. "yes that's correct" but if the child had not answered correctly or had not understood the question, they gave the correct response. The child's correct responses are totaled to give a raw score (i.e., number of correct answers). The BAS was normed on 1480 children aged 3–8 years 11 months (Elliot et al., [<reflink idref="bib23" id="ref87">23</reflink>]) and used with 800 children in the Effective Preschool Provisions, Northern Ireland (1998–2005) study. The BAS-II has demonstrated high test-retest reliability (Elliott, [<reflink idref="bib24" id="ref88">24</reflink>]). The BAS has excellent internal consistency for achievement scores and good internal consistency for cognitive scores (Elliott & Smith, [<reflink idref="bib25" id="ref89">25</reflink>]). The overall reliability for early number concepts is reported as 0.95 and it has good temporal stability is reported at 2-to-7-week test–retest intervals (Elliott & Smith, [<reflink idref="bib25" id="ref90">25</reflink>]).</p> <hd id="AN0186129863-24">Data analytic strategy</hd> <p>The data were analyzed with Latent Growth Curve Modelling (LGM), using the Mplus (Version 8.1) statistical package for structural equation modeling (Muthén & Muthén, 1998–2012). LGM provides estimates of key aspects of change, such as the status of individuals at the initial measurement point, their growth over time, and the amount of individual variability at the initial point and in rates of growth (Hancock & Buehl, [<reflink idref="bib31" id="ref91">31</reflink>]). LGM was used to analyze the children's growth trajectories for mathematical achievement measured by BAS. The degree to which the four domain-specific skills (i.e., cardinal principle knowledge, digit recognition, numeric ordering, and symbolic-quantity mapping), two domain-general skills (i.e., verbal working memory and auditory sustained attention), and receptive vocabulary were predictive of initial mathematical achievement and growth over the three time points were explored.</p> <p>First, the development of mathematical achievement was investigated in a univariate latent growth curve model, called the unconditional model. We controlled for age in our conditional model. Next, the conditional model including the predictor variables were fit to the data for outcome measure (i.e., mathematical achievement). Models were evaluated based on overall model fit, which is indicated by a specific set of fit indices that address different aspects of the model under investigation. The longitudinal nature of this study strengthens the appropriateness of using LGM with the current sample size, as the 106 participants responded 3 times giving a total number of 318 data points. Therefore, LGM was deemed appropriate for the sample size of this study</p> <hd id="AN0186129863-25">Results</hd> <p>Below, we present the descriptive statistics and correlations between outcomes and predictors, and the results from the growth curve model.</p> <hd id="AN0186129863-26">Descriptive statistics and correlations</hd> <p>The correlations between all the outcomes at the three time points (i.e., T1, T2, and T3) and the seven time-invariant predictors are depicted in Table 1. Table 2 presents the descriptive statistics (i.e., mean, standard deviations (SD), minimum and maximum, kurtosis and skewness) for the measures used in the final LGM of the current study. The normality of the data was explored, and all measures were within the acceptable limits of skewness of less than 3 and kurtosis of less than 4 (Kline, [<reflink idref="bib40" id="ref92">40</reflink>]). One exception was the auditory sustained attention measure with its high kurtosis value (i.e., eight participants did not fall within 3 SD of the mean). Further, auditory sustained attention was not correlated with mathematical achievement therefore, auditory sustained attention was excluded as a predictor.</p> <p>Table 1. Correlations among the study variables.</p> <p> <ephtml> <table><thead><tr><td /><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td></tr></thead><tbody><tr><td>1. Age</td><td /><td /><td /><td /><td /><td /><td /><td /><td /><td /></tr><tr><td>2. Cardinal Principle Knowledge</td><td>.365**</td><td /><td /><td /><td /><td /><td /><td /><td /><td /></tr><tr><td>3. Digit Recognition</td><td>.270**</td><td>.751**</td><td /><td /><td /><td /><td /><td /><td /><td /></tr><tr><td>4. Numerical Ordering</td><td>.327**</td><td>.494**</td><td>.512**</td><td /><td /><td /><td /><td /><td /><td /></tr><tr><td>5. Symbolic-Quantity Mapping</td><td>.279**</td><td>.494**</td><td>.453**</td><td>.340**</td><td /><td /><td /><td /><td /><td /></tr><tr><td>6. Verbal Working Memory</td><td>.401**</td><td>.454**</td><td>.339**</td><td>.393**</td><td>.426**</td><td /><td /><td /><td /><td /></tr><tr><td>7. Attention Sustained Attention</td><td>−.126</td><td>−.099</td><td>.032</td><td>.057</td><td>−.317**</td><td>−.178*</td><td /><td /><td /><td /></tr><tr><td>8. Receptive Vocabulary</td><td>.351**</td><td>.609**</td><td>.430**</td><td>.393**</td><td>.522**</td><td>.425**</td><td>−.414**</td><td /><td /><td /></tr><tr><td>9. Mathematical Achievement 1</td><td>.347**</td><td>.773**</td><td>.634**</td><td>.480**</td><td>.550**</td><td>.493**</td><td>−.210*</td><td>.662**</td><td /><td /></tr><tr><td>10. Mathematical Achievement 2</td><td>.248**</td><td>.646**</td><td>.544**</td><td>.448**</td><td>.436**</td><td>.350**</td><td>−.149</td><td>.524**</td><td>.694**</td><td /></tr><tr><td>11. Mathematical Achievement 3</td><td>.333**</td><td>.636**</td><td>.530**</td><td>.389**</td><td>.492**</td><td>.494**</td><td>−.171*</td><td>.504**</td><td>.678**</td><td>.714**</td></tr></tbody></table> </ephtml> </p> <p>1 Note. *<emph>p</emph> < 0.05, **<emph>p</emph> < 0.01 (2-tailed).</p> <p>Table 2. Descriptive statistics of the outcome measures at three time points and the time-invariant predictors.</p> <p> <ephtml> <table><thead><tr><td /><td><italic>M</italic></td><td><italic>SD</italic></td><td>Min.</td><td>Max.</td><td>Skew.</td><td>Kurt.</td></tr></thead><tbody><tr><td><bold>Outcome measures</bold></td><td /><td /><td /><td /><td /><td /></tr><tr><td>Mathematical Achievement T1</td><td>16.51</td><td>5.79</td><td>2.00</td><td>30.00</td><td>−0.42</td><td>−0.41</td></tr><tr><td>Mathematical Achievement T2</td><td>21.90</td><td>4.67</td><td>7.00</td><td>30.0</td><td>−0.86</td><td>0.67</td></tr><tr><td>Mathematical Achievement T3</td><td>26.64</td><td>4.15</td><td>14.00</td><td>34.00</td><td>−0.55</td><td>−0.22</td></tr><tr><td><bold>Predictors</bold></td><td /><td /><td /><td /><td /><td /></tr><tr><td>Cardinal Principle Knowledge</td><td>.49</td><td>.35</td><td>0</td><td>1.00</td><td>−0.15</td><td>−1.52</td></tr><tr><td>Digit Recognition</td><td>.41</td><td>.31</td><td>0</td><td>1.00</td><td>0.26</td><td>−1.08</td></tr><tr><td>Numerical Ordering</td><td>.34</td><td>.30</td><td>0</td><td>1.00</td><td>0.66</td><td>−0.68</td></tr><tr><td>Symbolic-Quantity Mapping</td><td>.64</td><td>.21</td><td>0</td><td>1.00</td><td>−0.37</td><td>−0.23</td></tr><tr><td>Verbal Working Memory</td><td>4.04</td><td>2.92</td><td>0</td><td>13.00</td><td>0.76</td><td>0.37</td></tr><tr><td>Attention Sustained Attention</td><td>3.29</td><td>5.79</td><td>0.60</td><td>51.04</td><td>5.89</td><td>41.42</td></tr><tr><td>Receptive Vocabulary</td><td>57.29</td><td>16.93</td><td>14.00</td><td>98.00</td><td>−0.05</td><td>−0.23</td></tr></tbody></table> </ephtml> </p> <p>2 Note. Min./Max. = Observed minimum/maximum. Skew = Skewness, Kurt = Kurtosis.</p> <hd id="AN0186129863-27">Growth curve models</hd> <p>To test our hypothesis about the pattern of longitudinal growth (i.e., whether there would be a linear growth (slope) in mathematical achievement across time), we performed a repeated measure ANOVA (Analysis of Variance), with Time (i.e., three time points (T1, T2 and T3)) as the within-subjects factor. Since the assumption of sphericity was violated, the degrees of freedom were corrected using Greenhouse-Geisser estimates. As expected, there was a main effect of Time, <emph>F</emph>(1.874, 196.781) = 344.130, <emph>p</emph> < 0.001, <emph>η</emph><subs><emph>p</emph></subs><sups>2</sups> =.77. There was (on an individual level) a linear increase across the three time-points in mathematical achievement (see Figure 1). Tests of polynomials indicated a significant main effect of linear terms of time (<emph>F</emph>(<reflink idref="bib1" id="ref93">1</reflink>, 105) = 590.679, <emph>p</emph> < 0.001, <emph>η</emph><subs><emph>p</emph></subs><sups>2</sups> =.85), and a non-significant main effect of quadratic terms of time (<emph>F</emph>(<reflink idref="bib1" id="ref94">1</reflink>, 105) = 1.438, <emph>p</emph> = 0.233, <emph>η</emph><subs><emph>p</emph></subs><sups>2</sups> =.014). As expected, there was a linear growth trend for mathematical achievement.</p> <p>Graph: Figure 1. Individual mathematical achievement scores across the three time-points.</p> <p>Latent linear growth modeling was performed to estimate the contribution of the predictors in explaining variation in the initial status and change in mathematical achievement over three periods in time. First, an unconditional growth model (i.e., without predictors) was specified to identify an appropriate model that accurately explained the development of participants mathematical achievement. This model included initial status (i.e., intercept) and growth (i.e., slope) latent factors. The mean of the intercept latent variable represents the average mathematical achievement score for the sample, and the variance of the latent variable estimates the individual variability in initial status. The mean of the slope latent variable represents the linear change (slope) for the sample and the variance indicates the individual variability on the slopes. For the intercept to represent initial status the factor loadings of that latent variable were fixed to 1. For the slope latent variable, we fixed the factor loadings according to the unequal spaces between three time points (i.e., T1 to T2 = 8 months and T2 to T3 = 15 months). Thus, the first factor loading was fixed to 0 to represent initial status and the other two factor loadings were fixed to 0.8 and 1.5, respectively (Figure 2).</p> <p>Graph: Figure 2. Graphical representations of the unconditional model, with non-standardised intercept and slope loadings.</p> <p>The fit indices and the corresponding fit criteria of interest on the unconditional and conditional (i.e., with predictors) models for each of these can be found in Table 3 (Browne & Cudeck, [<reflink idref="bib8" id="ref95">8</reflink>]; Byrne, [<reflink idref="bib10" id="ref96">10</reflink>] ; Hu & Bentler, [<reflink idref="bib36" id="ref97">36</reflink>] ; Schermelleh-Engel et al., [<reflink idref="bib66" id="ref98">66</reflink>]). The linear growth model without predictors demonstrated good fit to the data (see Table 3 for model fit indices), with an RMSEA value of 0.000 (90% CI:.000,.153) and a CFI value of 1.000. However, given that the model has a single degree of freedom, these results should be interpreted with caution (see the Limitation section for a discussion on this). The mean mathematical achievement scores at T1 (intercept) was was 16.55 with significant variance (<emph>σ</emph><subs><emph>i</emph></subs><sups>2</sups><emph>=</emph> 21.908, <emph>p</emph> < 0.001), indicating significant individual differences in initial level. There was significant growth with a mean of the slope latent variable of 6.77. These data imply that there was substantial gain in mathematical achievement over the period of 15 months. The variance of the slope factor was not significant (<emph>σ</emph><subs><emph>s</emph></subs><sups>2</sups><emph>=</emph> 0.807, <emph>p</emph> = 0.762), suggesting homogenous growth across the group. The covariance of the intercept and slope latent variables was not significant (Cov<subs>i,s</subs> is = −4.157, se = 2.908, <emph>p</emph> = 0.153) indicating that individual initial status was not associated with the rate of change.</p> <p>Table 3. Fit indices on the unconditional and conditional (i.e., with predictors) latent growth models (LGMs) and the corresponding fit criteria.</p> <p> <ephtml> <table><thead><tr><td>Fit indices</td><td><p><graphic href="hjcd_a_2434036_ilm0001.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msup><mrow><mrow><mi mathvariant="bold-italic">x</mi></mrow></mrow><mn>2</mn></msup></mrow></math></p></td><td><italic>df</italic></td><td><p><graphic href="hjcd_a_2434036_ilm0002.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msup><mrow><mrow><mi mathvariant="bold-italic">x</mi></mrow></mrow><mn>2</mn></msup></mrow></math></p>/. <italic>df</italic></td><td>RMSEA</td><td>CFI</td><td>TLI</td><td>SRMR</td></tr></thead><tbody><tr><td /><td><bold>Mathematical Achievement</bold></td></tr><tr><td>Unconditional</td><td>0.076</td><td>1</td><td>0.076</td><td>.000</td><td>1.000</td><td>1.017</td><td>0.004</td></tr><tr><td>Conditional</td><td>10.344</td><td>8</td><td>1.293</td><td>.048</td><td>0.993</td><td>0.978</td><td>0.049</td></tr><tr><td /><td><bold>Fit Criteria</bold></td></tr><tr><td /><td /><td /><td>0 ≤ <p><graphic href="hjcd_a_2434036_ilm0003.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msup><mi>x</mi><mn>2</mn></msup></mrow></math></p>/df ≤ 5</td><td><0.05</td><td>≥0.95</td><td>≥0.95</td><td>0 ≤ SRMR ≤ 0.05</td></tr></tbody></table> </ephtml> </p> <p>3 Note.</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></math> </ephtml> = chi-square value; <emph>df</emph> = degrees of freedom; CFI = Comparative Fit Index; TLI = Tucker – Lewis Index, RMSEA = Root Mean Square Error Approximation; SRMR = Standardised Root Mean Square Residual.</p> <p>A conditional model was then specified and estimated including predictors measured at T1 (i.e., age, cardinal principle knowledge, digit recognition, numeric ordering, symbolic-quantity mapping, verbal working memory and receptive vocabulary; Figure 3). We ran the model with the predictors included for both the intercept and growth factors, but we only interpret the relationship for intercept, due to the non-significant variance in growth identified in the unconditional model.</p> <p>Graph: Figure 3. Graphical representations of the conditional model, with non-standardised intercept/slope loadings and standardised regression coefficients for significant predictors of the intercept.</p> <p>The model with predictors fit the data well (see Table 3 for the fit indices of the conditional model). Table 4 depicts the standardized regression coefficients of these predictors of children's T1 mathematical achievement. Results showed that cardinal principle knowledge and receptive vocabulary were predictors of the intercepts, accounting for 83% (<emph>SE</emph> = 0.07, <emph>p</emph> < 0.001) of the variance in the intercept factor. The other predictors (digit recognition, numeric ordering, symbolic-quantity mapping, verbal WM) did not explain unique variance in the intercept.</p> <p>Table 4. Standardized regression coefficients for intercept factors of mathematical achievement.</p> <p> <ephtml> <table><thead><tr><td /><td>Intercept (Initial Status)</td></tr><tr><td /><td>β</td><td>SE</td></tr></thead><tbody><tr><td>Cardinal Principle Knowledge</td><td>.48***</td><td>.10</td></tr><tr><td>Digit Recognition</td><td>.12</td><td>.09</td></tr><tr><td>Numerical Ordering</td><td>.08</td><td>.07</td></tr><tr><td>Symbolic-Quantity Mapping</td><td>.11</td><td>.07</td></tr><tr><td>Verbal Working Memory</td><td>.08</td><td>.07</td></tr><tr><td>Receptive Vocabulary</td><td>.27***</td><td>.08</td></tr><tr><td>Age</td><td>−.02</td><td>.06</td></tr></tbody></table> </ephtml> </p> <p>4 Note. *<emph>p</emph> < 0.05, ***<emph>p</emph> < 0.001.</p> <hd id="AN0186129863-28">Discussion</hd> <p>Our findings indicate that children displayed substantial growth in mathematical achievement over a period of 15 months, during which they transitioned from preschool to primary school. Notably, this growth in mathematical achievement was linear and there was little variability in children's rate of growth for the timespan in the current study. In contrast, there was substantial variance in initial mathematical achievement, and this was explained by cardinality understanding and receptive vocabulary. Below, we first discuss the implications of these findings for the development of mathematical achievement, followed by considering the key underpinnings of mathematical skills in early childhood.</p> <hd id="AN0186129863-29">RQ1:</hd> <p>How does children's mathematical achievement change across time?</p> <p>The current findings indicate that there was substantial gain in mathematical achievement over the period of 15 months and that this growth was linear, which is in line with our hypothesis. This suggests that there was no "schooling effect" or seasonal learning differences on the development of mathematical achievement; that is, rates of growth were not steeper during the school year for mathematical achievement. Instead, the findings suggest that there is stability in mathematical achievement growth over the course of the transition. A meta-analysis (including 8 studies) investigated the effects of summer vacation on learning loss in mathematics and found that summer vacation influenced students negatively, resulting in primary school students (<emph>N</emph> = 6 studies) experiencing more loss in mathematics than secondary school students (<emph>N</emph> = 2 studies; Baş, [<reflink idref="bib5" id="ref99">5</reflink>]). However, this area of research has found mixed results with some studies showing that summer learning loss increases with grade level (Alexander et al., [<reflink idref="bib2" id="ref100">2</reflink>]). The current study shows no "schooling effect" on the development of mathematical achievement on the earliest school transition (preschool to primary school), which is a unique and novel finding due to the small number of longitudinal studies that investigate growth within this age range. Interestingly, in a longitudinal study involving a latent transition analysis with four indicators of basic numeracy skills (i.e., Cahoon et al., [<reflink idref="bib11" id="ref101">11</reflink>]) found that two transitions between profiles were observed at school-entry, demonstrating that many children make substantial learning gains once they enter school. Therefore, perhaps there is development in basic mathematical skills knowledge but not variability in growth in mathematical achievement during transition.</p> <hd id="AN0186129863-30">RQ2:</hd> <p>Does children's initial mathematical knowledge relate to development of their mathematical achievement across time?</p> <p>The current findings indicated a lack of variability in children's rate of growth, but there was substantial variance in initial mathematical achievement. The variation in initial mathematical achievement exhibited across individuals serves as an example of the complexity of childhood development (Vereijken, [<reflink idref="bib76" id="ref102">76</reflink>]), as achievement gaps have been shown to grow or shrink as children progress due to various reasons (Davis-Kean & Jager, [<reflink idref="bib15" id="ref103">15</reflink>]). The <emph>Matthew effect</emph> refers to the view that the rich get richer while the poor get poorer (Rigney, [<reflink idref="bib63" id="ref104">63</reflink>]). With regards to mathematics, the <emph>Matthew effect</emph> suggests that children with existing high mathematical skills develop better mathematical skills more quickly than those with low mathematical skills who therefore continue to fall further behind (Cahoon et al., [<reflink idref="bib11" id="ref105">11</reflink>]; Garon-Carrier et al., [<reflink idref="bib27" id="ref106">27</reflink>]). However, the current study suggests that there is stability in individual developmental pathways, irrespective of the level of children's mathematics skills. We suggest that children who start with higher levels of mathematical achievement may not always show faster growth, this perhaps depends on various factors (such as classroom instruction they are exposed to, timespan across the measures, how mathematical ability is measured).</p> <hd id="AN0186129863-31">RQ3:</hd> <p>Which cognitive skills predict children's initial mathematical knowledge and their growth in mathematical achievement over time?</p> <p>The current study demonstrated substantial variance in initial mathematical achievement, which was explained by cardinality understanding and receptive vocabulary. Digit recognition, numeric ordering, symbolic-quantity mapping, verbal working memory, and auditory sustained attention did not uniquely predict initial mathematical achievement. Due to the non-significant variance in growth in mathematical achievement, we were unable to identify predictors for this aspect. Based on these results, we will now discuss cardinal principle knowledge and receptive vocabulary in detail as predictors of initial mathematical achievement.</p> <hd id="AN0186129863-32">The importance of cardinality</hd> <p>Children's understanding of the counting system, captured through cardinal principle knowledge was a predictor of initial mathematical achievement. This could be anticipated as cardinality gives number words meaning, the starting point of children's semantic representation of symbolic number (Batchelor et al., [<reflink idref="bib6" id="ref107">6</reflink>]; Lyons et al., [<reflink idref="bib45" id="ref108">45</reflink>]; Sarnecka & Wright, [<reflink idref="bib65" id="ref109">65</reflink>]; Wagner & Johnson, [<reflink idref="bib77" id="ref110">77</reflink>]). Cardinality has been shown to be the first numeracy skill that develops (Cahoon et al., [<reflink idref="bib11" id="ref111">11</reflink>]; Slusser & Sarnecka, [<reflink idref="bib68" id="ref112">68</reflink>]) with age-of-acquisition of cardinal knowledge being fundamental for later number system knowledge, broader mathematical development, and school readiness (Geary et al., [<reflink idref="bib28" id="ref113">28</reflink>]). The findings from the current study support previous findings in that cardinality explained variance in initial mathematical achievement. This emphasizes the importance of providing children with opportunities to develop cardinality understanding, for example through activities and games with sets of objects.</p> <hd id="AN0186129863-33">The importance of general language skills</hd> <p>Children's receptive vocabulary scores were predictive of initial mathematical achievement status. Strong evidence suggests that language has a role in mathematical learning (Peng et al., [<reflink idref="bib59" id="ref114">59</reflink>]; Purpura et al., [<reflink idref="bib60" id="ref115">60</reflink>]) and that challenges with arithmetic, literacy, and language frequently co-occur (Mann Koepke & Miller, [<reflink idref="bib47" id="ref116">47</reflink>]). Previous research has reported that individuals with reading or language problems perform poorly on arithmetic tasks (e.g., multiplication and fact retrieval) compared to those without these problems (De Smedt & Boets, [<reflink idref="bib16" id="ref117">16</reflink>]; Gobel & Snowing, [<reflink idref="bib29" id="ref118">29</reflink>]; Moll et al., [<reflink idref="bib52" id="ref119">52</reflink>]). Hence, it is not surprising that receptive vocabulary scores were predictive of initial mathematical achievement. An important outcome from this study is that although cardinality involves number words, language is an independent predictor of initial mathematical achievement. Therefore, even at four years old (the mean age of children in this study), these skills (i.e., language and cardinality) are independent but important skills for mathematical achievement. This study shows that early exposure to these types of mathematical[<reflink idref="bib47" id="ref120">47</reflink>] concepts (i.e., cardinality) and language (even general language) is an important implication for mathematical achievement.</p> <hd id="AN0186129863-34">Working memory in early childhood</hd> <p>Verbal working memory did not explain unique significant variance in mathematical achievement at this stage of development (i.e., 4 years old). As stated previously, longitudinal studies investigating working memory and mathematical achievement are lacking. The longitudinal studies in the area usually investigate the relationship at the beginning of formal schooling (e.g., 6 years 8 months: De Smedt & Boets, [<reflink idref="bib16" id="ref121">16</reflink>]; mean age 6 and above:; H. L. Swanson, [<reflink idref="bib70" id="ref122">70</reflink>]). One study that investigated this relationship at school-entry (i.e., 4 years 6 months: Bull et al., [<reflink idref="bib9" id="ref123">9</reflink>]) found verbal working memory to show a significant yet small contribution to mathematical skills. Therefore, more research may be necessary to understand the longitudinal nature of the impact of working memory on mathematical achievement during early years (i.e., 0–5 years old).</p> <hd id="AN0186129863-35">Future research</hd> <p>The current study supports previous evidence that young children vary substantially in their mathematical skills in during their early years (i.e., 3–5 years), and that some children are more prepared to learn mathematics than their peers before school-entry (Duncan et al., [<reflink idref="bib21" id="ref124">21</reflink>]; Manolitsis, Georgiou, & Tziraki, [<reflink idref="bib48" id="ref125">48</reflink>]). To understand the etiology of individual differences in mathematical achievement, we need to examine the development of numerical concepts and procedural skills (alongside environmental and social factors) as they influence each other in a complex fashion from domain to domain at an early age (3–5-years-old; Nunes et al., [<reflink idref="bib55" id="ref126">55</reflink>]). In future work, microgenetic studies could address many questions about the changes in different domains over time and idenitify possible causes of change (e.g., cardinality; Chetland & Fluck, [<reflink idref="bib12" id="ref127">12</reflink>]; Flynn & Siegler, [<reflink idref="bib26" id="ref128">26</reflink>]) to aid understanding into the etiology of individual differences in mathematical development and to understand the path of development.</p> <hd id="AN0186129863-36">Limitations</hd> <p>Not all cognitive skills were measured in this study, but other precursor cognitive skills (e.g., spatial skills) which could be important for mathematical development should be considered in future research. Nevertheless, 83% of the variance was explained in intercepts with the measures that were included. Future research could include more precursor cognitive skills for further understanding into mathematical achievement. Within the current study, we were not able to examine the role of predictors (i.e., domain-specific, domain-general, and language skills) on the development of mathematical achievement due to lack of variance in children's mathematical growth, perhaps due to the small sample and/or short timespan in the current study.</p> <p>Additionally, our unconditional model has only one degree of freedom, which was expected given its simple structure. However, this simplicity presents some challenges. With a small sample size, the reliability of statistical tests may be compromised. Specifically, for models with few degrees of freedom and small sample sizes, the confidence interval (CI) for fit indices like RMSEA can be quite wide (Kenny et al., [<reflink idref="bib39" id="ref129">39</reflink>]), making the.000 RMSEA as observed in our study value potentially misleading. While such models can be useful for focused analysis, relying solely on single degree-of-freedom tests may oversimplify the data.</p> <p>The seemingly perfect model fit observed in our unconditional model could also be a result of the small sample size. To address this, future studies could benefit from using larger, more diverse samples, and incorporating different measures of mathematics achievement. In addition, assessing mathematical achievement at differential time intervals may provide insight into the subtilties of development processes over time. The data was not entered electronically on an item-by-item basis to allow us to calculate Cronbach's alpha for this sample. As this project has finished, we do not have the resources to reenter data. We therefore report this information from other studies that use the same tasks were possible. However, the psychometric information for the working memory task is unavailable, therefore results involving working memory should be treated with caution. For the numeric ordering task, the Cronbach's alpha was 0.48 as reported in Batchelor et al. ([<reflink idref="bib6" id="ref130">6</reflink>]) which is below acceptable levels however, Batchelor et al. ([<reflink idref="bib6" id="ref131">6</reflink>]) version of the task had fewer trials which reduces reliability. Therefore, we added more trials, so to improve performance of the task.</p> <hd id="AN0186129863-37">Conclusion</hd> <p>Using a latent growth curve approach, we were able to investigate children's initial status in mathematical achievement and development during the preschool to primary school transition when children are 3–5 years old, emphasizing the importance of cardinality and mathematical language at this early age. Therefore, longitudinal methods have the capacity to provide invaluable information for understanding the developmental change affecting young children. These early differences point to the importance of exposure to mathematical language and concepts in early childhood to ensure the development of broader mathematical skills.</p> <hd id="AN0186129863-38">Disclosure statement</hd> <p>No potential conflict of interest was reported by the authors.</p> <ref id="AN0186129863-39"> <title> References </title> <blist> <bibl id="bib1" idref="ref68" type="bt">1</bibl> <bibtext> Alexander, K. L., Entwisle, D. R., & Dauber, S. L. (1993). First‐grade classroom behavior: Its short‐and long‐term consequences for school performance. Child Development, 64 (3), 801 – 814. https://doi.org/10.2307/1131219</bibtext> </blist> <blist> <bibl id="bib2" idref="ref100" type="bt">2</bibl> <bibtext> Alexander, K. L., Entwisle, D. R., & Olson, L. S. (2007). Lasting consequences of the summer learning gap. 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Items – Name: Title
  Label: Title
  Group: Ti
  Data: The Cognitive and Numerical Predictors of Early Mathematical Achievement: A Latent Growth Curve Analysis
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Abbie+Cahoon%22">Abbie Cahoon</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0001-7587-6670">0000-0001-7587-6670</externalLink>)<br /><searchLink fieldCode="AR" term="%22Emine+Simsek%22">Emine Simsek</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0003-1679-1276">0000-0003-1679-1276</externalLink>)<br /><searchLink fieldCode="AR" term="%22Camilla+Gilmore%22">Camilla Gilmore</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-5879-2683">0000-0002-5879-2683</externalLink>)<br /><searchLink fieldCode="AR" term="%22Victoria+Simms%22">Victoria Simms</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0001-5664-6810">0000-0001-5664-6810</externalLink>)
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22Journal+of+Cognition+and+Development%22"><i>Journal of Cognition and Development</i></searchLink>. 2025 26(3):443-463.
– Name: Avail
  Label: Availability
  Group: Avail
  Data: Routledge. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 21
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2025
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research
– Name: Audience
  Label: Education Level
  Group: Audnce
  Data: <searchLink fieldCode="EL" term="%22Elementary+Education%22">Elementary Education</searchLink><br /><searchLink fieldCode="EL" term="%22Preschool+Education%22">Preschool Education</searchLink><br /><searchLink fieldCode="EL" term="%22Early+Childhood+Education%22">Early Childhood Education</searchLink>
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Mathematics+Achievement%22">Mathematics Achievement</searchLink><br /><searchLink fieldCode="DE" term="%22Preschool+Children%22">Preschool Children</searchLink><br /><searchLink fieldCode="DE" term="%22Elementary+School+Students%22">Elementary School Students</searchLink><br /><searchLink fieldCode="DE" term="%22Individual+Differences%22">Individual Differences</searchLink><br /><searchLink fieldCode="DE" term="%22Achievement+Gains%22">Achievement Gains</searchLink><br /><searchLink fieldCode="DE" term="%22Child+Development%22">Child Development</searchLink><br /><searchLink fieldCode="DE" term="%22Foreign+Countries%22">Foreign Countries</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Skills%22">Mathematics Skills</searchLink><br /><searchLink fieldCode="DE" term="%22Predictor+Variables%22">Predictor Variables</searchLink>
– Name: Subject
  Label: Geographic Terms
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22United+Kingdom%22">United Kingdom</searchLink>
– Name: SubjectThesaurus
  Label: Assessment and Survey Identifiers
  Group: Su
  Data: <searchLink fieldCode="SU" term="%22British+Ability+Scales%22">British Ability Scales</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1080/15248372.2024.2434036
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 1524-8372<br />1532-7647
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Longitudinal studies are essential for understanding causes of developmental change and growth rates of mathematical achievement. One hundred and twenty-eight UK-based children (M[subscript age] = 4 years; SD[subscript age] = 3.3 months; age range 43-54 months; 70 female) were tracked for 15 months, from the beginning of preschool until the end of the first year of primary school (i.e., across 7 preschools to 18 primary schools) and were assessed at three time points. At the beginning of preschool, data were collected from parents and children, including background demographics, domain-specific mathematical skills, domain-general cognitive skills, and language skills. Mathematical achievement was assessed once during preschool and at two time points during the first year of primary school. Using a latent growth model, we examined the contribution of the predictors to the growth patterns in mathematical achievement and the stability of initial individual differences during preschool to school transitions. Results showed that over a period of 15-months, children displayed substantial growth in mathematical achievement. This growth in mathematical achievement was linear and there was little variability in children's rate of development. In contrast, there was substantial variance in initial mathematical achievement, and this variance was explained by children's cardinality understanding and receptive vocabulary. These early variations highlight the importance of exposure to mathematical language and concepts in early childhood to ensure the development of broader mathematical skills.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2026
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1494947
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1494947
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1080/15248372.2024.2434036
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 21
        StartPage: 443
    Subjects:
      – SubjectFull: Mathematics Achievement
        Type: general
      – SubjectFull: Preschool Children
        Type: general
      – SubjectFull: Elementary School Students
        Type: general
      – SubjectFull: Individual Differences
        Type: general
      – SubjectFull: Achievement Gains
        Type: general
      – SubjectFull: Child Development
        Type: general
      – SubjectFull: Foreign Countries
        Type: general
      – SubjectFull: Mathematics Skills
        Type: general
      – SubjectFull: Predictor Variables
        Type: general
      – SubjectFull: United Kingdom
        Type: general
      – SubjectFull: British Ability Scales
        Type: general
    Titles:
      – TitleFull: The Cognitive and Numerical Predictors of Early Mathematical Achievement: A Latent Growth Curve Analysis
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Abbie Cahoon
      – PersonEntity:
          Name:
            NameFull: Emine Simsek
      – PersonEntity:
          Name:
            NameFull: Camilla Gilmore
      – PersonEntity:
          Name:
            NameFull: Victoria Simms
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          Dates:
            – D: 01
              M: 01
              Type: published
              Y: 2025
          Identifiers:
            – Type: issn-print
              Value: 1524-8372
            – Type: issn-electronic
              Value: 1532-7647
          Numbering:
            – Type: volume
              Value: 26
            – Type: issue
              Value: 3
          Titles:
            – TitleFull: Journal of Cognition and Development
              Type: main
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