Early childhood is a time when children are first acquiring basic foundations of numerical skills on which later math learning will be built. Children who enter kindergarten with better basic math knowledge are more likely to succeed across a variety of academic domains including math, reading and science (Claessens & Engel, 2013; Duncan et al., 2007; Romano et al., 2010). Similarly, EF skills are undergoing rapid development in early childhood. Children who enter kindergarten better able to control or regulate their cognition and behavior tend to adapt more easily to the classroom environment (Clark et al., 2002; Nesbitt et al., 2019; Neuenschwander et al., 2012), and tend to display stronger academic achievement (e.g., Best et al., 2011; Cameron et al., 2019; Shaul & Schwartz, 2014). Furthermore, low-income children tend to underperform in terms of both EF and math skills (Hackman & Farah, 2009; Hair et al., 2015; Jordan & Levine, 2009). Research focused on understanding the strength of the EF↔Math relation and how it can be leveraged to support the simultaneous development of both skills, can inform intervention efforts that aim to close the school readiness gap.

While there are multiple existing meta-analyses on the relation between EF and math (Friso-van den Bos et al., 2013; Peng et al., 2016; Yeniad et al., 2013), none have yet focused specifically on early childhood. Some studies included preschool and kindergarten-aged children in their analyses (Friso-van den Bos et al., 2013; Jacob & Parkinson, 2015; Yeniad et al., 2013), but they did not isolate the EF↔Math relation specifically among children in this younger age group. Given the practical and theoretical import of EF and math in early childhood (preschool and kindergarten years), a meta-analysis providing a detailed examination of how these constructs relate to one another specifically in this age group is warranted. Thus, the current meta-analysis aims to fill this gap. First, we test whether the strength of the EF↔Math relation in early childhood depends on how EF and math are measured. Second, we test whether the early-childhood EF↔Math relation is stronger or weaker among children of lower vs higher socio-economic status (SES). Third, we examine the strength and direction of longitudinal relations between EF and math in early childhood.

Both EF and math are, in truth, shorthand appellations for complex sets of cognitive processes. At a given point in development, not all researchers agree on what the most important subskills are in each domain. Even for a given subskill, the specific task by which it is measured may vary across research groups. It is thus important to systematically examine how variability in measurement may affect the EF↔Math relation. While it is important to acknowledge that a complete treatment of all possible measurement variations is beyond the scope of a single paper, below we motivate the various measurement categories we considered in this paper (see also Table 1, Section 1: ‘Measurement’).

Prior research suggests that low-income preschoolers tend to fall behind their higher-income peers in terms of both EF and math skills (Hair et al., 2015; Noble et al., 2007; Reardon, 2011). One possibility is that the lags seen among low SES children in EF and math may be linked, in which case one might expect the EF↔Math relation in early childhood to depend on SES. In the event of a discovered SES-dependency, this might contribute to (A) our understanding of the value of examining this relation in economically-diverse samples, as well as to (B) adjusting expectations about the potential for feedback between EF and math skills, for instance, in early childhood. We thus test whether the EF↔Math relation is moderated by SES. For a summary of SES categories considered here, see Table 1, Section 2 (‘SES’).

There is currently a relative lack of evidence regarding the direction of influence between EF and early math skills. To date, much of the research on EF↔Math relations has tended to assume – tacitly or explicitly – a unidirectional relation whereby early EF primarily predicts gains in math outcomes (EF→Math: e.g., Clark et al., 2010; Fitzpatrick & Pagani, 2012; Mulder et al., 2017). As such, a common occurrence in the literature is to collect an assessment of EF at an earlier time point (T1) and an assessment of math skills at a later time point (T2). An association between EF and math is then interpreted as evidence that EF influences math. However, only a few studies have collected both measures at both timepoints. Without doing so, a given study could not control for initial skill levels, and thus could not accurately estimate the relation between early EF and changes in math skills. It is thus likely that studies computing an EF(T1)↔Math(T2) association without controlling for Math(T1) are reporting inflated associations (Bailey et al., 2018; Jacob & Parkinson, 2015). Also left largely unexamined is the possibility of the reverse: that early math predicts changes in EF skills.

Only somewhat recently have researchers begun to implement repeated measures designs (sometimes referred to as cross-lagged longitudinal designs) to better estimate bidirectionality between EF and math (Chu et al., 2016; McKinnon & Blair, 2019; Welsh et al., 2010). Overall, this small, but growing, body of longitudinal work has converged to suggest that the longitudinal relation between EF and math in early childhood may well be bidirectional, with early math abilities predicting change in EF to a similar or even greater extent than initial EF abilities predict change in math (Fuhs et al., 2014; McKinnon & Blair, 2019; Welsh et al., 2010). To our knowledge, two papers, Jacob and Parkinson (2015) and Nguyen et al. (2019), have examined the prevalence and magnitude of longitudinal EF-Math relations across multiple studies, specifically in early childhood. When investigating the longitudinal relations between EF and math, Jacob and Parkinson relied more on a narrative approach, and did not use a formal meta-analytic framework. Nguyen et al. (2019), given their stated theoretical aims, were concerned primarily with decomposing (or not) EF subcomponents in their relation to math. Further, given the intervening years between 2019 and the present day, we are able to include roughly twice as many relevant studies and over 170 individual longitudinal effect-sizes in our meta-analytic dataset. That is not to say these earlier papers are not valuable; rather, we believe the current work provides an important update and extension of this prior work. In sum, we tested for longitudinal relations between EF and math in early childhood after controlling for initial performance, and for evidence of an asymmetry between potential directional relations. For a summary of longitudinal comparisons considered here, see Table 1, Section 3 (‘Longitudinal Relations’).

We identified relevant articles in two ways: first, by conducting an online search of common databases (i.e., Google Scholar and PsycInfo) using the following search terms: math; math achievement; math performance, numeracy; calculation; arithmetic; problem-solving; executive function/ing; working memory; updating; response inhibition; inhibitory control; attention shifting; set shifting; cognitive flexibility; domain-general cognition; self-regulation; preschool; early childhood; early childhood education; kindergarten. Second, we scanned the reference lists of relevant reviews and previously identified studies. The search included papers published prior to August 2023.

We assessed the relevance of each study from the initial search by scanning titles and abstracts for keywords. Studies were deemed relevant if they included any of the key terms listed above. Relevant studies were then scanned more thoroughly and included in the meta-analysis if they met the following inclusion criteria: (a) reported at least one correlation (r-value) between EF and math; (b) had a sample consisting of preschool age children (2-5yrs) and/or kindergarten children (5-6yrs)¹1

For longitudinal studies, the age restrictions only applied to age at Time 1. Age at Time 2 could fall outside of the age range of interest.

For a complete summary of the coded categories, as well as the relevant comparisons motivated in the Introduction above, see Table 1. All categories were double-coded; there were no discrepancies between coders (r = 1).

We used Pearson’s r-values to index EF↔Math relations. Because r-values are standardized (expressed as relative changes in units of standard-deviation), the standard-error of a given r-value (se_r) can be calculated, given the sample-size used to estimate that r-value (N_r): ser=1Nr-3.

Meta-analytic values were estimated in SPSS [IBM: version 29.0.1.1 (244)]. Estimated average r-values and corresponding confidence-intervals were generated using a random-effects (RFX) model (effect-sizes nested within studies), restricted maximum likelihood (REML) estimator, and Knapp-Hartung adjusted standard-errors (Langan et al., 2019). We selected RFX models because heterogeneity tests (Q-statistics) were highly significant. Weights were computed using fully inverted variance, such that they included both within- and between-study variance (Pustejovsky & Tipton, 2022). Specific estimates (e.g., Low vs Mid-High vs Mixed SES) were calculated in a similar manner via stratified subgroup analyses (also via RFX as specified as above).

Direct comparisons between pairs of estimated average r-values (e.g., comparing the average EF↔Math relation among Low SES samples vs the same among Mid-High SES samples) were computed via random-effects meta-regression, again using Knapp-Hartung adjusted REML estimates. Note that the number and breadth of available studies did not allow for a complete assessment of all combinations of potential sub-moderators (e.g., comparing Low vs Mid-High SES only for studies using a direct, lab-based, non-numerical, composite EF measure and a basic numeracy math measure). Hence, we isolated a single comparison category (e.g., Low vs Mid-High SES, or Lab-Based vs Naturalistic EF measures) at a time. In doing so, we sought to lean on the methodological strength of a meta-analytic approach, which seeks to identify broad trends across a range of studies.

In this regard, comparison categories were yoked to the theoretical questions outlined in the Introduction (see also Table 1 for a summary). We believe this approach to be more theoretically informative relative to an atheoretical consideration of all possible combinations. In addition, this approach maintains a more reasonable sample-size for each comparison, and it reduces the total number of comparisons being computed – both factors are important for increasing the generalizability and replicability of the results being presented here. Finally, despite our arguably more circumspect approach, we nevertheless tested a fair number of direct comparisons; hence, we report both the traditional threshold of p < .05, as well as Dunn-Šidák corrected thresholds (Šidák, 1967).

A Trim-and-Fill evaluation of publication bias indicated minimal evidence of publication bias (Shi & Lin, 2019). We used Egger’s regression test to determine imputation side (right). Imputation required only 14 additional effect-sizes (relative to 1022 observed effect-sizes). Trim-and-Fill results indicated a very slight underestimate of the overall effect (r = .357 instead of .350). Thus, we make no further adjustments for apparent publication bias.

Average effects for each measurement subcategory are shown in Figure 1. Exact values for reference purposes can be found in the Appendix (Table A1). Contrast results are given in Table 2. We conducted 16 measurement-related contrasts (Table 2), so the corrected threshold for this section was α < .0032.

Here we examined the effect of SES sample composition on EF↔Math relations. Average effects for each SES subcategory are shown in Figure 2. Exact values for reference purposes can be found in the Appendix (Table A2). Contrast results are given in Table 3. We conducted 3 SES-related contrasts (Table 3), so the corrected threshold for this section was α < .0170.

The highest average effect was observed for Low SES samples (r = .431), which was significantly higher (ps < .001) than those observed for either Mid-High SES (r = .317) or Mixed SES (r = .336). There was no significant difference between Mid-High and Mixed SES (p = .292).

Average effects for each measurement subcategory are shown in Figure 3. Exact values for reference purposes can be found in the Appendix (Table A3). Contrast results are given in Table 4. We conducted 4 contrasts comparing two longitudinal effects, so the corrected threshold for this section was α < .0127.

How one measures various constructs can have a substantial impact on the correlations one observes between those constructs. When considering the correlation between EF and math in early childhood, we found that all operationalizations of EF and math considered here yielded a significant average correlation. This speaks to the overall robustness of this relation, even in relatively young children, for whom measurement can sometimes prove challenging. On the other hand, we also found that the strength of the association did vary significantly with some – though not all – choices with respect to how EF and math were measured.

In terms of SES, Mixed-SES samples, which contributed 77% of the effects included here, yielded an average effect (r = .336, 95% CI [ .324, .348]) close to the overall average (r = .350). However, examining the subset of samples that isolated specific portions of the SES distribution revealed this average effect camouflages important SES-related differences in the EF↔Math relation. In particular, the relation was significantly stronger for samples comprised exclusively of low-income children (r = .431, 95% CI [.398, .464]) and the weakest for those without such children (i.e., consisting of primarily middle-to-high income samples: r = .317, 95% CI [.255, .378]. One interpretation is that low-income children may rely upon their EF skills when performing math to a greater extent than their higher-income peers. This finding could be a reflection of differential access to resources that support math learning in the home. For instance, high-income parents tend to engage in more math talk with their children compared to lower-income parents (Vandermaas‐Peeler et al., 2009), which, in turn, is associated with stronger early math skills (Susperreguy, 2013; Susperreguy & Davis-Kean, 2016). As such, EF may play less of a role in the math performance of high-income children as they are more likely to receive additional supports for early math learning in the home. However, this is just a hypothesis and more empirical work is needed to further understand the moderating role of SES on the relation between EF and math in early childhood.

The stronger EF↔Math relation for low-SES children may also have implications for school-readiness. Specifically, low-income children demonstrate poorer early math knowledge than their higher-income peers, and the gaps in this knowledge at school-entry are thought to be mediated by income-based discrepancies in EF (Dilworth-Bart, 2012; Fitzpatrick et al., 2014; Lawson & Farah, 2017). Further, gains in low-income preschoolers’ early math skills are predictive of improved EF capacities upon kindergarten entry (McCoy et al., 2019). The current meta-analysis is the first to directly contrast the magnitude of the EF↔Math relation in early childhood across the SES spectrum. In doing so, we find support for the claim that fostering math and EF skills in early childhood education can be mutually beneficial. Moreover, efforts to integrate EF and math in early childhood education may yield the greatest benefits in schools serving children from low-income families, where perhaps not coincidentally, children often lag the furthest behind their peers in both types of skills. A more nuanced understanding of how the relation between EF and math differs as a function of SES can help to inform targeted interventions designed to improve school readiness skills among low-income children specifically, though more work is certainly needed to further test this idea.

The current paper sought to systematically document the evidence for (or against) directional relations between EF and math within a formal meta-analytic framework. ‘Directional’ here refers not to causality, but to capacity to predict change (see below for a more detailed note on causality). Overall, we observed that the average adjusted longitudinal relation from EF to math is less than half the size of the average unadjusted longitudinal relation between EF and subsequent math performance (EF→ΔMath: r = .190, EF→Math: r = .419, respectively; see Table A3), suggesting that unadjusted relations are inflated by about 120%. A similar pattern was seen for longitudinal relations between math and subsequent EF skills (Math→ΔEF: r = .240, Math→EF: r = .420; see Table A3), suggesting an inflation factor of about 75%. Together, these results indicate that failing to adjust longitudinal effects between EF and math leads to substantially inflated estimates of the extent to which these effects are truly directional – that is, the extent to which a given variable predicts change in the outcome.

On the other hand, even after adjusting these effects for initial performance, we nevertheless found significant effects in both directions (ps < .001, ds > 1; see bottom two rows of Table 4, and dark pink bars in Figure 3). Together, these results support the notion that the relationship between EF and math is bidirectional (Clements et al., 2016). More broadly, our results support a view wherein both EF and math skills are dynamic, and the relationship between them is transactional (Miller-Cotto & Byrnes, 2020). That said, while average adjusted longitudinal effects were robustly positive, these average effects were relatively modest in magnitude. In terms of longitudinal effects in the literature, this means we would expect some portion of studies to find null or inconsistent results (e.g., Barnes et al., 2016). Further, we would expect only reasonably well-powered studies (i.e., sample-sizes in the hundreds or more) to be capable of detecting effects of the magnitude found here (e.g., Zhang et al., 2023); but even with this power, detection is not a foregone conclusion (Willoughby et al., 2019). With these points in mind, our results may thus help to contextualize some of the apparently inconsistent findings in the literature: null results, especially in smaller samples, are not entirely unexpected, and while unadjusted effects are likely to overstate true underlying longitudinal relations, we should nevertheless expect the average (adjusted) effect to be positive in both directions.

Another point worth emphasizing is that the directional relation from math to change in EF skills was just as strong, if not modestly stronger (p = .010, d = 0.39), than the generally more celebrated relation from EF to change in math skills. Specifically, many researchers describe EF as a set of skills that are foundational to the acquisition of math knowledge (e.g., Cragg & Gilmore, 2014; Nguyen et al., 2019; Passolunghi & Lanfranchi, 2012). However, findings from the current study suggest that rather than thinking about EF primarily as a precursor to math learning, we would do well to think about EF and math as complementary skills that each play a role in the development of the other. Reciprocity in the relation between EF and math may be a product of the fact that complex math problems require the integration of multiple EF skills. Moreover, acquiring math skills is a complex process requiring one to marshal multiple existing cognitive and neural subsystems to think about and manipulate the world in novel ways – a task that EF skills are uniquely suited to facilitate. Conversely, if one sees EF not as fixed but malleable given experience and input (Miller-Cotto & Byrnes, 2020), then learning math is in many ways an ideal context within which to practice and refine the efficient deployment of EF skills (Clements et al., 2016; Miller-Cotto & Byrnes, 2020).

On a more applied level, the fact that the longitudinal relation between EF and math, once adjusted for baseline performance, is weaker than what is commonly reported in the literature suggests that interventions designed to improve early math skills that focus exclusively on EF training may yield at best modest results. Indeed, this appears to be the case in several extant studies (Barnes et al., 2016; Fuchs et al., 2022; though see also DePascale et al., 2024). Our data indicate that a more accurate view of the developmental relation between EF and math is a bidirectional one, implying that math skills are just as – if not slightly more – predictive of EF development as EF skills are for math development. Based on this, we suggest that efforts to improve both math and EF performance prior to formal schooling should focus on how math activities can be leveraged to include more explicit EF support (Barnes, 2023; Clements et al., 2016; DePascale et al., 2024; Miller-Cotto & Byrnes, 2020).

One important limitation of the current work is that, given the relative scarcity of studies assessing the relation between EF and changes in math (and vice versa), we were unable to look at moderators of the bidirectional longitudinal relations between EF and math. For instance, longitudinal links, and their potential implications for interventions in particular, may depend on overlap between the specific subprocesses. Such subprocesses may include the EF subcomponents examined here, or they may involve other cognitive domains relevant for both EF and math, such as spatial thinking.

In another vein, it would be informative to know if the reciprocity in the relation between EF and math is specific to preschool-aged children. Current research on the topic has reported inconsistent results in terms of whether this reciprocity persists as children get older. For example, Schmitt et al. (2017) found evidence for a bi-directional relation between EF and math in preschool, but not in kindergarten. The authors observed that once children started formal schooling, EF predicted gains in math but not the other way around. However, a more recent study observed bi-directional relations between EF and math across kindergarten and into first grade (McKinnon & Blair, 2019). Understanding how reciprocity in the relation between EF and math may change with age is important for informing theories on the developmental progression of EF and its relation to math. Further, it may also be important for understanding how best to design developmentally-appropriate curricula and interventions aimed at improving EF and math skills. As such, future research should investigate how the bidirectional relations between EF and math may change after the onset of formal schooling.

The present meta-analysis examined whether the strength of the EF-Math relation in preschool and kindergarten depends on EF and math measurement factors, socio-economic status (SES), and the nature and direction of longitudinal relations. Several results of note emerged. [1] Composite and Achievement measures of EF and Math led to higher average estimates of the EF-Math relation relative to measures that attempt to isolate EF subprocess or specific math skills; however, more work may be needed to determine whether this is due to measurement or developmental factors. [2] We found that EF measures using numerical stimuli inflate estimates of the EF-Math association by roughly 40%, and thus recommend researchers avoid using numerical stimuli in their EF measures when their goal is to estimate the magnitude of EF↔Math associations. [3] In terms of SES, the strongest average EF-Math association was found for low SES samples – a finding with important implications for preschool programs targeted at this population, such as Head Start and many state pre-K programs. Considering longitudinal associations, [4] those that do not adjust for Time-1 measurement of the outcome variable lead to substantially (as much as 120%) inflated estimates of directional associations. After making these adjustments, we nevertheless found [5a] significant, albeit reduced bidirectional relations between EF and math. We also found [5b] that math is a stronger predictor of future change in EF than the reverse, and thus recommend that math instruction serve as a foundation for embedding EF-building skills rather than isolating EF-fostering activities from the math curriculum. In sum, we hope that the results of this work contribute to theoretical models of the interaction between EF and math in early childhood, to increasing understanding of the inter-connected developmental courses of growth in math and EF capacities, and to practical attempts to foster growth in children’s EF and math skills, whether in the lab, classroom or living room.