The roles of executive functioning and oral language skills in young Chinese children's arithmetic competence

Learning and Individual Differences 77 (2020) 101810

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Learning and Individual Differences journal homepage: www.elsevier.com/locate/lindif

The roles of executive functioning and oral language skills in young Chinese children's arithmetic competence

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Catrina Liua, Sum Kwing Cheunga, Kevin Kien Hoa Chunga, , Catherine McBrideb, Chun Bun Lama, Xiaomin Lia a b

Department of Early Childhood Education, The Education University of Hong Kong, Hong Kong, China Department of Psychology, The Chinese University of Hong Kong, Hong Kong, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Executive functioning Oral language skills Number-fact problems Story problems

This study investigated the relative contributions of executive functioning (EF) and oral language skills to performance on number-fact and story problems in Chinese children. A total of 280 kindergarten children in Hong Kong participated in this study. Children were assessed on their EF, phonological awareness, morphological awareness, receptive vocabulary, performance on number-fact and story problems. Results of path analysis showed that when children's age and parental education were controlled, children's EF, phonological awareness, and morphological awareness were correlates of performance on both number-fact and story problems, whereas receptive vocabulary was not. After further controlling for performance on number-fact problems, morphological awareness was the only variable under investigation that linked to performance on story problems. These findings underscore the importance of taking the development of EF and oral language skills into consideration when guiding children's arithmetic learning.

1. Introduction A key indicator of the level of success of children's mathematics learning is whether they know how to organize and apply what they learn to solve problems in everyday life activities (Kilpatrick, Swafford, & Findell, 2001). In particular, verbal arithmetic problems, including the forms of number-fact problems and story problems, are common problem types in early mathematics learning (Lesh, Landau, & Hamilton, 1983). The importance of the domain general skills, such as executive functioning (EF) and language skills, to arithmetic competence in young children has been found in some studies (e.g., LeFevre et al., 2010; Purpura, Hume, Sims, & Lonigan, 2011; Purpura, Schmitt, & Ganley, 2017; Zhang, 2016; Zhang et al., 2014). For example, to solve verbal arithmetic problems, EF is believed to be essential for maintaining and manipulating verbal information in mind (e.g., Espy et al., 2004; Harvey & Miller, 2017). Language skills are also required for decoding and comprehending the problem statements (e.g., Duncan et al., 2007; Fuchs et al., 2006). Nonetheless, existing studies have mostly focused only on the relationships of one or two aspects of the cognitive and language skills to performance on one type of verbal arithmetic problems (e.g., Purpura et al., 2011; Purpura et al., 2017; Purpura & Ganley, 2014). Furthermore, because many of these studies

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were conducted in Euro-American settings, much less is known about the Chinese context, where the number word system is distinct from other language systems (Ng & Rao, 2010). In the present study, therefore, we sought to investigate the relative importance of EF and a range of oral language skills to performance on number-fact and story problems among Chinese-speaking children. Examining children's ability to solve verbal arithmetic problems is worthwhile, because such ability is important for their real-life problem solving and can facilitate their acquisition of more advanced mathematical knowledge (e.g., Fuchs et al., 2014; Geary, Hoard, Nugent, & Bailey, 2013). It not only enhances understanding of the unique nature of these different problems, but can also yield practical implications for the teaching of arithmetic. 1.1. Arithmetic competence: Number-fact problems and story problems Number-fact problems, which are made up with a combination of number words and arithmetic operators (e.g., “how much is three plus two?”), are free of manipulations of elements and taps factual verbal arithmetic competence. Similarly, the story problem scenario is composed of story figures, elements or objects, quantity of numbers, and transformation of their relations (e.g., add, take away). For example, “Lisa has three cookies, daddy gives her two more. How many cookies

Corresponding author at: B2-1F-35, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong, China. E-mail address: [email protected] (K.K.H. Chung).

https://doi.org/10.1016/j.lindif.2019.101810 Received 20 January 2019; Received in revised form 2 December 2019; Accepted 10 December 2019 1041-6080/ © 2019 Elsevier Inc. All rights reserved.

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McBride, 2019).

altogether does she have now?”. Unlike print-related arithmetic tasks, verbal arithmetic tasks reduce the demands on word reading skills and written knowledge for young kindergarteners. Therefore, verbal number-fact and story problems can reflect children's early experiences in perceiving and understanding the relational changes around them. Story problems involve the integration of number-fact processing and linguistic story comprehension (Ostad, 1998). It is evident that children have to create mental images for the characters and objects mentioned in the story context, decipher more complex verbal problem statements, and construct their own number sentences (Wong & Ho, 2017). In this study, identical number words and arithmetic operations (e.g., addition or subtraction) were used to compare the general cognitive and language skills required for solving number-fact and story problems. Thus far, little attempt has been made to investigate the roles of general cognitive and language skills in young kindergarten children's performance on different types of arithmetic problems in one model.

1.3. Oral language skills and arithmetic competence Language and arithmetic skills are moderately correlated in Englishspeaking children (e.g., Fuchs et al., 2014; Purpura et al., 2011, 2017), as well as in Chinese-speaking children (e.g., Zhang et al., 2014; Zhang & Lin, 2015, 2018). Several neuroimaging studies have further indicated that solving arithmetic tasks activates the left angular gyrus in the brain, which is linked to language processing (e.g., Dehaene, 2011; Rivera, Reiss, Eckert, & Menon, 2005). Thus, a range of oral language skills, such as phonological awareness, morphological awareness, and vocabulary knowledge, were included in the present study to examine their independent associations with verbal arithmetic competence. Solving number-fact problems requires the activation, manipulation, and retrieval of phonological representations in response to number words in the long-term memory. Thus, phonological awareness may constrain the quality of phonological codes during the manipulation of number facts. In other words, more distinct phonological representations in memory may bring more efficient activation and retention of the phonological codes for numbers, which could free up more cognitive resources to calculation processes (Bull & Johnston, 1997). Weak phonological processing, on the other hand, may impair performance on number-fact problem as it leads to difficulties in recalling number facts (De Smedt, Taylor, Archibald, & Ansari, 2010; Simmons & Singleton, 2008) and decreases the storage of the problem or answer (Geary, 1993). Within the theoretical model for early arithmetic development (Krajewski & Schneider, 2009), phonological awareness might aid the acquisition of number words in the precise sequence for basic numerical processing, which in turn can facilitate higher-level arithmetic skills. To solve story problems, conscious sensitivity to the phonological structure of a language is more heavily involved to differentiate number words from other daily discourses. Being able to understand the content or intention of a verbal scenario is the first step to finding a correct solution. Indeed, significant associations between phonological awareness and performance on verbal arithmetic problems have been found in some studies (e.g., Alloway et al., 2005; Bjork & Bowyer-Crane, 2013; Simmons, Singleton, & Horne, 2008), but not in others (e.g., Purpura et al., 2011; Zhang et al., 2014). Such discrepancies may be partly attributable to differences in the range of other cognitive-linguistic skills included and the mode of presentation of arithmetic problems (written vs. verbal). This study extends prior studies by examining the relative contributions of phonological awareness and other language skills to Chinese children's performance on verbally presented number-fact and story problems. The logic of morphology is inherently embedded in the Chinese number word system. In particular, compounding morphology, which involves the combinations of two or more morphemes to construct a new word (e.g., snowman), is the dominant one in the Chinese language. The single-digit numerals (i.e., 1–9) are represented by single and short syllables (i.e., morphemes) in Chinese (e.g., 5 “wu”, 8 “ba”). However, the distinct feature of Chinese number word system is the systematic base-10 structure (Ng & Rao, 2010). It serves as the root for the learning of multi-digit number words by consistently mapping onto their corresponding place-value within the numeral system (Ho & Fuson, 1998). For example, the teen quantities such as 11 (shi-yi) and 18 (shi-ba) are literally “ten one” and “ten eight”, respectively. Therefore, it is both linguistically (e.g., three-ten-six) and numerically (e.g., 3 ∗ 10 + 6) transparent to generate new numbers (e.g., 36) following Chinese morphological structures. That is, Chinese number words are often combined with multiple morphemes (from 1 to 9) and they are placed before and after the base-10 multiplier to construct a meaning. Compounding morphological awareness therefore plays an important role in number words learning and further number-fact problem solving in Chinese children. Story problem solving, however, goes beyond what is linguistically

1.2. EF and arithmetic competence EF is a set of top-down mental processes required when individuals pay attention, plan and regulate thoughts, emotions and behaviors (Diamond, 2013). In adulthood literature, EF is found to be a multifaceted cognitive process which involves working memory (i.e., holding multiple pieces of information and mentally working with them), inhibitory control (i.e., suppressing irrelevant information and a prepotent response), and cognitive flexibility (i.e., switching between tasks) (e.g., Miyake et al., 2000). However, as EF is influenced by the development of the prefrontal cortex, which undergoes rapid growth in early childhood (e.g., Best & Miller, 2010), the EF structure tends to be unstable and less differentiated in early school years (e.g., Lee, Bull, & Ho, 2013; Willoughby, Wirth, & Blair, 2012). Similar to prior studies (e.g., Chung, Lam, & Cheung, 2018), we employed the unitary construct of EF task to extend our conceptual understanding of its relationship with performance on number-fact and story problems in young kindergarten children. Generally, acquiring arithmetic competence requires learning the associations in visual symbols, verbal sounds and meanings, and EF skills are crucial to memorize, manipulate and compare those quantities and shift flexibly across different modalities (e.g., from verbal to visual) (see McClelland & Cameron, 2019). When solving number-fact problems, individuals have to first hold a number word in mind, actively work on the incoming arithmetic relations (addition or subtraction), and then incorporate a new numerical representation into working memory to obtain an arithmetic solution. All of these processes involve EF skills. If the number-fact problems involve multi-digit numbers, EF may also support the borrowing or carrying procedures by recalling previous steps when considering the next step (Harvey & Miller, 2017). In support of these arguments, empirical findings have shown that EF skills (e.g., working memory and inhibitory control) were unique correlates of performance on numberfact problems (e.g., Bull & Scerif, 2001; Espy et al., 2004; Welsh, Nix, Blair, Bierman, & Nelson, 2010). Yet, the extent to which EF may independently predict performance on number-fact problems among other cognitive skills remains unclear. Meanwhile, story problems are more complex than number-fact problems in terms of the problem statements. To be specific, EF perhaps facilitates the translation of the problem statements to mathematical sentences, the integration of different pieces of information stored in working memory as the story progresses, the planning as to what number words and operators should be assigned, and finally the execution of the constructed addition or subtraction operations (Mayer & Hegarty, 1996). During these processes, EF may also be helpful in shifting across different story contexts and rule sets, and in inhibiting the tendency to focus on the semantic properties of the problem statements. Emerging evidence showed that there are significant associations between EF and performance on story problems in young children (e.g., Purpura et al., 2017; Yang, Chung, & 2

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et al., 2014), and parental education were included as potential covariates in this study.

required for number-fact problem solving. In story problems, individuals must interpret and translate the problem statements, which are rich in semantic and syntactic structures, before performing computational processing. Morphology taps the meaning structure of a language, and it may facilitate more precise understanding of the constructions underlying story problems. The additional demands of semantic relations underlying story problem solving is supported by the findings among English third graders that the composite language skills, including morphological knowledge, uniquely predicted performance on story problems but not arithmetic fact problems (Fuchs et al., 2006). Likewise, Zhang and Lin (2015) showed that Chinese children's morphological awareness as assessed in the second year of kindergarten (K2) uniquely predicted their performance on story problems in the third kindergarten year (K3). It remains unclear to what extent morphological awareness contributes to performance on story problems when performance on number-fact problems is also considered. Previous studies have usually included vocabulary knowledge as a proxy for general language ability in the domain of mathematics (e.g., Harvey & Miller, 2017; LeFevre et al., 2010; Purpura & Ganley, 2014). In essence, the numbers per se are represented and stored as symbolic words (i.e., vocabulary words) in memory (Chow & Ekholm, 2019). LeFevre et al. (2010) proposed a linguistic pathway to early numeracy skills, postulating that receptive vocabulary may reflect children's abilities in acquiring vocabulary knowledge in the number system. Indeed, Harvey and Miller (2017) found that receptive vocabulary accounted for unique variance in children's early mathematical knowledge over and above EF skills. Furthermore, many more vocabulary words are used to represent objects (e.g., “nine toy cars”) and relations (e.g., “take away” “more than”) to create story problems. Negen and Sarnecka (2012) reported a strong relation between general vocabulary and children's number-word knowledge, because children's vocabulary knowledge (e.g., nouns) is helpful to pick out the referents of number words. For example, when hearing “nine toy cars”, children who already know “toy cars” likely have a better chance to figure out what “nine” means. Thus, vocabulary knowledge is essential for understanding and constructing meaningful schematic representations delivered by words, sentences and quantitative relations.

2. Method 2.1. Participants Participants were 280 (160 boys, 57.1 %) third-year kindergarten children (mean age = 69.63 months, SD = 3.95) from 13 kindergartens across the four regions in Hong Kong. All children were native Chinese speakers and learned traditional Chinese characters in kindergarten. Parents of the participants also provided family demographic information, including parental education level, child gender and age. Parents varied in education attainments (6 % completed only elementary school, 64 % completed only high school, 19 % completed only higher diploma or associate degree programs and 11 % completed degree programs or above). 2.2. Measures 2.2.1. Executive functioning The computerized Heart and Flower task (Calderon, Jambaque´, Bonnet, & Angeard, 2014; Davidson, Amso, Anderson, & Diamond, 2006) was used to measure EF. There were three conditions: Block 1 (congruent, 12 trials, indicating working memory); Block 2 (incongruent, 12 trials, indicating inhibitory control) and Block 3 (mixed, 20 trials, indicating cognitive flexibility). In the congruent block, children were asked to press the button on the same side as the heart or flower. In the incongruent block, children were asked to inhibit a behavioral tendency and press the button on the opposite side of the flower or heart. In the mixed block, children were required to flexibly switch between these two rules, same or opposite side, in response to the stimuli. Children were asked to fixate on the cross located at the center of the computer screen for 500 ms. It was followed by a presentation of either a heart-shaped stimulus or a flower-shaped stimulus (2500 ms). The inter-stimulus interval was 500 ms. Thus, the total duration for one trial was 3000 ms. The response buttons for the left and right side were “z” and “m,” indicated by colored stickers. There was a between block break of 20 s. All conditions were presented after a practice session (four trials), and no feedback was given during testing trials. If a child failed to achieve 80% accuracy during the practice trial, additional practice was run automatically with a maximum of three trials (for all blocks). In each block, a point was awarded for each correct trial, and a composite score was calculated by summing the scores obtained across trials. The composite score was then standardized. Given that the underlying structure of EF tends to be unitary in early school years (e.g., Miyake & Friedman, 2012; Welsh et al., 2010), principal component analysis was conducted to assess the common entities of the three blocks. The results showed that only one component with eigenvalues > 1 was extracted, explaining 58.22 % of the variances. The factor loadings for the congruent, incongruent and mixed blocks were 0.76, 0.79, and 0.74, respectively. Therefore, one composite score was used to represent individuals' EF abilities and this score was calculated by averaging the standardized scores of the three blocks. This task has been used in previous studies with Hong Kong kindergarten children and showed good reliability and validity (cf. Chung et al., 2018).

1.4. The present study The present study focused on the relationships between EF, language skills, and performance on number-fact and story problems in Chinese-speaking children. Although Arabic numerals and operators are universal across languages, the Chinese-based system of number words is relatively transparent and regular compared to alphabetic languages like English. As a result, when processing arithmetic problems, spare cognitive resources can be devoted to more complex arithmetic procedures without conscious and effortful activation of numbers (Chan, 2014). The simple and user-friendly Chinese number system may facilitate children's number learning and arithmetic development (e.g., Huntsinger, Jose, Liaw, & Ching, 1997; Miller, Kelly, & Zhou, 2005). The aims of this study were twofold. The first aim was to investigate to what extent EF and oral language skills were associated with children's performance on number-fact and story problems. The second aim was to examine to what extent EF and oral language skills uniquely contribute to performance on story problems when performance on number-fact problems was considered. Based on literature review (e.g., Espy et al., 2004; Zhang & Lin, 2015), we hypothesized that EF and oral language skills (i.e., phonological awareness, morphological awareness and receptive vocabulary) would uniquely predict performance on number-fact and story problems. Furthermore, following prior studies (Fuchs et al., 2006; Ostad, 1998), we also hypothesized that EF and oral language skills (i.e., phonological awareness, morphological awareness and receptive vocabulary) would predict performance on story problems when performance on number-fact problems was controlled. Children's age, gender (boys sometimes outperform girls, see Zhang

2.2.2. Oral language skills 2.2.2.1. Phonological awareness. The 51-item syllable deletion task (Chung, McBride-Chang, Cheung, & Wong, 2013) was used to measure phonological awareness. Children were asked to say aloud what was left after deleting the first, the middle, or the last syllable from each word they heard. Based on the levels of difficulty, the items were grouped into six blocks, each of which consisted of seven to ten items. If a child failed on six or more items within one of the first three 3

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Table 1 Descriptive statistics for and correlations among variables under investigation (N = 280). Variables (max. possible score) 1. 2. 3. 4. 5. 6. 7. 8.

Age in months (−) Parental education (5)a Executive functioning (−)b Phonological awareness (51) Morphological awareness (48) Receptive vocabulary (60) Number-fact problems (12) Story problems (12)

M 69.63 2.35 −0.02 16.30 10.51 33.14 4.26 2.17

SD 3.95 0.79 2.30 7.89 5.70 7.52 3.51 2.34

Min 63 1 −8.96 0 0 8 0 0

Max 82 5 2.08 40 28 51 12 10

Reliability – – 0.78 0.90 0.87 0.85 0.88 0.78

Skewness 0.48 1.11 −1.64 0.09 0.18 −0.21 0.47 1.16

Kurtosis −0.30 1.23 2.28 −0.50 −0.39 −0.14 −0.93 0.80

2 0.03 – – – – – – –

3 0.03 −0.03 – – – – – –

4

5 ⁎

0.14 0.16⁎⁎ 0.31⁎⁎⁎ – – – – –

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0.14 0.09 0.23⁎⁎⁎ 0.52⁎⁎⁎ – – – –

6

7

8

0.09 0.09 0.27⁎⁎⁎ 0.40⁎⁎⁎ 0.54⁎⁎⁎ – – –

0.11 0.11 0.24⁎⁎⁎ 0.45⁎⁎⁎ 0.46⁎⁎⁎ 0.27⁎⁎⁎ – –

0.20⁎⁎ 0.17⁎⁎ 0.22⁎⁎⁎ 0.40⁎⁎⁎ 0.42⁎⁎⁎ 0.21⁎⁎⁎ 0.63⁎⁎⁎ –

Note. a Parental education: 1 = Elementary school, 2 = Secondary school, 3 = Higher diploma or associate degree, 4 = Bachelor degree, 5 = Postgraduate degree. b The composite score of three z-scores across block 1, 2, and 3. The reliabilities represent the internal consistency (Cronbach's alpha) of each task in the present sample. ⁎ p < .05. ⁎⁎ p < .01. ⁎⁎⁎ p < .001.

answer an arithmetic question embedded in a story told by the test administrator. The types of story problems used came from the study of Ostad (1998), and the number operations involved were the same as those in the number-fact problems. Two sample items were “Lei Lei had seven cookies. Then Dad gave her five more cookies. How many cookies does Lei Lei have now?” and “Sin Sin has nine blocks. Her brother has four blocks. How many more blocks does Sin Sin have than her brother?” Each correct answer was worth one point.

blocks, or failed on four or more items within one of the last three blocks, the whole task was terminated. One point was given when children pronounced the word correctly, and zero otherwise. 2.2.2.2. Morphological awareness. The 48-item morphological construction task (Cheung et al., 2010; McBride-Chang, Shu, Zhou, Wat, & Wagner, 2003) was used to assess morphological awareness. A scenario was presented orally by the examiner for each item and children were asked to construct new words for objects or concepts based on the given information. For example, there is a kind of web (網), which is made by a spider (蜘蛛), that we call a “spider web” (蜘蛛 網), what would we call a kind of web which is made by an ant? (螞蟻) (the correct response would be “ant web” (螞蟻網). Based on the levels of difficulty, the items were organized into eight blocks, each of which consisted of five to seven items. For the initial five blocks, if a child failed on four or more items within one block, the whole test was terminated. If the child successfully reached block 6, s/he would be required to complete the remaining three blocks regardless of errors. One point was given for the correct answer and zero otherwise.

2.3. Procedure Parental permission for child participation was obtained prior to data collection. Data collection was conducted from May to July in 2017. Each child was tested individually within 50 min in a quiet room in her/his own kindergarten, with short breaks between tests. The sequence of the tasks was randomized in this study. All tests were administered by trained research assistants and university student helpers. The current study was approved by the Institutional Review Board of the University.

2.2.2.3. Receptive vocabulary. This Chinese version of the Peabody Picture Vocabulary Test (3rd ed.; PPVT-III, Dunn & Dunn, 1997) task developed by Chung and McBride-Chang (2011) was used to assess children's receptive vocabulary because it showed good reliability in previous studies on Hong Kong children (e.g., Cronbach's alpha = 0.86 in Chung & McBride-Chang, 2011). The 60-items of the task were translated and chosen from the Version B of the Peabody Picture Vocabulary Test (3rd ed.; PPVT-III, Dunn & Dunn, 1997) based on second- and third-year kindergarten children in Hong Kong. For each item, children were presented four black-and-white pictures on a computer screen, and were asked to select the picture best representing the word given orally by the test administrator. Children were required to answer all items. One point was given when children selected the correct picture and zero otherwise.

3. Results Data screening pre-analysis (e.g., normality) was conducted before carrying out statistical analysis. EF and performance on story problems were only mildly skewed according to Tabachnick and Fidell (2007), thus no further transformation procedures were conducted. Table 1 presented the descriptive statistics and correlations for the variables under investigation, as well as the reliabilities of all measures in the present sample. Age and parental education had significant positive correlations only with performance on story problems but not numberfact problems. The correlations, however, were weak (rs = 0.20 and 0.17 respectively, both ps < 0.01). To verify whether boys and girls differed in their performances, independent samples t-tests were conducted. The results showed no significant differences in gender on all measures in this study (ps > 0.05), thus gender was not considered further in subsequent analyses. EF and oral language skills had significant correlations with performance on both number-fact and story problems. The strength of the relations ranged from weak to moderate (rs ranged from 0.21 to 0.46, ps < 0.001). As the participants were recruited from 13 kindergartens across the four regions in Hong Kong, we tested the null model from the multilevel framework to account for the potential nesting structure of our data. The model constraint command was used to create new parameters in Mplus 8.3 (Muthén & Muthén, 1998-2019) so as to estimate the standard errors and significance levels of the interclass correlations (ICCs). The degree of ICC shows the dependence of observations and is typically an indicator of the level of “clustering” of the dependent variable.

2.2.3. Arithmetic competence 2.2.3.1. Number-fact problems. This task consisted of 12 items and resembled those of previous studies (e.g., Cheung & McBride, 2017; Cheung, Yang, Dulay, & McBride, 2018; Jordan, Hanich, & Kaplan, 2003). Six items were addition problems (7 + 5, 5 + 8, 9 + 7, 15 + 8, 24 + 9, 39 + 6). The other six were subtraction problems (6–4, 8–5, 9–4, 12–9, 26–7, 34–8). For each item, children were required to tell the answer to the question presented verbally by the test administrator. One point was given for each correct answer. 2.2.3.2. Story problems. There were 12 items (six addition and six subtraction problems) in this task. For each item, children had to orally 4

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Fig. 1. Standardized path estimates of the linkages between EF, oral language skills and number-fact problems and story problems. Note. Significant paths were shown by solid arrows with standardized coefficients. The numbers in the brackets indicate standard errors. Non-significant paths were shown by dashed arrows. Only significant covariances between the error terms were shown. ^p < .08, *p < .05, **p < .01, ***p < .001.

Our results showed that the variable specific ICCs were small in magnitude (all ICCs < 0.08), but the number-fact problems indicated significant nestedness (p < .01). The models were then run in multilevel fashion, and the pattern of relationships and significance levels regarding the prediction of performance on story and number-fact problems by EF and language skills did not change. Thus, the Level 2 (i.e., kindergarten) structure was not taken into consideration further (cf. Muthen & Satorra, 1995). We then performed path analysis with Mplus 8.3 using a full information maximum likelihood estimation with nonnormality standard errors (MLR) to examine the relations of EF and language skills with performance on number-fact and story problems. The covariances between the error terms of exogenous variables were allowed to correlate with each other. The model was a saturated model with perfect model fit. As shown in Fig. 1, EF (β = 0.10, p < .05), phonological awareness (β = 0.26, p < .001), and morphological awareness (β = 0.31, p < .001) uniquely predicted performance on number-fact problems. For performance on story problems, EF (β = 0.10, p < .05), phonological awareness (β = 0.21, p < .001), and morphological awareness (β = 0.31, p < .001) were significant predictors in this study. The comparison of model coefficients showed that phonological awareness showed stronger association with performance on number-fact problems (β = 0.05, p = .019), while EF and morphological awareness were equal predictors of performance on number-fact and story problems. Receptive vocabulary was not a significant predictor of performance on neither type of problem. Age and parental education were significant covariates in predicting performance on story problems (β = 0.13, p < .05; β = 0.11, p < .08), but not number-fact problems. Overall, the model accounted for 28.2 % of variance in children's performance on number-fact problems and 26.3 % of variance on story problems. Furthermore, the relative strengths of EF and oral language skills in predicting performance on story problems were examined with age, parental education and performance on number-fact problems controlled (see Fig. 2). Morphological awareness was shown to be a unique precursor of performance on story problems, whereas EF, phonological awareness and receptive vocabulary were not. The model explained 45.3 % of variances in story problem solving.

Fig. 2. Standardized path estimates of the linkages between EF, oral language skills and story problem solving after controlling for number-fact problems. Note. Significant paths were shown by solid arrows with standardized coefficients. The numbers in the brackets indicate standard errors. Non-significant paths were shown by dashed arrows. ^p < .08, *p < .05, **p < .01, ***p < .001.

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4. Discussion

fact problems by facilitating the efficient and accurate representation and retrieval of arithmetic facts from long-term memory. In other words, children with poor phonological awareness appear to have difficulties in differentiating and retrieving arithmetic facts from semantic memory. Furthermore, individual differences in phonological awareness may also affect performance on number-fact problems through basic numerical skills (such as knowledge of number words and number sequence) (Krajewski & Schneider, 2009). For example, good phonological awareness may support the active use of counting or other “think aloud” strategies to recall problem solving protocols (Logie & Baddeley, 1987). In contrast, Purpura et al. (2011) showed that phonological awareness did not have unique effects on calculation in preschoolers. This probably happened because the calculation subtest used by Purpura et al. (i.e., Woodcock-Johnson III) was conducted in paper and pencil format and numeracy skills were included in their regressions. Similarly, Moll et al. (2015) reported nonsignificant associations between phonological awareness and written arithmetic competence in young English-speaking children. The presentation of our arithmetic problems (i.e., verbal) puts higher demands on phonological processing than the written arithmetic form. For example, it is required to activate and retrieve the mental phonological and semantic representations of number words and arithmetic operators in oral arithmetic problems. Contrary to our expectation, phonological awareness showed significantly stronger association with performance on number-fact problems, as compared to story problems. Also, phonological awareness was not significantly further associated with performance on story problems once performance on number-fact problems was considered. Fuchs et al. (2006) argued that phonological processing may underlie only number-fact skills. As phonological awareness taps the sensitivity to the sound structure of a language, the familiar problem statements and simple instructions of story problems may not necessarily involve effortful phonological processing beyond number-fact processing. It is possible that phonological awareness may not be able to detect its association with the low-performed and less sensitive story problem task. These results should warrant further research to examine the relationships between other aspects of phonological processing (e.g., listening comprehension) and performance on more complex story problems in older children. Aligned with our hypothesis, morphological awareness accounted for robust variance in performance on number-fact and story problems even after controlling for other skills. This finding can be explained partly by the unique characteristics of the Chinese number word system. As noted, compounding morphology is embedded in Chinese number words and this morphologically structured naming system gives Chinese children a bootstrapping advantage in learning number words (Ng & Rao, 2010). Moreover, morphological awareness facilitates the conceptual understanding of number words so that children with good morphological awareness possess a much better chance to deduce the numerical meanings of number words. For example, children who are able to construct novel words (e.g., “ant web”) from familiar morphemes (e.g., “spider web”) are more likely to acquire and understand new number words (e.g., “three-ten-six”, 36) from previously learned numbers (e.g., “two-ten-six”, 26). Indeed, emerging evidence indicates that compounding morphological awareness in Chinese is linked to counting ability in kindergarten children, thus facilitating the learning of numeracy and performance on number-fact problems (Liu, Lin, & Zhang, 2016). As noted, solving story problems requires the integration of numberfact skills and linguistic comprehension (Ostad, 1998). In our study, performance on number-fact problems, to a large extent, accounted for performance on story problems. Furthermore, morphological awareness remained to be a unique predictor of performance on story problems with EF, other language skills and performance on number-fact problems controlled. Indeed, morphological awareness is important for the comprehension of semantic information and linguistic structures couched in story problems (Fuchs et al., 2006; Zhang & Lin, 2015). This is

This study sought to examine the correlates of performance on number-fact and story problems in Chinese kindergarten children. Our results showed that EF, phonological awareness, and morphological awareness were unique correlates of performance on number-fact and story problems, whereas receptive vocabulary was not. Furthermore, morphological awareness was the only significant predictor of performance on story problems after controlling for age, parental education and performance on number-fact problems. 4.1. The relationships between EF and arithmetic competence Consistent with our initial speculation and previous research (e.g., Cameron et al., 2012; Espy et al., 2004; Moll, Snowling, Göbel, & Hulme, 2015), EF was a unique correlate of performance on the two types of verbal arithmetic problems in this study. On one hand, EF skills are crucial for basic numerical skills, such as number word acquisition and counting (e.g., Lan, Legare, Ponitz, Li, & Morrison, 2011; Zhang, 2016). A longitudinal study by Moll et al. (2015) reported that EF at three years old contributes to arithmetic competence at six years old by counting and number knowledge at four years old. On the other hand, EF skills are also required in performing more sophisticated arithmetic procedures. In particular, the number-fact problems used in this study involved borrowing or carrying procedures and required multiple steps to obtain a solution. For example, to perform “7 + 5 =?” and “12–9 =?”, children are required to maintain number words online for subsequent adding or subtracting processing and also flexibly apply those different procedures (Espy et al., 2004; Lan et al., 2011). However, EF did not make further contribution to performance on story problems in this study. As noted in bivariate correlations, EF was similarly correlated with performance on number-fact problems (r = 0.24) and story problems (r = 0.22). There are two possible explanations for this finding. First, the number-fact problems and story problems used in the present study were parallel to each other (i.e., identical numerosities and operations), and no extraneous information was included in story problems. It was thus not necessary for children to draw upon extra EF resources to find the solution when the same quantitative problem was presented in a familiar story context. Second, the story problems used in this study involved various relational types (e.g., combination of two sets of objects, comparison of two quantities). Children may need to construct different problem solving schemas when solving different types of problems. Since our participants were young kindergarten children, their EF development may not be able to support the demanding process underlying a verbal story statement. In fact, the low average score and limited variation of performance on story problems (M = 2.17; SD = 2.34) meant that a good number of children in our sample had scored zero in this task (n = 89, 31.8 %), a pattern suggesting that the task might be too difficult for children at this age in this particular cultural context. Purpura and Ganley (2014) also showed that EF skills (e.g., working memory) was not a significant predictor of performance on story problems in 4 to 6-year-old children when calculation skills was also included in the model. Future researchers should use other EF and story problem solving tasks to capture the natural variation in problem solving at this age. 4.2. The relationships between oral language skills and arithmetic competence Phonological awareness, to some extent, also depends on EF skills to maintain and manipulate speech sounds in mind in order to generate what is left after deleting one syllable or sound. Nonetheless, our results showed that phonological awareness had significant linkages with performance on number-fact and story problems even with EF and other language skills considered. According to Geary (1993), phonological awareness may directly contribute to performance on number6

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5. Limitations and conclusions

because children are required to listen to the story scenarios, work conceptually with numbers, and construct their semantic relational networks which are determined by language (e.g., combine or compare). During these processes, access to the meaning structure of a language is beneficial to detect more precise relations of the story structures (e.g., performing “subtraction” when the relational term is “more than”), rather than comprehending some superficial cue words, to build the problem solving model (Fuchs et al., 2006; Lewis & Mayer, 1987). It would be valuable for future studies to examine the relations of different types of morphological awareness to performance on more developmentally appropriate story problems in different languages. However, morphological awareness exhibited equal strength to performance on number-fact and story problems in this study. Story problems, relatively rich in semantic information and relational structures though, did not ask for additional influence of morphological awareness in this study. However, Zhang and Lin (2015) showed in Chinese kindergarten children that compounding morphological awareness significantly contributed to performance on verbal story problems, but not written number-fact problems. Arguably, morphological awareness, which taps the sound meaning structure of a language, may exert more significant impact on performance on verbal arithmetic problems. However, the difficulty level of arithmetic demands and the complexity of story contexts underlying story problems in this study may not be able to differentiate the strength of different skills required for solving number-fact and story problems. Although vocabulary knowledge was positively correlated with performance on both number-fact and story problems in this study, their associations failed to be retained when other language skills were also included in statistical analyses. This finding is similar to one from Chow and Ekholm (2019), who found that vocabulary did not make unique contribution to calculations and problem solving among other language skills. It is possible that the relation of vocabulary knowledge may be overridden by other salient cognitive-linguistic precursors in this study, especially given its close associations with morphological awareness (r = 0.54) and phonological awareness (r = 0.40). It seems that receptive vocabulary taps in basic number word learning involving hearing and expressing number words (LeFevre et al., 2010; Zhang, 2016), whereas its power may be attenuated for more advanced number-fact problems in this study. The receptive vocabulary task simply requires the output of acquired knowledge (Purpura et al., 2017), whereas most arithmetic tasks in our study seem not to rely on the direct retrieval of stored information. Regarding the relation between vocabulary knowledge and performance on story problems, there are at least three possible explanations for the nonsignificant findings. First, the advantages of vocabulary development (e.g., facilitating children to figure out the referents of number words) may fade when short and familiar vocabulary words and instructions are used in the problem statements. Second, it may be the math-specific vocabulary words (e.g., “more”, “less”) rather than the general vocabulary words (e.g., cookies) that account for performance on story problems. Compared with most nominal vocabulary words, math-specific vocabulary items tend to be more abstract and also more crucial for solving story problems. Purpura and Reid (2016) indicated that math-specific vocabulary words accounted for unique variances in numeracy skills over and above general vocabulary knowledge. Third, our task simply measured the size of receptive vocabulary, but what matters more for solving story problems may be the depth of concept understanding. Significant associations have previously been found, for example, between expressive vocabulary (Purpura & Ganley, 2014), definitional vocabulary (Purpura et al., 2011) and performance on story problems in young children. Future studies should explore how different types of vocabulary knowledge are related to performance on different types of verbal arithmetic problems.

Some limitations of the current study are worth mentioning. First, we only used one child task measure to assess EF. Prior research has indicated that child task measures tend to capture specific aspects of EF, especially compared to adult report measures (e.g., Sherman & Brooks, 2010). Future researchers should use both child task and adult report measures to triangulate EF to better explore its association with arithmetic abilities. Second, this study only addressed the roles of domain-general skills (including EF and oral language skills) in performance on verbal arithmetic problems. It would be interesting to investigate the relative contributions of domain-general skills and math-specific skills (e.g., counting, knowledge of number sequence, and quantity comparison skills) to performance on verbal arithmetic problems among kindergarten children. Third, story problems simultaneously measured children's understanding of semantic meaning and syntactic structures. It remains unclear how these aspects of understanding are related to language skills and EF. Additional research is needed to distinguish between children's understanding of semantic meaning and syntactic structures and differentially link them to language skills and EF. To conclude, the present findings have expanded the existing studies by showing that EF and oral language skills could correlate with children's arithmetic competence. First, within a multivariate framework, our findings suggest that EF, phonological awareness and morphological awareness provide the building blocks for solving number-fact and story problems. Practically, EF is reported to be a malleable skill and different kinds of game-playing (e.g., drama pretend play) can refine EF skills (Diamond, 2013; Zhang, 2016). Furthermore, parents and educational practitioners are encouraged to expose children to the sound and meaning structure of a language, rather than the surface level of knowing “what” only. In other words, the ability to reflect on the representational structure of a language may be transferred to other factors (e.g., symbolic number skills) influencing subsequent arithmetic performance (Vukovic & Lesaux, 2013). Second, it is worthwhile to pay particular attention to compounding morphological awareness in Chinese, which remained to be a significant predictor of performance on story problems after considering performance on number-fact problems in the framework. In any case, receptive vocabulary seems not to be an independent predictor of verbal arithmetic competence when a wide range of cognitive and language skills are included. Funding sources The work described in this article was fully supported by a grant from the Standing Committee on Language Education and Research (SCOLAR), Hong Kong, China [EDB(LE)/P&R/EL/164/1] to Kevin Kien Hoa Chung. Declaration of competing interest None. References Alloway, T. P., Gathercole, S. E., Adams, A., Willis, C., Eaglen, R., & Lamont, E. (2005). Working memory and phonological awareness as predictors of progress towards early learning goals at school entry. British Journal of Developmental Psychology, 23(3), 417–426. https://doi.org/10.1348/026151005X26804. Best, J. R., & Miller, P. H. (2010). A developmental perspective on executive function. Child Development, 81(6), 1641–1660. https://doi.org/10.1111/j.1467-8624.2010. 01499.x. Bjork, I. M., & Bowyer-Crane, C. (2013). Cognitive skills used to solve mathematical word problems and numerical operations: A study of 6-to 7-year-old children. European Journal of Psychology of Education, 28(4), 1345–1360. https://doi.org/10.1007/ s10212-012-0169-7. Bull, R., & Johnston, R. S. (1997). Children’s arithmetical difficulties: Contributions from processing speed, item identification, and short-term memory. Journal of Experimental Child Psychology, 65, 1–24. https://doi.org/10.1006/jecp.1996.2358.

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