SES is a composite measure of family economic and social position, commonly assessed through indicators such as parental education level, occupational status, and household income (Li et al., 2020). According to ecological systems theory, children's development is shaped by multi-level influences, including the micro-system (e.g., family), meso-system (e.g., school and community), and macro-system (e.g., societal policies and culture). The closer these systems are to the individual, the greater their impact on development (Bronfenbrenner and Morris, 1998). Among these, SES is a key factor influencing cognitive development, particularly in the realm of mathematics (Fan et al., 2022; Mao, 2022).

The impact of SES on children's mathematical skills is evident even before formal schooling begins (Klibanoff et al., 2006). Differences in economic resources between social classes can significantly affect children's early math learning, as SES is associated with varying access to learning opportunities and social experiences that support mathematical development (Jordan and Levine, 2009). Research by Denton et al. (2002) demonstrated that children from higher SES backgrounds perform significantly better in math during preschool and early elementary years compared to those from lower SES backgrounds. Huang (2006) further suggested that parental education and the material resources provided by the family are related to children's performance on tasks involving the CP, written number-symbol representation, and arithmetic operations. The family stress model proposed by Conger and Donnellan (2007) holds that a lower socioeconomic status is associated with increased economic stress. This pressure can lead to family conflicts, reduce parent-child interaction and decrease academic guidance. Such an environment limits children's opportunities for cognitive enrichment and learning guidance, further hindering their early academic development, including mastering concepts such as CP. More recent studies have reinforced this notion, highlighting that children from higher SES families, who are exposed to frequent math-related activities and abundant resources, outperform their lower SES peers in understanding and applying mathematical concepts and skills (Ramani and Siegler, 2008; Silver and Libertus, 2022). Based on these findings, we propose the following hypothesis:

The ANS is a cognitive mechanism used by humans and other animals to estimate quantities quickly and intuitively, allowing us to compare two sets of objects' relative quantities without precise counting or the use of symbols (Negen and Sarnecka, 2015; Cochrane et al., 2019). While ANS has traditionally been viewed as an innate cognitive system (Dehaene, 2011; Starr and Brannon, 2015), recent studies have shown that its sensitivity may be influenced by environmental factors, suggesting the system's plasticity (Tosto et al., 2014).

Although SES is a well-established environmental factor that affects children's development, there is limited consensus in the literature regarding the relationship between SES and ANS (McNeil et al., 2011; Purpura and Simms, 2018; Slusser et al., 2019; Wei et al., 2020; Bachman et al., 2022). For example, a longitudinal study by Wei et al. (2020) with 173 children (mean age = 67.25 months, SD = 3.67) found no correlation between SES (family income and parental education level) and performance on ANS tasks (e.g., Dot Comparison, Number Line Estimation). In contrast, Bachman et al. (2022) reported a significant positive correlation between SES and ANS accuracy in a community-based longitudinal study involving 178 children aged 4 years and their parents.

Numerous studies have emphasized the importance of ANS in the early development of mathematical abilities (Libertus et al., 2011; Bonny and Lourenco, 2013; Vogel and De Smedt, 2021). Cognitive neuroscience research has identified the intraparietal sulcus as a key brain region involved in numerical cognition, suggesting a link between ANS and CP (Dehaene et al., 1996). Despite some opposing views (Libertus et al., 2014; Carey et al., 2017; Schröder et al., 2022), empirical studies largely support the role of ANS in children's mastery of early mathematical concepts, including CP (Libertus et al., 2011; Inglis and Gilmore, 2014; Fuhs et al., 2016; vanMarle et al., 2018). Children with higher ANS sensitivity are more likely to grasp symbolic number knowledge, including cardinality and arithmetic knowledge (Chu et al., 2016; Geary et al., 2013; van Marle et al., 2014).

Given the relationships between SES, ANS, and CP, we hypothesize that ANS may mediate the effect of SES on CP development. Although limited research has directly examined this hypothesis, exploring this possibility will deepen our understanding of how environmental factors influence mathematical development through ANS.

The development of early mathematical skills cannot be considered in isolation from other cognitive and academic abilities. Increasing evidence underscores the significant role of language skills in children's acquisition of early mathematical knowledge (Purpura et al., 2019; Erath et al., 2021). RVS, one of the foundational components of language development (Wilkins, 1972), refer to the words that an individual can understand and respond to, even if they are not able to produce these words themselves (Burger and Chong, 2011). RVS is recognized as a crucial tool in the development of mathematical concepts in children (Negen and Sarnecka, 2012).

Enhanced receptive vocabulary skills not only facilitates children's ability to acquire mathematical language through social interactions, but it also enriches their mathematical terminology and deepens their understanding of mathematical concepts (Klibanoff et al., 2006; Susperreguy and Davis-Kean, 2016). As noted by Barner et al. (2009), language skills play a decisive role in children's acquisition of cardinality knowledge for small numbers. Negen and Sarnecka (2012) further found that the larger the receptive vocabulary, the more profound children's understanding of the CP becomes. Similarly, Pixner et al. (2018) reported a significant correlation between children's vocabulary skills and their understanding of small number concepts.

Numerous studies have confirmed that children from higher SES backgrounds tend to outperform their peers from lower to middle SES backgrounds in terms of vocabulary skills (Fernald et al., 2013; Salminen et al., 2021; Xiao et al., 2020; Ma et al., 2024). These linguistic disparities can have significant implications for mathematical learning, particularly for children from low-SES families, for whom limited language development may pose a barrier to acquiring mathematical knowledge (Abedi and Lord, 2001). Further research has demonstrated that preschool children from middle-SES families perform better on tasks involving oral number combinations and mathematical story comprehension, possibly because they have more exposure to numerically relevant vocabulary (Jordan et al., 1992, 1994, 2006). Mastery of number-related vocabulary aids in the completion of mathematical tasks (Mix, 2008), and the lack of this vocabulary in low-SES preschoolers may place them at a disadvantage in their mathematical skill development (Bao, 2014).

Based on the above discussions, we propose the following hypothesis:

Numerous studies have highlighted the critical role of ANS and RVS in children's numerical cognition development. Specifically, as the underlying mechanism for children's intuitive perception of quantities in early childhood, ANS enables children to perceive the differences in the quantity of objects, such as distinguishing which side of two piles of apples has more. This non-verbal sense of quantity based on ANS lays the foundation for children's subsequent digital cognitive development. However, for children to transform this intuitive perception of quantity into precise mathematical concepts, RVS plays a crucial mediating role (Spelke, 2003; Purpura and Logan, 2015; Barner, 2017; Spelke, 2017). As Spelke (2003) pointed out, language skills play a key mediating role in the process in which children refine their basic quantitative perception into accurate numerical representations. For example, when children learn to say “one apple” and “two apples,” they begin to use language symbols to gradually transform this non-verbal sense of quantity based on ANS into precise mathematical concepts (Barner, 2017).

As discussed in Sections 1.2 and 1.3, children from higher-SES families tend to have access to richer learning environments, creating more opportunities for interaction (Bradley and Corwyn, 2002; Evans, 2004). This environment helps children's ANS development, enhancing their sensitivity to quantity-related information and increasing their exposure to a broader vocabulary, thereby effectively boosting their RVS (Fuhs and McNeil, 2013; Golinkoff et al., 2019). Given these observations, the rich learning environment provided by a higher socioeconomic status (SES) not only benefits the cultivation of children's ANS, enabling them to perceive quantitative differences keenly but also creates favorable conditions for children to establish connections between this non-verbal quantitative intuition and precise numerical symbols through RVS.

Therefore, we propose the following hypothesis:

The development of early mathematical skills, particularly mastery of the CP, plays a crucial role in determining later mathematical achievement and academic success. Among the many factors influencing children's CP development, SES, ANS, and RVS have attracted significant attention. However, the specific relationships and underlying mechanisms between these variables remain unclear. Therefore, we propose a chain mediation model to examine whether SES indirectly influences children's CP development through ANS and RVS. Empirical data will be collected from children across various socioeconomic backgrounds, assessing their sensitivity to ANS, level of receptive vocabulary, and understanding of CP. The SPSS PROCESS macro will be used to perform chain mediation effect analysis to test the proposed hypotheses. The results of the study will elucidate the intrinsic links and action pathways between SES, ANS, RVS, and CP.

A visual representation of the proposed model is shown in Figure 1.

The participants in this study were children from Urumqi City and Changji Prefecture in the Xinjiang Uygur Autonomous Region of China. Throughout the research process, we strictly adhered to ethical guidelines, ensuring that all children's guardians provided informed consent prior to participation. The inclusion criteria for participants were as follows: (1) children were able to understand and communicate in Mandarin Chinese; (2) participants did not have any intellectual or hearing impairment; and (3) children had normal vision or could achieve normal visual acuity through appropriate corrective measures. Additionally, the age range of participants was limited to children between the ages of 3 and 7 years.

Thanks to the active support of the management, teachers, and parents at the participating preschools, we successfully recruited 120 children and their families for the study. Despite challenges such as a flu outbreak, scheduling conflicts, and incomplete data collection, we ultimately collected a complete and high-quality dataset of 110 participants, which provided a solid foundation for our analysis. The sample consisted of an equal number of boys and girls (Mean age = 67.53 months, SD = 7.415 months). Table 1 provides detailed information on the participant characteristics.

Previous research has shown that both gender and age are significant predictors of the ANS, RVS, and the CP in children (Holloway and Ansari, 2009; Sekuler and Mierkiewicz, 1977; Sumit Kumar et al., 2022; Barsotti et al., 2023; Hornburg et al., 2018; Das et al., 2022). Therefore, in the current study, children's gender and age were included as covariates in the chain mediation analysis.

Data were analyzed using SPSS version 29.0. Descriptive statistics and correlation analyses were first conducted to assess the overall performance of the participants on SES, ANS, RVS, and CP, as well as to examine the relationships between these variables. Following this, we employed path analysis using PROCESS macro version 4.1 (Model 6) to test whether ANS and RVS mediate the relationship between SES and children's CP, controlling for children's gender and age. The significance of the mediation effects was evaluated using a bootstrap procedure with 5,000 resamples, with a 95% confidence interval (CI). A significant mediation effect was indicated if the bootstrap CI did not contain zero (Hayes, 2013).

To explore the relationships between SES, ANS, RVS, and children's CP, we conducted descriptive statistics and correlation analyses. The results indicated significant positive correlations between SES (r = 0.443, p < 0.001), ANS (r = 0.447, p < 0.001), RVS (r = 0.560, p < 0.001), and CP. Additionally, SES (r = 0.446, p < 0.001) and ANS (r = 0.467, p < 0.001) were positively correlated with RVS, while SES (r = 0.304, p < 0.01) was also significantly positively correlated with ANS. These results provide preliminary support for the validity of our hypotheses. Means, standard deviations, and correlation coefficients for all variables are presented in Table 2.

To further investigate the relationships between SES, ANS, RVS, and children's CP, we conducted a chain mediation analysis using PROCESS macro (Model 6) developed by Hayes, with children's gender and age included as covariates. The results (see Table 3) indicated that SES significantly associated with ANS (β = 0.264, p < 0.01). Furthermore, both SES (β = 0.281, p < 0.01) and ANS (β = 0.338, p < 0.001) significantly associated with RVS. Additionally, SES (β = 0.164, p < 0.05), ANS (β = 0.191, p < 0.05), and RVS (β = 0.272, p < 0.01) significantly associated with CP. These results suggest that SES influences CP through both direct and indirect pathways via ANS and RVS.

Next, we used bootstrapping (5,000 resamples, 95% CI) to further test the chain mediation model. As shown in Table 4 and Figure 2, the total effect of SES on children's CP was 0.315. The mediation effects were decomposed into three distinct pathways: Path 1 (SES → ANS → CP) had an effect size of 0.051 [95% CI (0.0014, 0.3993)], Path 2 (SES → RVS → CP) had an effect size of 0.076 [95% CI (0.0249, 0.1373)], and Path 3 (SES → ANS → RVS → CP) had an effect size of 0.024 [95% CI (0.0053, 0.0581)]. The mediation effects accounted for 16.04%, 24.19%, and 7.67% of the total effect, respectively. The total indirect effect was 0.151, which accounted for 47.91% of the total effect of SES on children's CP. Since none of the 95% confidence intervals for these indirect effects contained zero, all mediation effects were statistically significant.

The results of this study indicate that SES has a significant positive effect on children's mastery of the CP, confirming Hypothesis 1. This finding is consistent with previous research (Rogelberg et al., 2021; Silver and Libertus, 2022; DePascale et al., 2023), which suggests that children from higher SES families tend to perform better in numerical domains. Conger and Donnellan (2007) proposed two influential models, the Family Investment Model and the Family Stress Model, to explain the “social causality” between SES and individual development. According to the Family Investment Model, parents from higher SES backgrounds have greater access to resources (Susperreguy et al., 2020) and are able to invest more material resources in their children's development (Conger and Dogan, 2007; Dearing and Taylor, 2007; Kaushal et al., 2011). Additionally, these parents are more likely to dedicate time and resources to engaging in enriching and enjoyable mathematical learning activities with their children (Roubinov and Boyce, 2017; Carneiro and Ginja, 2016), which effectively promotes the development of children's mathematical abilities.

In contrast, children from lower SES backgrounds typically have fewer opportunities to experience high-quality educational settings and demonstrate lower performance in early mathematics tasks and standardized math tests (Elliott and Bachman, 2018; Lyu et al., 2019; Short and McLean, 2023). From the perspective of the Family Stress Model, lower SES is associated with increased economic pressure, which can lead to family conflict, reduced parent-child interaction, and less academic guidance (Masarik and Conger, 2017). Economic stress may also limit the time parents can spend with their children, and their inability to invest in creating an optimal learning environment for their children can result in insufficient learning resources and a lack of attention to home-based learning activities (Conger et al., 2002). Such an environment restricts children's access to cognitive enrichment and learning guidance, further hindering their early academic development, including the mastery of concepts such as CP.

The present study also found that the ANS partially mediates the relationship between SES and children's mastery of the CP, supporting Hypothesis 2. ANS is an innate numerical system that provides preschool children with an intuitive sense of quantity, allowing them to process object quantities before fully mastering symbolic number knowledge. This ability facilitates the understanding of the order and cardinality of numbers (Bethany et al., 2017; Chu et al., 2016). Previous studies have shown that the accuracy of the ANS is positively correlated with children's ability to master the CP; the greater the precision of the ANS, the better the children's understanding and mastery of CP (Huntley-Fenner and Cannon, 2000; Mussolin et al., 2012; van Marle et al., 2014; Wagner and Johnson, 2011).

Moreover, the level of family SES directly influences the quality of children's early mathematical experiences, and early math education plays a critical role in the development of ANS sensitivity (Pica et al., 2004; Fuhs and McNeil, 2013). Specifically, children from higher SES families are more likely to have access to high-quality educational resources and parent-child interactions, with more opportunities to engage in mathematical activities during these interactions (Bonny and Lourenco, 2013; Jiang and Wang, 2024). This enhances their ANS accuracy, which in turn facilitates a quicker mastery of the CP. In contrast, children from lower SES families, who lack effective mathematical guidance and resources (Fuhs and McNeil, 2013; Ramani and Siegler, 2008; Votruba-Drzal, 2003), may experience limited development of their ANS, which directly impacts their understanding and mastery of the CP. In addition, parents with a higher level of education tend to have higher expectations and more participation in their children's mathematics learning. This positive parent-child interaction and support further promoted the development of ANS (Keijer, 2021), thereby enhancing children's understanding of the CP. Consequently, the ANS acts as a bridge between SES and CP, partially explaining the differences in CP abilities observed in children from high and low SES families.

Our findings also support Hypothesis 4, suggesting that RVS play a partial mediating role in the relationship between SES and children's mastery of the CP. This result aligns with previous research by Abedi and Lord (2001), which indicated that insufficient language development can be a barrier to children's early mathematical achievements, especially for those from low-income backgrounds. Vygotsky's theory posits that although thought and language originate from different processes, they ultimately converge, with language structures serving as the foundation for cognitive structures. This convergence limits cognitive development, particularly in the domain of mathematics, where language skills constrain mathematical thinking (Vygotsky, 1962). In this context, language skills are seen as a limiting factor in the development of mathematical abilities, such as understanding the CP.

Neurocognitive research has found that vocabulary skills and numerical cognition share common brain regions (Baldo and Dronkers, 2007; Arsalidou and Taylor, 2011), suggesting that children with stronger vocabulary skills tend to perform better in numerical tasks (Aunola et al., 2004). Furthermore, parents with a higher level of education tend to be able to provide a more language-rich environment (Bradley and Corwyn, 2002; Evans, 2004; Linver et al., 2002), where parents engage in more frequent verbal interactions with their children, use a more diverse vocabulary and complex sentence structures, and ask questions that stimulate conversation. This environment promotes the development of RVS (Golinkoff et al., 2019) and increases children's exposure to mathematical vocabulary, thereby laying a foundation for understanding the CP.

In contrast, children from lower SES families, due to resource constraints and differences in parental involvement, are likely to experience lower levels of language interaction and problem-solving activities, which restrict the development of their RVS (Bradley and Corwyn, 2002; Burchinal et al., 2008; Hoff, 2006). This limitation in RVS development, in turn, restricts their understanding of CP. Thus, RVS serves as a mediator between SES and CP, explaining part of the variation in CP mastery between children from different SES backgrounds.

An important finding of this study is that both the ANS and RVS play a chain-mediating role in the relationship between SES and children's mastery of the CP. ANS, an innate ability to approximate quantities, provides children with an initial, albeit vague, sense of number. As children's language skills, particularly their RVS, develop, they begin to link their intuitive sense of quantity with the vocabulary in language, transforming their vague number sense into a more precise representation of quantity (Piaget, 1977; Spelke, 2003). It is noteworthy that children's mathematical knowledge often develops through interactions that involve mathematical language (Klibanoff et al., 2006). For example, during everyday family conversations, parents and children might discuss shopping (e.g., “We need to buy 2 boxes of milk and 3 packs of cookies”), cooking (e.g., “We need 3 cups of flour to make a cake”), or organizing toys (e.g., “We have 5 stuffed animals and 4 toy cars—can you help me put them in different boxes?”). These language-based interactions help children connect their intuitive number sense to real-world contexts, thereby enhancing their understanding of CP (Barner, 2017; Spelke, 2017; Gibson et al., 2020; O'Rear and McNeil, 2019).

In higher SES environments, interactions between children and adults, especially those involving language and mathematics, tend to be more frequent (Bonny and Lourenco, 2013; Jiang and Wang, 2024; Li et al., 2023). These frequent interactions foster the development of a more precise ANS and further contribute to children's mastery of CP. However, parents engaged in physical labor or low-skilled jobs may find it challenging to provide their children with equally rich and high-quality learning stimuli at home due to the limitations of their working hours and energy and the influence of their educational background. This may lead to relatively lagging children in the development of ANS, RVS, and CP. Thus, both ANS and RVS play a critical, interconnected role in mediating the effects of SES on children's understanding of mathematical concepts.