Compensatory Relation Between Executive Function and Fluid Intelligence in Predicting Math Learning

Abstract

Math learning is a key educational goal, and one marked by substantial individual differences even in the earliest grades. Although considerable research has examined the extent to which domain-general processes, such as executive functions and fluid intelligence, contribute to this variability, there is a notable gap in understanding how they may interact to predict early math learning. In particular, prior work had not examined potential moderating effects whereby the relation between executive functions and math outcomes depends on a child’s fluid intelligence, and vice versa. The current study addressed this gap by examining the math skills in Russian first-graders (N = 160) as a function of fluid intelligence (measured with Raven’s matrices) and various components of executive functions. Consistent with prior research, the results revealed the main effects of Raven’s scores, verbal working memory, and the control component of executive function (a composite of inhibition and cognitive flexibility scores) on math growth. Importantly, extending previous research, the study found that both memory and control components of executive function interacted with fluid intelligence. Specifically, executive function had a stronger positive effect on math learning for children with lower levels of fluid intelligence. The implications for intervention research and educational practice are discussed.

1. Introduction

Children’s mathematical skills in early elementary school predict their later academic achievement, which is, in turn, associated with broader professional and life outcomes (Aubrey et al., 2006; Jordan et al., 2009; Scammacca et al., 2020; Watts et al., 2014). Not surprisingly, thus, there has been great interest among researchers and educators in identifying the processes that contribute to variability in early math learning. In one line of research, investigators examine environmental factors involved in math learning, including family resources, home activities, and math talk (e.g., LeFevre et al., 2010; Huntsinger et al., 2016; Ramani et al., 2015). While acknowledging the role of environmental input, the current paper focused on cognitive processes that serve as essential tools of incorporating that input. Specifically, we investigated fluid intelligence and executive functions as predictors of the growth of first-graders’ math skills.

Fluid intelligence and executive functions represent domain-general mechanisms that are included in major theoretical models of math learning (Cragg et al., 2017; L. S. Fuchs et al., 2010; Geary, 2011). While they are closely related and work in concert during complex cognitive tasks, these domain-general processes are conceptually distinct. Fluid intelligence involves the capacity to extract regularities and engage in abstract reasoning that allows one to solve novel problems, whereas executive functions involve the capacity to regulate thinking and behavior that allows one to achieve goal-directed tasks. The relation between these domain-general processes in the context of learning has been primarily investigated as additive. In other words, each process has been studied as an independent contributor to learning outcomes (Demetriou et al., 2023). Little attention has been paid to a potential interaction between them, which limits nuanced understanding of their roles in math learning. The present study examined whether individual differences in math learning can be partly explained by an interaction between these domain-general processes. Specifically, we tested whether the relation between children’s executive functions and math outcomes depends on their fluid intelligence, and vice versa.

1.1. Fluid Intelligence and Executive Functions in Relation to Math Skills

Historically, studies of cognitive predictors of math learning focused on fluid intelligence (Floyd et al., 2003; McGrew & Hessler, 1995). Because math is a complex system of knowledge that relies heavily on abstract thinking, one could expect fluid intelligence to be essential for the mastery of this academic domain. For example, children need to rely on abstraction and pattern recognition to acquire such foundational concepts as numerical magnitude (understanding the relation between the number and the quantity it represents) or the hierarchical nature of the base-10 system (understanding the recursive structure of multi-digit numbers). Indeed, fluid intelligence has been established as a predictor of math learning across different age groups and educational contexts (Geary, 2011; Lin & Powell, 2021; Peng et al., 2019; Stevenson et al., 1976). It shows a particularly strong link to performance on problem-solving tasks that require applying abstract rules to novel contexts, compared to simpler tasks that can be solved using memorized facts (Deary et al., 2007; L. S. Fuchs et al., 2010; Primi et al., 2010).

In the contemporary research on general cognitive predictors of math learning, the main focus has shifted to executive functions (Carriedo et al., 2024; Clark et al., 2013; Finders et al., 2021; Fung et al., 2020; Nesbitt et al., 2015; Ten Braak et al., 2022). The construct of EF comprises several interrelated components. Although there is an ongoing discussion about the structure of this multifaceted construct, many researchers follow the Miyake et al. (2000) model, whereby EF encompasses working memory (holding and manipulating information), inhibition (resisting distractions and overriding dominant responses), and cognitive flexibility (shifting attention between tasks and perspectives). Similarly to fluid intelligence, multiple studies established a positive relation between EF skills and math outcomes (Blair & Razza, 2007; Brock et al., 2009; M. W. Fuchs et al., 2014; Fung et al., 2020; Nesbitt et al., 2015; Ribner et al., 2017; Wei et al., 2020; Willoughby et al., 2012). At the same time, as data accumulate, researchers are becoming increasingly aware of various nuances in this relation. Among them, several investigations reported that components of EF (e.g., working memory versus inhibition) vary in how strongly they predict specific math skills (Cragg et al., 2017; Lan et al., 2011). Such findings underscore the importance of simultaneously considering different aspects of EF in relation to math learning, yet most studies that have examined this relation included one or two EF components or used composite measures that do not tease apart unique contributions of different components.

Another nuance in understanding the relation between EF and math learning has to do with the possibility that this relation varies depending on the level of other cognitive skills, such as fluid intelligence. Although the importance of considering the role of EF in conjunction with intelligence is widely acknowledged (e.g., Carriedo et al., 2024; Dowker, 2019; Geary, 2011), the issue of a potential moderation of one of these processes by the other has been hardly addressed. Many studies exploring the role of EF in math learning include a measure of intelligence; however, it is often used as a control variable to determine whether EF predicts math skills above and beyond intelligence (e.g., Iglesias-Sarmiento et al., 2023; Ribner et al., 2017). Other studies have examined intelligence and EF skills to compare their relative strengths in predicting math achievement (Alloway & Alloway, 2010; Arán Filippetti & Richaud, 2017; Kyttälä & Lehto, 2008). These studies provide evidence that the constructs of EF and fluid intelligence, while interrelated, are distinct and each contributes to math learning. What cannot be inferred from these studies is whether and how the domain-general processes interact to support math learning.

1.2. Considering an Interaction Between Fluid Intelligence and Executive Functions

When two processes work together to support learning, the question arises about the nature of their “collaboration.” One possibility, of course, is that they independently contribute to the learning outcome. In other words, their relation may be additive, as shown in Figure 1a: higher levels of each predictor are associated with greater math learning, but the influence of one does not depend on the level of the other. Another possibility is that there is an amplifying interaction, whereby the impact of one process (such as EF) becomes more pronounced at higher levels of the other (such as intelligence). This could happen, for example, if the low ability to extract patterns from stimuli makes the learning task so difficult that it constrains the potential benefits of stronger EF skills. This possibility is illustrated in Figure 1b, where the positive effect of EF on math performance is more pronounced at higher levels of intelligence.

Alternatively, a multiplicative relation between domain-general cognitive processes may follow a compensatory pattern, as illustrated in Figure 1c. That is, EF skills may play a particularly strong role in math learning of students with lower fluid intelligence. As these students might have difficulty with abstract reasoning and problem-solving essential to math, they may rely more on EF skills to focus on task, break down problems into manageable parts, and hold multiple pieces of information in mind while working through them. Children with higher levels of fluid intelligence, while certainly requiring EF skills to learn math, may not need as much support. This dynamic is consistent with the psychological compensation framework (Bäckman & Dixon, 1992), suggesting that deficits in one area can be offset by the investment of greater effort or the use of other skills. Within this framework, higher EF might be expected to have a stronger positive effect on math learning in students with lower fluid intelligence, and vice versa.

At present, little research has explored the interactions between domain-general processes, making it difficult to favor one theoretical scenario over another. To our knowledge, only one study has examined the interaction between EF and fluid intelligence in relation to academic outcomes (Carriedo et al., 2024). This cross-sectional study included students from different grade levels, with all the predictor and outcome measures collected at a single time point. The investigators identified distinct EF profiles (e.g., low inhibition and high working memory) in their participants and examined the relation between fluid intelligence and math skills within each profile. Although the study did not statistically compare the strength of these relations across profiles, the examination of the reported regression coefficients suggests a compensatory relation between EF and fluid intelligence in elementary school students.

Further, there is a body of research that, while not directly examining the interaction between EF and intelligence, offers insights relevant to making predictions about it. Specifically, several studies have shown that interventions, including those focusing on EF, have a stronger positive effect on children who begin with lower levels of academic skills (e.g., D. Fuchs et al., 2019; Torgesen, 2004). Additionally, there is evidence that the role of EF becomes more significant on more challenging tasks (e.g., Laski & Dulaney, 2015) and among children who have learning disabilities (e.g., Henik et al., 2015; Peng et al., 2016). Building on these findings, one might hypothesize that children with lower levels of fluid intelligence may benefit more from strong EF skills, as these skills provide essential resources for engaging in task analysis, pattern recognition, and problem-solving.

1.3. Present Study

The primary goal of the present study was to examine whether and how fluid intelligence and the components of EF interact to predict children’s math learning. Additionally, this study was designed to address certain limitations of prior research. In particular, it included measures of different EF components, unlike many earlier investigations that assessed only one or two components, most often working memory, as predictors of children’s math performance. Further, rather than assessing math skills at a single time point, the present study examined growth in math competence over the course of a school year. We predicted that the amount of math growth would be related to both EF and fluid intelligence, and that each of these predictors would remain significant when controlling for the other. We also hypothesized that the components of EF would interact with fluid intelligence in a compensatory way, such that the role of EF would become more pronounced at lower levels of fluid intelligence.

2. Materials and Methods

2.1. Participants

The study included 160 children (50% boys, 50% girls) recruited from public schools in a large Russian city. Children were assessed twice over the course of their first-grade year: at Time 1, the mean age was 82 months (range: 76–90 months); the assessment at Time 2 was eight months later. Background demographic information on family education and income was collected. All parents reported their educational level, but only 18% reported income; because of a high rate of missing data, this variable was excluded from analysis. Education was reported on a 4-point scale from 1 (high school diploma) to 4 (graduate degree): 6% of parents had a high school diploma as the highest level of education, 30% had 1–2 years of professional training after high school, 55% held a 4- or 5-year college degree, and 9% had a graduate degree.

All participating children were first-graders at the time of the study. In Russia, children typically enter first grade between 6.5 and 7.5 years old; thus, the mean age was slightly older than that of first-graders in many Western countries. All the schools from which participants were recruited followed the same curriculum and pedagogical guidelines. Interviews with teachers indicated that classroom instruction included number skills–symbolic number knowledge (number recognition and comparison), arithmetic (addition and subtraction), and mathematical reasoning (word problems), as well as spatial skills such as linear measurement and geometric shapes.

2.2. Materials

2.2.1. Math Assessment

To evaluate math skills, we used the International Performance Indicators in Primary Schools. This instrument, originally developed in English, demonstrated high reliability and predictive validity (Tymms et al., 2015). In creating a Russian version, back-translation was used to check the equivalence of items and Rasch analytic techniques were used to confirm the comparability of scales across languages (Ivanova et al., 2018). The present study focused on children’s performance on the math section, which assessed key numerical skills: counting (up to 100), number recognition and comparison (single- and multi-digit), and addition and subtraction within 20 (number facts and word problems). The items in this section are organized in the order of increasing difficulty, and, hence, were administered in the same fixed order to all children. At Time 1, participants started from the first item; they were stopped after making three errors in a row. At Time 2, the start item varied depending on where the participants were previously stopped; the stop rule was the same. At each time point, the math score was computed as the percentage of correct responses (out of all items). The results showed high internal consistency of the math assessment at both time points (Cronbach’s alpha = 0.89 and 0.90, respectively).

2.2.2. Executive Functions

To assess components of the EF, we utilized instruments that have been widely used in research with same-aged children and showed strong correlations with other tests measuring corresponding EF components. Working memory (WM) was measured with a Russian-language equivalent of the listening and spatial recall tasks (Alloway et al., 2005, 2006; L. S. Fuchs et al., 2010; Lee et al., 2012; Pickering & Gathercole, 2001). Inhibition and cognitive flexibility were assessed using the dots task (Davidson et al., 2006; Diamond, 2013; Ramani et al., 2020). WM tasks were administered first, with half the sample beginning with listening recall and the other half with spatial recall, followed by the dots task.

Listening Recall. In this task, children listened to sets of spoken sentences and recalled the last word of each sentence in the order presented. The sets varied in size: the first set contained one sentence per trial, and each subsequent set added one more sentence (up to six). Each set included three trials. During each trial, children first judged whether each sentence was true or false. After completing all sentences within a trial, they had to recall the last word of each sentence in the correct order. For example, in a trial within a three-sentence set, a child would hear three sentences (e.g., “Dogs can fly”, “Ice is cold”, “Birds have feathers”), make a judgment after each sentence (false, true, true), and then recall the last words in order (“Fly”, “Cold”, “Feathers”). The task was stopped if the child failed all trials within a given set size. One point was awarded for each correct word recalled in the correct order. The total number of points served as a measure of verbal WM.

Visuospatial Recall. In this task, participants were presented with sets of two cartoon figures (distinguishable by the color of their hats) both of whom held a ball. The sets varied in size: the first set included one pair per trial, and each subsequent set added one more pair (up to six). Each set included three trials. During each trial, children were asked whether the characters in each pair held the ball in the same hand—with the task complicated by the fact that one of the characters was presented at different degrees of rotation relative to the other. Once they made this judgment for all the pairs within a trial, children had to recall the locations of the ball held by one of the characters, in the order of presentation. The total number of points awarded for each correct location recalled in the correct sequence served as a measure of spatial WM.

Dots Task: Inhibition and Cognitive Flexibility. In this task, a circle (“dot”) appeared on a computer screen either on the left or on the right side. The task was presented in three blocks (20 trials per block), with the dot location (left vs. right) randomly chosen for each trial. In the first block (congruent), participants were instructed to press the button on the same side where the dot was shown. In the second block (incongruent), they were instructed to press the button on the opposite side from the dot. For half of the sample, the dot that required a congruent response was solid gray and the dot that required an incongruent response had stripes, whereas for the other half of the sample, the stimuli were reversed (i.e., striped dot on congruent trials and solid gray on incongruent trials). In the final block (mixed), congruent and incongruent trials were mixed in a random order. Prior to this block, the rules of the previous two blocks were reinforced: “Gray dot means the same side; striped dot means the opposite side.”

Whereas memory is required on all trials (to remember the rules), the incongruent trials impose an additional demand for inhibiting a prepotent response (pressing the button on the same side), thus providing a test of inhibition. The mixed block introduces a further demand for switching the rules depending on the type of stimulus (solid gray versus striped), thus providing a test of cognitive flexibility. Note that the dots task was used to measure inhibition and cognitive flexibility, whereas another instrument was used to assess WM, because the memory demands in the dots task are quite limited.

Two scores were computed for each block: accuracy and reaction time (RT). Following the established protocol (Davidson et al., 2006), trials with an RT < 200 ms, considered too fast to be in response to the stimulus, were excluded from analysis. Thus, a trial was coded as correct if RT > 200 ms and the response was consistent with the rule for a given stimulus. For each participant, accuracy was computed as percent of correct responses out of 20 trials within a block. The time measure was also computed within each block as the median RT on accurate responses (median RT is typically used on this task to reduce the effect of outliers). To take into account the baseline performance and to capture the “cost” of additional demands imposed in Blocks 2 and 3, percentage change scores were computed for each child both for accuracy and reaction time. For example, percentage change in RT on incongruent trials was computed as [(Incongruent condition RT − Congruent condition RT)/Congruent condition RT] * 100. Change scores computed separately for the incongruent and switch blocks served as measures of inhibition and cognitive flexibility, respectively.

2.2.3. Fluid Intelligence

Raven’s Colored Progressive Matrices, which are specifically designed and normed for children aged 5–11 years, were used to assess fluid intelligence (Raven et al., 1990). The task includes 36 items divided into three sets, with 12 items per set. Each item presents a pattern of geometric designs with a missing piece. Children were asked to select which of six options was the piece that would complete the visual pattern. For each correct responses, the child was awarded one point. The total number of points earned served as a measure of fluid intelligence.

2.3. Procedure

Data for the present study were obtained as part of a larger investigation conducted by one of the authors in Russia in 2017. Children were tested individually in a quiet space in their schools by graduate psychology students. Time 1 assessments occurred at the start of the school year and involved two 30 min sessions: the first session included measures of executive function and fluid intelligence; the second included the assessment of math skills. Time 2 assessments occurred at the end of the school year during a single 30 min session that included the measures of math skills.

2.4. Analysis

Our main analytic approach involved regression analysis, testing fluid intelligence, different components of EF, and the interaction between intelligence and EF as predictors of children’s math scores at the end of the school year, while controlling for their math scores at the start of the year. There was no reason to expect that the cognitive processes examined here would operate differently in Russian children compared to same-aged children from Western countries who have comprised most samples in previous research. However, to be sure, we conducted additional checks to compare mean scores and correlations obtained in the present study to those reported in the prior literature. One advantage of our sample for the purposes of the present study is the uniformity of the first-grade math curriculum across schools and classrooms, unlike the varied pedagogical approaches and curricular materials characteristic of schools in the US. This consistency in instructional input—at least, in terms of school instruction—helps reduce variability in educational experience, which is an important consideration when studying cognitive predictors of learning.

Before presenting results, several preliminary points should be noted. First, initial analyses were conducted using both accuracy and RT measures of inhibition and cognitive flexibility. The patterns of results were equivalent; below, we present the findings related to RT, which is more common in the literature. Second, to facilitate interpretability, the inhibition and cognitive flexibility scores (calculated as RT incongruent/switch trials − RT congruent trials) were inverted by multiplying them by −1, so that higher scores indicate greater EF. Finally, the child’s age (in months), sex, and parental education level were used as covariates in all inferential analyses.

3. Results

3.1. Correlation Analyses

Table 1. Descriptive statistics and correlations among study variables.

Variable	Mean (SD)	1	2	3	4	5	6
1. Raven’s score	25.0 (5.6)	--
2. WM verbal	8.8 (3.3)	−0.05	--
3. WM spatial	9.8 (3.0)	0.17 *	0.35 **	--
4. Inhibition	31.2 (12.1)	0.44 **	0.31 *	0.28 *	--
5. Cognitive flexibility	54.4 (13.7)	0.48 **	0.35 *	0.24 *	0.76 **	--
6. Math, Time 1	50.5 (9.5)	0.42 **	0.18 *	0.24 *	0.45 **	0.43 **	--
7. Math, Time 2	60.1 (10.9)	0.63 **	0.17 *	0.30 *	0.48 **	0.50 **	0.80 **

Note. * p < 0.05, ** p < 0.001.

provides the means, standard deviations, and correlations for key study variables. The components of EF (WM, inhibition, and cognitive flexibility) were intercorrelated. Of note is a particularly high correlation between inhibition and cognitive flexibility. Further, nonverbal intelligence was correlated with all the components of EF, except for verbal WM. Finally, all cognitive predictors were related to math scores at both test points, which, in turn, were highly correlated with each other. Among the control measures, child’s sex and age were not related to any of the variables, whereas parent education was positively correlated with the growth of children’s math scores (r = 0.31, p < 0.05).

3.2. Regression Analysis

Following descriptive analyses, we tested regression models with the math scores at Time 2 as the outcome and the math scores at Time 1 as a covariate. The first model included fluid intelligence and all the components of EF as the key predictors, while controlling for child’s age, sex, and parent education. The initial test resulted in a multicollinearity warning (tolerance statistics < 0.01), which was caused by the high correlation between two predictors: inhibition and cognitive flexibility. To address the problem, we created a composite measure from these variables (henceforth, referred to as EF control). The results of analyses that included the composite measure are presented in

Table 2. Regression models predicting growth of math scores in first-graders.

Variable	Model 1			Model 2
	$\mathit{\beta }$	t	$\mathit{p}$	$\mathit{\beta }$	t	$\mathit{p}$
Age	0.05	0.43	0.67	0.05	0.45	0.65
Sex	0.04	0.98	0.33	0.03	0.74	0.46
SES	0.06	1.79	0.07	0.07	1.85	0.06
Math, Time 1	0.69	17.47	<0.001	0.74	17.81	<0.001
Raven’s	0.32	8.15	<0.001	0.29	7.33	<0.001
WM verbal	0.19	2.83	0.005	0.18	2.72	0.007
WM spatial	0.07	1.72	0.08	0.05	1.18	0.20
EF control	0.17	4.31	<0.001	0.19	4.76	<0.001
Ravens × WM verbal				−0.08	−2.05	0.04
Ravens × WM spatial				−0.003	−0.06	0.90
Ravens × EF control				−0.09	−2.34	0.02

, Model 1.

Next, we added to Model 1 three terms capturing the interaction between each of the EF components and Raven’s score. Adding these interaction terms produced a significant increase in the amount of explained variance, compared to Model 1 (p = 0.002). Note that we originally included additional terms to examine interactions between EF components, but these terms were nonsignificant and given that they were not central to the goals of the study and that the number of predictors approached the recommended maximum for our sample size, we eliminated these interaction terms. The final model (Model 2) explained a large amount of variance in math scores at the end of the school year ($R^2$ = 0.83). The results are presented in Table 2, Model 2.

Consistent with prior research, the strongest predictor of children’s math performance at Time 2 was their math performance at Time 1. Further, fluid intelligence and EF control were significant predictors of math scores at Time 2, even after controlling for the initial math level and demographic variables. Critical to our hypothesis, there was a significant interaction between Raven’s and EF control scores, as well as between Raven’s and verbal WM. The negative β for these interaction terms suggests that the impact of one of the predictors on the outcome increases at lower values of the other predictor (e.g., the positive effect of EF on math scores becomes stronger as the fluid intelligence decreases).

To further explore and illustrate the nature of this interaction, we examined the relation between each predictor and math growth (Time 2—Time 1) at different levels of the other predictor. First, we computed correlations between EF control and math growth for children whose Raven’s scores were more than one standard deviation below versus above the mean. This analysis showed that EF was related to math growth more strongly in children with lower Raven’s scores (N = 30, r = 0.59) than in children with higher Raven’s scores (N = 32, r = 0.14). Using Fisher’s z-transformation, we confirmed that these correlation coefficients were significantly different (z = 2.01, p = 0.03). Next, we examined correlations between verbal WM and math growth for children whose Raven’s scores were more than one standard deviation below versus above the mean. The results showed that verbal WM was related to math growth more strongly in children with lower Raven’s scores (N = 30, r = 0.64) than in children with higher Raven’s scores (N = 32, r = 0.24); the two correlation coefficients are marginally different from one another: z = 1.85, p = 0.05. Figure 2 illustrates the interaction between each of the two EF variables (EF control and verbal WM) and fluid intelligence in predicting first-graders’ math growth over the school year.

4. Discussion

The current study examined the extent to which the variability in first-graders’ math learning was associated with two major domain-general processes, fluid intelligence and executive functions. Unlike previous studies that either considered the role of each process in isolation or examined both processes assuming an additive relation between them, this study tested a moderation model that included interaction terms between fluid intelligence and components of executive functions. As discussed below, the results both replicated and extended previous findings, providing more nuanced understanding of the way in which cognitive processes are involved in math learning.

4.1. Replicating Prior Findings: EF and Fluid Intelligence as Predictors of Math Learning

Given that this study was carried out with children from a different cultural context than most other studies exploring cognitive predictors of math learning, one of the issues to be addressed is to what degree the findings are parallel to those reported in prior work. As noted earlier, the Russian first-graders in the present study were about a year older than first-graders in the US. Comparing our participants’ cognitive skills to those of same-aged children from other studies, we noticed substantial similarity in terms of the means scores on EF and fluid intelligence tasks and their relation to math outcomes. Specifically, the average scores on Raven’s matrices, the dot task, and the two working memory tasks were close to the scores of same-aged North American, European, and East Asian children tested with the same tasks (Alloway et al., 2005; Davidson et al., 2006; Lee et al., 2012; Nadler & Archibald, 2014).

Also similar to prior work, the components of EF (WM, inhibition, and cognitive flexibility) were related to each other and to fluid intelligence, and both types of cognitive predictors (EF and fluid intelligence) were associated with math growth. Among these relations, several interesting findings emerged that may benefit from a closer look. In particular, two aspects of EF—inhibition and cognitive flexibility—were so highly correlated that it was impossible (due to multicollinearity) to examine them as separate predictors within a single regression model. In interpreting this finding, it should be noted that there is an ongoing discussion of whether there is a detectable distinction between EF components among children (Espy et al., 2011; Hughes et al., 2009; Wiebe et al., 2011). Several developmental studies showed that EF becomes more differentiated with age. For example, a study by Lee et al. (2013) that examined multiple measures of EF yielded a two-factor model in 5–6-year-olds, with inhibition and cognitive flexibility loading on the same factor. Other studies produced a parallel two-factor structure in children up to 9 years of age (McAuley & White, 2011; Van der Ven et al., 2012), with a three-factor structure emerging between 11 and 15 years (Lee et al., 2013; Lehto et al., 2003; Rose et al., 2012). Thus, our finding that the measures of inhibition and cognitive flexibility were redundant in predicting math growth is likely due to insufficient differentiation between these two components of EF in 7–8-year-olds.

Unlike the two EF measures representing executive control aspects, the two WM measures showed a different pattern of association with math learning. While both measures were correlated with math scores at Times 1 and 2, the regression analysis showed that verbal WM was a significant predictor of math growth, whereas spatial WM was not. Prior research indicates that the structure of WM is relatively stable across development, with verbal and visuospatial components being separable even in young children (Alloway et al., 2006; Engle et al., 1999). However, findings on how these components contribute to math outcomes are inconsistent. Some studies have shown that verbal WM is more strongly related to math than spatial WM (Bayliss et al., 2003; Friso-van den Bos et al., 2013), whereas others report the opposite (Andersson & Östergren, 2012; Foley et al., 2017; Szűcs et al., 2014). One explanation for this inconsistency is the type of math skill under investigation: it has been suggested that verbal WM is more crucial for acquiring foundational math knowledge, whereas spatial WM could play a greater role in more advanced math skills (Cragg et al., 2017). Our findings align with this perspective, as the math assessment used in the present study focused on early numerical concepts and arithmetic skills.

4.2. Extending Prior Findings: Interaction Between EF and Fluid Intelligence

A key finding of the present study is the interaction between EF and fluid intelligence in predicting math learning outcomes. Both the control component of EF (a composite of inhibition and cognitive flexibility) and the verbal WM component exhibited a compensatory relation with fluid intelligence: the positive impact of EF skills on math learning was particularly pronounced at lower levels of fluid intelligence. This pattern is unlikely to be explained by a ceiling effect among high-performing students, as none of the participants approached 90% accuracy on the math task at either pre- or post-test. The results suggest that EF serves as a critical cognitive resource for children who find it more challenging than their peers to extract patterns from data and apply them in novel contexts, enabling them to grasp new math concepts and skills.

To illustrate this phenomenon, consider the acquisition of a key first-grade math concept in both Russian and US schools: the base-10 structure of two-digit numbers. Mastering this concept requires children to detect a recursive pattern from multiple examples, form an abstract understanding of place value—where a digit’s position determines its value—and apply this understanding to interpret any two-digit number. For children with lower fluid intelligence, EF plays a crucial role in supporting these processes by helping retain the value of one unit while analyzing the other, inhibit the tendency to treat each digit as a single unit, and flexibly shift between interpreting numeric symbols as tens versus ones. In contrast, children with higher fluid intelligence, who more readily detect patterns and form abstract concepts, may require less EF support when learning the base-10 system as it may take them less cognitive effort to process information and infer generalizable rules. This does not imply that they do not need EF; rather, variability in EF may exert less impact on their learning outcomes.

The present findings suggest that EF is likely to play a more prominent role in children’s math learning when their mental recourses are strained, whether due to instructional or individual factors. Evidence supports this notion. For example, it has been shown that EF skills are stronger predictors of math performance on tasks with greater cognitive demands. Laski and Dulaney (2015) compared kindergartners’ improvement on a number line task under two training conditions: one with explicit cues to support the linear representation of magnitude and another without such cues. In both cases, children’s learning was linked to their inhibitory skills, but the relation was much stronger in the second condition that placed greater demands on the learner. These findings, combined with the current study, suggest that EF interacts in a compensatory way with other cognitive processes.

Broader psychological research shows that compensatory mechanisms operate across different domains and age groups. In fact, much of the evidence for these mechanisms comes from research on aging. For example, as memory capacity declines, older adults compensate by using memory strategies more systematically; importantly, it has been shown that they benefit from mnemonic strategies more than younger adults with stronger memory capacities (Bäckman & Dixon, 1992; Salthouse et al., 1984). Although these studies differ in design from the current investigation, they reveal a consistent pattern of findings: the more cognitive resources are taxed, the more individuals benefit from “extra help” that can come from internal or external sources.

4.3. Implications

The current study offers insights for the design and evaluation of math interventions. First, it highlights the potential of programs that focus on enhancing EF in young students to support math learning. While a growing body of research demonstrates that EF is a malleable skill (e.g., Diamond et al., 2019; Gunzenhauser & Nückles, 2021), fluid intelligence may be less amenable to change (Schneider et al., 2014). In light of the compensatory effects observed in the present study, the findings suggest that by focusing educational efforts on enhancing EF, the strength of EF can be harnessed to compensate, at least partly, for lower fluid intelligence, ultimately fostering math development.

Moreover, the results highlight the importance of considering not only the main effects but also interactions when predicting or promoting academic skills. Much of the current intervention research focuses on the direct impacts of training and the indirect effects via mediational mechanisms. The few studies that have examined interaction effects typically focus on children’s initial skills as potential moderators, while controlling for domain-general skills, such as fluid intelligence. Yet, analyzing interactions involving both domain-specific and domain-general factors allows for a better understanding of heterogeneous effects, helping to identify the subgroups of students who would benefit the most from targeted interventions. By addressing both main and interaction effects, researchers can develop more targeted and effective educational practices tailored to the diverse cognitive profiles of students.

4.4. Limitations and Future Directions

We acknowledge several limitations of the present study. First, its correlational design limits the ability to draw causal inferences. Although the current study controlled for potential confounding factors—most notably by including Time 1 math scores as a covariate in the analysis of Time 2 scores—the findings should be interpreted with caution due to the absence of experimental manipulation. To establish a clearer understanding of the causal role of domain-general processes (and their interaction) in math learning, future research should implement an interventional design. For example, studies could examine whether enhancing EF skills leads to differential math learning outcomes for children with higher versus lower fluid intelligence.

Further, the compensatory interaction observed between EF and fluid intelligence in this study may not generalize across all learning contexts. There are likely limits to how much strong EF can compensate for weaknesses in other cognitive skills, whether domain-specific (e.g., number sense) or domain-general (e.g., fluid intelligence). It is plausible that at particularly low levels of a cognitive skill, even strong EF may not be sufficient to bridge the gap. Conversely, when a learning task is especially demanding (for example, taxing analytic capacity in children with higher fluid intelligence), the interaction between EF and fluid intelligence may follow an amplification pattern, whereby EF plays an even greater role in students with higher levels of intelligence. Future research should systematically investigate the boundaries of this compensatory effect to determine under what conditions, and to what extent, EF can buffer the effects of cognitive challenges.

4.5. Conclusions

This study examined the interplay between fluid intelligence and executive functions in predicting math learning among first-grade students in Russia. While both fluid intelligence and EF were independently associated with math growth, the study’s most important contribution lies in uncovering a compensatory interaction between them. The compensatory pattern aligns with theoretical models proposing that when one cognitive capacity is lower, another can assume a greater role to support learning. The results underscore the importance of considering how cognitive processes interact, rather than work in isolation, to influence learning outcomes. From an applied perspective, the findings suggest that interventions aimed at improving EF may be especially beneficial for students who struggle with fluid reasoning, offering a promising avenue for reducing early disparities in math achievement.