Introduction

Patterning Skills and Mathematics Achievement

Patterning, defined as the ability to detect and generalize regularities in ordered sequences (Miller et al., 2016), is theorized to play a foundational role in early mathematical development. The Early Math Trajectories Model (Rittle‐Johnson et al., 2017) identifies six core early math competencies that form the foundation for later mathematical achievement, with patterning emerging as a particularly unique and powerful predictor. This model positions patterning as more than just another math skill: it serves as a cognitive bridge that connects concrete perceptual experiences to abstract mathematical reasoning. Through training children with patterning activities that involve repeating patterns and growing patterns, they develop thinking skills essential for multiple mathematical domains. These patterning types require children to recognize regularities, deduce rules, and generalize structures, processes that mirror the cognitive demands of more advanced math concepts.

The model explains how early patterning ability supports mathematical development in several pathways. First, patterning work reinforces fundamental skills, such as one-to-one correspondence and ordinality which are crucial for counting and early arithmetic. Second, the abstract reasoning involved in patterning, such as identifying core units and predicting subsequent elements, paralleled the cognitive processes required for symbolic mapping and calculation. Third, patterning fosters relational thinking, the ability to recognize and apply abstract rules across contexts, which becomes increasingly important as mathematics grows more complex. This explains why children with strong early patterning skills tend to perform better later in areas ranging from basic number sense to algebra readiness.

Empirical studies partially support this theoretical claim. For instance, correlational research demonstrates positive associations between patterning skills and mathematical achievement, with evidence showing that children’s proficiency in repeating and growing patterns predicts number sense, arithmetic fluency, and algebraic reasoning (Mulligan et al., 2020; Wijns et al., 2021a, 2021b). These findings align with the hypothesis that pattern recognition underpins mathematical conceptualization by strengthening decomposition, rule generalization, and part-whole reasoning. However, the patterning-mathematics correlation is not consistent. Some studies have reported non-significant or context-dependent associations, particularly when outcomes involve conceptual understanding rather than procedural skills (Fyfe et al., 2017; Kidd et al., 2013). For example, Borriello et al. (2023) found no direct link between patterning and fraction knowledge in older children, suggesting that the relation may diminish as mathematical demands shift toward higher-order competencies.

The intervention studies also demonstrated controversial conclusions. For example, experimental research revealed that targeted pattern training can effectively improve children’s patterning skills, with evidence from multiple studies indicating significant gains in repeating and growing pattern tasks after structured interventions (McKnight et al., 2021; Wijns, 2021; Zippert et al., 2021). These findings suggested that patterning skills are malleable and responsive to instruction. However, the transfer of these gains to broader mathematical knowledge remains inconsistent. While two early, less rigorous studies reported tentative benefits for numeracy outcomes (Hendricks et al., 2006; Papic et al., 2011), more recent and methodologically robust studies with preschoolers and kindergarteners have failed to replicate these effects (McKnight et al., 2021; Wijns et al., 2021a, 2021b; Zippert et al., 2021). For instance, despite significant improvements in patterning abilities, intervention groups showed no advantage over control groups on measures of numeracy, arithmetic, or broader mathematics achievement. This discrepancy has led some researchers to question the practical value of patterning instruction for enhancing early mathematics learning, particularly for children aged 4 to 5 years (McKnight et al., 2021).

These inconsistencies highlighted the need to synthesize existing evidence through a meta-analysis. A quantitative synthesis can clarify the overall strength of the patterning-mathematics association while concerned about differences across studies. Furthermore, theoretical models, for example, the Early Math Trajectories Model implicitly assume moderating factors that may explain divergent findings.

Potential Moderators of the Patterning-Mathematics Association

The relation between patterning and mathematics is not consistent; it may be moderated by key variables that shape its strength. Below, we examine how age, patterning type, element type, measure and study design moderate this association.

Age

The role of patterning in mathematical development shifts as children grow, which reflects broader cognitive changes. The link between patterning skills and mathematical achievement appears contingent on developmental and task-specific factors. Age moderates this association, which reflects shifts in cognitive priorities across childhood. The association between patterning and mathematics evolves dynamically with age, which reflects shifts in how cognitive systems interact with mathematical learning. During early childhood (ages 3 to 5), patterning serves as a foundational scaffold for numeracy, with repeating sequences (e.g., ABAB) leveraging children’s reliance on perceptual regularity and visuospatial processing. These activities strengthen skills such as one-to-one correspondence and ordinality, as demonstrated by Rittle-Johnson et al. (2017), who showed how repetitive patterning tasks help young children internalize numerical relationships. Interventions targeting patterning, such as Zippert et al.’s (2021) kindergarten study, revealed a sensitive period where patterning training significantly boosted number sense, which highlighted its direct and pronounced role in early mathematical development.

However, as children progress into later grades, the association between patterning and mathematics decreases, due to executive functions (e.g., working memory, inhibitory control) increasingly mediating complex mathematical tasks, such as fraction operations or multistep problem-solving (Borriello et al., 2023). Thus, age moderates the link by altering the mechanisms through which patterning contributes to learning. Younger children depend heavily on patterning’s perceptual scaffolding, whereas older children’s mathematical success increasingly hinges on how well executive functions organize and extend pattern-based knowledge. Therefore, the link between patterning and mathematics may weaken as children mature, not because patterning becomes irrelevant, but because its contributions are increasingly integrated with executive functions and symbolic systems that dominate advanced mathematical learning. Thus, exploring age as a moderator clarifies how patterning transitions from a foundational scaffold to a latent component of complex mathematical reasoning (Borriello et al., 2023; Zippert et al., 2021). This developmental trajectory underscored that patterning’s role was not static but adapted to cognitive and curricular demands. Having established how age shapes the patterning-mathematics link, we now turn to how the specific demands of patterning tasks moderate this relation.

Patterning Type

Patterning skills are often assessed with two patterning types, repeating and growing. Repeating patterns have “a cyclic structure that can be generated by the repeated application of a smaller portion of the pattern” (e.g., red-blue-red-blue, ABC-ABC). Growing patterns consist of sequences of elements that increase (or decrease) systematically (e.g., AB-ABB-ABBB, 3–6–9–12) (Papic et al., 2011). In addition to these two classic types, our study includes a mixed type of patterns that integrate repeating and growing patterns, and in some cases, include rotating patterns. Notably, within the mixed type included in our study, we identified a special pattern: rotating patterns, which are the rotations of an object (Kidd, 2014). Specifically, among the studies we included, 23 incorporated rotating patterns in the mixed type, while 15 did not.

Empirical studies support the idea that repeating patterns are easier than growing ones because growing patterns are more cognitively complex (Gadzichowski, 2012; Warren, 2005). Repeating patterns may have a stronger influence on foundational math skills (Wijns et al., 2021a, 2021b; Zippert et al., 2020), while growing patterns may play a greater role in predicting more complex mathematical reasoning and problem-solving abilities (Borriello et al., 2023; Wijns et al., 2021a, 2021b). Rotating patterns may be related to mental rotation skills that are associated with better mathematics performance, especially in arithmetic and geometry (Cheng & Mix, 2014; Hawes et al., 2015). While some interventions targeting mental rotation can improve specific math skills, the overall transfer to general math achievement is controversial (Georges et al., 2019; Hawes et al., 2015). This suggests that the type of patterning skill could moderate the association between patterning abilities and mathematical outcomes.

Beyond patterning types, the type of elements used in patterning tasks may also influence their mathematical relevance.

Element

The materials used in patterning tasks, symbolic (e.g., colored blocks) or non-pictorial (e.g., numbers), may shape how patterning skills translate to mathematical competence. Symbolic materials are tangible, perceptually distinct items that rely on sensory recognition. Non-pictorial materials involve the use and manipulation of mathematical symbols (e.g., letters, numbers) and are characterized by the absence of corresponding visuospatial depictions. Symbolic materials (e.g., colored blocks) enhance early pattern learning by linking sensory experiences to numerical concepts (e.g., counting), whereas non-pictorial elements (e.g., numbers) promote rule generalization and algebraic reasoning in older children (Papic et al., 2011). The materials used in patterning tasks, symbolic (e.g., colored blocks) or non-pictorial (e.g., numbers), further shape how patterning translates to mathematical competence.

Symbolic elements dominate early patterning due to their perceptual salience and alignment with young children’s embodied cognition. Manipulating colored blocks in repeating sequences grounds numerical concepts in multisensory experiences, reducing cognitive load and scaffolding skills such as counting and cardinality (Papic et al., 2011). For instance, Rittle-Johnson et al. (2019) found that 75% of preschoolers could duplicate symbolic patterns, with observational studies noting children’s spontaneous engagement in such activities (Waters, 2004). However, symbolic materials risk perceptual anchoring: overly salient features (e.g., colors) may distract children from underlying relational rules.

The association between patterning and mathematics may weaken with non-pictorial elements (e.g., numbers), likely because their higher cognitive demands may reduce children’s task performance, while their actual math skills remain unaffected. Thus, element may moderate the link by altering the accessibility of patterning tasks, with symbolic materials providing a “bridge” that abstract materials may lack for younger learners (Papic et al., 2011; Uttal & Cohen, 2012). These moderators illustrate that the patterning-mathematics link may be influenced by developmental and material factors.

Mathematical Measure

In addition to the variables mentioned above, measures may also moderate the relation between patterning skills and mathematical performance. Mathematical performance has been measured in various ways, such as standard achievement test scores, record-card grades, teacher ratings, or self-reports of achievement. In the mathematical area, it has two main types: standardized tests and researcher-based tests. Standardized tests include state and district assessments, national surveys, and assessments. Conversely, researcher-based test has been defined as measures developed by study authors or teachers. Previous studies have widely tested the function of these two kinds of measures, and a meta-analysis has revealed that the average effect size of researcher-based tests is higher than that of standardized tests (Wolf, 2021). One of the reasons why students may perform better in researcher-based tests is that researcher-based measures are narrower compared with independent measures; that is, researcher measures capture constructs on a small domain, whereas standardized tests capture constructs on a broad domain or multiple domains (de Boer et al., 2014). Additionally, the use of researcher and developer measures is confounded with greater implementation fidelity because researchers who develop their own measures are more invested in implementation in those studies, which could lead to higher average effect sizes (Li & Ma, 2010).

Study Design

Study design may play a crucial role in shaping the relation between patterning skills and mathematical performance, as well as the various measures of mathematical performance. Research design can be categorized into intervention and non-intervention designs (including cross-sectional and longitudinal studies). A review of 14 studies involving children aged 4 to 7 years old examined the causal effects of pattern training on mathematics outcomes. Eight of these studies, including the most rigorous ones, reported improvements in patterning skills but found no evidence of transfer to mathematics knowledge (Fyfe & Borriello, 2025). In contrast, non-intervention often reports significant associations between early patterning skills and later mathematical achievement. For instance, large-scale naturalistic and longitudinal studies have found that patterning ability uniquely predicts later mathematics achievement, even after controlling for other cognitive factors (Bojorque et al., 2021; Di Lonardo Burr et al., 2022; Larkin et al., 2024). The findings from intervention studies are more likely to reveal non-significant relations compared to those from non-intervention designs. This may suggest that study design may moderate the association between patterning skills and mathematical performance (Table 1).

Table 1 A Classification of Element Categories
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Research Gaps and Current Objectives

Despite its theoretical prominence, patterning remains underexplored in meta-analytic research compared to other correlates of mathematical achievement (e.g., gender, and attitudes). Existing inconsistencies such as conflicting findings on repeating and growing patterns (Wijns et al., 2019) underscore the need to quantify the patterning-mathematics association while systematically testing moderators such as age, patterning type, element, mathematical measure, and study design.

To address these gaps, this meta-analysis examines two research questions:

  • What was the overall strength of the correlation between patterning skills and mathematical achievement?

  • How did age, patterning type (repeating, growing and mixed), element (symbolic, non-pictorial, and mixed), mathematical measure (researcher-based and standardized) and study design (intervention and non-intervention) moderate this association?

Methods

Inclusion Criteria

The current meta-analysis established rigorous inclusion criteria to ensure methodological consistency across studies. First, regarding population characteristics, all selected studies were required to involve children specifically identified as preschoolers, kindergarteners, or elementary school students. Studies focusing on adolescent populations, adult participants, animal models, or other non-human subjects were systematically excluded from consideration.

Second, the operationalization of patterning skills followed strict parameters. Eligible studies needed to explicitly measure children’s ability to identify, extend, or generate either repeating patterns (e.g., ABAB sequences) or growing patterns (e.g., ABBABBB sequences). This focus on sequential patterning abilities led to the exclusion of investigations examining non-sequential pattern recognition (such as spatial symmetry tasks) or abstract rule-learning tasks unrelated to mathematical development (including linguistic or musical patterning paradigms).

Third, with respect to mathematical achievement, included studies were required to assess domain-specific, direct competencies within well-defined mathematical domains, explicitly excluding indirect or non-mathematics-specific measures (e.g., grade point average, graduation rates, school disciplinary data). Eligible mathematical outcomes encompassed number sense (including counting abilities and magnitude comparison tasks), arithmetic skills (ranging from basic calculation fluency to complex problem-solving), early numeracy (incorporating subitizing and ordinality assessments), algebraic thinking (particularly rule generalization tasks), geometry (focusing on shape properties and spatial relationships), and fraction understanding (including proportional reasoning). This comprehensive approach ensured coverage of core mathematical skills developed during childhood.

Fourth, methodological considerations dictated that only empirical studies employing experimental, correlational, or longitudinal designs were included. This criterion guaranteed that all analyzed data met the minimum standards of scientific rigor and appropriate research design. Finally, practical constraints necessitated that all included studies be peer-reviewed articles published in English. Studies using neuroimaging techniques (such as fMRI or EEG), machine learning methodologies, or interventions targeting non-mathematical outcomes were specifically excluded to maintain a focus on behavioral and educational correlates of patterning skills and mathematical development.

Literature Search

The systematic literature search process is illustrated in Fig. 1. We conducted comprehensive searches in ProQuest, ERIC, and Web of Science using Boolean operators (“AND”/“OR”) with key terms: (“patterning skill*” OR “patterning” OR “repeating patterning” OR “growing patterning” OR “growth patterning”) AND (“math*” OR “arithmetic*” OR “calculation*” OR “num*” OR “fraction*” OR “geometr*” OR “algebr*”), following search protocols established by Peng et al. (2016). The wildcard symbol (*) accounted for morphological variations (e.g., “mathematical”/“mathematics”, “geometric”/“geometry”). Our search spanned publications from 1988 through 2025 to balance historical depth with contemporary findings. To ensure comprehensive coverage, we supplemented database searches with backward reference checking of included studies and forward citation tracking of seminal works, while also manually examining reference lists from prior relevant reviews to identify any additional eligible studies. The search was restricted to peer-reviewed English-language publications focusing on children aged 3 to 12 years.

Fig. 1
figure 1

The Flow Chart of Literature Screening

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The initial search yielded 315 records, which underwent two phases of review. In the first phase, duplicates were removed, reducing the records to 278. Subsequent title and abstract screening excluded 220 records based on eligibility criteria (e.g., qualitative studies, non-English manuscripts, or non-peer-reviewed publications). This resulted in 58 records proceeding to full-text review.

In the second phase, 52 full-text articles were assessed for eligibility. Of these, 33 were excluded for the following reasons: 20 lacked sufficient data to convert effect sizes to Pearson’s r, six did not measure mathematical performance, six were non-empirical studies, two lacked a patterning measure, one lacked target patterning skills. Ultimately, 17 studies that included 81 effect sizes met all inclusion criteria and were retained for analysis.

Coding Procedure and Effect Size Computation

A systematic approach was used to identify and code correlation coefficients, sample sizes, and moderators from the included studies. The coding sheet (see Appendix Table 9) included citation, study ID, effect size ID, year of publication, effect sizes (correlation coefficient, Standardized Mean Difference), sample sizes, participant ages, and moderators. These moderators covered patterning type (e.g., repeating, growing, mixed), element (symbolic, non-pictorial, and mixed). Study characteristics, age, study design (intervention or non-intervention) and mathematical measure were also coded. Considering the rotating type as a special pattern, we also take it into the exploration to examine the moderation (see Supplementary). For studies requiring conversion of correlation coefficients (as presented in Table 10), the Standardized Mean Difference (SMD) was converted to Cohen’s d. Then, Cohen’s d was transformed to Fisher’s z (Borenstein et al., 2021). Overall, r was transformed to Fisher’s z (Fisher, 1992).

Analytical Procedure

The analytical procedure comprised four sequential steps to systematically evaluate the association between patterning skills and mathematical achievement, addressing moderators, statistical dependencies, heterogeneity, and collinearity.

First, we calculated the overall weighted average correlation between patterning skills and mathematical achievement. Pearson correlation coefficients (r) were transformed into Fisher’s z scores for standardization. For studies reporting mean differences, Cohen’s d was computed directly from the Standardized Mean Difference and converted to Fisher’s z.

Second, meta-regression analyses were conducted to test the moderating effects. A multilevel random-effects model was specified, with age as a continuous moderator and patterning type, element, mathematical measure and study design as categorical moderators. Study ID and effect ID were included as random intercepts to account for within-study statistical dependencies. Then, for moderator effects, post-hoc pairwise comparisons were conducted using Holm-Bonferroni adjusted contrasts (Eichstaedt et al., 2013; Holm, 1979).

Heterogeneity was quantified via Q-statistics, I2, and τ2. A Begg-Mazumdar analysis was used to test the small-study effect (Begg & Mazumdar, 1994; Hunsberger et al., 2022). All analyses were conducted in R using packages such as {metafor} and {dplyr}, which enabled effect size computation, model estimation, and result visualization.

Justification of the Overall Number of Models

The pre-specification of our primary model, which estimates one main effect, five key moderators, and one exploratory moderator, is strongly justified by the methodological evidence from López-López et al. (2017). First, the study’s data structure (17 studies, 81 effect sizes, average 4.7 effect sizes per study) aligns with the “dependent effect size” scenarios the authors focused on, and their simulations confirmed that both three-level mixed-effects models and robust variance estimation (RVE) models. These two models can reliably handle multiple moderators without introducing meaningful bias. Specifically, López-López et al. (2017) demonstrated that all valid methods for dependent effect sizes yield nearly unbiased slope estimates for moderator effects across various scenarios, including when moderators are assessed at either the study or effect size level. Furthermore, our use of REML estimation for these models is consistent with the study’s implicit emphasis on robust variance component estimation. REML avoids the bias in heterogeneity variance (τ2) estimates that can arise with other methods, which is critical for maintaining reliability when testing multiple moderators.

Evaluation of Publication Bias

Funnel Plot Analysis and Egger’s Test for Funnel Plot Asymmetry

Egger’s linear regression test was performed to statistically quantify funnel plot asymmetry. Results showed a non-significant intercept estimate (b = 0.533, p =.190, 95% CI [−0.26, 1.33]) and a non-significant slope (b = −0.153, p =.908, 95% CI [−2.76, 2.46]). These non-significant findings (both p >.05) provided no statistical evidence of publication bias, aligning with the visual symmetry of the funnel plot.

The funnel plot was constructed to visually assess potential publication bias, with observed outcomes plotted against their standard errors (see Fig. 2). The plot revealed that studies were distributed without severe asymmetry across the standard error spectrum. Although there was some visual scatter, the quantitative test (e.g., Egger’s linear regression test) did not indicate significant publication bias or small-study effects, which suggested that the observed pattern was likely due to natural variability or heterogeneity rather than systematic bias. This supports the robustness of the meta-analytic findings.

Fig. 2
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Funnel Plot

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Trim and Fill Adjustment

The Trim and Fill method (Duval & Tweedie, 2000) was further applied to adjust the pooled effect size for potential unreported studies. The original pooled effect size was 0.486 (95% CI [0.37, 0.60]). After adjustment for hypothetical missing studies, the revised effect size was 0.486 (95% CI [0.37, 0.60]). Notably, the method did not detect a need to impute additional studies (SE = 2.48), which indicated no substantial underrepresentation of non-significant or small-sample studies.

Results Robustness Test

To examine whether the observed high heterogeneity (I2 = 78.25%) was confounded by small-study effects, a Begg-Mazumdar analysis was conducted (Begg & Mazumdar, 1994; Hunsberger et al., 2022). The analysis used effect sizes as the dependent variable and the inverse of effect size variance (serving as a proxy for approximate sample size, where a larger inverse variance indicates a larger sample size and higher precision) as the independent variable. A significant correlation may suggest potential small-study bias, while a non-significant correlation may indicate that such bias had minimal impact on the heterogeneity and pooled effect size. Statistical significance was determined using the α = 0.05 threshold. The results of the Rank Correlation Test for Funnel Plot Asymmetry revealed a weak correlation (Kendall’s tau = 0.136, p =.081 >.05), which indicated no statistically significant small-study bias. This suggested that small-study effects or publication bias had minimal impact on the robustness of the pooled results and observed heterogeneity.

A leave-one-out sensitivity analysis was conducted to evaluate whether the pooled effect size was disproportionately influenced by any single study. The results showed that after excluding any single study, the pooled effect size fluctuated within a narrow range of 0.464 to 0.512, with minimal deviation from the original model (Fisher’s z = 0.486). Additionally, all p-values of the models remained < 0.05, and none of the 95% CIs included 0. These findings indicate that no single study unduly influenced the significance of the overall effect, demonstrating the high robustness of the core conclusions.

Result

The Relation between Patterning Skill and Mathematics Achievement

This meta-analysis synthesized 81 effect size estimates to explore the correlation between patterning skill and mathematical achievement. As shown in Table 2, the results indicated a significant positive association between patterning skills and mathematical achievement. Fisher’s z = 0.486 (p <.01), with a 95% CI [0.37, 0.60], r = 0.451 (p <.01), with 95% CI of [0.36, 0.54].

Table 2 The Relation between Patterning Skills and Mathematics Achievement
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Heterogeneity across studies was evaluated using the Q statistic and I2 statistic. A total of 17 independent studies (k = 17) with 81 effect sizes were included, and the degrees of freedom (df) for the Q test were calculated as 16. The results showed that the Q test was not statistically significant (Q = 73.556, df = 16, p >.05), while the I2 statistic was 78.25%, which indicated high heterogeneity. This pattern suggested that true differences across studies contributed substantially to the total variance, and the non-significant Q test might be due to limited statistical power.

The Moderation Effect

As presented in Table 3, the moderator analysis was performed to explore potential sources of heterogeneity in the correlation between patterning skill and mathematical achievement. Significant moderator effects were identified for two variables: element (QM = 8.051, df = 1, p =.005, σ2between study = 0.015, 95% CI = [0.00, 0.07]), study design (QM = 13.560, df = 1, p <.001, σ2between study = 0.008, 95% CI = [0.00, 0.05]).

Table 3 Meta-regression of the Moderation Analysis on the Relation between Patterning Skill and Mathematics Achievement
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Non-significant moderator effects were observed for age (QM = 3.404, df = 1, p =.065, σ2between study = 0.022, 95% CI = [0.00, 0.08]), mathematical measure (QM = 0.000, df = 1, p =.999, σ2between study = 0.034, 95% CI = [0.01, 0.11]), and patterning type (QM = 1.008, df = 2, p =.604, σ2between study = 0.031, 95% CI = [0.01, 0.10]). Building on the overall findings summarized in Table 3, the moderation models presented in Table 7 (mathematical measure) and Table 8 (patterning type) further indicated that the association between patterning skills and mathematical achievement remains consistent across different patterning types (repeating, growing, mixed) and mathematical measures (researcher-based, standardized). All analyses included 81 single cells unless otherwise specified.

Table 4 summarizes the results of individual moderator meta-regression models, with Holm-Bonferroni correction implemented to account for multiple comparisons. Element exhibited a significant effect (original p =.005), which remained statistically significant following Holm-Bonferroni correction (Holm p =.018). This finding suggested that the type of elements used in patterning tasks (i.e., symbolic vs. mixed) may significantly moderate the relation. Study design emerged as a statistically significant moderator of the core effect. After correction, the Holm p-value was.001 (original p <.001), which indicated that the magnitude of the core effect differed significantly across distinct study designs. Patterning type, measure, and age had no evidence of moderating effects, with uncorrected and adjusted p-values all exceeding.05. These results support the generalizability of the core effect, as its magnitude remained stable across variations in patterning type, mathematical measure, and age groups.

Table 4 Results of Moderator Tests with Holm-Bonferroni Correction
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As presented in Table 5, a significant moderator effect of element was observed. With “Mixed” as the reference group, the intercept was statistically significant (β = 0.270, SE = 0.088, p =.002, 95% CI [0.10, 0.44]). The “Symbolic” subgroup showed a significant positive estimate compared to the reference group (β = 0.292, SE = 0.103, p =.004, 95% CI [0.09, 0.49]). Given that the “non-pictorial” subgroup comprised only a single effect size, it was excluded from the moderation analysis.

Table 5 The Significant Moderator: Element
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As presented in Table 6, study design emerged as a significant moderator. Using “Non-intervention” as the reference group, the intercept was highly significant (β = 0.561, SE = 0.048, p <.001, 95% CI [0.47, 0.66]). The “Intervention” subgroup exhibited a significantly weaker correlation between patterning skill and mathematical achievement than the non-intervention group (β = −0.349, SE = 0.095, p <.001, 95% CI [−0.54, −0.16]).

Table 6 The Significant Moderator: Study Design
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Discussion

This association remained robust despite substantial heterogeneity, which highlighted the need to examine potential moderating factors. Moderator analysis revealed that factors including age, patterning types, element, mathematical measure, and study design. Element and study design significantly moderated this relation after collection. Specifically, patterning tasks using symbolic elements (e.g., shapes, colored blocks) had a stronger association with mathematical achievement compared to those using mixed elements (e.g., combining numbers and shapes), while non-pictorial elements were excluded due to insufficient sample size. For study design, the patterning skills in the non-intervention design showed a stronger association with mathematical achievement compared to the intervention design. These findings suggested the important role of patterning in early mathematical development (Fyfe & Borriello, 2025; Rittle‐Johnson et al., 2017) and offered insights into how study design and patterning task design can optimize the transfer of patterning skills to mathematics achievement.

The Relation between Patterning Skill and Mathematics Achievement

The results indicated a positive association between patterning skill and mathematics achievement, consistent with previous studies (Liu et al., 2022; Vanluydt et al., 2021). It may reflect their shared cognitive foundation in structured rule-based reasoning. Patterning skills, often termed the “precursor language of mathematics,” fundamentally involve identifying and applying regularities (Borriello et al., 2023; Fyfe & Borriello, 2025). Mathematics, particularly in arithmetic, algebra, and geometry, serves as the symbolic and systematic formalization of these regularities (Clements & Sarama, 2020). For example, the concept of “units” established through patterning tasks (e.g., recognizing “AB” as a repeating unit in ABAB patterns) supported children’s understanding of “chunks” and “multiples” in mathematics. Similarly, the “incremental rules” mastered in pattern-extension tasks create cognitive mappings to arithmetic operations and number sequence reasoning (Vanluydt et al., 2021; Wijns, 2021).

The Role of Patterning Type

The different patterning types have not significantly moderated the relation between patterning skill and mathematical achievement, which is consistent with previous studies (Borriello et al., 2023; Wijns et al., 2021a, 2021b). This consistency across task forms supports the idea that nurturing patterning skills, regardless of whether they involve repeating, growing, or mixed patterns, is an important way to foster children’s mathematical achievement (Rittle‐Johnson et al., 2017). These findings underscore the importance of patterning skills as a foundational component of mathematical development.

However, these findings are based on a relatively small number of studies for certain pattern types, particularly growing patterns, which may limit the generalizability of these results. Future research with larger and more diverse samples is needed to further investigate how different patterning types contribute to mathematical development.

The Role of Element

Element showed a significant moderating effect, which demonstrated that the relation between patterning skill and mathematical achievement remained significant across different elements. It is important to clarify that while non-pictorial elements were excluded in the formal analysis, the small sample size was insufficient to support meaningful inferences about their independent effect. Thus, the following interpretation focuses on the contrast between symbolic elements and mixed elements, which combine symbolic and non-pictorial components. This trend offers targeted insights into how different elements may moderate the patterning-mathematics relation.

One plausible interpretation is that both symbolic and non-pictorial elements, despite their compositional differences, collectively tax limited working memory resources. This aligns with Baddeley’s work, which notes that the nature of this cognitive load varies between element types. Symbolic elements act as cognitive scaffolds in foundational patterning tasks (Baddeley, 2012). Mixed elements integrate symbolic and non-pictorial components, which can reduce over-reliance on specific symbolic forms that hinder transfer effects (Uttal et al., 2009). Different element types may also engage distinct statistical learning mechanisms. Research has indicated that symbolic elements rely on sensory regularities such as color or shape repetition that are easy to detect (Bulf et al., 2011). Building on this finding and consistent with the notion that element composition shapes cognitive processing (Collins & Laski, 2015), mixed elements (e.g., blocks paired with numbers) probably require integrating these sensory regularities with symbolic regularities, a process that may activate deeper rule-induction processes. This difference may explain the significant trend observed in the unadjusted analysis.

Notably, the analysis of the effect of the non-pictorial subgroup is not examined due to the small sample size, which may also obscure potential effects. These limitations highlight that future studies should adopt larger and more balanced samples across element type subgroups. Additionally, incorporating task demand could help clarify how element type interacts with task characteristics to shape the patterning-mathematics link.

The Role of Age

Age did not emerge as a significant moderator in single-factor analyses. However, this non-significant result is likely constrained by the narrow age range of the current sample (4.5 to 6.85 years), which may have obscured the moderating role of age. Past studies have outlined a clear developmental trajectory: in early childhood (approximately 3–5 years), patterning serves as a foundational scaffold for numeracy. During this stage, children rely on symbolic elements and low-complexity tasks to internalize numerical relationships through perceptual-motor engagement (Rittle-Johnson et al., 2019; Zippert et al., 2021). As children grow older (6 years), the role of patterning in mathematics shifts, not because patterning becomes less relevant, but because mathematical reasoning increasingly demands executive functions that extend beyond perceptual representations (Borriello et al., 2023). This progression is consistent with cognitive developmental theories, which describe a transition from concrete, perception-based reasoning to more abstract, rule-based logic (Rittle-Johnson et al., 2017).

Our sample, however, primarily captures children in the “transition phase” (5 to 6 years), with limited representation of younger (3 to 4 years) and older (7 + years) groups. Within this narrow age range, the developmental shifts in patterning-related cognitive mechanisms may be too subtle to emerge as a significant moderating effect. For example, both younger and older children in our sample may already possess basic perceptual-motor patterning skills, while the more advanced abstract processing associated with older age is underrepresented. It underscores the importance of future studies with a broader age range to capture the full developmental trajectory.

The Role of Mathematical Measure

Mathematical measure did not emerge as a significant moderator, a finding that aligns with a prior systematic review about between visuospatial working memory and mathematical performance (Allen et al., 2019) and further reinforces the robustness of the patterning-mathematics link. Allen et al. (2019) have found that while standardized measurement tools tend to yield more stable effect size estimates, the positive association between patterning skills and mathematical performance persists regardless of the type of tool used. This is consistent with our non-significant moderation result.

In practical terms, this non-significant moderation result carries an important implication for researchers and educators. The positive link between patterning skills and mathematics achievement is not dependent on the specific tool used to measure mathematical performance. This may reduce concerns about “researcher tool-dependent” findings and enhance confidence in the generalizability of the patterning-mathematics association across diverse assessment contexts. Whether using standardized tests for broader applicability or researcher-developed tools for tailored investigations, the consistent positive link between patterning and mathematics emphasizes its importance for early mathematical development.

The Role of Study Design

Study design emerged as a significant moderator, with the intervention subgroup having a weaker relation than the non-intervention group. Past studies have supported that interventions can quickly improve children’s patterning skills, and some studies show these gains can predict later math performance (Fyfe & Borriello, 2025; Hendricks et al., 2006; Papic et al., 2011). However, the variability in the quality of interventions may influence the transfer to the children’s mathematical performance, as noted in prior research (Fyfe & Borriello, 2025). Not all patterning interventions use rigorous methods, which may inflate short-term patterning performance while failing to promote cognitive transfer to mathematics, which could weaken the observed correlation between post-intervention patterning skills and mathematical outcomes. This emphasizes the importance of intervention quality in maximizing the benefits of patterning instruction.

For educators, these findings point to actionable implications. The design of interventions plays a crucial role in the patterning skills program. By implementing high-quality interventions, educators can leverage the malleability of patterning to promote enduring mathematical competence and better prepare children for future academic success.

Limitations and Future Directions

The meta-analysis revealed substantial heterogeneity across studies but found no evidence of publication bias. This heterogeneity likely reflects variation in samples, task implementations, and procedures (Higgins & Thompson, 2002). Although element emerged as a significant moderator, subgroup evidence is limited: the symbolic subgroup included only six effect sizes, and the non-pictorial subgroup was excluded because it contributed a single effect size. Consequently, these subgroup findings should be interpreted cautiously, and future research should investigate additional moderators.

The current synthesis could not account for key cognitive variables (e.g., executive functions) that may influence the relation between patterning skills and mathematical achievement. To clarify underlying mechanisms, future research should employ multi-method assessments that combine behavioral patterning tasks with standardized cognitive batteries and process-level measures (e.g., eye-tracking during pattern generalization) (Zhang, 2025).

The predominance of correlational designs constrains causal inference, and the absence of neurocognitive evidence leaves biological mechanisms unexplored. Addressing these gaps will require studies using diverse designs (including experimental and longitudinal methods) and larger, more representative samples.

Finally, exploratory analyses suggested pattern-type effects: rotation-free tasks showed a stronger patterning–mathematics association than rotating tasks. Future work should systematically investigate specific pattern types to refine theory and guide the design of instructional materials.

Conclusion

This meta-analysis contributes to our understanding of the relation between patterning skills and mathematical achievement by identifying study design and element as significant moderators, which provides insights into the cognitive and task-related processes that may be associated with this relation.

First, a positive association was identified between patterning skills and mathematics achievement, aligning with the theoretical model (Rittle‐Johnson et al., 2017). This relation reflected shared cognitive processes, such as rule induction and structural generalization, which supported both pattern recognition and mathematical competence.

Second, study design and element emerged as the significant moderators after Holm-Bonferroni correction. Non-intervention studies (including cross-sectional and longitudinal designs) showed a stronger link between patterning skills and mathematical achievement compared to intervention studies. This result highlights the malleability of patterning skills, targeted interventions can effectively improve children’s pattern performance, but also underscores that the transfer of these gains to broader mathematics outcomes may be limited by factors like intervention quality. It emphasizes the need for rigor in intervention design to maximize the practical value of patterning instruction. Similarly, the tasks that used symbolic elements were associated with a stronger patterning-mathematics link compared to using mixed-element tasks, likely because mixed element increases children’s cognitive load, further emphasizing the importance of task design in optimizing the benefits of patterning activities. Notably, the subgroup distribution for the “element” moderator was imbalanced, with the non-pictorial subgroup including only one study and therefore excluded from the moderation model. Even the symbolic subgroup, with only six effect sizes, had limited statistical power, which highlighted the need for future research with larger and more balanced samples.

Third, age, patterning type and mathematical measure did not significantly moderate the association. The non-significant effect of age is likely due to the narrow age range of the sample, which masked developmental shifts in how patterning supports mathematics. The stability across patterning types and measures further confirms that the value of patterning skills transcends task form and assessment method.

Finally, the study identified methodological limitations that guide future research directions. High heterogeneity across studies suggests true differences in effect sizes may stem from variations in sample characteristics, task implementations, or procedures. Future studies should adopt larger and age-diverse samples, pre-registered confirmatory designs, and multi-method assessments (e.g., combining behavioral tasks with cognitive or neurocognitive measures) to clarify the causal mechanisms and contextual boundaries of the patterning-mathematics relation.