# The role of inhibition in the development of arithmetic skills-A cross-sectional study in 5- and 7-year-old children

Sixtine Omont-Lescieux, André Knops, Ilse E J I Coolen

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## To cite this version:

Sixtine Omont-Lescieux, André Knops, Ilse E J I Coolen. The role of inhibition in the development of arithmetic skills-A cross-sectional study in 5- and 7-year-old children. Child Development, In press, 10.1093/chdev/aacaf009. hal-05497500

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Submitted on 6 Feb 2026

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# The role of inhibition in the development of arithmetic skills—A cross-sectional study in 5- and 7-year-old children

Sixtine Omont-Lescieux<fnref n="1" />,<fnref n="2" />, André Knops<fnref n="1" /> and Ilse E.J.I. Coolen<fnref n="1" />,<fnref n="3" />,<fnref n="4" />,<fnref n="5" />*

## Abstract

Inhibition has been suggested to contribute to symbolic and nonsymbolic quantity processing, but conclusions remain inconsistent. Using Structural Equation Modelling, the structure of inhibition and its contributions to symbolic and nonsymbolic arithmetic are explored in predominantly White, high-SES French 5- and 7-year-olds (N = 331, 169 females). Data were collected in Paris schools between January 2020 and March 2021. Results suggest a lack of support for a unitary or binary (Response Inhibition and Distractor Suppression) inhibition construct and highlight a link in both age groups between the Stop-Signal Task and symbolic arithmetic, and a link between the Flanker task and nonsymbolic arithmetic only in 7-year-olds. This study dissects the differential contributions of inhibition facets to arithmetic development in a critical time window.

**Keywords** inhibition, development, numerical cognition, arithmetic

## Lay summary

Inhibition has been suggested to contribute to symbolic (Arabic numbers) and nonsymbolic (dot sets) arithmetic, but conclusions remain inconsistent. This study tests whether these inconsistencies arise from differences in age (preschool or primary school), inhibition type, and arithmetic format. Using multiple inhibition tasks, we tested whether they reflect distinct inhibition skills and how these relate to arithmetic in 5-7-year-olds. Results showed that inhibition tasks do not measure a single underlying skill. Inhibiting automatic responses only related to symbolic arithmetic, whereas ignoring irrelevant information related to nonsymbolic arithmetic, but only in primary school children and only for one of the tasks. In sum, the link between inhibition and arithmetic depends on the inhibition task, arithmetic format, and the age tested.

The ability to count, to understand numbers, and to be proficient in mathematics is related to socioeconomic status in later life (Ritchie & Bates, 2013). Numerous studies point out the importance of basic numerical skills in early education (i.e., understanding of quantity and numbers for basic arithmetic) for success in mathematics later on (Bull & Lee, 2014; Geary et al., 2018). These basic numerical skills, in particular mental arithmetic with nonsymbolic quantities and symbolic numbers, require domain-general skills like inhibition to select appropriate strategies and suppress related but irrelevant information. Inhibition is an umbrella term that describes the cognitive capacity to do so and is

defined as the ability to adapt to conflicting situations (Diamond, 2013) and block or suppress memorized or habitual responses, automatic processes and irrelevant or disruptive stimuli in order to avoid interference with the correct response or process (Lee & Lee, 2019). Indeed, inhibition has been shown to be associated with symbolic arithmetic (Cragg & Gilmore, 2014), notably when ignoring inappropriate strategies (e.g., adding instead of subtracting in addition exercises; Bull & Lee, 2014), or when choosing the best strategy to answer an arithmetic problem (e.g., rounding up or down in two-digit additions; Lemaire & Lecacheur, 2011). Furthermore, the processing of nonsymbolic quantities has been

<footnote n="1">¹LaPsyDÉ, UMR CNRS 8240, Université Paris Cité, Paris, France</footnote>

<footnote n="2">²Laboratoire de Neuroanatomie et Neuroimagerie Translationnelles, UNI-ULB Neuroscience Institute, Université libre de Bruxelles, Brussels, Belgium</footnote>

<footnote n="3">³Department of Cognitive Neuroscience, Donders Institute for Brain, Cognition, and Behaviour, Radboud University Medical Center, Nijmegen, The Netherlands</footnote>

<footnote n="4">⁴Faculty of Psychology and Educational Sciences, KU Leuven, Leuven, Belgium</footnote>

<footnote n="5">⁵Department of Experimental Psychology, University of Oxford, Oxford, United Kingdom</footnote>

<footnote>*Corresponding author: LaPsyDÉ, UMR CNRS 8240, Université Paris Cité, 46 rue Saint-Jacques, Paris 75005, France. Email: coolenilse@gmail.com</footnote>
suggested to require inhibition, when perceptual properties such as total area and density of the quantities need to be ignored to make numerical judgments (Clayton et al., 2015). However, the particular role that inhibition plays during the development of early symbolic and nonsymbolic number processing remains unclear as the literature comprises many inconsistent findings (Gilmore et al., 2015), which this study aims to unravel.

These inconsistencies and ambiguities can be attributed to several plausible factors. Firstly, (a) distinct types of inhibition emerge and evolve throughout development (Diamond, 2013; Friedman & Miyake, 2004; Rey-Mermet et al., 2018), with potential different roles in mathematics development for distinct inhibition types; second (b) inhibition is known to rapidly develop in early childhood (Best & Miller, 2010), leading to potential changes in the association between inhibition and mathematics depending on age; and finally, (c) mathematics comprises different subskills (Gilmore, 2023), and previous studies have not consistently examined the same subskills when exploring potential associations with inhibition (e.g., rational number: Coulanges et al., 2021, or cardinality principle: Geary et al., 2018). Each of these factors will be discussed in turn.

## Inhibition as multicomponential construct

Inhibition has been suggested to be a non-unitary construct with two types of inhibition that appear during development (Diamond, 2013): Distractor Suppression (or attentional inhibition; resistance to interference from the distractor) and Response Inhibition (or action inhibition; inhibition of a prepotent response). However, according to Friedman and Miyake (2004), these two forms of inhibition are highly correlated and depend on one and the same factor. To date, there is no clear consensus on the structure of inhibition, but the most commonly used model is composed of Response Inhibition (RI) and Distractor Suppression (DS) (Friedman & Miyake, 2004; Rey-Mermet et al., 2018; Wiebe et al., 2008). Response Inhibition corresponds to exerting behavioral control over desires, impulses, or prepotent and habitual responses, and is usually measured by tasks such as the Stop-Signal Task (Friedman & Miyake, 2004). In such tasks participants must intermittently suppress a motor response following the presentation of a conditional cue (e.g., press a key as quickly as possible, unless an auditory tone is presented; Verbruggen et al., 2013, 2019). Distractor Suppression corresponds to direct-ing attention towards desired environmental stimuli or cognitive processes and away from irrelevant distractors (Diamond, 2013; Friedman & Miyake, 2004). Distractor Suppression is often meas-ured by tasks such as the Flanker task or a Shape matching task, in which participants have to attend to a target stimulus, while ignoring irrelevant stimuli or features (Friedman & Miyake, 2004).

The potential multifactorial organization of inhibition shows how important it is to use different tasks to measure the various subtypes of inhibition to better understand their involvement dur-ing development. Indeed, a recent study by Iglesias-Sarmiento and colleagues (2023) demonstrates the involvement of differ-ent types of inhibition (including Distractor Suppression and Response Inhibition) in mathematical success at the age of 7 years, compared to 11 and 15 years. Therefore, in this study we take into account the two types of inhibition that are most often found in the literature (Response Inhibition and Distractor Suppression).

Despite the conceptual distinction between Response Inhibition and Distractor Suppression, it should be noted that inhibition tasks can suffer from task impurity; for example, they can involve multiple executive functions or other cognitive pro-cesses (Friedman & Miyake, 2004). For instance, inhibition tasks may rely on working memory to hold task rules in mind, or on attentional control to monitor for infrequent signals. Different inhibition tasks may involve distinct sources of impurity, reflecting varying contributions of other cognitive processes. Additionally, what is often described as task impurity might also reflect differences in how inhibition is recruited depending on the specific demands and content of an inhibition task (Wilkey, 2023). This questions whether inhibition is a single, general abil-ity, or whether it depends on the type of inhibition required for the task at hand (Wilkey, 2023). If inhibition is truly unitary and independent of the task, we would expect individuals to perform similarly across different inhibition tasks. However, if inhibition is dependent on the task as suggested by Wilkey (2023), then an individual's performance would vary more between different inhi-bition tasks, as well as how each inhibition task would relate to a mathematical task. In our study, we measured inhibition with three different tasks (Stop-Signal Task—Response Inhibition; Flanker task and a Bivalent Shape Task—Distractor Suppression) previously used in research relating inhibition to mathematical skills. As the Flanker task was embedded in a broader attention task (Attentional Network Task) and might thus have more task impurity and different demands, we opted to include a second task reflecting Distractor Suppression, the Bivalent Shape Task. Including the Flanker task allows us to investigate whether dif-ferent task demands and impurity contribute to inhibition per-formance. By including multiple inhibition tasks with potentially differing demands, contents, and impurity, we aimed to address the multicomponential nature of inhibition.

## The maturation of inhibition

The maturation of inhibition starts in infancy (Best & Miller, 2010) with substantial enhancements in inhibition observable in chil-dren aged 3 to 4 years (Hughes, 1998). Nevertheless, the develop-ment of inhibition skills extends into the initial years of primary school until the age of 8 years (Best & Miller, 2010; Hughes, 1998). After the age of 8 years, it has been suggested that there is only marginal progress in inhibition, and it is generally believed to pla-teau for easy inhibition tasks, in terms of significant improvements after the age of 12 years (Best & Miller, 2010). However, improve-ments have been found in reaction times and accuracy in tasks such as the Stop-Signal Task and the Flanker task until the age of 15 years (Huizinga et al., 2006). Rapid improvements in preschool and early primary school (Allan et al., 2014) are likely to impact their association with mathematics. Moreover, mathematics rap-idly evolves across preschool and primary school, which results in an increasing complexity. For example, while early preschool mathematics consists of one-digit calculations, children progres-sively encounter more multidigit calculations, involving multiple steps and additional procedures such as carrying and borrowing (i.e., transferring a value between units and tens to complete an operation, e.g., 17 + 16). Inhibition is likely to contribute more as mathematics becomes more complex as children need to stay focused on a single step at a time and avoid irrelevant and distracting information or maladaptive procedures (Bull & Lee, 2014). Therefore, the contributions of inhibition to mathematics
could increase over time, in particular with the transition from
preschool to primary school, as this transition is a crucial time
window for the development of both inhibition and mathematics
learning (Allan et al., 2014). For this reason, we tested preschool
children in the final year before their transition to primary school
(5–6 years) and in their second year of primary school (7–8 years).
Nevertheless, potential changes to this trajectory could depend
on the type of mathematics skill (e.g., simple vs. multistep) and
the developmental trajectory of specific inhibition skills (e.g.,
whether they are fully developed early on or continue their devel-
opment). In view of the potential impact of the maturation of inhi-
bition, we aim to test different inhibition tasks (Stop-Signal Task,
Flanker task, and Bivalent Shape Task), potentially developing at
different rates, and to compare findings between children in pre-
school (5 years) and at the start of primary school (7 years).

Mathematics as multicomponential
construct

Similarly to inhibition, mathematics also has a multicomponen-
tial nature with many subskills such as arithmetic, geometry,
algebras, probability, etc., but most studies fail to differentiate
these different subskills (Gilmore, 2023), which may each be dif-
ferently linked to inhibition. Several studies have indeed shown
significant correlations between national mathematics scores
and measures of inhibition in preschool children transitioning to
formal school (Bull et al., 2008) as well as primary school children
at ages 6–7 years (Mazzocco & Kover, 2007) and at age 11 years (St
Clair-Thompson & Gathercole, 2006). However, few studies have
looked at specific mathematics subskills while a mediating role
has been found for subskills (e.g., numerical fact knowledge and
procedural arithmetic skills) on the relation between inhibition
and mathematical achievement (Gilmore, 2023). Therefore, to
better understand the specific contributions of inhibition, we will
investigate this in a single aspect of mathematics, arithmetic, as
foundational skill to more complex mathematics curriculum (Bull
& Lee, 2014).

Another important factor to take into account in mathematics
acquisition is the presence of different formats, a symbolic for-
mat (e.g., Arabic numerals, 1, 2, 3, ... ) and a nonsymbolic format
(e.g., groups of objects). Nonsymbolic numerical processing, such
as nonsymbolic arithmetic, relies on an approximate intuitive
sense of quantity, without relying on formal symbols (Dehaene,
2011; Halberda et al., 2008). On the other hand, around the age of
3 years, the learning of numerical symbols starts through educa-
tion, after which the association of quantities with symbols needs
to be learned providing a basic skill set for later mathematics
(Lipton & Spelke, 2006). While nonsymbolic and symbolic skills
seem ultimately linked, they tend to differ in the way they relate
to later mathematics (De Smedt et al., 2013) as well as the way
domain-general skills such as inhibition tend to relate to symbolic
and nonsymbolic skills (Guan et al., 2021). The particular contri-
bution of inhibition to different mathematical facets depends on
the assumed subprocesses that are involved.

Inhibition and symbolic mathematics

The literature suggests that inhibitory mechanisms play a role in
the selection of the appropriate symbolic arithmetic operation,
thus helping to prevent errors like performing multiplications
when the intended operation is an addition (Bull & Lee, 2014).

Moreover, some studies have revealed that when solving a sym-
bolic arithmetic problem such as a + b – b or a + b – c, inhibition
(as measured through Response Inhibition tasks such as the Go-
No-go task and the Stop-Signal Task) was a reliable predictor for
choosing the most suitable strategy (Eaves et al., 2022; Robinson
& Dubé, 2013). Therefore, in the case of symbolic arithmetic, inhi-
bition (in particular Response Inhibition) contributes by suppress-
ing inappropriate strategies (Bull & Lee, 2014).

Inhibition and nonsymbolic mathematics

Nonsymbolic magnitude tasks such as the commonly used mag-
nitude comparison task (i.e., which out of two dot arrays has
more dots) or a nonsymbolic arithmetic task (as the one tested
in our study; see Figure S1 in supplementary material; e.g., Knops
et al., 2014) have previously been suggested to reflect and meas-
ure more general cognitive abilities such as inhibition (Fuhs &
Mcneil, 2013; Gilmore et al., 2013). In nonsymbolic magnitude
processing tasks, such as the nonsymbolic arithmetic task, the
properties of perceptual cues (e.g., dot size, density, total area)
influence numerosity estimates (i.e., the visually larger dot array
is also the more numerous array; Clayton et al., 2015). Magnitude
processing tasks usually take this into account by implementing
congruent trials (e.g., continuous perceptual and numerical var-
iables match—the visually largest set is the most numerous set)
and incongruent trials (e.g., continuous perceptual and numerical
variables differ—the visually smallest set is the most numerous
set). Performance in terms of speed and accuracy tends to differ
with longer response times and higher error rates for incongruent
items than for congruent items, corresponding to a congruency
effect (Gilmore et al., 2013). This congruency effect in magnitude
processing tasks mimics inhibitory mechanisms similar to those
found in inhibition tasks (e.g., Stroop task or a Flanker task, Fuhs
& Mcneil, 2013; Gilmore et al., 2013). For example, in a Flanker
task, a participant would see a line of five fish, and has to indicate
which way the middle fish is swimming (i.e., left or right). In con-
gruent items all fish swim in the same direction (i.e., >>>>>), and
in incongruent items the middle fish swims in the opposite direc-
tion to its neighbors (i.e., >><<>). The participant must there-
fore inhibit irrelevant visual information about the surrounding
fish (Distractor Suppression). In a similar vein, incongruent trials
of a magnitude processing task are characterized by visual infor-
mation that needs to be ignored in order to make a correct numer-
ical judgment, suggesting a link between nonsymbolic processing
and Distractor Suppression (Gilmore et al., 2013). Nevertheless,
inconsistencies in the literature persist, as some studies have
shown no link between inhibition and total area controlled trials
in a nonsymbolic magnitude task (Keller & Libertus, 2015).

Owing to theoretically different roles for inhibition in sym-
bolic and nonsymbolic arithmetic, it is important to differentiate
between these two formats. Indeed, the role of inhibition in the
development of symbolic and nonsymbolic arithmetic between
the ages of 5 and 7 years remains unclear in the literature (Bellon
et al., 2016; Guan et al., 2021).

The current study

In this study, we investigate the role of inhibition in solving sym-
bolic and nonsymbolic arithmetic problems (addition and sub-
traction) during preschool and primary school, representing a
crucial period in mathematics learning. We aim at addressing
the main challenges in the current literature in three ways: (a) we
will compare all hypothesized models and relations between two cohorts (preschool and primary school) to test potential changes due to the development of inhibition and mathematics skills; (b) we will measure inhibition in three different tasks, including two Distractor Suppression tasks (Flanker task, Bivalent Shape Task) and one Response Inhibition task (Stop-Signal Task); and finally (c) we will focus on a single crucial mathematics subskill, arithmetic, taking into account the potential impact of symbolic and nonsymbolic formats.

We will first (1) probe the idea of an abstract construct of inhibition during early development in preschool (5–6-year-olds) and primary school (7–8-year-olds) children using Confirmatory Factors Analyses, contrasting a unitary factor structure with a two-factor structure or the absence of any structure. Then, (2) using Structural Equation Modelling (SEM), we will study the associations between the construct of inhibition (based on the best-fitting model in step 1) and symbolic and nonsymbolic arithmetic. Finally, (3) this study includes data from two cohorts (5–6-year-old preschoolers and 7–8-year-old primary school children), allowing us to examine the differences in the structure of inhibition and its respective relations to symbolic and nonsymbolic arithmetic between both age groups.

## Hypotheses

We hypothesize that (1) while all inhibition measures will correlate, the two types of inhibition (Response Inhibition and Distractor Suppression) will be somewhat separable. Therefore, we would expect a unitary factor structure to fit the data adequately, but a dual-factor model to fit the data better, compared to the unitary factor structure.

Second, (2) in the case where factor analyses confirm a unitary structure of inhibition, we assume that the combined concept of inhibition will relate to both symbolic and nonsymbolic arithmetic. In the case of a dual-factor structure, we expect Distractor Suppression to be more closely linked to nonsymbolic arithmetic (Gilmore et al., 2013) and Response Inhibition to symbolic arithmetic, following findings in the literature (Eaves et al., 2022; Robinson & Dubé, 2013).

Finally, (3) we hypothesize that inhibition will contribute more to arithmetic in older children as mathematics increasingly demands more complex processes, although there might be differences depending on the symbolic or nonsymbolic format as they related differently to inhibition and have different developmental trajectories. Moreover, we hypothesize the separation of Distractor Suppression and Response Inhibition to be more pronounced in older children.

## Method

### Participants

This study is part of a wider longitudinal research project investigating the contribution of domain-general functions to the development of numerical cognition (blinded for review purposes), and we were particularly interested in the role of inhibition in the development of symbolic and nonsymbolic arithmetic skills. We tested 331 predominantly White children in French preschool (Grande Section Maternelle [GSM]) and in the second year of French primary education (Cours Elémentaire 1 [CE1]) in

a cross-sectional study between January 2020 and March 2021: 174 children aged 5 years (M = 5.58, SD = 0.30; 74 males) and 157 aged 7 years (M = 7.48, SD = 0.39; 87 males). Parents were from high SES backgrounds, 61.65% had a university degree or higher, 19% had a secondary school degree, and 5.5% had a primary school degree, with 13.8% missing responses. We excluded 23 participants, owing to COVID-19 school closures (3) and because they did not succeed the Stop-Signal Task (SST) (20). As a result, in our final sample we have 156 children aged 5 years (M = 5.59, SD = 0.29; 68 males) and 152 aged 7 years (M = 7.48, SD = 0.39; 83 males). Children were recruited from 10 different school settings in Paris. Parents (or legal guardians) gave written consent for the children, and children orally agreed to participate in this study. The ethics committee (Comité D'Etique de la Recherche U-Paris) approved this study (IRB - 00012020-10).

## Material

### Inhibition measures

Within the present study we chose three inhibition tasks testing two forms of inhibition, the Stop-Signal Task measuring the Response Inhibition and two tasks measuring Distractor Suppression, the Flanker task and the Bivalent Shape Task (BST). These tasks were chosen to best fit with the debate in the literature on the potential unitary or binary structure of inhibition in the period of preschool and at the beginning of primary education.

#### Stop-Signal Task

We used the Stop-Signal Task, allowing us to measure the child's cognitive ability to stop motor responses (Verbruggen et al., 2013, 2019). The SST implemented with the E-Prime software (E-Prime, Psychology Software Tools) featured two training blocks followed by an experimental block. The first training block included 16 Go trials (i.e., press the button) to train participants in the response modalities. Participants were instructed to press the left arrow (key “L” of an Azerty keyboard) as quickly as possible if the arrow pointed left, and the right arrow (key “/” of an Azerty keyboard) if the arrow pointed right. A second training block of 16 trials (12 Go trials and 4 Stop trials (i.e., do not press the button)) allowed participants to practice and become familiar with the stop signals. If a tone (i.e., the Stop Signal, 750 Hz, 100 ms) sounded after a variable delay (i.e., the Stop Signal Delay [SSD]) following the presentation of the arrow, the participants were instructed to withhold their response. The experimental part encompassed 64 trials, with 16 Stop trials and 48 Go trials. Each block was systematically balanced with respect to stimulus orientation (50% of arrows pointing to the left and 50% pointing to the right). The task was adaptive in difficulty, with an initial SSD set at 350 ms and the duration extended by 50 ms following each successful Stop trial, heightening the challenge in subsequent trials, up to a maximum of 1,000 ms. Conversely, it was reduced by 50 ms after each unsuccessful Stop trial, making the subsequent trial easier, to a minimum of 0 ms (Logan & Cowan, 1984). Participants had 2,500 ms to respond. The measure used in the analyses was the Stop-Signal Reaction Time (SSRT) based on the recommendations from Verbruggen et al. (2013, 2019), who recommended to estimate the SSRT with a block-based integration method with replacement of go omissions as this produced the most reliable and least biased SSRT estimates. Following Congdon et al.
(2012), SSRTs were not calculated for subjects with a probability of responding on stop trials [p(respond/signal)] less than .25 or exceeding .75, for subjects with a probability of failing to respond for the go trials higher than .2, and for subjects with a mean RT on go trials longer than 2,000ms. Consequently, 20 participants were deleted.

### Flanker task

We adapted the child-friendly version of the Attention Network Test (ANT) from Rueda et al., 2004 measuring both spatial attention and inhibition (through a Flanker task). For the purpose of this study only the inhibition measure from the Flanker part was used. The Flanker task represents Distractor Suppression and has previously been suggested to be an inhibition task closely related to nonsymbolic processing (Wilkey, 2023). The task was designed with the E-Prime software (E-Prime, Psychology Software Tools), and composed of one training block (7 trials) followed by an experimental block (54 trials). The Flanker task consisted of a horizontal row of five fish corresponding to 18 congruent and 18 incongruent trials. The fish flanking the one in the middle were oriented in either the same (i.e., congruent trials) or the opposite (i.e., incongruent trials) direction as the central fish (i.e., target). Participants were told that the middle fish was hungry and were asked to feed the fish by indicating the direction of the central fish by pressing the “q” or “m” key (respectively showing a left- and right-swimming fish) on an Azerty keyboard as rapidly and accurately as possible. The target display was presented until a response was made or up to 2,500 ms. The final measure was produced from the congruency score for RT and the Error Rate (ER) (difference between incongruent and congruent trials). We combined both ER and RT in one final measure $P=RT(1+2 ER)$, following Lyons et al. (2014).

### Bivalent Shape Task

The second task used to measure Distractor Suppression was the Bivalent Shape Task (BST; Esposito et al., 2013), available in version 1.0 of the PEBL Test Battery (Mueller & Piper, 2014). The aim of this task was for the participant to determine whether the shape in the middle of the screen was a circle or a square. If the middle shape was a circle, participants had to press the answer button (i.e., left shift key on an Azerty keyboard) on the left side of the keyboard depicting a red circle, but if it was a square, participants had to press the answer button depicting a blue square on the right side of the keyboard (i.e., right shift key). There were visual cues below the central shape (i.e., on the left a red circle, on the right a blue square), to indicate which side to respond to according to the visual stimuli without children having to look at the response keys. The central target stimuli shape was either colored (red or blue) or an unfilled black outline (neutral trial). To correctly respond to each trial, the participant must take into account only the shape and not the color. In congruent trials, color and shape of the central stimuli match the color and shape of the right or left visual cue (i.e., a red circle or a blue square; 30 trials). In incongruent trials, color and shape of the central stimuli mismatch with that of the right or left visual cue (i.e., a blue circle or a red square; 30 trials). Children completed 90 trials, preceded by six practice trials. The final measure was produced from the congruency scores for the RT and the ER. We combined both measures, ER and RT in one final measure, $P=RT(1+2 ER)$, following Lyons et al. (2014).

## Arithmetic measures

We have focused on addition and subtraction problems as arithmetic measures fitting to the age ranges tested, which we have tested in symbolic and nonsymbolic formats.

### Nonsymbolic addition and subtraction tasks

Nonsymbolic addition and subtraction problems (Figure S1 in supplementary material) were presented in two separate blocks. Each trial started with a central fixation cross for 500 ms, followed by a black background and a grey box positioned on the left side of the screen. Subsequently, the first group of dots, presented in white, appeared at the left top of the screen above the box and dropped down until they disappeared in the box. In the addition task, a second group of dots appeared at the top of the screen and also dropped down to disappear into the box. In the subtraction task, a second group of dots, smaller than the first, dropped out of the box and disappeared at the bottom of the screen. The duration of the dot movements (from the appearance to the disappearance) was 1,000 ms. Next, in both tasks a third group of dots appeared to the right of the box, and children were asked to decide whether there were more dots in the box (click on the left response button situated underneath the box; “left ctrl key”) or next to the box (click on the right response button situated underneath the group of dots displayed next to the box; “right ctrl key”). Children had a maximum of 10 s to select the correct answer. A practice consisting of two trials preceded the testing phase, with no time limit for the response, and feedback was provided by the experimenter. Percentage of correct answers was used to calculate an accuracy score.

For each task, the response alternatives (RA) were varied as a function of their ratio with the correct result (CR). The division of CR by RA leads to 6 different approximate ratios (0.33, 0.5, 0.75, 1.25, 2.0, 3.0). Operands, CR, and RA are presented in Table S1 in supplementary material. The nonnumerical properties of the stimuli were adjusted so that the sum or difference of the total area occupied, density, convex hull or diameter of individual dots were either congruent (50% of the time) or incongruent (50% of the time) with the CR.

### Symbolic addition and subtraction task

To test symbolic arithmetic, we used the arithmetical operation subset from the TEDI Math Battery (Test for Diagnostic Assessment of Mathematical Disabilities; Van Nieuwenhoven et al., 2001). This battery provides an assessment of basic mathematical skills, with the arithmetical subtest including 12 visually presented symbolic addition problems and 10 subtraction problems. Only the first example of the additions and the first example of the subtractions were accompanied by a verbal presentation of the problem. Participants are asked to orally respond to these simple addition and subtraction problems. For the addition operations, 8 were composed of single digits and 4 of double digits. For the subtraction operations, 5 were composed of single digits and 5 of double digits. Cohort 1 (5–6-year-olds) completed only the initial 5 trials for both arithmetic operations to adapt the difficulty to their level following the guidelines of Van Nieuwenhoven et al. (2001). Subtraction problems were not recommended for the 5-year-old children; however, for consistency and because many studies highlight that preschool children begin to successfully perform subtraction problems (Baroody et al., 2009), they were presented to both cohorts. Note that this resulted in a floor effect, with 46
participants on the symbolic addition task and 68 participants on the subtraction task in the youngest cohort scoring 0 on the task. This is likely to have an impact on the reliability of the current findings concerning symbolic arithmetic in the 5-year-olds. The percentage of correct responses for addition problems and subtraction problems separately were used as arithmetic measures.

## Design and procedure

In this study, we divided the children's tasks into two 20-30 min sessions. In the first session, there were the following tasks: symbolic addition and subtraction tasks, nonsymbolic addition and Flanker; and in the second the following tasks: nonsymbolic subtraction, SST, and BST. We carried out the tasks at school, testing the children individually in a quiet room (maximum 5 children at the same time in the same room, playing the games with an experimenter). To motivate the children, we designed a treasure map with the different games and for each game, the child stuck on a sticker. At the end, they received a diploma for their participation in the research. The data and the analysis script are available on OSF and can be accessed at https://osf.io/m4bde/.

## Results

### Analysis plan

For the main analyses (confirmatory), we first conducted a Confirmatory Factor Analyses testing the simplest model including three factors (Inhibitory Control [IC] and symbolic [S] and nonsymbolic [NS] arithmetic). To establish whether the factors represent the same construct in both cohorts, measurement invariance was tested on this model. Next, we performed a comparison of model fit of the structure of IC as a single construct (IC defined by the three inhibition measures: SST, BST, and Flanker task) or a dual construct (Response Inhibition defined by the SST measure and Distractor Suppression defined by the BST and Flanker task measures) using multigroup (i.e., cohort) CFA. A three-factor multigroup CFA model (IC, S arithmetic, and NS arithmetic) was pitted against a four-factor multigroup CFA model (RI, DS, S arithmetic, and NS arithmetic). We also tested a third model including the S arithmetic factor, the NS arithmetic factor, and each IC measure as separate observed variable, based on the lack of significant correlations between the IC measures as found in preliminary correlational analyses. Good model fits were defined as CFI values larger than or equal to .90, TLI values larger than or equal to .95, and RMSEA values below .08. We compare model fits between models based on the Akaike information criterion (AIC), with lower AIC indicating a better fit. We also report the difference in CFI with a difference of -0.01 or more between models indicating a lower model fit (Cheung & Rensvold, 2002). Age was added as covariate in all CFA and SEM models, but findings remained the same when omitting this covariate. In a second step, based on the best-fitted CFA model, we investigated the relations of inhibition with symbolic and nonsymbolic arithmetic. For this, we used multigroup Structural Equation Modelling, using multiple unidirectional linear regressions to capture the intercorrelations between latent factors. The decision to use SEM was motivated by the need to confirm the latent structure of IC as a unitary or dual construct, to evaluate

multiple relations between latent factors at the same time from a multigroup perspective to test developmental differences, and to control for measurement error in a way that regressions cannot (Nunkoo & Ramkissoon, 2012).

Missing data represented 5.3% in total, which was judged as missing at random (i.e., did not participate or was interrupted during a task), apart from the missing data from the SST (which were deleted owing to not being missing at random). While a Little's test was significant for cohorts 1, $\chi^2$ (36) = 52.03, *p* = .041, and cohort 2, $\chi^2$ (38) = 62.82, *p* = .007, indicating that data was not missing completely at random, the reasons for missing data were explored and judged as missing at random. Indeed, missing data concerned data from tasks administered in the second testing session, which was most affected by unforeseen COVID-19 school closures (see Table S2 in supplementary material). The data missing at random were handled by using a full information maximum likelihood estimation for the CFA and SEM analyses, allowing for a better fit of the models compared to alternative methods (Köse, 2014). For CFA and SEM analyses, latent variables were standardized allowing free estimation of all factor loadings. Prior to testing data in CFAs and SEM, we reversed the negative scores of all inhibition measures (BST, Flanker task, and SST) to simplify interpretations in subsequent analyses. This makes all scores positive, meaning that the higher the score, the better the performance. Next, we applied an arcsine transformation to the arithmetic data (symbolic and nonsymbolic) to stabilize the variance of these percentage data.

To test the multivariate normality and the homogeneity of variance of the data, we conducted a Mardia multivariate normality test. Mardia Skewness test (516.45), *p* < .001, and Mardia Kurtosis test (9.73), *p* < .001, show that our data do not follow a symmetric and a normal distribution. Therefore, we will use a nonparametric test in our correlations (Spearman) and a Robust Maximum Likelihood (MLR) estimator in our CFA and SEM. The Robust Maximum Likelihood estimator has been shown as one of the better estimators to account for nonnormality and adjusts standard errors accordingly, allowing for valid statistical inference even in the presence of skewed or kurtotic data as is our case (Li, 2016).

Preliminary data analyses consisted of task validity checks and task reliability checks; see supplementary material. All tasks were judged to have valid indicators for inhibition and were reliable. Data were analyzed using Lavaan version 0.6-12 (Rosseel, 2012) in R version 4.0.2.

### Preliminary analyses

#### Descriptives

Descriptive statistics per cohort for age in months and every task measured are displayed in Table 1.

#### Correlational analyses

We used Spearman correlations to compute the correlations between all tasks per cohort; see Table 2. We observe that the three IC scores (SST, BST, and Flanker) do not correlate with each other at the age of 5 years (SST and BST, *r* (142) = .07, *p* = .43; SST and Flanker, *r* (137) = -.01, *p* = .93; BST and Flanker, *r* (138) = -.01, *p* = .88). However, at the age of 7 years, we observe a significant positive correlation between both DS measures, BST and Flanker, *r* (131) = .20, *p* < .05; but not for the other IC measures, *p*<sub>s</sub> > .62.
<table><caption>**Table 1** Descriptive statistics per task for each cohort.</caption><thead><tr><th>Tasks</th><th>Cohort</th><th>N</th><th>M</th><th>SD</th><th>Mdn</th><th>Min</th><th>Max</th><th>Range</th></tr></thead><tbody><tr><td rowspan="2">Age</td><td>C1</td><td>156</td><td>67.05</td><td>3.50</td><td>67.25</td><td>58</td><td>73</td><td>15</td></tr><tr><td>C2</td><td>152</td><td>89.76</td><td>4.66</td><td>89.00</td><td>77</td><td>108</td><td>31</td></tr><tr><td rowspan="2">BST</td><td>C1</td><td>143</td><td>347.66</td><td>129.47</td><td>365.35</td><td>0.00</td><td>922.26</td><td>922.26</td></tr><tr><td>C2</td><td>132</td><td>372.83</td><td>88.47</td><td>374.49</td><td>3.85</td><td>554.52</td><td>550.67</td></tr><tr><td rowspan="2">Flanker</td><td>C1</td><td>147</td><td>971.88</td><td>301.56</td><td>977.55</td><td>154.69</td><td>2,096.64</td><td>1,941.96</td></tr><tr><td>C2</td><td>147</td><td>922.53</td><td>183.56</td><td>946.14</td><td>0</td><td>1,649.02</td><td>1,649.02</td></tr><tr><td rowspan="2">SST</td><td>C1</td><td>142</td><td>0.72</td><td>0.22</td><td>0.76</td><td>0</td><td>1.10</td><td>1.10</td></tr><tr><td>C2</td><td>130</td><td>0.89</td><td>0.12</td><td>0.91</td><td>0.25</td><td>1.16</td><td>0.91</td></tr><tr><td rowspan="2">NS add</td><td>C1</td><td>156</td><td>0.74</td><td>0.11</td><td>0.77</td><td>0.28</td><td>0.96</td><td>0.67</td></tr><tr><td>C2</td><td>149</td><td>0.82</td><td>0.08</td><td>0.83</td><td>0.50</td><td>0.98</td><td>0.48</td></tr><tr><td rowspan="2">NS sub</td><td>C1</td><td>142</td><td>0.73</td><td>0.11</td><td>0.75</td><td>0.30</td><td>0.96</td><td>0.66</td></tr><tr><td>C2</td><td>132</td><td>0.80</td><td>0.10</td><td>0.82</td><td>0.48</td><td>0.96</td><td>0.48</td></tr><tr><td rowspan="2">S add</td><td>C1</td><td>156</td><td>0.44</td><td>0.39</td><td>0.40</td><td>0</td><td>1</td><td>1</td></tr><tr><td>C2</td><td>151</td><td>0.83</td><td>0.18</td><td>0.92</td><td>0</td><td>1</td><td>1</td></tr><tr><td rowspan="2">S sub</td><td>C1</td><td>156</td><td>0.19</td><td>0.26</td><td>0.17</td><td>0</td><td>1</td><td>1</td></tr><tr><td>C2</td><td>151</td><td>0.67</td><td>0.25</td><td>0.70</td><td>0</td><td>1</td><td>1</td></tr></tbody></table>

Note: add = addition, BST = Bivalent Shape Task, C1 = cohort 1 (5 years), C2 = cohort 2 (7 years), Max = maximum, Mdn = median, Min = minimum, NS = nonsymbolic, S = symbolic, SST = Stop-Signal Task, sub = subtraction.

<table><caption>**Table 2** Correlational matrix reporting Spearman coefficients between all measures per cohort.</caption><thead><tr><th></th><th>SST</th><th>BST</th><th>Flanker</th><th>NS add</th><th>NS sub</th><th>S add</th><th>S sub</th></tr></thead><tbody><tr><td><strong>C1</strong></td><td>(142)</td><td>(143)</td><td>(147)</td><td>(156)</td><td>(142)</td><td>(156)</td><td>(156)</td></tr><tr><td>SST</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>BST</td><td>0.07</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Flanker</td><td>-0.01</td><td>-0.01</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>NS add</td><td>0.18*</td><td>0.11</td><td>-0.08</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>NS sub</td><td>0.07</td><td>-0.03</td><td>-0.03</td><td>0.47***</td><td>—</td><td>—</td><td>—</td></tr><tr><td>S add</td><td>0.18*</td><td>-0.04</td><td>-0.01</td><td>0.30***</td><td>0.35***</td><td>—</td><td>—</td></tr><tr><td>S sub</td><td>0.07</td><td>-0.18*</td><td>-0.02</td><td>0.18***</td><td>0.37***</td><td>0.64***</td><td>—</td></tr><tr><td><strong>C2</strong></td><td>(130)</td><td>(132)</td><td>(147)</td><td>(149)</td><td>(132)</td><td>(151)</td><td>(151)</td></tr><tr><td>SST</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>BST</td><td>-0.04</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Flanker</td><td>0.01</td><td>0.20*</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>NS add</td><td>0.09</td><td>0.13</td><td>0.26***</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>NS sub</td><td>0.02</td><td>0.01</td><td>0.27***</td><td>0.45***</td><td>—</td><td>—</td><td>—</td></tr><tr><td>S add</td><td>0.26**</td><td>0.05</td><td>0.18*</td><td>0.27***</td><td>0.24**</td><td>—</td><td>—</td></tr><tr><td>S sub</td><td>0.22**</td><td>-0.02</td><td>0.10</td><td>0.25***</td><td>0.30***</td><td>0.64***</td><td>—</td></tr></tbody></table>

Note: Significance levels: * < .05; ** < .01; *** < .001. Value in parentheses corresponds to N. add = addition, BST = Bivalent Shape Task, C1 = cohort 1, C2 = cohort 2, NS = nonsymbolic, S = symbolic, SST = Stop-Signal Task, sub = subtraction.

All symbolic and nonsymbolic arithmetic measures are positively correlated with each other, $p_s < .02$.

We found significant positive correlations between the SST and S and NS addition at the age of 5 years (SST and NS add, $r (142) = .18, p < .05$; SST and S add, $r (142) = .18, p < .05$) and a significant negative correlation between BST and S sub, $r (143) = -.18, p < .05$. For all other links between IC measures and arithmetic, there were no significant correlations at age 5, $p_s > .20$.

At the age of 7 years, we observed significant positive correlations between the SST and S arithmetic (SST and S add, $r (129) = .26, p < .01$; SST and S sub, $r (129) = .22, p < .01$); between the Flanker task and NS arithmetic (SST and NS add, $r (146) = .26, p < .001$; SST and NS sub, $r (131) = .27, p < .001$); and between the Flanker task and S addition, $r (146) = .18, p < .05$. The other correlations between IC measures and arithmetic were not significant, $p_s > .14$.
# Main analyses

## Measurement invariance

Before comparing the multigroup CFA models, we tested for measurement invariance to establish factorial validity across both cohorts. Measurement invariance was tested on the 3-factor model including a single IC factor, as this is the simplest and most complete model (excluding potential single-indicator factors). Configural invariance was assessed through model fit, which indicated a good fit ($\chi^2(30) = 38.403, p = .140$, Robust CFI = .978, Robust TLI = .959, Robust RMSEA = .039, AIC = 6,401.8). Subsequently, metric invariance was assessed using the chi-squared difference test to compare between the metric and the configural invariance models. No significant difference was observed between the configural and metric fit ($\chi^2\text{diff}(4) = 2.8533, p = .583$); however, AIC indicated that the metric fit improved the model. The metric invariance model fit was good ($\chi^2(34) = 39.808, p = .227$, Robust CFI = .983, Robust TLI = .972, Robust RMSEA = .032, AIC = 6,398.1). Metric invariance was met on the basis of the goodness of fit and the AIC.

This means that the tasks' measures and their underlying constructs are the same in both cohorts.

## Confirmatory factor analyses

First, we tested the model fit of three multigroup (cohorts) models: model 1 (three factors including IC [SST, BST, Flanker], S arithmetic [S add, S sub], and NS arithmetic [NS add, NS sub]), model 2 (four factors including RI [SST], DS [BST, Flanker], S arithmetic, and NS arithmetic), and model 3 (including all three IC measures separately as single-indicator factors and the factors S arithmetic and NS arithmetic). These models are displayed in Figure 1. We found, on the basis of the AIC, that the best model was model 1 and that model 3 fit the data least. However, the fit indices per separate model contradict this finding (model 1: AIC = 6,401.761, $\chi^2(30) = 38.403, p = .140$, Robust CFI = .978, Robust TLI = .959, and Robust RMSEA = .039; model 2: AIC = 6,402.429, $\chi^2(24) = 23.571, p = .486$, Robust CFI = 1.000, Robust TLI = 1.003, and Robust RMSEA = .000; model 3: AIC = 6,412.101, $\chi^2(18) = 20.075, p = .329$, Robust CFI = .994, Robust TLI = .981, and Robust RMSEA = .026), yet without balancing model complexity and model fit. With regard to our fit indices, the CFI difference between model 2 and 3 indicated no difference ($\Delta\text{CFI} < -.01$), while model 1 had a lower fit compared to model 3, exceeding $-.01$ ($\Delta\text{CFI} = .022$). In model 2 we found a factor loading as well as the TLI exceeding 1, which indicates a potential problem with the model. This high factor loading potentially resulted from low correlations between the observed variables loading onto the DS factor and the use of a single-variable factor for RI. Consequently, this model was not a suitable fit for our data and will not be considered for subsequent SEM. We further compared model fits between all three models. Model 1 fit the data significantly better than model 2, $\chi^2\text{diff}(6) = 17.68, p < .01$, but not model 3, $\chi^2\text{diff}(12) = 19.28, p = .082$. Models 2 and 3 were not significantly different, $\chi^2\text{diff}(6) = 3.00, p = .809$. Given that the single inhibitory construct (model 1) best fit the data based on the AIC, this model was used in subsequent Structural Equation Modelling. Note that model 3 included 3 single-indicator factors, essentially functioning like observed variables rather than traditional latent factors. This deviates from the typical latent factor structure in CFA and is not ideal as it can lead to artifacts and therefore lower reliability of the results (Fornell

& Larcker, 1981). Nevertheless, as none of our IC measures correlate, as the factor loadings of the IC measures were low (ranging from -.08 to .34; see Figure 1), and as model 3 had the best fit indices based on the CFI, we decided to explore the SEM including each IC measure as observed variable predicting S and NS arithmetic (exploratory analysis).

On the basis of these results, we can conclude that IC measures do not represent a unitary or dual structure. Indeed, factor loadings that indicate how strongly they represent the underlying construct are very low and do not fit together. Moreover, a model that allows for separated IC measures shows a better model fit.

Tables S3, S4, and S5 in supplementary material present standardized factor loadings of models 1, 2, and 3, respectively, for each variable, as well as the covariance between latent variables.

## Structural equation modelling

Two SEM models were created, SEM model 1 included three factors (IC, S arithmetic, and NS arithmetic), with regressions from IC to S and NS arithmetic. SEM model 2 consisted of two factors (S arithmetic and NS arithmetic) with regressions from each IC measure (SST, BST, Flanker) to S and NS arithmetic. The AIC values for both models were identical.

### SEM model 1

The SEM model 1, displayed in Figure 2, fit the data well, AIC = 6,401.761, $\chi^2(30) = 38.404, p = .140$, Robust CFI = .978, Robust TLI = .959, and Robust RMSEA = .039. As reflected in the CFA model 1, the inhibition indicators did not load well on the same IC factor. The IC factor did not relate to S arithmetic (cohort 1, $\beta = .53, p = .303$; cohort 2, $\beta = .70, p = .732$), or NS arithmetic (cohort 1, $\beta = .26, p = .564$; cohort 2, $\beta = -.30, p = .714$). The only significant covariance in the model was between S and NS arithmetic in cohort 1 (0.50, $p < .05$), but not in cohort 2 (0.94, $p = .711$). There were no significant relations between the factors and the covariate age (see Table S6 in supplementary material).

This means that no relation was found between the unitary construct of inhibition and symbolic or nonsymbolic arithmetic.

### SEM model 2

SEM model 2, displayed in Figure 3, fit the data well, AIC = 6,401.514, $\chi^2(24) = 21.073, p = .634$, Robust CFI = 1.000, Robust TLI = 1.020, and Robust RMSEA = .000, although TLI values exceeded 1, likely reflecting the use of single-indicator factors. In both cohorts, S arithmetic was significantly predicted by the SST abilities (cohort 1, $\beta = 0.18, p < .05$; cohort 2, $\beta = .27, p < .01$), thus indicating that better SST performance is related to better S arithmetic scores. In cohort 2, but not in cohort 1 ($\beta = -.07, p = .437$), NS arithmetic was significantly predicted by the Flanker task, $\beta = .31, p < .01$, thus indicating that a higher Flanker task score is related to higher NS arithmetic performance. The covariance between NS arithmetic and S arithmetic was significant for both cohorts (cohort 1, .54, $p < .001$; Cohort 2, .49, $p < .001$). In addition, the covariances in cohort 2, between age and SST, .28, $p < .001$, as well as age and S arithmetic, .19, $p < .05$, were also significant, indicating that performance on the SST and S arithmetic increases with age in children in cohort 2 (see Table S7 in supplementary material).

This means that the Stop-Signal Task was related to symbolic arithmetic in both cohorts and that the Flanker task was related to nonsymbolic arithmetic, but only in cohort 2 (7-year-olds), the
<figure><img src="image_3.png" /><figcaption>**Figure 1** Diagram of the three different multigroup CFA models. With model 1 (A), model 2 (B), and model 3 (C), including the factor loading for each variable for cohort 1 and cohort 2. Note: Bold lines = significant correlation. SST = Bivalent Shape Task; C1 = cohort 1; C2 = cohort 2; SST = Stop-Signal Task. Significance levels: \*p\* < .05, \*\*p\* < .01, \*\*\*p\* < .001.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 2** SEM model 1 with standardized regression estimates (beta value) for both cohorts. Note: Bold lines = significant link. C1 = cohort 1; C2 = cohort 2. Significance levels: \*p\* < .05, \*\*p\* < .01, \*\*\*p\* < .001.</figcaption></figure>

older children. The Bivalent Shape Task was not related to symbolic or nonsymbolic arithmetic.

## Discussion

The aim for this study was to investigate the structure of inhibition and its role in solving symbolic and nonsymbolic arithmetic problems during preschool (5-year-olds) and early primary school (7-year-olds).

## Inhibition as a construct

We hypothesized that all inhibition measures would correlate, but that a dual structure representing Distractor Suppression and Response Inhibition would fit the data better than a unitary
<figure><img src="image_5.png" /><figcaption>**Figure 3** SEM model 2 with standardized regression estimates (beta value) for both cohorts. Note: Bold lines = significant correlation. BST = Bivalent Shape Task; C1 = cohort 1; C2 = cohort 2; SST = Stop-Signal Task. Significance levels: \*p < .05, \*\*p < .01, \*\*\*p < .001.</figcaption></figure>

Inhibitory Control construct. We further expected the separation of Distractor Suppression and Response Inhibition to be more pronounced in older children.

Contradictory to what we expected, while the Confirmatory Factor Analyses revealed a good fit with a single inhibitory construct (i.e., Stop-Signal Task, Bivalent Shape Task, and a Flanker task taken together) in both 5- and 7-year-old children when including symbolic and nonsymbolic arithmetic factors in the same model, we did not find a single or dual inhibitory construct. Instead, the best fitting model separated the three inhibition measures. It should be noted that the adequate fit of all models was largely due to the strong factor loadings for the symbolic and nonsymbolic factor and that models separating the inhibition factor included single-indicator factors (i.e., Response Inhibition factor defined only by SST; Distractor Suppression defined by BST and Flanker task) potentially contributing to lower fit indices (Aguirre-Ureta & Hu, 2019). Indeed, opposing our hypothesis, none of the inhibition measures correlated with each other in both cohorts, with the exception of a small positive correlation between both Distractor Suppression tasks (Flanker task and BST) in the 7-year-olds. It is still possible that a clearer separation between Distractor Suppression and Response Inhibition occurs later in development, reflecting the increasing refinement with age. Indeed, the emergence of a small correlation between both Distractor Suppression tasks in the older cohort potentially suggests that the onset of the formation of two inhibition factors is more likely to occur around or after the age of 7 years (Wiebe et al., 2008). Furthermore, Iglesias-Sarmiento et al. (2023) recently found that inhibition task scores (RI and DS tasks) correlated more and more with each other over the course of development between the ages of 7, 11, and 15 years. From a developmental point of view, the brain undergoes progressive increases in activation and interregional connectivity within inhibition networks resulting in fewer or lower correlations between tasks in early age (Rubia, 2013). However, the more the brain develops, regions that support similar cognitive processes become more connected, which might result in stronger correlations between tasks representing similar cognitive processes (Rubia, 2013).

Nevertheless, the nearly entire absence of correlations between all inhibition tasks (SST, Flanker task, and BST) raises the question of the very existence of an inhibition construct in 5- and 7-year-olds. The psychometric concept of inhibition has previously been

called into question in adults, where low to no correlations and factor loadings were found between inhibition measures (Rey-Mermet et al., 2018; Shilling et al., 2002), highlighting that the degree of correlations between tasks depends on the similarities between the tasks and the indices used within a task (e.g., congruency effect). Indeed, Rey-Mermet et al. (2018) found low correlations between 11 different inhibition measures regardless of whether they were testing Response Inhibition or Distractor Suppression. Moreover, Shilling et al. (2002) even found very low correlations between different variants of one particular inhibition task, the Stroop task. Rey-Mermet et al. (2018) suggested that inhibition tasks do not measure a common underlying construct, which is supported by our findings. Therefore, research using specific inhibition measures should clarify that findings only hold true for one specific paradigm and not for the construct of inhibition. In the present study we aimed to address this challenge by testing the link between a factor of inhibition with arithmetic, but instead found that such a factor does not reliably link different inhibition measures (SST, Flanker task, BST).

On a theoretical level, we should question what makes different inhibition tasks more or less similar to each other and whether these similarities go beyond the pure inhibitory nature of them (e.g., different types of inhibition). One potential account relates to the fact that inhibition tasks suffer impurities (Friedman & Miyake, 2004) and different inhibition tasks are likely to reflect distinct sources of impurity (i.e., different underlying cognitive sub-processes). A second account, although not necessarily mutually exclusive, relates to the suggestion that different forms of inhibition are required depending on the specific demands and content of the tasks (Wilkey, 2023).

In line with the first account, on a factorial level, inhibition has previously not been found as separate from other executive function measures (Friedman et al., 2008), suggesting that other cognitive processes such as other executive functions might contribute to inhibition tasks (Rey-Mermet et al., 2018). The low correlations between our inhibition tasks potentially reflect differences in additional demands or strategies used that are specific to each inhibition task. For example, the difference between the Bivalent Shape Task and the Flanker task could in part reflect the additional attentional demand of the Flanker task as it was embedded in an Attentional Network Task. Moreover, while in the Stop-Signal Task the participants are instructed to respond as quickly as possible on go-trials, this is not an advantageous strategy to respond correctly to the stop-trials, where participants are often inclined to wait in anticipation for the stop-trial (Verbruggen et al., 2013). Furthermore, task rules are required to be maintained in working memory while participating in an inhibition task. While the Bivalent Shape Task attempts to lower working memory demands by displaying visual cues to indicate which side the participants have to press (Esposito et al., 2013), the Stop-Signal Task and the Flanker task do not repeat instructions. Instead, participants are required to remember to not press the button when hearing a tone in the Stop-Signal Task and to only look at the middle fish in the Flanker task. These findings support the account that different inhibition tasks reflect distinct sources or levels of impurity.

In line with the second account, Starling-Alves et al. (2024) put forward low correlations between two inhibition tasks with different contents (numerical and nonnumerical), but with similar structures and performed by the same participants. Similar low correlations were found between different variants of the Stroop task (Shilling et al., 2002). Wilkey (2023) suggests that the
inhibitory process used could be dependent on the demands and contents of a given task. The author argues that an inhibitory process cannot exist without a specific context to inhibit, which would explain why, for example, inhibition measures with a non-numerical content are less correlated to mathematics compared to inhibition measures with a numerical content (Starling-Alves et al., 2024). In our study, we used three diverse tasks with potential different demands and contents. While the Flanker task has irrelevant flanking information to the target stimuli (i.e., external to the target stimuli), the Bivalent Shape Task has irrelevant information within the target stimuli (i.e., internal). On the other hand, the Stop-Signal Task required participants to inhibit an auditory tone rather than a visual cue.

While our findings are not well suited to identify which sources of impurity and which specific demands differ between the tasks, they do support that both accounts are likely causes of the lack of an underlying inhibitory construct.

## Inhibition and its relations to symbolic and nonsymbolic arithmetic

In terms of the link between inhibition and arithmetic, we hypothesized that the construct of inhibition would relate to both symbolic and nonsymbolic arithmetic, with an increasing strength in these relations with age, but that differences would emerge for different inhibition types and arithmetic formats. Indeed, we expected that while Response Inhibition would relate more to symbolic arithmetic (Eaves et al., 2022; Robinson & Dubé, 2013), Distractor Suppression would relate more to nonsymbolic arithmetic (Gilmore et al., 2013). The absence of a reliable inhibition construct has led us to also explore the relation between each inhibition measure and arithmetic, as differences are likely to arise from the use of different paradigms.

While the model with the unitary inhibition factor showed no associations between inhibition and symbolic and nonsymbolic arithmetic, the model with the separate inhibition measures shows links with symbolic and nonsymbolic arithmetic, similar to those observed in the literature. Indeed, at 5 and 7 years, we observe a unique link between the Stop-Signal Task and symbolic arithmetic, supporting previous findings (Eaves et al., 2022; Robinson & Dubé, 2013). At 5 years, there was no link between the various inhibition measures and nonsymbolic arithmetic. However, at 7 years, an association emerged between the Flanker task and nonsymbolic arithmetic. Again, this type of link has previously been described in the literature, suggesting that the Flanker task might reflect the Distractor Suppression skills required to inhibit nonnumerical features in nonsymbolic processing in order to make numerical judgments (Gilmore et al., 2013).

In terms of symbolic arithmetic, Response Inhibition is potentially recruited to ignore responses automatically activated (e.g., 7-3≠10; Bull & Lee, 2014), an error frequently observed in the symbolic subtraction performance in our sample. Indeed, Response Inhibition has previously been related to the need for children to inhibit existing and erroneous knowledge when new similar information needs to be learned—symbolic subtractions vs. additions (e.g., McNeil et al., 2017). Another role that Response Inhibition is likely to play in symbolic arithmetic is to inhibit less efficient strategies such as counting on fingers (Siegler & Jenkins, 1989), which we observed as being discouraged by teachers in the sample tested, but still frequently used especially by preschool

children. An additional observation is that although the Stop-Signal Task is related to symbolic arithmetic in both cohorts, the strength of the association does seem to increase with age as we expected, thus suggesting that inhibition contributes more to symbolic arithmetic in older children (Bull & Lee, 2014; Gilmore et al., 2015; Lemaire & Lecacheur, 2011). While the youngest cohort only received single-digit calculations appropriate for their age, multidigit calculations have been shown to rely more on inhibition (Hodzik & Lemaire, 2011) as was supported by the increasing relation between inhibition and symbolic arithmetic in the 7-year-olds who also received multidigit calculations.

In terms of nonsymbolic arithmetic, at the age of 7 years an association emerged between the Flanker task and nonsymbolic arithmetic. In the Flanker task, Distractor Suppression is required in incongruent trials where the irrelevant flankers' orientation is incongruent with the target's orientation. In the context of our nonsymbolic arithmetic task, the same Distractor Suppression seems to be required in order to inhibit the irrelevant nonnumerical features of dots arrays (i.e., dot size, density, total area; Fuhs & Mcneil, 2013; Gilmore et al., 2013). However, no link was found between inhibition and nonsymbolic arithmetic in 5-year-olds. Potential explanations lie in the developmental trajectories of inhibition and nonsymbolic arithmetic and in particular the extent to which inhibitory control limits performance at different ages. While inhibition is likely required to solve nonsymbolic arithmetic tasks across development, performance in 5-year-olds may be influenced by a wider range of factors that reduce the observable role of inhibition. Children at the age of 5 years, in contrast to the age of 7 years, might not fully ignore the irrelevant nonnumerical information in order to obtain a numerical judgment, in line with literature highlighting that young children might not fully comprehend the question “which has more dots” as a pure numerical one (Krajcsi et al., 2024). For instance, 5-year-olds may not yet consistently interpret the task as requiring comparison of numerical quantity, as reflected by their lower performance. Instead, they might rely on perceptual cues such as dot size, total area, or spacing, especially when these cues conflict with number. As a result, even children with strong inhibition skills, as reflected in Flanker performance, may not show better nonsymbolic arithmetic outcomes, simply because they are not yet applying those skills in this context. A 7-year-old, on the other hand, may know they should attend to number but differ in how successfully they can ignore conflicting visual cues, allowing individual differences in inhibition to play a clearer role in performance.

It should be noted that 5-year-olds outperform 7-year-olds on the Flanker task, although not significantly, while this is reversed in the nonsymbolic arithmetic task. It is possible that 5-year-olds employed strategies better adapted to the specific demands of the Flanker task, even though they do not use these inhibitory strategies in the nonsymbolic arithmetic task. Due to rapid developmental changes in inhibition around the ages of 5 to 7 years, 5-year-olds could potentially outperform 7-year-olds in periods of rapid evolution of inhibition skills. Moreover, in line with inhibition task impurity, differences in underlying cognitive subprocesses between inhibition tasks have also been shown to differ depending on age, potentially providing a benefit for younger children. Chatham et al (2009) showed that 3-year-olds engage inhibition reactively (i.e., in response to the environment), while 8-year-olds apply proactive recruitment of inhibition (i.e., planned and in anticipation of its future utility). This provided a potential benefit for younger children to respond more quickly
in a reactive way, while older children take longer to anticipate and proactively prepare for a predictable future. In the case of a Flanker task, anticipating the future is not possible and thus not beneficial, and rapid reactive responses are required. This reflects the use of a qualitatively different form of inhibition depending on age (Chatham et al., 2009), potentially impacting how strongly the Flanker measure relates to arithmetic in different ages.

Contradictory to what we expected, no link was observed between the Bivalent Shape Task and either nonsymbolic or symbolic arithmetic in both cohorts, despite the task showing good split-half reliability and a valid congruency effect. Despite the Flanker task, the Bivalent Shape Task, and the nonsymbolic arithmetic task all having to rely on inhibiting distracting information, the Bivalent Shape Task could potentially rely more on visual and attentional processes (Mueller & Esposito, 2014). Alternative accounts include differences in complexity (Best & Miller, 2010) or sensitivity (Lee & Lee, 2019) of the Bivalent Shape Task compared to both the Flanker task and the nonsymbolic arithmetic task, which could be more dissimilar in the Bivalent Shape Task. Nevertheless, this nonsignificant result does support previous findings by Keller and Libertus (2015), who did not find a relation in 5-6-year-old children between a nonsymbolic processing task (i.e., total area controlled trials of a magnitude comparison task) and an inhibition subtest of the NEPSY (they say “circle” when a square is displayed or they say “up” for each down arrow), partly reflecting a verbal version of the Bivalent Shape Task. While the absence of a distractor in the form of the shape’s color results in these tasks being different to each other, both tasks do have similar stimuli (i.e., squares and circles) and have both been tested in a similar age range. In correspondence with our hypothesis, the Response Inhibition task used by Keller and Libertus (2015) in their second experiment was also not significantly related to total-area controlled trials in nonsymbolic magnitude processing, highlighting that the contributions of inhibition in nonsymbolic processes are mainly through Distractor Suppression when assessed with a Flanker task.

Nevertheless, taking into account the low correlations between the inhibition measures, likely caused by task impurity or different task demands and contents (Wilkey, 2023), our findings confirm that the link between inhibition and arithmetic differs according to the task content or underlying processes (i.e., both of the inhibition task and the mathematical task) and the age tested.

## Limitations

Some limitations come into play in the design of this study that should be taken into account. First, our study design was not optimized to test a separate Response Inhibition latent factor because we only included a single Response Inhibition measure, the Stop-Signal Task. The use of SEM was required to test the factor structure of inhibition and to evaluate multiple relations simultaneously; nevertheless, pitting a model with a single inhibition factor against a model where this factor is split into three single-indicator factors, essentially representing observed variables, is generally not advisable owing to issues with model parsimony resulting in a less reliable model. This can explain why, while nearly none of the inhibition measures are correlated with each other or load well on one factor, the model with the single inhibition factor still had the lowest AIC. Therefore, we advise to be cautious of this finding and urge future research to take into account

that there is likely no single inhibition construct, but rather various forms of inhibition as represented by specific paradigms.

Second, in our study we investigate symbolic and nonsymbolic arithmetic skills in the form of additions and subtractions, which are skills that have not yet been acquired fluently in our age ranges, specifically in the 5-year-olds (Evans et al., 2016). Therefore, many children, especially the 5-year-olds, show a floor effect on symbolic arithmetic, in particular subtractions. For reasons of comparability, we decided to keep the same statistical model for both cohorts, hence not opting for potentially more appropriate models that could take floor effects into account. This is likely to influence the reliability of the findings, in particular in 5-year-olds concerning the contributions of inhibition to symbolic arithmetic, and result should be interpreted with this in mind. Nevertheless, as SEM models measurement error and we used the Robust Maximum Likelihood estimator, we believe that our analytical approach remains valid. Indeed, the Robust Maximum Likelihood estimator has been shown as one of the better estimators to account for non-normality and allows for valid statistical inference even in the presence of skewed and kurtotic data (Li, 2016).

Finally, there are methodological differences between our symbolic and nonsymbolic arithmetic tasks. For example, while the nonsymbolic arithmetic task is a forced-choice task, the symbolic one requires a production of the response. Future research should replicate the current findings when the task designs of both the symbolic and the nonsymbolic tasks are more similar in order to more conclusively show that the findings are due to the format (symbolic vs. nonsymbolic) and not due to other task differences.

## Conclusion

In sum, the findings of this study challenge the existence of a unified inhibition construct in both 5- and 7-year-olds, as none of the inhibition measures loaded well onto a single inhibition factor or correlated with each other, except for a small positive correlation observed between both Distractor Suppression tasks in the older cohort. The absence of intercorrelations raises questions about the applicability of a broad inhibition construct in young children, echoing debates in adult research (Rey-Mermet et al., 2018), suggesting that findings from studies using specific inhibition paradigms cannot be generalized to other inhibition paradigms. Nevertheless, some hypothesized relations between inhibition and arithmetic have been confirmed, demonstrating a role for the Stop-Signal Task in symbolic arithmetic at ages 5 and 7 years. Furthermore, the Flanker task was found to associate to nonsymbolic arithmetic at the age of 7 years, but not at 5 years. On the basis of these results, we can conclude that (a) the role of inhibition varies depending on the mathematical format tested, (b) the contributions of inhibition change depending on the ages tested, and (c) the contributions of inhibition are highly depending on the inhibition measure used as they do not measure one and the same underlying inhibition construct in preschool and early primary school. This study has refined our cognitive understanding of the (absence of a) psychometric construct of inhibition and the contributions of inhibition measures to the development of arithmetic.

## Future directions

More work is required to replicate and expand task-specific contributions of different inhibitory processes to mathematical
components to investigate which inhibitory process, if any, contributes to which mathematical component and how. Indeed, while computational modelling (i.e., factor analysis) allows us to create an inhibitory construct taking into account measurement error and thus mitigating issues arising from task differences, the lack of a unified inhibitory construct argues against the use of such techniques. However, as different inhibition tasks are not measuring the same thing, single inhibition measures are likely not a valid representation of the theoretical construct of inhibition. Instead, inhibition could be seen as an umbrella term for a variety of different inhibitory processes depending on task demand and maturation. Fine-grained experimental designs could shed more light on the exact processes underlying specific inhibition tasks, their similarities or dissimilarities, and their contributions to mathematics.

## Supplementary material

Supplementary material is available at *Child Development* online.

### Data availability

The data and the analysis script supporting the findings are available on OSF and can be accessed at https://osf.io/m4bde/.

### Author contributions

Sixtine Omont-Lescieux (Data curation, Formal analysis, Investigation, Project administration, Validation, Visualization, Writing—original draft, Writing—review and editing), André Knops (Conceptualization, Funding acquisition, Methodology, Project administration, Supervision, Writing—review and editing), and Ilse E.J.I. Coolen (Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Project administration, Supervision, Validation, Writing—original draft, Writing—review and editing)

### Funding

This work is supported by a collaborative (PRC) grant of the French National Research Agency (ANR-18-CE28-0003) to Dr. André Knops in support of Dr. Ilse Coolen and Dr. Sixtine Omont-Lescieux. Dr. Ilse Coolen is funded by the European Union from 2024 (HORIZON EUROPE Marie Sklodowska-Curie Actions, project SPATHS, grant number 101150769, under the HORIZON-MSCA-2023-PF-01-01 call). Views and opinions expressed are, however, those of the authors only and do not necessarily reflect those of the European Union or the European Research Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

### Conflicts of interest

None.

### Acknowledgments

The authors would like to thank all involved children, parents, and teachers and all the interns who helped with data collection, in particular Melissa Ozkaynar. Special acknowledgments are extended to research engineers Lorna Le Stanc and Julie Vidal for

providing assistance with the Stop-Signal Task extraction (Lorna) and creation (Julie).

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