The association between visual-spatial skills and preschoolers’ arithmetic ability: the mediating role of patterning ability and the moderating role of executive function

Abstract

As a foundational element of early childhood education, preschoolers’ arithmetic ability eases the later arithmetic learning in grades. However, the mechanisms underlying young children’s arithmetic ability remain unclear. To this end, the present study focuses on the link between visual-spatial skills and arithmetic ability among young children, with a particular emphasis on the mediating role of patterning ability and the moderating role of executive function. The study surveyed 233 preschool children in their final year and their parents in Fuyang, a city in central China, between November and December 2021. Results showed that visual-spatial skills positively correlated with preschoolers’ arithmetic ability, that patterning ability partially mediated the link between visual-spatial skills and arithmetic ability, and that executive function negatively moderated the link between visual-spatial skills and patterning ability. These findings illuminate the pedagogical strategies, such as the cultivation of visual-spatial skills and pattern recognition awareness, especially for those with varying levels of executive function in early math education.

1 Introduction

Arithmetic is widely recognized as a core and foundational component of early mathematics learning, including young children’s developing understanding of fundamental numerical concepts and relationships, such as counting, symbol-magnitude mapping, and calculation (Rittle-Johnson et al., 2017; Zhang and Lin, 2015; Zippert et al., 2020). Early arithmetic ability is typically assessed through children’s performance on various types of problems, including non-symbolic and symbolic arithmetic calculations, as well as arithmetic word problems (Huttenlocher et al., 1994; Rittle-Johnson et al., 2017; Zhang and Lin, 2015; Zheng et al., 2024; Zippert et al., 2020). Fostering the development of these foundational arithmetic skills represents a crucial objective in early mathematics education (Göbel et al., 2014; Zheng et al., 2024). Ample empirical evidence highlights the contribution of early arithmetic abilities to, such as, their strong predictive power for later school mathematics achievement and to the establishment of a necessary foundation for acquiring more advanced mathematical knowledge (Jordan et al., 2007; Rittle-Johnson et al., 2017; Yang et al., 2019; Yang and Meng, 2020; Zhang and Lin, 2015; Zheng et al., 2024). Conversely, young children who exhibit weak initial arithmetic abilities or experience arithmetic learning difficulties tend to progress more slowly in subsequent developmental stages, subsequently leading to increasingly pronounced disparities from their peers over time (Geary et al., 2012). Given this importance mentioned above of this early mathematical foundation, the present study aimed to illuminate the underlying cognitive mechanisms that influence preschoolers’ arithmetic ability.

Previous study has shown that domain-general cognitive factors (e.g., visual-spatial skills and executive function, Fung et al., 2020; Yang et al., 2019; Zhang, 2016) and domain-specific cognitive factors (e.g., patterning ability, Borriello et al., 2023; Burr et al., 2022; Fyfe et al., 2017; Rittle-Johnson et al., 2019; Wijns et al., 2019, 2021; Zhang et al., 2014; Zippert et al., 2019, 2020) are tightly correlated with arithmetic ability of preschoolers. However, most of these studies focused on independently contributive roles of domain-general and -specific cognitive factors (e.g., visual-spatial skills, executive function, and patterning ability) in this arithmetic ability. According to Pathways to Mathematics Model (LeFevre et al., 2010) proposed by cognitive ability (i.e., spatial attention) exerts an impact on arithmetic performance through early numeracy knowledge. To be specific, the present study aimed to examine the association between visual-spatial skills and children’s arithmetic ability through patterning ability (Borriello et al., 2023; Zhang et al., 2014), while focusing on the interactive effects of general cognitive factors, i.e., visual-spatial skills and executive function, across different domains (Bäckman and Dixon, 1992; Collins and Laski, 2015). We will in detail introduce relevant background as follows.

1.1 The relationship between visual-spatial skills and arithmetic ability

Visual-spatial skills refer to a kind of spatial ability for creating, retrieving, and transforming visual images (McGrew, 2009; Xie et al., 2020). It is not a single concept, encompassing basic spatial perception (Linn and Petersen, 1985; Uttal et al., 2013; Zhang and Lin, 2015), complex mental rotation (Newcombe and Shipley, 2014; Uttal et al., 2013), and spatial visualization (Newcombe and Shipley, 2014; Uttal et al., 2013).

Many studies have demonstrated strong correlations between visual-spatial skills and young children’s arithmetic ability (Amland et al., 2025; Atit et al., 2022; Fung et al., 2020; Gunderson et al., 2012; Xie et al., 2020; Yang et al., 2019, 2021; Zhang, 2016; Zhang and Lin, 2015). According to the Mental Number Line Theory, human brain may represent a mental number line, where digits are typically arranged in a left-to-right spatial sequence (Dehaene et al., 1990; Hawes and Ansari, 2020; Zhang, 2016), when performing the arithmetic problems. By this, visual-spatial skills enable children to mentally represent and manipulate numerical information (Fung et al., 2020; Gunderson et al., 2012), thereby facilitating their effective resolution of arithmetic problems.

Moreover, much empirical evidence has shown that visual-spatial skills (e.g., spatial perception ability, mental rotation skills, and spatial visualization) are positively correlated with children’s arithmetic performance. For instance, spatial perception ability correlates positively with arithmetic ability among preschool children aged 3–6 (Fung et al., 2020; Yang et al., 2019, 2021; Zhang and Lin, 2015); the mental rotation skills of 5-year-old preschool children are significantly and positively correlated with arithmetic ability (Gunderson et al., 2012); spatial visualization is significantly and positively correlated with arithmetic ability of preschool children aged 3–6 (Barnes et al., 2011; Yang et al., 2021). Although associations between various types of visual-spatial skills and young children’s arithmetic abilities were established, younger children often perform at a chance level in complex visual spatial tasks (e.g., mental rotation, spatial visualization) (Zhang, 2016), which may indicate the difficulty of these complex tasks. In contrast, standard spatial perception tests (more details see section 2: Visual-Spatial Skills Scale) are generally simpler and more accessible to young children (Zhang, 2016). Therefore, in this study, we chose this simpler task to assess the level of visual-spatial skills in young children.

1.2 The mediating role of patterning ability

Although many studies have documented the positive association between visual-spatial skills and young children’s arithmetic ability, its underlying mechanism remains still unclear. Among factors, the ability of recognizing the pattern of task (i.e., patterning ability) in question is very important for successfully solving this arithmetic problem. A pattern is a sequence that develops according to certain rules (Burr et al., 2022), in which there exists a specific relationship among the items or elements (MacKay and De Smedt, 2019), featuring repetitiveness and predictability (MacKay and De Smedt, 2019; Rittle-Johnson et al., 2017). In a sequence, elements consist of either letters or numbers or non-numerical components, such as colors, shapes, sizes, and images (Burr et al., 2022). These elements can alternate in repetition (e.g., red-blue-red-blue) or follow systematic patterns of increase or decrease (e.g., 2–4–6) (Burr et al., 2022; MacKay and De Smedt, 2019; Wijns et al., 2019, 2021).

On the one hand, many studies demonstrated that patterning ability, referring to the ability to notice and use predictable sequences (Burr et al., 2022; Wijns et al., 2021; Zippert et al., 2019), was associated with preschoolers’ arithmetic learning. Patterning ability emphasizes the identification and application of rules and structural relationships in things. It can be assessed through repetitive pattern tasks (Borriello et al., 2023; Burr et al., 2022) and growth pattern tasks (Borriello et al., 2023; Wijns et al., 2019, 2021; Zhang et al., 2014) or one-dimensional (Borriello et al., 2023; Burr et al., 2022) and multidimensional pattern tasks (Borriello et al., 2023; Collins and Laski, 2015). Conceptually, counting and arithmetic principles are essentially complex relationships connected by rules, similar to patterns (Burr et al., 2022; Zippert et al., 2019). In other words, the numerical sequence knowledge (such as 1, 2, 3, …) and computational rules in arithmetic are essentially abstract mathematical patterns. The core of arithmetic lies in identifying and applying fixed rules within numerical relationships (e.g., natural number sequences follow a continuous progression of “+1”). Thus, both patterning ability and arithmetic ability conceptually involve the perception and application of rules and structural relationships. Furthermore, empirical studies also demonstrated a significant correlation between patterning ability and arithmetic ability (Borriello et al., 2023; Burr et al., 2022; Wijns et al., 2021; Zippert et al., 2020). For example, a cross-sectional study involving 4-year-olds demonstrated a significant positive correlation between repetitive patterning ability and arithmetic ability, but their performance in growth tasks showed inconsistent results with arithmetic ability, even exhibiting a significant negative correlation in some items (Wijns et al., 2019). Other studies revealed that both types of patterning ability were significantly positively correlated with arithmetic ability (4–6-year-olds: Wijns et al., 2021; 4–9-year-olds: Borriello et al., 2023). However, these studies rarely use nonlinear or multi-level structural patterns.

Although tasks in various patterns all involve discovering relationships and predicting the next upcoming elements, the connection between patterning ability and arithmetic ability can vary depending on the type of stimuli used, such as shapes and numbers. Numerical pattern tasks, which employ abstract mathematical symbols (e.g., numbers), allow children to more easily utilize the rules of the number system for operational reasoning (e.g., 1+2 = 3) (Zhang et al., 2014). Therefore, the ability to complete such tasks is more directly related to arithmetic ability. In contrast, non-numerical pattern tasks (here, i.e., visual pattern task) typically do not involve explicit numerical knowledge (Zippert et al., 2019). Children can discover general rules in sequences by recognizing and generalizing the surface features of non-symbolic visual materials (Burr et al., 2022; Wijns et al., 2021; Zippert et al., 2019). This process supports children’s understanding of the rules underlying number systems and indirectly facilitates their arithmetic learning. Existing research on patterning ability in young children has predominantly focused on visual patterns (Borriello et al., 2023; Burr et al., 2022; Wijns et al., 2021; Zippert et al., 2020), while relatively less attention has been paid to numerical patterns (Wijns et al., 2019; Zhang et al., 2014) and multidimensional numerical tasks. Therefore, this study will examine multidimensional numerical pattern tasks to verify and enrich the evidence regarding the domain-specific connection between pattern ability and arithmetic ability in young children.

On the other hand, visual-spatial skills also were positively related to young children’s patterning ability (Rittle-Johnson et al., 2019; Wijns et al., 2021; Zhang et al., 2014). In processing pattern tasks, children are required to identify elements and their positional information. Sometimes, they also need to decompose these visual-spatial information into “units” or chunks to better recognize the structure and rules of the pattern (Mulligan and Mitchelmore, 2009; Wijns et al., 2021). That is to say, visual-spatial skills can function as assisting children in rapidly identifying spatial arrangements of elements in pattern tasks and structural (e.g., repeating core units in repeating patterns, fixed intervals between numbers in numerical growth patterns), thereby providing essential perceptual input for rule extraction (Collins and Laski, 2015; Zippert et al., 2019; Mulligan and Mitchelmore, 2009; Wijns et al., 2021). Moreover, empirical evidence has showed that visual-spatial skills are positively correlated with both repetitive (Rittle-Johnson et al., 2019; Wijns et al., 2019, 2021) and growth pattern recognition (Wijns et al., 2019, 2021) in children aged 4–6. Given that multidimensional pattern tasks (e.g., matrix) involve more complex rules requiring identification of patterns across multiple directional sequences, their completion is likely closely linked to visual-spatial skills.

According to the Pathways to Mathematics Model developed by LeFevre et al. (2010), domain-general cognitive abilities (such as spatial attention, linguistic) can establish correlations with early mathematical development through domain-specific skills (such as early numeracy knowledge) (Burr et al., 2022; LeFevre et al., 2010). Borriello et al. (2023) further proposed that patterning abilities may mediate the relationship between learner characteristics (e.g., spatial skills) and children’s arithmetic abilities. Empirical studies have shown that special abilities relevant to mathematical abilities (such as number sequence knowledge, numerical skills) can mediate the relation between spatial abilities (e.g., visual-spatial skills) and children’s mathematical achievements (Cirino, 2011; Cragg et al., 2017; Hawes et al., 2019; Krajewski and Schneider, 2009; LeFevre et al., 2010; Zhang et al., 2014). Based on the theoretical support and empirical evidence, this study proposes that patterning ability mediates the relationship between visual-spatial skills and arithmetic ability among preschoolers.

1.3 The moderating role of executive function

Executive function is hypothesized to play a moderating role in the link between visual-spatial skills and patterning ability among preschoolers. Executive function¹ refers to these higher mental processes underlying goal-oriented behaviors (Welsh, 2002). The current study measured two core components, working memory and inhibition, as recommended in the previous study (Thorell and Nyberg, 2008). Working memory is a limited capacity system that enables the retention and simultaneous processing of information during complex cognitive activities (Baddeley, 1983), such as storing and manipulating elements, the core units, structures, or rules of a recognized numerical sequence (Simmering, 2016). Inhibition involves the suppression of prepotent response and interference control function (Barkley, 1997). To be more specific, it may involve helping children filter out irrelevant distractions (e.g., colors, shapes), focus on underlying structure of patterns, avoid erroneous responses (Collins and Laski, 2015), or inhibit inappropriate strategies (Pelegrina et al., 2024). For instance, when encountering “2, 3; 5, ?,” they focus on the relationship between 2 and 3 while temporarily inhibiting the relationship with 5.

According to Bäckman and Dixon’s (1992)Psychological Compensation Framework, individuals activate compensatory mechanisms when facing “skill-demand” mismatches. In pattern tasks, such mismatches may arise from increased task complexity or imbalances in cognitive resources. Children are less likely to simply combine visual-spatial skills with executive functions, rather dynamically choose dominant cognitive strategies based on task requirements and their own resource availability (Collins and Laski, 2015) . In the same vein, when encountering pattern tasks, individuals must first meet the basic demands of the task on executive functions (such as working memory and inhibition). More specifically, when processing the pattern tasks, individuals are required to hold the items in working memory, while retrieving the relevant information through inhibiting the irrelevant information. Under such circumstances, only when individuals’ level of executive function is lower, can they be dependent on the visual-spatial skill when performing the pattern tasks. In contrast, if individuals’ level of executive function is higher, sufficient cognitive resources allow them to employ alternative strategies to complete the pattern tasks. Collins and Laski (2015) held the similar view that visual-spatial skills provide fundamental representational support for pattern recognition, while executive functions (here, i.e., inhibition and working memory) serve as key moderating resources for processing abstract and transformational rules required for pattern processing (Miller et al., 2016).

1.4 The present study

Based on existing empirical and theoretical support, the present study examined the relation between visual-spatial skills and arithmetic ability, with a particular emphasis on the mediating role of patterning ability and the moderating role of executive function. The purpose of this study is to provide references for the early arithmetic learning and teaching of children, especially during the COVID-19 pandemic. Based on the aforementioned findings, the present study proposed hypotheses that:

H1: Visual-spatial skills are positively correlated with arithmetic ability among preschoolers.

H2: Patterning ability mediates the relationship between visual-spatial skills and arithmetic ability.

H3: Executive function moderates the relationship between visual-spatial skills and patterning ability among preschoolers. Specifically, the relationship becomes weaker as the level of executive function increases.

2 Methods

2.1 Participants

The sample size for this study was determined based on previous studies (n = 185, Borriello et al., 2023; n = 66, Collins and Laski, 2015; n = 66, Zippert et al., 2019; n = 109, Zhang, 2016). Also, we employed G*POWER 3.1 (Faul et al., 2009) to determine the minimum sample size for our model. The analysis was conducted with slightly below the threshold for a medium effect size (f² = 0.10), a significance level (α = 0.05), and a power of 0.8 (1-β). The model comprised of one independent variable, one moderating variable, one mediator, two control variables, and one interaction term. The calculation indicated that a minimum sample size of 143 was required to achieve adequate power. Therefore, the sample size in this study met the statistical power requirements.

A total of 280 paper copies of the informed consent forms were distributed to parents of senior kindergarten children from four public kindergartens in Fuyang City in central China, with 238 parents agreeing to participate in the study. Five children failed to complete half of the tests, which leads to a final sample size of 233. Among them, there were 141 boys and 92 girls, with a mean age of 68.5 months (SD = 4.48 months). Additionally, the family economic backgrounds of these children surveyed were comparable. Both the teachers and the children’s parents provided written informed consent, while the children themselves gave oral informed consent. This study was approved by the ethics committee of the first author’s institution (jyxy-2021-10-8-1), ensuring that all procedures adhered to ethical guidelines and safeguarded the rights and interests of the participants involved.

2.2 Measurements

2.2.1 Visual-spatial skills

The Gardner’s Revised Visual-Spatial Skills Scale (Gardner, 1996) was employed to assess preschoolers’ visual-spatial skills, comprising 17 items, which includes 1 non-scored example item and 16 formal test items. For each test item, there are five black-and-white figures presented. Among them, only one figure has a distinct orientation or a different orientation of some internal parts compared to the others. The children’s task is to identify this unique figure. For example, there are 5 shapes, 4 of which are vertical lines and one left is a horizontal line. The child needs to distinguish between the horizontal line and the other vertical lines. A correct answer earns one point, while an incorrect answer receives zero points. The test was terminated when one child answers four out of five consecutive questions incorrectly. The total score for this set of test questions is 16 points. The study revealed a high level of internal consistency of this scale (Cronbach’s α = 0.89).

2.2.2 Executive function

The Parental Version of the Children’s Executive Function Scale (CHEXI; Thorell and Nyberg, 2008) was used to assess the executive function levels of preschoolers. The scale demonstrates good test-retest reliability (Thorell and Nyberg, 2008) and has been translated into various languages, including Spanish, French, and Chinese.² The scale comprises 24 items, with 13 specifically assessing working memory (e.g., “Has difficulty remembering lengthy instructions”) and 11 evaluating inhibition (e.g., “Rarely seems to motivate himself/herself to do something he/she doesn’t want to do”). All items use a five-point scale ranging from 1 (completely inconsistent) to 5 (completely consistent), employing a reverse scoring system where higher scores indicate more pronounced executive function deficits. Therefore, the raw scores underwent conversion prior to analysis. The study revealed a high level of internal consistency of this scale (Cronbach’s α = 0.88).

2.2.3 Patterning ability

The subscale of the “Number Matrix” scale within the Woodcock-Johnson IV Achievement test (Schrank et al., 2014) was used to assess the patterning abilities of senior preschoolers. Based on the test guidelines, the first 12 questions of this scale were chosen, with the first question serving as an example and the 11 questions as formal test items. In these assessment tasks, numbers are presented in a 2 × 2 matrix format (e.g., ), requiring young children to identify visual Arabic numeral symbols, discover numerical arrangement patterns, and verbally report the missing number. The test begins with a practice item where examiners guide children to observe how numbers change from left to right in the first row or from top to bottom in the first column. Children are then asked to apply the same pattern to the second row or column to determine the missing number. The process requires children to infer missing values by comparing the differences between the numbers in the rows and columns. If children provide incorrect answers or fail to respond, examiners must clearly explain the numerical changes and guide them to reanalyze the pattern. Even when correct answers are given, raters must still clarify the numerical logic. During formal testing, examiners only encourage children to answer questions without providing correct or incorrect answers or explanations. In the present task, for young children, a correct answer earns one point, while an incorrect answer receives zero points. The test was terminated when one child answers six consecutive questions incorrectly. The study revealed a high level of internal consistency of this scale (Cronbach’s α = 0.78).

2.2.4 Arithmetic ability

The subscale of the “Applied Problems” scale in the Woodcock-Johnson IV Achievement test (Schrank et al., 2014) was used to assess the arithmetic ability of senior kindergarten children. Instead of the recommended first 8 items, the first 20 test items were chosen in this study, given the fact that Chinese children generally exhibit a higher arithmetic proficiency compared to Western children (Miller et al., 2005), and that the ceiling effect was avoided in the testing process (Yang et al., 2019). Among the test questions, 16 are accompanied by visual pictures. For example, a math problem presents a picture of a bench with three birds perched on it. During testing, the tester is required to point and swipe across the picture while verbally asking the child: “Three birds were sitting on the park bench. One flew away. How many birds were left?” Each correct answer earns one point, and incorrect answers receive zero points. The test was terminated when a child answers five consecutive questions incorrectly. The maximum score for this test project is 20 points. The study revealed a high level of internal consistency of this scale (Cronbach’s α = 0.79).

2.3 Procedure

The questionnaire primarily consists of two parts: a parent-completed section and a child-answered section. All tests were performed in a quiet environment. For Parent-Completed Section, the experimenter distributed the “Children’s Executive Function Scale (Parent Version)” to the class teachers. The teachers then handed out the questionnaires to the parents of the selected children based on a pre-determined randomized list of participants. For Child-Answered Section, the assessments of children’s patterning ability, arithmetic ability and visual-spatial skills were conducted individually by the experimenter such that their responses were recorded by the experimenter. The entire test duration for each participant was approximately 15–25 min, and there are short breaks during the test based on the child’s needs. Note: Data collection was conducted during the COVID-19 pandemic, such that the data collection at the early phase only involved direct assessments of preschool children’s visual-spatial skills, patterning ability, and arithmetic ability. Subsequently, with the escalation of pandemic control measures, the originally planned direct assessment of executive functions was modified to be measured using the Parental Version of the Children’s Executive Function Scale.

2.4 Data analysis

Descriptive statistics and correlation tests were performed using SPSS Statistics 30.0, with the PROCESS (Hayes, 2013) 4.0 plugin applied to evaluate a moderated mediation model. As CHEXI is a reverse-scoring scale, all item scores were inverted prior to statistical analysis. For mediation effect analysis, Model 4 was selected in the software. In the moderated mediation analysis, Model 7 was employed and the data from the moderated model were used for simple slope testing.

3 Results

3.1 Common methods bias test

Given that data from multiple sources were collected from the same group of young children, the data may suffer from a risk of common method bias. To address this bias, we implemented strict controls in the research procedure: (1) the data were collected separately from families and the children themselves, and (2) the experimenters underwent rigorous training. Meanwhile, we also performed the Harman’s single-factor test to assess the extent of the common method bias. The results showed that six factors had eigenvalues > 1, with the largest factor explaining 30.13% of the variance (<40%). This result suggested that there is no significant common method bias in the present study.

3.2 Descriptive statistics

The descriptive results and correlations are presented in Table 1. Correlational results showed that there were significant positive correlations between visual-spatial skills, patterning ability and arithmetic ability among preschoolers (ps < 0.001), and that there was a significant association between executive function and patterning ability (p < 0.05). However, the correlational relationships between the remaining variables were not significant.

3.3 Test for the mediation model

This study aimed to verify the extent to which visual-spatial skills affect patterning ability, and to promote arithmetic ability of young children through patterning ability. A mediation analysis (Model 4) using the PROCESS add-on in SPSS was conducted. Visual-spatial skills was the independent variable, patterning ability was the mediator, and arithmetic ability was the dependent variable, with gender and age as covariates. The results showed that with controlling for gender and age demonstrated that visual-spatial skills can positively predict the arithmetic ability of preschoolers (B = 0.22, SE = 0.03, t = 6.58, p < 0.001). When patterning ability and visual-spatial skills simultaneously entered the regression model, both visual-spatial skills (B = 0.15, SE = 0.03, t = 4.58, p < 0.001) and patterning ability (B = 0.45, SE = 0.07, t = 6.63, p < 0.001) positively predicted arithmetic ability. Visual-spatial skills significantly and positively predict patterning ability (B = 0.16, SE = 0.03, t = 5.31, p < 0.001) (see Table 2). The bootstrap method was used to test the mediating effect, with 5000 repeated sampling (Hayes, 2013). If the 95% confidence interval of the indirect effect does not contain 0, it indicates that the mediating effect is significant. The mediating effect of patterning ability was significant (B = 0.07, 95% CI [0.04, 0.12]). The confidence interval did not include zero, indicating that patterning ability significantly mediated the relationship between visual-spatial skills and children’s arithmetic ability. Furthermore, after controlling for patterning ability, the direct effect of visual-spatial skills on arithmetic ability remained significant (B = 0.15, 95% CI [0.09, 0.22]), which did not include zero, indicating that visual-spatial skills continued to be a significant predictor of children’s arithmetic ability. These results suggested that patterning ability partially mediated the association between visual-spatial skills and arithmetic ability. In particular, the size of the direct effect of visual-spatial skills on children’s arithmetic ability is 68.18%, and the size of the mediating effect is 31.82% (see Table 3).

3.4 Test for the moderated mediation models

Based on the theoretical and empirical assumptions mentioned above, we examined the moderating role of executive function in the association between visual-spatial skills and patterning ability (see Table 4). The moderation effect analysis (model 7) with controlling for age and gender revealed that visual-spatial skills significantly and positively predicted patterning ability (B = 0.16, SE = 0.03, t = 5.16, p < 0.001), and executive function also significantly and positively predicted patterning ability (B = 0.02, SE = 0.01, t = 2.00, p < 0.05). More importantly, we found that the interaction between visual-spatial skills and executive function significantly and negatively predicted patterning ability (B = -0.01, SE = 0.00, t = -2.83, p < 0.01), indicating that executive function significantly moderates the relationship between visual-spatial skills and patterning ability. Furthermore, patterning ability significantly and positively predicted arithmetic ability (B = 0.14, SE = 0.04, t = 3.98, p < 0.001), and visual-spatial skills significantly predicted arithmetic ability (B = 0.48, SE = 0.07, t = 6.79, p < 0.001). Since patterning ability partially mediated the relationship between visual-spatial skills and arithmetic ability, executive function significantly moderated the relationship between visual-spatial skills and patterning ability in this mediated model.

To further break down the source of how the executive function moderated the relationship between visual-spatial skills and patterning ability, we divided the standardized executive function scores into high and low groups based on one standard deviation above and below the mean. We then conducted a simple slope test on the moderation effect (see Figure 1). The results revealed that executive function negatively moderated the relationship between visual-spatial skills and patterning ability. Specifically, for children in the low executive function group, visual-spatial skills significantly increased with higher levels of patterning ability (B = 0.25, SE = 0.04, t = 5.93, p < 0.001), while for those in the high executive function group, the increase in visual-spatial skills with higher levels of patterning ability was not significant (B = 0.08, SE = 0.05, t = 1.60, p > 0.05).

FIGURE 1

4 Discussion

The present study aimed to examine links between visual-spatial skills and arithmetic ability in preschoolers, specifically examining the mediating role of patterning ability and the moderating role of executive function. Our findings provide valuable insights into the cognitive mechanisms underlying early mathematical development, through integrating visual-spatial skills, patterning ability, and executive functions within a single complex model.

4.1 The positive link between visual-spatial skills and arithmetic ability

Consistent with our hypothesis and extensive previous research (Fung et al., 2020; Gunderson et al., 2012; Yang et al., 2019, 2021; Zhang and Lin, 2015), the results demonstrated a significant and positive correlation between visual-spatial skills and preschoolers’ arithmetic ability. According to the Mental Number Line Theory, individuals process numerical information through mental number lines, and visual-spatial skills support children’s arithmetic performance by constructing spatial representations of quantities and facilitating mental simulation operations (Hawes and Ansari, 2020; Zhang, 2016). In this study, preschoolers solved arithmetic context problems accompanied by illustrations. Successful problem-solving largely relied on visual-spatial skills to mentally represent the perceived and extracted spatial and numerical information, such as by forming dynamic mental imagery or constructing spatial-numerical mappings on a mental number line. These internal representations provided cognitive support for executing subsequent arithmetic operations. For example, when dealing with object addition/subtraction (e.g., “Three birds were sitting on the park bench. One flew away. How many birds were left ?”), children may use visual-spatial abilities to form dynamic mental images, tracking the initial position, movement process, and final state of objects to infer quantitative changes. Children can also convert visualized quantitative relationships into numerical connections, then utilize mental number lines for spatial representation and manipulation—for instance, moving leftward by one unit from a spatial starting point representing “3” to ultimately locate the spatial position of “2.” The inclusion of picture-based arithmetic tasks in this study further underscores the importance of visual-spatial skills. This study further extends prior longitudinal findings by demonstrating this predictive relationship cross-sectionally in a sample of Chinese preschoolers. In addition, the findings of this study supplement previous studies primarily based on Western samples (e.g., Gunderson et al., 2012) and emphasizing visual-spatial skills as a key cognitive cornerstone for the development of arithmetic ability.

4.2 The mediating role of patterning ability

A finding of this study is that patterning ability partially mediated the link between visual-spatial skills and arithmetic ability. This confirms our hypothesis and is consistent with previous studies suggesting that numerical skills or patterning ability can serve as mediators between broader cognitive factors, including spatial abilities, and mathematical achievement (Cirino, 2011; Cragg et al., 2017; Hawes et al., 2019; Krajewski and Schneider, 2009; LeFevre et al., 2010; Zhang et al., 2014). Our study also provided empirical evidence for the hypothesis proposed by Borriello et al. that patterning ability mediates the relationship between visual-spatial skills and arithmetic ability.

The results showed that there was a significant positive correlation between visual-spatial skills and patterning ability, indicating that children with higher visual-spatial skills might have higher patterning ability. Previous studies have predominantly examined this association using either non-numerical or numerical patterns in a single-dimensional form (horizontal: Borriello et al., 2023; Burr et al., 2022; Fyfe et al., 2017; Miller et al., 2016; Rittle-Johnson et al., 2019; Wijns et al., 2019, 2021; Zhang et al., 2014; Zippert et al., 2019, 2020; or vertical: Borriello et al., 2023; Miller et al., 2016; Zippert et al., 2019). Departing from these approaches, this present study adopts a numerical growth pattern with two-dimensional structure, in which Arabic numerals were treated as the elements. In this task, children have to accurately identify numerical symbols and their spatial positions, while processing numerical relationships in sequences from both row and column dimensions. Taking the item “” as a numerical matrix example, young children first need to recognize that “2” is in the upper left, “3” in the upper right, and “4” in the lower left. Next, they need to decompose the matrix into sequences in both horizontal and vertical dimensions, and analyze the numerical relationships in the horizontal dimension (between 2 and 3) and the vertical dimension (between 2 and 4), respectively. During this process, they may use mental number lines to understand dimensional differences, such as a 1-unit gap between 2 and 3 in horizontal dimension and a 2-unit gap between 2 and 4 in vertical dimension. By integrating both horizontal and vertical information, they can derive the rule that each unit increase horizontally corresponds to a 2-unit increase vertically. These core cognitive processes, including locating numbers, deconstructing matrix structures, and identifying rules of numerical changes across different dimensions, all rely on support from visual-spatial skills. Therefore, the stronger a child’s visual-spatial skills, the more efficiently they can locate numerical positions, decompose matrices, and distinguish the numerical relationships between rows and columns. This skill thereby facilitates the recognition and application of multidimensional complex pattern rules. Conversely, preschoolers with lower visual-spatial skills may struggle to quickly identify and locate numerical symbols, distinguish the spatial structure of matrix rows and columns, or recognize numerical variation patterns. This difficulty in abstracting pattern rules often leads to failure in pattern task performance.

Moreover, patterning ability also positively correlated with preschoolers’ arithmetic ability. This indicates that strong patterning ability helps children effectively identify inherent structure and patterns in numerical sequences, and apply these patterns to solve arithmetic problems (Zippert et al., 2019; Zhang et al., 2014). The arithmetic ability tasks in this study, although not involving pure numerical operations, still require young children to extract quantitative information from experimenter’s verbal descriptions, manipulate mental representations, and then perform corresponding calculations. Therefore, both matrix and arithmetic tasks share similar cognitive processes in understanding numerical relationships, grasping numerical variation rules, and applying these rules to solve problems (Zippert et al., 2019). Thus, children who demonstrate strong rule-based reasoning and transfer skills in matrix tasks are more likely to quickly identify numerical relationships and variation patterns in applied problems, and transfer numerical rule knowledge to arithmetic problem-solving, thereby improving their arithmetic performance. For example, when faced with the question “There are 3 birds on the chair. After 1 bird flies away. How many remain?”, children need to convert it into the arithmetic expression “What is three minus one?,” and use numerical variation rules (such as “2 is one less than 3”) to arrive at the answer. This demonstrates that patterning ability provides crucial cognitive support for young children to understand and manipulate numerical relationships and develop early arithmetic thinking.

In conclusion, this study elucidates the specific pathways through which visual-spatial skills influence arithmetic ability: Children with stronger visual-spatial skills demonstrate greater efficiency in integrating multidimensional information during two-dimensional pattern tasks, thereby forming clear rules. This ability to recognize multi-dimensional numerical relationships enables them to more acutely perceive correlations between numbers in arithmetic computations, providing core support for solving arithmetic problems. This mechanism explains why preschoolers with superior visual-spatial skills perform better in tasks requiring spatial representation, such as arithmetic word problems and sequence pattern problems (Zhang et al., 2014). Essentially, visual-spatial skills serve as a critical cognitive scaffold by facilitating patterning ability, thereby supporting the processing of complex relationships in arithmetic problems. This study extends pattern testing tasks to two-dimensional numerical matrices, addressing previous research gaps in multidimensional spatial pattern analysis while deepening our understanding of the mediating role of patterning ability between visual-spatial skills and arithmetic ability, particularly in elucidating the specific mechanisms underlying the recognition of complex numerical symbol sequence rules.

4.3 The moderating role of executive function

Furthermore, the study revealed that executive function negatively moderated the link between visual-spatial skills and patterning ability: The relationship between visual-spatial skills and patterning ability was stronger for children with lower levels of executive function, whereas those with higher executive function levels show no significant association. More importantly, this finding not only validates the psychological compensation framework, which posits that individuals can compensate for skill deficits in specific domains by leveraging dominant cognitive skills, but also expands the application scenarios of this theory. Unlike Collins and Laski (2015), who focused on task-demand-driven compensation, this study approaches the issue from the perspective of individual resource differences, demonstrating significant variations in visual-spatial skills reliance across individuals with different executive function levels when performing the same task. These findings provide crucial evidence for understanding the individualized mechanisms of compensatory pathways. The results not only highlight the core moderating role of executive function in coordinating cognitive resource allocation but also further enrich the empirical foundation of the psychological compensation theory from the perspective of individual differences.

Our study demonstrates that young children with lower executive function, due to limited working memory capacity and weaker inhibition, tend to confuse spatial arrangement information with numerical relationships when viewing number matrix. In such cases, children with stronger visual-spatial skills can more clearly identify the spatial distribution patterns of numbers. This clear spatial representation supports their ability to recognize patterns between numbers in different positions. For example, when encountering a matrix task containing “,” one child with strong visual-spatial skills can clearly perceive that “2 is in the upper left, 3 in the upper right, and 4 in the lower left.” Based on this clear spatial framework, children can further understand the relationships between elements in different positions, such as “What’s the relationship between the upper left and upper right? (add 1) What’s the relationship between the upper left and lower left? (add 2) Could the upper right and lower right also be add 2?”. Conversely, children with weaker visual-spatial skills find it difficult to rapidly recognize spatial arrangement rules between rows and columns, resulting in difficulties in solving problems. This suggests that developing a clearer visual-spatial representation of numerical relationships and spatial positioning in matrices may help children with weaker executive functions improve their performance in pattern tasks. Therefore, differences in visual-spatial skills have a more pronounced impact on pattern task performance in young children with lower executive function levels. In other words, exceptional visual-spatial skills are particularly crucial for children with limited executive function capabilities.

Conversely, for preschoolers with higher executive function, the predictive role of different visual-spatial skills in patterning ability is not significant. Even those with relatively lower visual-spatial skill levels can demonstrate patterning abilities comparable to those with higher levels. This may be attributed to their strong working memory and inhibition, which enable them to simultaneously retain spatial and numerical information while effectively suppressing attention to the specific positions of numbers in the matrix, and instead focusing on analyzing abstract numerical relationships (e.g., from 2 to 3 is an increase of 1 unit, from 2 to 4 is an increase of 2 units). Additionally, studies on older children have shown that children with high executive function tend to employ more efficient memory retrieval strategies (such as directly retrieving arithmetic facts) when performing simple addition and subtraction calculations (Barrouillet and Lépine, 2005; Geary et al., 2012), rather than relying on visual-spatial strategies like mental number lines. Therefore, when analyzing numerical relationships, the preschoolers with higher executive function may not need to rely on intuitive spatial representations to infer numerical differences but instead grasp numerical variation rules through other non-visual strategies. Our results show that, although this cognitive process requires support from visual-spatial skills, such as perceiving numerical symbols and spatial positions, once children abstract pattern rules from visual imagery, executive function becomes dominant. In this context, the predictive role of visual-spatial skills may become less apparent, thereby mitigating the limitations of visual-spatial skills—particularly those at lower levels—on patterning ability. In other words, higher-level executive function provides an intrinsic compensatory factor, reducing the dependence of patterning ability on visual-spatial skills. However, it is noted that we should be cautious that the executive function moderated the relation between visual-spatial skills and patterning ability, given that the measure of preschoolers’ executive function was obtained from these children’s parents. We also discussed this point in the limitation section.

Collectively, our findings demonstrate the complex pathways through which basic cognitive abilities contribute to early arithmetic development of preschoolers. We demonstrate that visual-spatial skills support arithmetic not only directly but also indirectly through the mediating role of patterning ability, particularly the ability to recognize rules in numerical growth patterns. Importantly, we show that the strength of the link between visual-spatial skills and patterning ability is not consistent but is contingent on a child’s executive function level, suggesting that executive function plays a crucial role in how children leverage their visual-spatial strengths for pattern tasks. These results contribute to the theoretical understanding of the specific cognitive building blocks for early math and how they interact.

4.4 Implications and limitations

This study was performed during the COVID-19 pandemic provides a new idea for data collection during the epidemic period, and provides valuable experience data for the development of preschool children’s arithmetic ability during the epidemic period. More importantly, the above findings may suggest that in early mathematics education, developing children’s arithmetic ability should not only focus on visual-spatial skills but also consider individual cognitive differences to achieve more targeted outcomes. Moderation effect analysis reveals that when assessing children’s patterning abilities, both executive functions and visual-spatial skills must be considered, as their combined effects profoundly influence patterning ability, which in turn indirectly impacts arithmetic abilities. Therefore, the first step should be to identify the strengths and weaknesses of children’s visual-spatial skills and executive functions. For those with weaker executive functions, interventions should focus on improving the clarity and organization of visual-spatial information to facilitate compensation. This includes consciously guiding children to discover and summarize spatial relationships between numbers, helping them apply rules derived from concrete spatial information to abstract mathematical symbol relationships. For children with weaker visual-spatial skills, interventions should prioritize enhancing executive functions by guiding them to systematically observe, compare, and reason, thereby optimizing the efficiency of utilizing limited visual information. However, for children with higher executive function levels, simply improving visual-spatial skills may have limited impact on promoting patterning abilities.

Despite these valuable insights, the study has certain limitations. First, the cross-sectional study design failed to establish definitive causal relationships between variables, necessitating longitudinal research to track the developmental trajectory of these skills and their dynamic interactions. Second, the urban-based sample may limit the generalizability of findings to rural or children from diverse backgrounds, requiring future studies to expand sample coverage for enhanced replicability. Third, pandemic restrictions during the survey period constrained some offline experimental procedures. To mitigate infection risks, we employed the Parental Version of the Children’s Executive Function Scale instead of direct executive function testing in young children. Future research should consider conducting direct executive function assessments to rigorously validate their role in the model. Additionally, while focusing on core visual-spatial ability, patterning ability, and executive function, the model excluded other known factors related to mathematical ability (e.g., verbal language skills including vocabulary and language processing) (Zhang, 2016; Zhang et al., 2014). Future studies should develop frameworks to explore comprehensive cognitive mechanisms, particularly their potential role in the cognitive predictive network of early arithmetic development. These limitations guide future research while emphasizing the need for cautious interpretation of findings. Finally, we are inclined to treat patterning ability as a mediating variable in the relationship between visual-spatial skills. Of course, other types of relationships between these two variables may also exist; however, given that the present study is cross-sectional and based on current theoretical assumptions, we are unable to address this question definitively. Future researchers could further investigate this important issue.

In conclusion, this study performed during the COVID-19 pandemic shows the significant direct and indirect roles of visual-spatial skills in predicting preschoolers’ arithmetic ability, highlighting the mediating function of patterning ability. Crucially, it reveals that executive function negatively moderates the link between visual-spatial skills and patterning ability, suggesting that individual differences in cognitive control influence how visual-spatial strengths contribute to patterning ability. These findings contribute to understanding of the cognitive architecture supporting early math development and offer valuable guidance for targeted educational interventions, especially in the post-pandemic era.

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