<figure><img src="image_1.png" /></figure>

HAL
open science

# Causal Effects of an Ecologically Valid Home Numeracy Intervention on Preschoolers' Numeracy Skills

Cléa Girard, Angie de Lamper, Stien Callens, Davina van den Broek, Bert de Smedt

<figure><img src="image_2.png" /></figure>

## To cite this version:

Cléa Girard, Angie de Lamper, Stien Callens, Davina van den Broek, Bert de Smedt. Causal Effects of an Ecologically Valid Home Numeracy Intervention on Preschoolers' Numeracy Skills. Child Development, 2025, 10.1111/cdev.70010. hal-05355990

HAL Id: hal-05355990

https://hal.science/hal-05355990v1

Submitted on 9 Nov 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_3.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Causal effects of an ecologically valid home numeracy intervention on preschoolers'

numeracy skills

Cléa Girard¹,², Angie De Lamper², Stien Callens², Davina Van den Broek² & Bert De Smedt²

¹Laboratoire de Psychologie et Neurocognition (LPNC), CNRS UMR 5105, Grenoble Alpes
University, Grenoble, France

²Parenting and Special Education Research Unit, Faculty of Psychology and Educational
Sciences, KU Leuven, Leuven, Belgium

**Corresponding Authors:** C. Girard (clea.girard@cnrs.fr) or Bert De Smedt (bert.desmedt@kuleuven.be)

**Funding:** This work was supported by Fondation Fyssen, post-doctoral study grant to C.Girard

**Keywords:** home numeracy intervention; numeracy skills; preschool
**Abstract**

The home numeracy environment is suggested to influence children's numerical development, but causal evidence for this assertion remains limited. Addressing this gap, we randomly assigned 117 predominantly White 4- to 5-year-olds (M = 4.68 years, SD = 0.2, 47% girls) attending preschool in Flanders (Belgium) to either an experimental (numeracy) or an active control (language) condition. The 6-week intervention (pretest-March 2023; posttest-May 2023) consisted of an ecologically valid implementation, with parents integrating flexible activities into their routines. Pre- and post-intervention assessments measured children's numerical skills. Medium effects were observed on transcoding and ordering skills, providing causal evidence for the impact of the home numeracy environment on children's numeracy. This study highlights the potential of ecologically valid interventions to support early numeracy in daily life.

**Running head:** Home numeracy intervention on preschoolers
The Home Numeracy Environment (HNE) has been suggested to play an important role
in children’s early numeracy, as is evidenced by the correlation between the frequency of home
numeracy activities that parents provide and their children’s numeracy outcomes (Daucourt et
al., 2021, for a meta-analysis). However, a major limitation to the existing research is the lack
of evidence for the causal nature of this association. This causal nature can only be addressed
through rigorously controlled intervention studies. Although recent intervention studies aimed
at increasing home numeracy activities have shown some promising results (Libertus, 2024),
the findings across studies remain inconsistent. Some studies were limited by methodological
weaknesses, such as small sample sizes or inadequate active control conditions. Others relied
on highly structured designs involving close parental supervision, intensive training, or
repetitive tasks. While these approaches may improve experimental control (i.e. internal
validity), they may compromise ecological validity by limiting their relevance to families’
everyday routines (Ledford et al., 2016).

Ecological validity refers to the extent to which an intervention reflects real-world
conditions and aligns with the activities that families already carry out in their daily lives
(Chezan et al., 2022; Ledford et al., 2016). It is assumed that this alignment facilitates the
implementation of the intervention. A lack of ecological validity may hinder the generalizability
as well as the theoretical and practical relevance of findings. Recent studies have begun to
address this lack of ecological validity in HNE research. For instance, Leyva et al. (2023)
implemented a six-week program tailored to families’ routines, using video examples to
promote math-related interaction in natural contexts. While their intervention increased parental
math talk and home math activities, it did not lead to immediate improvements in children’s
numeracy. Similarly, Ribner et al. (2023) emphasized the importance for flexibility and
alignment of the intervention with daily life, contrasting their approach with earlier scripted
designs. These two studies illustrate the potential of ecologically grounded designs. Building
on these efforts, the present study aimed to evaluate whether a flexible, minimally supervised
intervention designed to enhance home numeracy activities can produce measurable gains in
the early numerical skills of 4- to 5-year-old children.

The HNE encompasses the various factors through which parents and caregivers
influence their children’s numeracy development (LeFevre et al., 2009; Skwarchuk et al.,
2014). It includes both formal activities explicitly designed to teach mathematical concepts
(e.g., reciting the number words sequence or practicing addition) and informal activities that
embed mathematical content into everyday interactions (e.g., cooking, playing board games or
reading number storybooks; Mutaf-Yıldız et al., 2018a). The HNE also encompasses parents’
expectations for their children’s numeracy learning and parents own attitudes towards
mathematics (Dowker, 2021; Napoli et al., 2021). A meta-analysis of 64 studies reported a small
positive association (r = 0.14) between HNE and children's numeracy (Daucourt et al., 2021),
with the strongest effects observed at the preschool age. However, these correlational findings
do not establish causality. It remains unclear whether a richer HNE actively promotes numeracy
development, whether children with stronger early numeracy skills elicit more numeracy
engagement from their caregivers (Ciping et al., 2015; Silinskas et al., 2020), or whether this
association is shaped by shared underlying factors such as genetic predispositions (Hart et al.,
2021). These questions can only be answered through intervention studies that manipulate the
HNE (Libertus, 2024).

There is now a growing body of intervention studies, which typically last 4 to 6 weeks,
that have attempted to enhance the HNE via parental activities (e.g., Gibson et al., 2020;
Purpura et al., 2021; Scalise et al., 2022). These interventions can be categorized into two
categories: interventions that involve storybook reading with books designed to promote
mathematical language (e.g., Purpura et al., 2021) and interventions that involve playful
interactions, such as numerical cards (Scalise et al., 2022), board games (Ribner et al., 2023;
Silver et al., 2024) or dice games (Niklas et al., 2016). Promising intervention effects have been reported on children's counting, ordering, transcoding, arithmetic, or mathematical language (e.g., Cheung & McBride-Chang, 2015; Purpura et al., 2021; Ribner et al., 2023). For example, in a study by Purpura et al. (2021), 3- to 5-year-old American children and their parents were asked to read three picture books with mathematical language for 4 weeks (4 times per week). Each page had three dialogic reading prompts, which parents could use to engage in mathematical content with their children. After the intervention, the children showed significant improvements in counting, ordering and arithmetic tasks, compared to children in an active control condition who received similar picture books without mathematical language (Purpura et al., 2021). Similar improvements in numeracy were observed in 4-year-old children from American families who played a numerical board game with their parents twice a week for eight weeks (Ribner et al., 2023), and in 3-to-5-year-old Filipino children who played a numerical card game with their parents for ten weeks (Cheung & McBride-Chang, 2015).

By contrast, several studies using similar designs, which manipulated the HNE, have failed to demonstrate significant improvements in children's numeracy skills (e.g., Ramani & Scalise, 2020; Sonnenschein et al., 2016). For instance, Sonnenschein et al. (2016) found that 4-year-old American children who played a number board game with their parents three times a week for five weeks did not improve in numeracy. Likewise, Ramani & Scalise (2020) showed that 3-to-5-year-old American children who played a magnitude comparison card game with their parents twice a week for six weeks did not improve in numeracy.

These inconsistent results may reflect the influence of moderating variables, such as differences in pretest skills, pre-existing HNE practices or the amount and quality of exposure to the intervention (Ramani & Scalise, 2020; Scalise et al., 2022). For example, Gibson et al. (2020) and Silver et al. (2024) reported that children with higher pretest numeracy skills benefited more from the HNE intervention. Ribner et al. (2023) recently found that the effect
of the parent-child number game condition on children’s skills was fully mediated by changes
in home numeracy practices.

In line with these inconsistent findings, two very recent meta-analyses (Cahoon et al., 2023; Nelson et al., 2024) highlighted the substantial heterogeneity in the effects of HNE interventions. Nelson et al. (2024) reported a significant effect (g = 0.26; 25 studies) for early math interventions conducted in informal learning environments, while Cahoon et al. (2023) found a small effect (d = 0.18; 10 studies) in interventions targeting preschool-aged children. Despite these encouraging overall findings, both meta-analyses emphasized considerable variabilities in outcomes across studies, likely due to differences in intervention design, implementation formats, and contextual factors. Nevertheless, it remains unclear to what extent existing HNE intervention studies truly reflect the causality of HNE on children's numeracy skills, as they face several limitations.

Firstly, previous studies were often based on applications of existing laboratory-based
interventions, implemented by researchers outside the home environment, rather than on
ecologically valid resources suited to the daily lives of families with preschoolers. Specifically,
existing HNE-interventions involved the strict implementation of a single, repetitive task over
several weeks. Moreover, some of these studies included substantial external support such as
close supervision, e.g. regular home visits or audio recordings of play sessions (Dulay et al.,
2019; Purpura et al., 2021; Ramani & Scalise, 2020), or intensive parent training components
(Cheung & McBride, 2017). However, these close supervision and training differ considerably
from the kinds of spontaneous, everyday activities in which parents typically engage with their
children. As a result, the existing HNE intervention literature suffers from limited ecological
validity (Chezan et al., 2022; Ledford et al., 2016). To more carefully demonstrate the potential
causal effect of the HNE on children’s numeracy, the manipulation of the HNE should closely
align with the kinds of numeracy-related interactions that parents naturally engage in during
their everyday activities. These types of interactions have been documented in correlational
studies using questionnaires. Such research, conducted in cultural contexts that align with the
context of the current study – namely in Belgium (Mutaf-Yıldız et al., 2018a, 2018b), France
(Girard et al., 2023), and Northern Ireland (Cahoon et al., 2017) – has shown that families with
preschoolers frequently engage in storybook reading and number-related games. Given the
prevalence of these activities in West-European contexts and in the country of this study in
particular, we focused on them in the intervention. As a result, our study was ecologically valid
in terms of its content for the families who participated.

Interestingly, a recent intervention by Leyva et al. (2023) has started to broaden the
focus of HNE interventions beyond a single type of activity, seeking to promote mathematical
engagement across a variety of family routines through a six-week program. The “Math Made
for Me” (MM4M) program tailored the intervention to each family’s typical routines, and
provided brief video examples of real families engaging in math talk during everyday activities.
This approach aligns with our aim of enhancing ecological validity in HNE interventions by
avoiding reliance on a single scripted activity or specific type of material. Importantly, Leyva
et al. (2023) reported promising effects on home math engagement: Parents increased both their
math talk and their involvement in home numeracy activities. However, the intervention did
not lead to immediate gains in children’s mathematical skills. This study illustrates both the
promise and the challenge of embedding math-related interactions into everyday family life - it
demonstrates the feasibility of fostering home math engagement, but also the difficulty of
detecting short-term effects on children’s numeracy outcomes. Extending this work, the present
study aimed to test whether an intervention embedded in everyday family life - without the
constraints of a single, repetitive activity or formal training - could support children’s early
numeracy.
*Secondly*, many existing HNE-intervention studies did not systematically include a matched active control condition - for example, a condition with equivalent activities without numeracy content. This makes the interpretation of the existing HNE-intervention studies challenging (e.g., no active control group in Cheung & McBride-Chang, 2015; or no control group in Leyva et al., 2018; Niklas et al., 2016).

*Thirdly*, most previous studies have been conducted with relatively small sample sizes, typically involving only (20 to 30 participants per group). These small sample sizes may limit statistical power and reduce the generalizability of findings.

*Fourthly*, the randomization of participants across conditions – and its verification (i.e., the absence of mean pretest differences) - has often been insufficiently documented in earlier studies, making the results difficult to interpret. Small samples and unclear randomization further challenge the generalizability of previous findings (Hornburg et al., 2021).

*Fifthly*, the majority of existing studies restricted their analysis to children's performance and did not measure home numeracy factors before and after the intervention (with a few exceptions, such as Niklas et al., 2015; and Ribner et al., 2023). As a result, the precise impact of the HNE-intervention on the HNE itself remains unclear. On the one hand, it is often unknown whether the experimental and control groups differed in their HNE before the start of the intervention. On the other hand, the lack of post-intervention measurements makes it difficult to evaluate how the intervention may have shaped the HNE. Notably, Ribner et al. (2023) reported a surprising decrease in the frequency of HNE activities following a targeted parental intervention (designed to improve parents approximate number sense). This finding underscores the importance of directly assessing how HNE interventions influence home practices before drawing conclusions about their broader impact.

*Sixthly*, none of the existing HNE intervention studies have been conducted with European children, which may limit the generalizability of their conclusions to other contexts.
The majority of participants in previous research were American (e.g., Gibson et al., 2020; Purpura et al., 2021; Ribner et al., 2023), with only a few studies including Filipino, Chinese or Australian participants (e.g., Cheung & McBride-Chang, 2015; Niklas et al., 2016; Cheung & McBride, 2017). Moreover, the available correlational evidence on the associations between HNE and children's numeracy has been predominantly American, while such associations have not always been observed in a European context (e.g., De Keyser et al., 2020). These limitations highlight the need to investigate the effects of HNE interventions in a variety of contexts.

The current study aimed to address the six limitations outlined above through a pre-
registered, randomized, pretest-posttest intervention design (https://osf.io/k5ea9). The main
objective was to test the causal role of the HNE in supporting children's numeracy by firstly
implementing an ecologically valid manipulation of a key component of the HNE—namely,
the frequency and diversity of parent-child numeracy activities in the home. To do so, we
provided families over a six-week period with a set of different numeracy activities, that have
been previously used in separate intervention studies (i.e., numerical cards, board and dice
games, numerical storybooks). Importantly, parents were given the flexibility to choose which
activities to engage in each week, thereby maintaining ecological validity. Secondly, our design
incorporated a matched active control condition, in which parents were asked to engage in
similar activities with no mathematical content. This design allowed us to isolate the specific
contribution of numeracy content, while controlling for general cognitive stimulation, parental
engagement, and implementation. Thirdly, based on a priori power calculations, we selected a
sufficiently large sample size to adequately test our hypotheses. Fourthly, we randomly
assigned children within the same preschool class to either the experimental or the control
condition. This method accounted for individual characteristics (i.e., preschool education,
children's initial numeracy skills, children's HNE before the start of the intervention) that might
influence the effects of the intervention. *Fifthly*, we assessed the effect of the intervention not only on children's numeracy outcomes, but also on parent-reported HNE factors, by asking parents to complete two HNE questionnaires before and after the intervention. *Finally*, the study was conducted in a West-European context, contributing data on HNE interventions outside of previously studied cultural settings.

Building on previous correlational and intervention research, the present study investigated the effects of our HNE intervention on a variety of numerical abilities. In line with findings from previous HNE interventions (e.g., Cheung & McBride-Chang, 2015; Purpura et al., 2021; Ribner et al., 2023), we investigated the effect of HNE on children's counting, ordering, transcoding, arithmetic and mathematical language skills in children aged 4 to 5 years. We pre-registered several hypotheses. Our main hypothesis was that children in the experimental condition, focusing on numeracy activities, would improve their numeracy more than children in the control condition, who focused on language activities. Additionally, we hypothesized that parents in the experimental condition would report greater changes in HNE factors after the intervention, compared to parents in the control condition. As an exploratory analysis, we predicted that the effects of the intervention might be moderated by children's initial numeracy level, parents' initial frequency of HNE activities, or measures of implementation monitoring.

# Method

## Participants

Families with four- to five-year-old children in the second year of preschool in the Flemish region of Belgium were invited for this intervention. Families were recruited through preschool teachers, who agreed to disseminate the study to the families and to receive the experimenters during class to test the children at pre- and posttests. Parents were contacted
through their child's school and they were provided with a detailed information letter that outlined the study's goals and procedures. This letter emphasized that participation entailed exploring a set of materials with their child over a six-week period, with activities designed to take no more than 10 minutes per day. A total sample of 117 4- to 5-year-olds (M = 4.68 years, SD = 0.29, range = 4.22 - 5.21, 47% girls) agreed to participate. The children came from 26 different schools (including 42 different classrooms) in the Flemish part of Belgium. In Flanders (Belgium), preschool education (3 years) is free and it is attended by the vast majority of children (98%; De Keyser et al., 2020). Preschool education follows government guidelines for specific areas of non-formal learning, focusing on play-based preparatory activities without formal mathematics instruction. All children were in year 2 of preschool education. This study provides insights into home numeracy practices within the Flemish Belgian context, where educational and socio-cultural factors may differ from other regions in Europe. According to the pre-registration, no children were excluded on the basis of disability.

Parents had to complete an online questionnaire about their home learning environment before and after the intervention: 96.8% completed the pre-intervention questionnaire (n = 113) and 71.8% (n = 84) completed the post-intervention questionnaire (3 families did not complete any questionnaire). The response rate for these questionnaires did not differ between the experimental and control conditions (at pretest, t(115) = -1.03, p = .31, at posttest, t(115) = 1.80, p = .07). Children were predominantly White (97%) and all children and parents included in the study were Dutch speakers, although 10.3% of parents also reported to speak an additional language at home. Most parents had a university-level education, which indicated that the participating families predominantly came from relatively advantaged socioeconomic backgrounds. Parents were equally and randomly distributed in the two conditions. The study was conducted between February and May 2023. Parents provided written informed consent prior to any data collection, as approved by the authors' institutional ethical review board (G-
2022-5836). Children and their parents were allowed to keep the intervention materials as a thank-you gift, and children received stickers to maintain attention and motivation throughout data collection and intervention implementation. Each family, regardless of their intervention condition, was also offered a lottery ticket with the opportunity to win a €30 voucher. At the end of the study, all families were offered the materials from the other condition as well, to ensure that every participant ultimately had access to both sets of resources.

## Pre-registration and justification of sample size

The sample size, materials, and the planned analyses of this study were pre-registered on the Open Science Framework (https://osf.io/k5ea9). The minimum sample size required to achieve a power (1 - β) of 0.80 was calculated prior to the study using G* Power 3.1. At the time of the pre-registration, we used the meta-analysis by Benavides-Varela (2020), which focused on the effectiveness of digital-based interventions on children's numeracy, to determine our expected mean effect size (*d* = .55). We selected this meta-analysis because we wanted to base our calculations on intervention studies and not on correlational work, and this meta-analysis was at that time the most recent one at our disposal.

Based on a repeated-measures ANOVA with two time points (pre- and post-test), an alpha level of .05, a power of .80, and an assumed correlation among repeated measures of .50, this resulted in a minimum sample size of 40 participants per condition to detect an effect of the intervention on children's numeracy skills. Anticipating a risk of participant attrition over the course of the intervention, we planned to recruit 60 participants per condition.

These calculations were not optimal, as Benavides-Varela et al. (2020) focused on mathematical learning difficulties and covered a wide age range. Since the time of our pre-registration, two meta-analyses more specifically focused on the effects of HNE interventions have been published (Cahoon et al., 2023; Nelson et al., 2024), which allowed us to obtain more precise
estimates of the effect size of HNE interventions. We therefore conducted additional power
analyses (non-pre-registered), using the same parameters as in the original calculation. Based
on the effect size of *f* = .13 reported by Cahoon et al. (2023), a sample size of 120 participants
would be required, while the effect size of *f* = .18 reported by Nelson et al. (2024) would require
64 participants. These updated calculations suggest that our final sample of 117 participants
was adequately powered to detect effects of the magnitude indicated by these more recent meta-
analyses.

Design and procedure

Children were assessed at their school, individually in a quiet room by an experimenter blind to the condition. We measured children's once before and once after the 6-week intervention (in March 2023 and in May 2023, respectively) with the same set of tests. Parents were asked to complete a questionnaire about their home environment before and after the intervention. Within a class (and a school), children were randomly assigned to either the experimental (Numeracy, n = 59) or the control condition (Language, n = 58) (Figure 1). As a result, the two conditions were matched in terms of their preschool educational environment.

**Figure 1**

Flow diagram of the progress through the phases of the intervention
<figure><img src="image_4.png" /></figure>

*Note.* Flow diagram according to CONSORT 2010 Statement (Schulz et al., 2010)

All materials, including booklets and activity sets, were distributed to the schools, where teachers handed them directly to the participating children to take home. All materials were provided remotely, ensuring flexibility and accessibility, and no parental training sessions were required. In both conditions, participants received a personalized package containing the following items 1) a printed booklet with general information that detailed instructions for each activity and that included examples of motivational incentives to use during play, 2) a suggested activity calendar designed to provide structure, while allowing families to adapt the schedule and complexity of these activities to their routines and 3) the complete set of activity materials specific to the experimental or control condition. Each booklet also contained a logbook for tracking the activities that parents completed with their child. Parents were asked to report this information through weekly online surveys. To make the intervention engaging and user-
friendly, a colorful planning document was provided. It showed the different activities per week and it included stickers to celebrate completed activities.

The intervention was designed to maximize ecological validity by aligning the content of the intervention with activities that families naturally do at home when they aim to support their children's learning. The activities were selected based on prior research conducted in the same country as in the current study, that is, Belgium (Mutaf-Yıldız et al., 2018a, 2018b) and similar neighboring West-European countries such as France (Girard et al., 2023) and Northern Ireland (Cahoon et al., 2017). These correlational studies, which used questionnaires to characterize the HNE, demonstrated that parents frequently engage in storybook reading, board games, and card games with preschool-aged children. These activities incorporated numeracy content in playful and interactive ways, making them familiar and relevant for families in West-European contexts. Families were encouraged to adapt the order, timing, and complexity of the activities to their child's preferences and family routines, allowing for flexibility. In contrast to prior interventions requiring extensive training or monitoring, this study adopted a minimally intrusive approach with brief instructions and low-cost, easy to access materials. A detailed description of the materials provided and the guidelines for each game can be found in the section 1.1 of the Supplementary Material (Tables S1 and S2).

In the experimental condition, participants were provided with a set of materials adapted from previous intervention studies, designed to elicit interactions about numeracy. These materials included a domino game, a card game, a board game and a mathematical language storybook. These games could be played with different modalities (e.g., symbolic, non-symbolic, or mixed) and at different levels of complexity, as detailed in the accompanying booklet. The booklet recommended a six-week schedule, and suggested that parents began with simpler versions and gradually increased the difficulty level. This schedule was intentionally
flexible, allowing parents to tailor activities to their child's developmental readiness and maintain their motivation throughout the intervention.

In the control condition, participants received equivalent materials - such as a verbal memory game, a visual memory game (pictures, letters, rhymes), a bingo game, and a storybook - without numerical content. Three activities per week were suggested in the booklet for both conditions, but families were free to decide which activities to play, how often, and how challenging to make them, thereby preserving ecological validity. This design ensured that both conditions were equally engaging and cognitively stimulating, with the primary difference between the conditions being the presence or absence of explicit numerical content.

Weekly videos (three per condition) were sent via email or text message, providing additional guidance and motivation. These videos aimed to influence parents' beliefs about their children's learning (in mathematics for the experimental condition, in language for the active control condition). They emphasized the importance of early skills and illustrated how parents could easily facilitate these skills in daily life (see **Figure S1**, for further details). To sustain engagement with the intervention and to document its implementation, these weekly messages also included a link to a brief online survey about the activities conducted each week.

The two conditions were matched to ensure they were equally challenging and stimulating from a cognitive perspective. Both conditions were also designed to be equally motivating and they offered diverse and adaptable activities for families. The primary and only difference between the two conditions was the inclusion of numerical content in the experimental condition and its absence in the control condition.

## Child measures

### Numerical abilities
Children's numerical skills were measured with five tasks previously developed by our research team and validated in studies with children of the same age range (Bakker et al., 2023; Rathé et al., 2022; Turan & De Smedt, 2023; Vandecruys et al., 2024).

**Counting.** Children's understanding of cardinality (Wynn, 1990) was assessed with an object counting task in which children had to enumerate up to 8 different sets of stones (i.e., 3, 4, 6, 8, 11, 14, 16, and 19 stones). The maximum score was 8 (1 point per correct answer). The task had good internal consistency (Cronbach's alpha at pretest = .84, Cronbach's alpha at posttest = .82).

**Transcoding.** Children's ability to identify symbolic numbers was assessed by asking them to name Arabic numbers. The task was divided into 10 blocks of increasing difficulty (the first 3 blocks consisted of numbers below 10, the next 4 blocks consisted of numbers between 10 and 100, and the last 3 blocks consisted of numbers above 100). The task was discontinued if the child was unable to identify any of the 3 items within one block, and the maximum score was 30 (1 point per correct answer). Internal consistency was excellent (Cronbach's alpha at pretest = .90, Cronbach's alpha at posttest = .91).

**Ordering.** Children's understanding of ordinality was assessed by a number ordering task, similar to Bakker et al. (2019) and Bakker et al. (2023). Each trial displayed three empty boxes in a row, with a target number placed in the central box. Children were asked to verbally identify the number that comes immediately before and after the target number. The task included two practice trials followed by two blocks of 8 trials each, with four “before” and four “after” prompts per block. The first block included numbers below 20 (3, 7, 12, 16), and the second block included numbers above 20 (24, 36, 68, 71). The second block was administered only if the child scored at least 50% on the first block (≥4 correct). Each correct answer scored 1 point, for a maximum of 16. Internal consistency was good (Cronbach's alpha at pretest = .86, Cronbach's alpha at posttest = .91).
**Arithmetic.** Children's early arithmetic skills (addition and subtraction) were assessed with a task with four blocks. In the first block (non-symbolic arithmetic), the examiner showed a set of stones, placed them into a non-transparent box, then added or removed stones while the box remained closed. The experimenter used number words to explain how many stones were in the box and how many more were added or subtracted. Children were asked to determine how many stones were finally inside the non-transparent box, relying on mental calculation rather than counting visible objects. The three subsequent blocks presented problems that were read aloud in symbolic form (e.g., "3 + 2"). The problems were displayed on cards and increased in difficulty level: numbers below 10 (block 2), above 10 (block 3), and between 10 and 30 (block 4). The task was discontinued if the child failed to answer any item correctly within a block. The maximum score was 18 (4 points for the non-symbolic block and 14 points for the symbolic blocks, 1 point per correct answer). Internal consistency of this measure at pretest was low (Cronbach's alpha = .446). This was because the task was rather difficult for children of this age. Despite this low reliability, the task was included to capture potential intervention effects on more advanced numerical skills. The reliability calculated on the basis of this measure at posttest was acceptable (Cronbach's alpha = .73).

**Mathematical language.** Children's mathematical language was assessed using the Preschool Assessment of the Language of Mathematics (PALM) from Purpura and Logan (2015) adapted by Turan and De Smedt (2023). The task included 24 items that measured children's understanding of quantitative and spatial language. All items were designed such that children could solve them without exact quantitative skills. Examples of questions were "Which glass has *more* water?" or point to the dot that is the *nearest* to the boy. This task had 12 items that measured children's knowledge of quantitative terms (e.g., equal, more, less) and 12 items that measured children's knowledge of spatial terms (e.g., near, far, before). The maximum correct
score was 24 (1 point per correct answer) and the internal consistency was good (Cronbach's
alpha at pretest = .80, Cronbach's alpha at posttest = .76).

Non-verbal reasoning

As a control measure, we included the Matrix Reasoning subtest of the Dutch Wechsler
Intelligence Scale for Children, Third Edition (Wechsler, 2011). Children had to indicate which
of four or five visually presented items would fit into the empty spot of an incomplete visual
pattern. The task was discontinued after 4 consecutive errors (or 4 errors for 5 consecutive
items). The maximum score was 29 (1 point per correct answer), and the internal consistency
was good (Cronbach's alpha at pretest = .87, Cronbach's alpha at posttest = .87).

Home environment

The questionnaire was adapted from previous studies that assessed the HNE (De Keyser
et al., 2020; Girard et al., 2023; LeFevre et al., 2009) and included additional items to obtain
numeracy and non-numeracy equivalents. The questionnaire measured the frequency of home
learning activities related to numeracy (n = 19) or language (n = 17) (e.g., "In the last month,
how often did you and your child play puzzle or construction games?"). Responses had to be
given on a six-point Likert scale ("Never", "Once a month", "Several times a month", "Once
per week", "More than once a week", "Daily"). For each participant, mean scores for numeracy
and language activities were computed separately by averaging the scores of all relevant items.
This part of the questionnaire had good internal consistency (At pretest, Cronbach's alpha for
numeracy = .86, Cronbach's alpha for language = .84; At posttest, Cronbach's alpha for
numeracy = .88, Cronbach's alpha for language = .80).

The questionnaire also measured parents' expectations about their children's learning in
numeracy (n = 11) and language (n = 7), using learning milestones parents expected their
children to reach before entering first grade. Parents were asked to rate the importance of their child acquiring a particular skill before starting first grade (e.g., "how important is it to you that your child will be able to write numbers up to 10 by the time he or she enters first grade?") using a four-point Likert scale ("Not at all important", "Not important", "Important", "Very important"). Similar to the scores for activities, mean expectation scores for numeracy and language were computed separately by averaging the scores of all relevant items. These items also demonstrated strong internal consistency (At pretest, Cronbach's alpha for items on numeracy expectations = .88, Cronbach's alpha for items on language expectations = .78; At posttest, Cronbach's alpha for numeracy = .92, Cronbach's alpha for language = .79).

The questionnaire also asked parents to rate their own attitudes towards numeracy (n = 4) and language (n = 4) (e.g., "I find mathematics fun") and their past school experiences, e.g., "When I was in school, I was good at mathematics"), using a five-point Likert scale (i.e., "Strongly disagree", "Disagree", "Neutral", "Agree", "Strongly agree"). Consistent with the other HNE scores, mean attitude scores for numeracy and language were computed separately by averaging the scores of all relevant items. These measures also showed acceptable internal consistency (Cronbach's alpha for numeracy attitude items = .78, Cronbach's alpha for language attitude items = .73).

Parents were additionally asked to report the subjective assessment of changes in their home activities, in their expectations for their children's learning, and in their own attitudes (one item each for numeracy and one item each for language). Because we only had one item for each of these aspects of HNE, it was difficult to determine reliability of these assessments. These measures were therefore not used in the subsequent analyses. Finally, the questionnaire also measured the educational level of the parents by asking their highest educational degree.

*Implementation monitoring*
Parents were asked to report their engagement with the intervention materials on a weekly basis, via online surveys sent by email or text message. These surveys consisted of questions on which activities from the proposed schedule were completed, how many times each activity occurred, and the average duration of the activities. Parents could also report whether they engaged in other activities than the materials they were provided with. We used these self-reports to calculate an implementation score for each family, using our preregistered criteria (https://osf.io/k5ea9) to examine implementation monitoring. Each family received an implementation score ranging from 0 to 6, based on the number of weekly surveys in which they reported engaging in at least one activity. A score of 0 indicated that no activities were reported for any of the six weeks, suggesting no evidence of intervention implementation for that family. A score of 6 indicated full adherence to the intervention across the six weeks. The weekly surveys were designed to be brief and accessible. They allowed parents to complete them without adding a significant burden to their daily activities. Short motivational videos were sent together with these surveys to encourage participation and to reinforce the importance of numeracy and language activities for children's learning.

## Analyses

Following our pre-registered analysis plan (https://osf.io/k5ea9), we first assessed the quality of our randomization by testing whether the conditions did not differ at pretest. This was done using Bayesian analyses to test for evidence of the absence of a difference between the two conditions. We also assessed whether the two conditions were similar in their adherence to the intervention, again using Bayesian analyses to test for evidence of the absence of a difference between the two conditions. To assess the implementation of the intervention, the implementation scores derived from the weekly surveys were analyzed. These scores provided a quantitative measure of parental engagement across the six-week period. Descriptive analyses
were conducted to examine the average frequency of reported activities, session durations, and variations in engagement between families. The question of whether implementation scores differed between the experimental and control conditions was explored to verify comparable engagement across groups.

We next tested our pre-registered hypotheses regarding the effects of the HNE intervention using ANOVAs. First, we expected that children's numeracy scores in the experimental condition (numeracy) would show greater increases from pre- to posttest compared to children in the active control condition (language). For our control measure, that is non-verbal reasoning, we did not expect a difference between the two conditions. Second, we also expected an effect of the intervention on the HNE itself. We hypothesized that, after the intervention, parents in the experimental condition would report to do more home numeracy activities with their children, to have higher expectations for their children's numeracy learning and to have more positive attitudes toward numeracy, as compared to parents in the control condition.

As exploratory analyses, we planned to test whether the intervention effect on children's numeracy skills was moderated by children's initial numeracy level, parents' initial frequency of home numeracy activities, or by measures of implementation monitoring. Finally, as an additional analysis, we planned to reanalyze the data with linear mixed models, accounting for the potentially nested structure of the data, that is children nested in schools.

The data were analyzed with Jamovi (version 1.6.3) and Jasp (version 0.17.3) using both frequentist and Bayesian statistics. Bayesian statistics allowed us to quantify the evidence in support of either the alternative hypothesis of a difference or the null hypothesis of no difference. Bayesian analyses were performed using the Jamovi Bayesian Methods package with default priors, specifically a stretched beta prior width set to 1. Following Jeffreys (1961), Bayes factors (BFs) less than 3 were considered as anecdotal evidence, those within the range
of 3 to 10 were considered as substantial evidence, 10 < BFs < 30 were considered as strong evidence, 30 < BFs < 100 as very strong evidence, and BFs > 100 were considered as extreme evidence that the data are more likely under the alternative hypothesis (i.e., BF<sub>10</sub>) or the null hypothesis (i.e., BF<sub>01</sub>). The integration of both frequentist and Bayesian statistical approaches allowed us to make informed decisions about rejecting or accepting the null hypothesis, while also providing a quantifiable measure of the strength of the evidence in favor for either the null or alternative hypothesis (which is, for example, critical for evaluating the efficacy of randomization between conditions). Raincloud plots were obtained using the Raincloudplots R package (Allen et al., 2021).

The digital materials and study data can be retrieved from the Open Science Framework: https://osf.io/f74t6.

## Results

### Descriptive statistics and implementation checks

Descriptive statistics for children's and parents' characteristics at pretest are presented in Table 1 and Table 2, respectively. Data on the intervention implementation are presented in Table 3.

### Table 1

#### Descriptive Statistics for Children's Variables at Pretest

<table><thead><tr><th rowspan="2"></th><th colspan="2">Experimental condition<br>(numeracy, n = 58)</th><th colspan="2">Active control condition<br>(language, n = 59)</th><th colspan="2">Group difference</th></tr><tr><th>M (SD)</th><th>range</th><th>M (SD)</th><th>range</th><th>p</th><th>BF<sub>01</sub></th></tr></thead><tbody><tr><td>Sex</td><td colspan="2">40.7% girls</td><td colspan="2">53.4% girls</td><td>.166</td><td>1.72</td></tr><tr><td>Age in years</td><td colspan="2">4.68 (0.28)</td><td colspan="2">4.68 (0.30)</td><td>.986</td><td>5.08</td></tr><tr><td>Numeracy</td><td colspan="6"></td></tr></tbody></table>
<table><tr><td>Counting</td><td>4.93 (2.36)</td><td>0-8</td><td>4.43 (2.31)</td><td>0-8</td><td>.248</td><td>2.78</td></tr><tr><td>Transcoding</td><td>7.61 (4.88)</td><td>0-22</td><td>7.53 (4.21)</td><td>0-18</td><td>.929</td><td>5.07</td></tr><tr><td>Ordering</td><td>3.07 (3.17)</td><td>0-16</td><td>3.41 (3.02)</td><td>0-15</td><td>.547</td><td>4.31</td></tr><tr><td>Arithmetic</td><td>1.90 (1.48)</td><td>0-8</td><td>1.71 (1.09)</td><td>0-4</td><td>.429</td><td>3.83</td></tr><tr><td>Mathematical language</td><td>19.68 (3.15)</td><td>12-24</td><td>19.93 (4.00)</td><td>2-24</td><td>.704</td><td>4.76</td></tr><tr><td>Non-verbal reasoning</td><td>9.20 (4.57)</td><td>0-18</td><td>9.64 (4.89)</td><td>1-21</td><td>.620</td><td>4.55</td></tr></table>

Note. BF<sub>01</sub> values indicate the strength of the evidence in favor of the null hypothesis of an absence of difference between the two conditions.

We first tested whether the randomization was optimal. We expected the two conditions not to differ, or H0 to be true. ANOVAs indicated that children in the experimental condition (numeracy) did not differ from children in the active control condition (language) on any of the characteristics measured at pretest (see Tables 1 and 2 for details). Bayesian analyses mostly provided substantial evidence supporting the absence of difference between the two conditions on any of the abovementioned characteristics (see Tables 1 and 2).

<caption>**Table 2**</caption>

Descriptive Statistics for Parent's and HNE Variables at Pretest

<table><thead><tr><th rowspan="2"></th><th colspan="2">Experimental condition<br>(numeracy, n = 58)</th><th colspan="2">Active control condition<br>(language, n = 55)</th><th colspan="2">Group difference</th></tr><tr><th>M (SD)</th><th>range</th><th>M (SD)</th><th>range</th><th>p</th><th>BF<sub>01</sub></th></tr></thead><tbody><tr><td>Sex</td><td>86.4% mothers</td><td></td><td>90.9% mothers</td><td></td><td>.453</td><td>5.20</td></tr><tr><td>Caregiver education</td><td>29% secondary school degree</td><td>18% secondary school degree</td><td>71% university degree</td><td>82% university degree</td><td>.166</td><td>1.46</td></tr><tr><td>HNE activities</td><td>1.98 (0.73)</td><td>0.47-3.50</td><td>1.93 (0.71)</td><td>0.58-3.67</td><td>.678</td><td>4.63</td></tr></tbody></table>
<table><tr><td>Parental expectations<br/>toward their child's<br/>mathematical learning</td><td>2.66 (0.47)</td><td>1.75-3.75</td><td>2.69 (0.52)</td><td>1.58-3.92</td><td>.798</td><td>4.86</td></tr><tr><td>Parental attitude<br/>toward numeracy</td><td>1.16 (0.72)</td><td>-0.75 - 2</td><td>1.02 (0.73)</td><td>-0.75 - 2</td><td>.326</td><td>3.23</td></tr></table>

*Note.* Parents who were not mothers were all fathers (3 parents in the control condition did not indicate their sex or education degree). Ratings: HNE activities, “Never” was coded 0, “Once a month” was coded 1, “Several times a month” was coded 2, “Once per week” was coded 3, “More than once a week” was coded 4. Parental expectations, “Not at all important” was coded 1, “Not important” was coded 2, “Important” was coded 3, “Very important” was coded 4. Parental attitude, “Strongly disagree” was coded -2, “Disagree” was coded -1, “Neutral” was coded 0, “Agree” was coded 1 and “Strongly agree” was coded 2. BF<sub>01</sub> values indicate the strength of the evidence in favor of the null hypothesis of an absence of difference between the two conditions.

Of all families, 103 completed at least one weekly survey. Of these, two families completed only one survey and reported no activities for that week. Consistent with our preregistered coding scheme, these two families received an implementation score of 0, indicating no evidence of implementing the intervention. Overall, 86.3% of the families (n = 51 in the experimental condition; n = 50 in the control condition) reported engaging in at least one activity in at least one week. The implementation scoring provided a global measure of participation in the intervention activities. The total number of intervention activities and average time spent on each were also recorded. None of these indicators differed significantly between conditions (Table 3).

**Table 3**

*Descriptive Statistics for Weekly Surveys Collected During the Intervention*
<table><thead><tr><th rowspan="2"></th><th colspan="2">Experimental condition<br>(numeracy, n = 59)</th><th colspan="2">Active control condition<br>(language, n = 58)</th><th colspan="2">Group difference</th></tr><tr><th><i>M</i> (SD)</th><th>range</th><th><i>M</i> (SD)</th><th>range</th><th><i>p</i></th><th><i>BF</i><sub>01</sub></th></tr></thead><tbody><tr><td>Number of weekly questionnaires<br>answered</td><td>3.78 (2.33)</td><td>0 - 6</td><td>4.29 (2.09)</td><td>0 - 6</td><td>.212</td><td>2.51</td></tr><tr><td>Total activities</td><td>26.53 (17.27)</td><td>3-99</td><td>26.36 (15.59)</td><td>7-98</td><td>.957</td><td>4.80</td></tr><tr><td>Average time spent by activity (in<br>minutes)</td><td>7.58 (2.91)</td><td>3.06-15</td><td>8.00(2.25)</td><td>3.21-14.50</td><td>.460</td><td>3.76</td></tr></tbody></table>

*Note.* Statistics for total activities and average time spent by activity are for participants with at least one weekly survey (numeracy, *n* = 51, language, *n* = 50). BF<sub>01</sub> values indicate the strength of the evidence in favor of the null hypothesis of an absence of difference between the two conditions.

## Intervention effects on children's numerical skills

The pretest and posttest assessments of the children were separated by on average 8.7-weeks (*M* = 60.81 days; *SD* = 6.11 days, range = 47–77 days). The number of days between pretest and posttest did not differ by condition (*M*<sub>Numeracy</sub> = 60.9 days, *M*<sub>Language</sub> = 60.7 days, *F*(1, 115) = 0.034, *p* = .855, η<sup>2</sup><sub>p</sub> < .001; BF<sub>01</sub> = 5.01). Repeated measures ANOVAs were conducted with time (pretest vs. posttest) as within-subject factor and condition (experimental vs. control) as between-subjects factor. There were significant effects of time on all measures (all *F*<sub>s</sub> > 20.468, all *p*<sub>s</sub> < .001, and η<sup>2</sup><sub>p</sub> > .151, see **Figure 2**).

## **Figure 2**

*Scores for all children at pretest and posttest scores (n=117)*
<figure><img src="image_5.png" /></figure>

Repeated measures ANOVAs also revealed significant interaction effects between time and condition on ordering and transcoding with medium effect sizes (Figure 3 and Table 4). There were no such interaction effects for the other numerical skills (counting, arithmetic and mathematical language) and for non-verbal reasoning.

<caption>Figure 3</caption>

Scores for children in experimental (numeracy) or control (language) condition at pretest and posttest
<figure><img src="image_6.png" /><figcaption>Table 4</figcaption></figure>

Repeated measures ANOVA on children's skills showing Time x Condition interaction effect

<table><thead><tr><th rowspan="3"></th><th colspan="2">Experimental condition (numeracy)</th><th colspan="2">Active control condition (language)</th><th rowspan="2">F</th><th rowspan="2">η<sup>2</sup>p</th><th rowspan="2">p</th></tr><tr><th colspan="2">Pretest</th><th colspan="2">Posttest</th></tr><tr><th>M (SD)</th><th>M (SD)</th><th>M (SD)</th><th>M (SD)</th><th colspan="3">(1,115)</th></tr></thead><tbody><tr><td>Counting</td><td>4.93 (2.36)</td><td>5.90 (2.14)</td><td>4.43 (2.31)</td><td>5.14 (2.13)</td><td>0.49</td><td>.004</td><td>.485</td></tr><tr><td>Transcoding</td><td>7.61 (4.88)</td><td>9.76 (5.50)</td><td>7.53 (4.21)</td><td>8.21 (4.15)</td><td>5.67</td><td>.047</td><td>.019</td></tr><tr><td>Ordering</td><td>3.07 (3.17)</td><td>5.15 (4.48)</td><td>3.41 (3.02)</td><td>3.81 (3.56)</td><td>10.60</td><td>.084</td><td>.001</td></tr><tr><td>Arithmetic</td><td>1.90 (1.48)</td><td>3.34 (2.56)</td><td>1.71 (1.09)</td><td>2.57 (1.75)</td><td>2.13</td><td>.018</td><td>.147</td></tr></tbody></table>
<table><tr><td>Mathematical language</td><td>19.68 (3.15)</td><td>20.81 (2.68)</td><td>19.93 (3.99)</td><td>20.81 (3.10)</td><td>0.45</td><td>.004</td><td>.505</td></tr><tr><td>Non-verbal reasoning</td><td>9.20 (4.57)</td><td>11.54 (5.15)</td><td>9.64 (4.89)</td><td>12.07 (4.72)</td><td>0.01</td><td>&lt;.001</td><td>.909</td></tr></table>

The time x condition interactions for ordering and transcoding were further analyzed in more detail. For ordering, children in the experimental condition improved from pretest to posttest (difference in condition means = 2.085, t(115) = 5.710, 95% CI [1.11, 3.07], Cohen's d =.578, $p_{bonf}$ < .001) while children in the control condition did not (difference in condition means = 0.397, (t(115) = 1.077, 95% CI [-0.59, 1.39], Cohen's d =0.11, $p_{bonf}$ = 1.00). For transcoding, children in the experimental condition improved from pretest to posttest (difference in condition means = 2.153, t(115) = 4.917, 95% CI [0.98, 3.33], Cohen's d =.456, $p_{bonf}$ < .001) while children in the control condition did not (difference in condition means = 0.672, t(115) = 1.523, 95% CI [-0.51, 1.86], Cohen's d =.142 $p_{bonf}$ = .783).

In an additional non-pre-registered exploratory analysis, we re-analyzed the data of the arithmetic task by separating non-symbolic and symbolic items. This was done to verify whether the impact of the intervention depended on the complexity of the task, i.e., non-symbolic (easy) vs. symbolic (difficult) items (Girard et al., 2021) (see section 2.1 of the Supplementary Material, **Figure S2**). Repeated measures ANOVAs revealed significant interaction effects between condition and time for symbolic items ($F(1,115) = 6.35, \eta^2p = .052, p = .013$) but not for non-symbolic items ($F(1,115) = 0.03, \eta^2p < .001, p = .863$). This significant interaction indicated that children in the experimental condition improved from pretest to posttest in symbolic arithmetic (difference in condition means = 0.966, t(115) = 5.723, 95% CI [0.51, 1.42], Cohen's d =.801, $p_{bonf}$ < .001) while children in the control condition did not (difference in condition means = 0.362, t(115) = 2.127, 95% CI [-0.10, 0.82], Cohen's d = .300 $p_{bonf}$ = .213).
*Intervention effects on the home numeracy environment*

In both intervention conditions, the home learning environment changed over the course
of the intervention by increasing the number of learning activities that parents shared with their
children. Over the 6-week period, parents reported doing a total of 25.7 intervention activities
with their child (see **Table 3** for details in each condition). For the subsample of parents who
completed the questionnaire both before and after the intervention (n = 38 in the experimental
condition, n = 45 in the control condition), we conducted repeated-measures ANOVA on
parental expectations (regarding their children's numeracy learning) and home numeracy
activities with time as within subject-factor and condition as between-subject factor (**Figure 4**).
Consistent with our preregistered plan, we did not impute or otherwise adjust for missing
posttest questionnaire data. As a non-preregistered exploratory check, we compared pretest
characteristics of the children whose parents returned the posttest questionnaire with those who
did not. There were no substantial differences between these two groups.

**Figure 4**

Pretest and posttest HNE scores (n = 83)
<figure><img src="image_7.png" /></figure>

There was a significant main effect of time on parental numeracy expectations ($F(1,81) = 5.775, p = .019, \eta^2p = .067$) but not on home numeracy activities ($F(1,81) = 2.894, p = .093, \eta^2p = 0.34$). There was no significant effect of condition on parental numeracy expectations ($F(1,81) = 0.142, p = .707, \eta^2p = .002$) and on home numeracy activities ($F(1,81) = 1.846, p = .178, \eta^2p = .022$). ANOVAs revealed no significant interaction effects between condition and time on the home numeracy activities ($F(1,81) = 2.423, p = .123, \eta^2p = .029$) and parental numeracy expectations ($F(1,81) = 3.334, p = .072, \eta^2p = .040$) (Figure 4, Table S3).

Additional non-preregistered exploratory analyses were done at the item level to provide information about changes in specific home numeracy activities (e.g., numerical board and card game activities, practicing reading numbers up to 10 or 20) and about parents' expectations of their children's numeracy skills over time. These data are reported in the supplementary materials section (Tables S4 and S5).
## Moderation analyses

We conducted exploratory moderation analyses on outcome variables for which the intervention had significant effects. This was done by investigating the three-way interactions of time x condition x moderator for transcoding, ordering and symbolic arithmetic. Three moderators were examined: children's overall pretest numeracy scores, pretest scores on home numeracy activities, or intervention implementation (the number of weekly surveys completed). For each of these moderators, the time x condition x moderator interactions were not significant (transcoding all *Fs* < 0.141, all *ps* > .708; ordering, all *Fs* < 1.575, all *ps* > .212; symbolic arithmetic, all *Fs* < 2.028, all *ps* > .157), while time x condition interactions remained significant.

## Sensitivity analyses

As pre-registered, we also included additional analyses that allowed us to account for the nested structure of the data. Linear mixed models on transcoding and ordering skills in children with repeated measures nested within participants (specified as random slope and random intercept) yielded exactly the same results as the aforementioned ANOVAs. Adding random intercept for school and classroom (random slopes were not included because they resulted in an overfitted model) did not change the results (transcoding: time x condition interaction, *F*(1,115) = 5.666, *p* = .019; ordering: time x condition interaction, *F*(1,115) = 10.599, *p* = .001).

We further conducted sensitivity analyses to ensure that the different response profiles from the parents to the questionnaires (pretest, posttest, and weekly) did not influence the results we obtained. To this end, we performed the same ANOVA analyses only for the participants
whose parents completed all the three measurement points. These analyses did not reveal any
differences from the abovementioned reported results.

Discussion

The HNE, and particularly the frequency of home numeracy activities, has been
identified as critical for children’s numeracy development (Daucourt et al., 2021). However,
most of the existing evidence is correlational, and although recent HNE intervention studies
have led to promising results, their overall findings remain mixed (Libertus, 2024).
Methodological and theoretical limitations across this body of research still hinder firm
conclusions regarding the causal impact of the HNE on children’s numeracy skills. The present
intervention was specifically designed to address these limitations. We examined the effect of
a more realistic and ecologically valid intervention designed to increase parent-child
engagement in numeracy activities at home—a core component of the HNE. This intervention
included a variety of activities and did not require extensive parental training. We compared
the effect of this intervention to a closely matched active control condition that had similar
cognitive demands but no numeracy content. This design enabled us to test the causal effect of
the HNE intervention on children’s numeracy skills. The study was adequately powered and
ensured careful randomization across the two conditions. We also examined the intervention’s
impact on HNE as reported by parents in a questionnaire, both to assess any changes in the
HNE and to verify that families were evenly distributed in terms of pre-existing HNE factors.
This study was conducted in a Western European context (Flanders, Belgium), broadening
findings beyond American contexts.

As hypothesized, children in the HNE intervention condition showed specific
improvements in their numerical skills compared to those in the control condition. Notably,
these effects were obtained through an ecologically valid design that closely reflected the type
of activities parents typically engage in at home. In intervention research, ecological validity
refers to the extent to which findings can be generalized to real-world settings and can be
implemented by typical caregivers (Ledford et al., 2016). In the present study, ecological
validity was ensured by the use of activities (such as storybooks, board games, card games) that
are already commonly used in many families (Girard et al., 2023; Mutaf-Yıldız et al., 2018a)
and that required minimal external supervision or parental training. This approach contrasts
with earlier HNE studies that relied on rigid, highly scripted interactions or repetitive tasks
conducted in controlled settings (e.g., Sonnenschein et al., 2016; Gibson et al. 2020).

Due to the lack of significant effects in earlier HNE interventions studies (Sonnenschein
et al., 2016), more recent research reporting positive outcomes has limited the nature of the
interventions to more laboratory-based settings with greater experimental control. While such
designs may enhance internal validity, they often compromise ecological validity by moving
away from real-world applicability (Ledford et al., 2016). To the best of our knowledge, our
study is the first to demonstrate positive effects of an HNE intervention on preschoolers'
numeracy skills under realistic, ecologically valid conditions – combining a variety of playful
activities with minimal supervision. The strong adherence to our intervention, reflected in a low
attrition rate of 13.7%, compared to 50% in Sonnenschein et al. (2016), further supports its
relevance to typical home practices (Chezan et al., 2022).

From a theoretical perspective, our findings support a causal role of the HNE in
children’s numeracy development. They demonstrate that flexible, parent-led interventions
with minimal constraints can significantly improve children’s numeracy learning. This aligns
with a growing body of evidence that highlights the value of interventions that align with
families’ daily routines (Leyva et al., 2023) and demonstrates that increasing home numeracy
activities can support children’s early mathematical learning. From an applied perspective, such
findings motivate the development of cost-effective and easily to implement interventions that
have the potential to reach diverse populations. An important strength of our study lies in the
use of an active control condition, in which families engaged in activities that were similar in
structure and level of engagement, but without any numeracy content. Participation and
adherence were equivalent across groups, which allows us to attribute the observed effects of
the HNE intervention specifically to its numerical content.

Our findings on ordering, transcoding and symbolic arithmetic build on previous
observational studies that reported associations between children's numeracy skills and the
frequency of home numeracy activities (Daucourt et al., 2021). They are also consistent with
previous laboratory-based HNE intervention studies showing positive effects on children’s
numeracy, particularly on transcoding or ordering (Cheung & McBride, 2017; Cheung &
McBride-Chang, 2015; Dulay et al., 2019). Moreover, our results highlight the importance of
ordering skills, which have been identified as predictors of later arithmetic development (Lyons
et al., 2014; Xu & LeFevre, 2021).

In contrast to previous HNE intervention studies (Cheung & McBride-Chang, 2015;
Purpura et al., 2021), we did not observe specific intervention effects on counting or
mathematical language. We also did not find effects on non-symbolic arithmetic skills. To our
knowledge, no prior study has tested the effect of an HNE intervention on non-symbolic
arithmetic skills. However, correlational findings on this question remain mixed, with some
studies reporting associations (Skwarchuk et al., 2014; Susperreguy et al., 2020), and others not
(e.g., Mutaf-Yıldız et al., 2018a). One possible explanation for these null findings is the
presence of ceiling effects for mathematical language and counting, as well as a floor effects
for non-symbolic arithmetic, which may have limited our ability to detect intervention-related
changes. Alternatively, these specific skills may not have been as directly or effectively targeted
by the intervention. For mathematical language, the absence of group differences might also be
explained by the fact that both intervention conditions involved storybook reading. This shared
linguistic component could have induced a general language effect, thereby reducing the
likelihood of detecting differences between groups on specific mathematical language.

We also examined the effects of the intervention on the characteristics of the HNE. Parents in the experimental condition did not report overall changes in their numeracy activities or in their expectations regarding their children's numeracy learning. While these results may initially appear disappointing, they are not entirely unexpected. The HNE questionnaire covers a wide range of activities and expectations, many of which fall outside the scope of the intervention.

Consequently, the total scores from this questionnaire may not have fully captured the specific activities promoted during the intervention period. Based on the descriptive item-level data, we tentatively suggest that the intervention may have influenced responses for specific activities more directly targeted – such as playing games with numbers. However, these subtle effects might not have been adequately captured by the questionnaire used, which lacked the granularity needed to capture specific small changes in narrowly defined aspects of the HNE.

Future studies should consider to use more detailed and context-sensitive measures to
systematically assess such targeted changes.

Another possibility is that by focusing on the specific activities proposed in the intervention, parents might have reduced the time spent on other numeracy-related activities they would otherwise have engaged in. In other words, their attention may have shifted towards the intervention content at the expense of other numerical activities, potentially explaining the lack of change in overall HNE scores. Moreover, the six-weeks duration may have been too short for families to develop lasting modifications in their HNE practices. While the intervention might have temporarily increased parents' awareness of numeracy-related interactions, it remains unclear whether these shifts can lead to enduring changes in the home environment. Further research is necessary to explore these potential mechanisms of change,
ideally using longer interventions and follow-up assessments to evaluate the stability and long-
term evolution of both the HNE and children's numeracy skills.

Differently from previous intervention studies, we examined potential moderators of the
intervention's effects. This is important given the likelihood of inter-individual variability in
response to intervention. In contrast to prior findings suggesting that children with higher
pretest numeracy skills benefit more from HNE interventions (Gibson et al., 2020; Silver et al.,
2024), we found no evidence that initial skill levels moderated the intervention's effects.
Similarly, the level of intervention implementation did not moderate children's outcomes, in
contrast with earlier studies (Ramani & Scalise, 2020; Sonnenschein et al., 2016). One possible
explanation lies in the high ecological validity and flexibility of our design, which allowed
families to choose from a range of activities without being bound by strict implementation
guidelines. This inclusive approach may have reduced variability in adherence and contributed
to a more consistent overall impact.

Two recent meta-analyses on the effects of home numeracy interventions provide valuable benchmarks for contextualizing our findings (Cahoon et al., 2023; Nelson et al., 2024). Nelson et al. (2024) reported a statistically significant effect ($d = 0.26$) for early math interventions conducted in informal learning environments in preschool to third grade and Cahoon et al. (2023) reported a small effect ($d = 0.18$), in home-based math interventions targeting preschoolers. The effect of our intervention was larger, within the 95% confidence intervals of these two meta-analyses, with medium effect sizes, that is $d = 0.44$ for transcoding and $d = 0.61$ for ordering. Interestingly, Nelson et al. (2024) emphasized that the most effective interventions often relied on affordable materials—such as board games and dice—and did not require costly digital tools. These features closely align with our intervention design and this might explain why we observed a larger effect than in the abovementioned meta-analyses.
Both meta-analyses emphasized the importance of the engagement with parents during the intervention as a key moderator of intervention effectiveness. Nelson et al. (2024) reported significantly larger effects when caregiver training was combined with follow-up support; Cahoon et al. (2023) suggested that limited caregiver involvement—often restricted to brief instructions or text messages—reduced the impact of some interventions. Different from these suggestions, the current intervention with minimal parental training and supervision yielded robust medium effects. This suggests that an alternative model—based on the autonomous use of engaging, developmentally appropriate, and flexible materials integrated to family routines—can be both feasible and effective in real-world contexts.

## Limitations and future directions

When interpreting these findings, some limitations of the current study should be considered. First, our study did not include a delayed posttest measure. As a result, we cannot determine whether the observed intervention effects were sustained over time. Future studies should include follow-up assessments to examine the sustainability of intervention effects.

Second, our moderation analyses were exploratory, and our sample size might have been too small to detect potential moderating effects. Replication with larger samples is necessary to more carefully investigate the factors that may influence intervention responsiveness.

Third, although we attempted to recruit a broad and diverse sample across different classes and schools, the final sample predominantly included families from relatively advantaged socioeconomic backgrounds. Despite our efforts, families from low socioeconomic backgrounds did not participate in our study. Therefore, we cannot rule out the possibility that the effects of the current intervention might differ when families from lower socioeconomic backgrounds are considered (Dulay et al., 2019). Future research should allocate
more resources to the targeted recruitment of these populations, for example through teachers or healthcare professionals (Mazzocco, 2016).

Fourth, our study was conducted in the Flemish region of Belgium, and the generalizability of the findings to the broader European context remains uncertain. Cultural, educational, and socio-economic factors vary considerably across European countries and may influence both the structure of the HNE (see Hannula-Sormunen et al., 2023 for an example) and its effects on children's numeracy development. Future research should seek to replicate this study across diverse European contexts to assess the broader applicability of the intervention.

Fifth, the design of our intervention did not allow us to isolate the effects of individual components (e.g., storybooks vs. games). Rather than relying on a single repetitive activity, we adopted a flexible, ecologically valid approach using varied activities that reflect realistic home practices. While this enhances ecological validity, it limits our ability to identify which specific elements were the most effective.

Sixth, the arithmetic task used in this study showed low reliability at pretest, likely due to its difficulty for this age group. The task was nonetheless included for multiple reasons. We used this difficult task to be able to detect intervention effects on a task that is routinely used in older children before the start of formal schooling (Jordan et al., 2006). Arithmetic skills are also an integral part of children's early numerical competencies, which are typically measured via the same task as the one we employed (e.g., Jordan et al., 2006; Purpura et al., 2013). Notably, the task's reliability improved significantly at posttest (Cronbach's alpha = .73), suggesting that it became more reliable as children's skills improved. Nevertheless, the low reliability of the task at pretest should be acknowledged and future studies might consider refining this task or exploring alternative measures of arithmetic to ensure consistent reliability across time points.
Seventh, at the time of our pre-registration, no meta-analysis specific to HNE interventions was available. We based our initial sample size calculation on a meta-analysis of digital interventions for children with math difficulties (Benavides-Varela et al., 2020), which was not fully aligned with our study's population or context. Since then, two meta-analyses have provided more directly relevant effect size estimates for HNE interventions (Cahoon et al., 2023; Nelson et al., 2024). Based on these more recent estimates, we conducted additional, non-pre-registered, power analyses, which indicated that our final sample of 117 participants was adequately powered to detect effects of the magnitude typically observed in HNE intervention studies– thereby supporting the robustness of our design.

Eighth, the classroom-level randomization in our study was implemented to control for potential confounding effects related to preschool experiences. We therefore ensured that children within the same classroom had comparable school environments. However, this approach could have introduced the possibility that parents from different groups (experimental vs. control) within the same classroom shared some of the activities and it was not possible to actively monitor or prevent such sharing.

Lastly, both recent meta-analyses (Cahoon et al., 2023; Nelson et al., 2024) suggested that more structured and controlled interventions may be more effective in improving children's numeracy skills than broader, ecologically valid formats. Because our study did not include a highly controlled condition, direct comparisons between these two types of interventions were not possible. Future research could consider comparing structured versus flexible intervention models to better understand the trade-offs between internal and ecological validity in promoting early numeracy.

**Conclusion**
Our pre-registered randomized pretest-posttest intervention study shows that an
ecologically valid HNE intervention can improve numeracy – specifically transcoding and
ordering skills – in 4-to-5-year-old children. While the intervention did not lead to significant
changes in broader HNE factors, likely because it targeted a specific subset of activities, our
findings support a causal role of the HNE in children’s numeracy. From an applied perspective,
these findings highlight the potential of low-cost, home-based numeracy interventions, centered
on playful and adaptable activities tailored to real-life situations, to meaningfully support the
development of children's numerical skills.

References

Allen, M., Poggiali, D., Whitaker, K., Marshall, T. R., van Langen, J., & Kievit, R. A. (2021).
Raincloud plots: A multi-platform tool for robust data visualization. *Wellcome Open
Research, 4*, 63. https://doi.org/10.12688/wellcomeopenres.15191.2

Bakker, M., Torbeyns, J., Verschaffel, L., & De Smedt, B. (2023). Longitudinal pathways of numerical abilities in preschool : Cognitive and environmental correlates and relation to primary school mathematics achievement. *Developmental Psychology, 59*(3), 442-459.
https://doi.org/10.1037/dev0001491

Bakker, M., Torbeyns, J., Wijns, N., Verschaffel, L., & De Smedt, B. (2019). Gender equality in 4- to 5-year-old preschoolers’ early numerical competencies. *Developmental Science, 22*(1), e12718. https://doi.org/10.1111/desc.12718

Benavides-Varela, S., Zandonella Callegher, C., Fagiolini, B., Leo, I., Altoè, G., & Lucangeli, D. (2020). Effectiveness of digital-based interventions for children with mathematical learning difficulties : A meta-analysis. *Computers & Education, 157*, 103953.
https://doi.org/10.1016/j.compedu.2020.103953

Cahoon, A., Cassidy, T., & Simms, V. (2017). Parents’ views and experiences of the informal and formal home numeracy environment. *Learning, Culture and Social Interaction, 15*(Supplement C), 69-79. https://doi.org/10.1016/j.lcsi.2017.08.002

Cahoon, A., Jiménez Lira, C., Estévez Pérez, N., Benavides Pando, E. V., Campver García, Y., Susana Paz García, D., & Simms, V. (2023). Meta-Analyses and Narrative Review of Home-Based Interventions to Improve Literacy and Mathematics Outcomes for Children Between the Ages of 3 and 5 Years Old. *Review of Educational Research, 00346543231212491*. https://doi.org/10.3102/00346543231212491

Cheung, S. K., & McBride, C. (2017). Effectiveness of parent–child number board game playing in promoting Chinese kindergarteners’ numeracy skills and mathematics interest. *Early Education and Development, 28*(5), 572-589.
https://doi.org/doi.org/10.1080/10409289.2016.1258932

Cheung, S. K., & McBride-Chang, C. (2015). Evaluation of a Parent Training Program for Promoting Filipino Young Children’s Number Sense with Number Card Games. *Child Studies in Asia-Pacific Contexts, 5*(1), 39-49. https://doi.org/10.5723/csac.2015.5.1.039
Chezan, L. C., McCammon, M. N., Drasgow, E., & Wolfe, K. (2022). The Ecological Validity of Research Studies on Function-Based Interventions in Schools for Children With Autism Spectrum Disorder. *Behavior Modification, 46*(1), 202-229.

https://doi.org/10.1177/0145445520964921

Ciping, D., Silinskas, G., Wei, W., & Georgiou, G. K. (2015). Cross-lagged relationships between home learning environment and academic achievement in Chinese. *Early Childhood Research Quarterly, 33*, 12-20. https://doi.org/10.1016/j.ecresq.2015.05.001

Daucourt, M. C., Napoli, A. R., Quinn, J. M., Wood, S. G., & Hart, S. A. (2021). The home math environment and math achievement: A meta-analysis. *Psychological Bulletin, 147*(6), 565-596. https://doi.org/10.1037/bul00000330

De Keyser, L., Bakker, M., Rathé, S., Wijns, N., Torbeyns, J., Verschaffel, L., & De Smedt, B. (2020). No Association Between the Home Math Environment and Numerical and Patterning Skills in a Large and Diverse Sample of 5- to 6-year-olds. *Frontiers in Psychology, 11*, 547626. https://doi.org/10.3389/fpsyg.2020.547626

Dowker, A. (2021). Home Numeracy and Preschool Children's Mathematical Development: Expanding Home Numeracy Models to Include Parental Attitudes and Emotions. *Frontiers in Education, 6*(575664). https://doi.org/10.3389/feduc.2021.575664

Dulay, K. M., Cheung, S. K., Reyes, P., & McBride, C. (2019). Effects of parent coaching on Filipino children's numeracy, language, and literacy skills. *Journal of Educational Psychology, 111*(4), 641-662. https://doi.org/10.1037/edu0000315

Gibson, D. J., Gunderson, E. A., & Levine, S. C. (2020). Causal Effects of Parent Number Talk on Preschoolers' Number Knowledge. *Child Development, 91*(6), e1162-e1177. https://doi.org/10.1111/cdev.13423

Girard, C., Bastelica, T., Léone, J., Epinat-Duclos, J., Longo, L., & Prado, J. (2021). The relation between home numeracy practices and a variety of math skills in elementary school children. *PLOS ONE, 16*(9), e0255400. https://doi.org/10.1371/journal.pone.0255400

Girard, C., Longo, L., Chesnokova, H., Epinat-Duclos, J., & Prado, J. (2023). To what extent do home numeracy practices and parental number talk relate to children's math skills? A pre-registered study in 5-year-old children. *Learning and Individual Differences, 106*, 102328. https://doi.org/10.1016/j.lindif.2023.102328

Hannula-Sormunen, M., Batchelor, S., Torbeyns, J., Simms, V., Nanu, C., Laakkonen, E., & De Smedt, B. (2023). Age group differences in SFON tendency and arithmetical skills of four to seven year olds in four countries with different school starting ages. *Cognitive Development, 66*, 101296.

Hart, S. A., Little, C., & Bergen, E. van. (2021). Nurture might be nature: Cautionary tales and proposed solutions. *Npj Science of Learning, 6*(1), Article 1. https://doi.org/10.1038/s41539-020-00079-z

Hornburg, C. B., Borriello, G. A., Kung, M., Lin, J., Litkowski, E., Cosso, J., Ellis, A., King, Y., Zippert, E., Cabrera, N. J., Davis-Kean, P., Eason, S. H., Hart, S. A., Iruka, I. U., LeFevre, J.-A., Simms, V., Susperreguy, M. I., Cahoon, A., Chan, W. W. L., ... Purpura, D. J. (2021). Next Directions in Measurement of the Home Mathematics Environment: An International and Interdisciplinary Perspective. *Journal of Numerical Cognition, 7*(2), 195-220. https://doi.org/10.5964/jnc.6143

Jordan, N. C., Kaplan, D., Nabors Oláh, L., & Locuniak, M. N. (2006). Number sense growth in kindergarten: A longitudinal investigation of children at risk for mathematics difficulties. *Child Development, 77*(1), 153-175. https://doi.org/10.1111/j.1467-8624.2006.00862.x

Ledford, J. R., Hall, E., Conder, E., & Lane, J. D. (2016). Research for Young Children With Autism Spectrum Disorders: Evidence of Social and Ecological Validity. *Topics in Early Childhood Special Education, 35*(4), 223-233. https://doi.org/10.1177/0271121415585956
LeFevre, J.-A., Skwarchuk, S.-L., Smith-Chant, B., Fast, L., Kamawar, D., & Bisanz, J. (2009). Home Numeracy Experiences and Children's Math Performance in the Early School Years. *Canadian Journal of Behavioural Science/Revue canadienne des sciences du comportement*, 41, 55-66. https://doi.org/10.1037/a0014532

Leyva, D., Davis, A., & Skorb, L. (2018). Math Intervention For Latino Parents and Kindergarteners Based on Food Routines. *Journal of Child and Family Studies*, 27(8), 2541-2551. https://doi.org/10.1007/s10826-018-1085-5

Leyva, D., Libertus, M. E., & McGregor, R. (2023). Math intervention targeting family routines increases parental math talk and math activities. *Journal of Applied Developmental Psychology*, 89, 101595. https://doi.org/10.1016/j.appdev.2023.101595

Libertus, M. E. (2024). Parent-Focused Interventions to Support Children's Early Math Learning. *Current Directions in Psychological Science*, 09637214231212806.
https://doi.org/10.1177/09637214231212806

Lyons, I. M., Price, G. R., Vaessen, A., Blomert, L., & Ansari, D. (2014). Numerical predictors of arithmetic success in grades 1–6. *Developmental Science*, 17(5), 714-726.
https://doi.org/10.1111/desc.12152

Mazzocco, M. (2016). Mathematics Awareness Month: Why Should Pediatricians be Aware of Mathematics and Numeracy? *Journal of Developmental & Behavioral Pediatrics*, 37(3), 251-253. https://doi.org/10.1097/DBP.0000000000000294

Mutaf-Yıldız, B., Sasanguie, D., De Smedt, B., & Reynvoet, B. (2018a). Frequency of Home Numeracy Activities Is Differentially Related to Basic Number Processing and Calculation Skills in Kindergartners. *Frontiers in Psychology*, 9, 340.
https://doi.org/10.3389/fpsyg.2018.00340

Mutaf-Yıldız, B., Sasanguie, D., De Smedt, B., & Reynvoet, B. (2018b). Investigating the relationship between two home numeracy measures: A questionnaire and observations during Lego building and book reading. *British Journal of Developmental Psychology*, 36(2), 354-370. https://doi.org/10.1111/bjdp.12235

Napoli, A. R., Korucu, I., Lin, J., Schmitt, S. A., & Purpura, D. J. (2021). Characteristics Related to Parent-Child Literacy and Numeracy Practices in Preschool. *Frontiers in Education*, 6. https://doi.org/10.3389/feduc.2021.535832

Nelson, G., Carter, H., Boedeker, P., Knowles, E., Buckmiller, C., & Eames, J. (2024). A Meta-Analysis and Quality Review of Mathematics Interventions Conducted in Informal Learning Environments with Caregivers and Children. *Review of Educational Research*, 94(1), 112-152. https://doi.org/10.3102/00346543231156182

Niklas, F., Cohrsen, C., & Tayler, C. (2016). Improving Preschoolers’ Numerical Abilities by Enhancing the Home Numeracy Environment. *Early Education and Development*, 27(3), 372-383. https://doi.org/10.1080/10409289.2015.1076676

Niklas, F., Tayler, C., & Schneider, W. (2015). Home-based literacy activities and children's cognitive outcomes: A comparison between Australia and Germany. *International Journal of Educational Research*, 71, 75-85. https://doi.org/10.1016/j.ijer.2015.04.001

Purpura, D. J., Baroody, A. J., & Lonigan, C. J. (2013). The transition from informal to formal mathematical knowledge: Mediation by numeral knowledge. *Journal of Educational Psychology*, 105(2), 453-464. https://doi.org/10.1037/a0031753

Purpura, D. J., Schmitt, S. A., Napoli, A. R., Dobbs-Oates, J., King, Y. A., Hornburg, C. B., Westerberg, L., Borriello, G. A., Bryant, L. M., Anaya, L. Y., Kung, M., Litkowski, E., Lin, J., & Rolan, E. (2021). Engaging caregivers and children in picture books: A family-implemented mathematical language intervention. *Journal of Educational Psychology*, 113(7), 1338-1353. https://doi.org/10.1037/edu0000662
Ramani, G. B., & Scalise, N. R. (2020). It's more than just fun and games : Play-based mathematics activities for Head Start families. *Early Childhood Research Quarterly*, 50, 78-89. https://doi.org/10.1016/j.ecresq.2018.07.011

Rathé, S., Torbeyns, J., De Smedt, B., & Verschaffel, L. (2022). Longitudinal associations between spontaneous number focusing tendencies, numerical abilities, and mathematics achievement in 4-to 7-year-olds. *Journal of Educational Psychology, 114*(1), 37-55.
https://doi.org/doi.org/10.1037/edu0000665

Ribner, A., Silver, A. M., Elliott, L., & Libertus, M. E. (2023). Exploring effects of an early math intervention : The importance of parent–child interaction. *Child Development, 94*(2), 395-410. https://doi.org/10.1111/cdev.13867

Scalise, N. R., DePascale, M., Tavassolie, N., McCown, C., & Ramani, G. B. (2022). Deal Me in : Playing Cards in the Home to Learn Math. *Education Sciences, 12*(3), 190.
https://doi.org/10.3390/educsci12030190

Schulz, K. F., Altman, D. G., & Moher, D. (2010). CONSORT 2010 Statement : Updated guidelines for reporting parallel group randomised trials. *Trials, 11*, 32.
https://doi.org/10.1186/1745-6215-11-32

Silinskas, G., Sénéchal, M., Torppa, M., & Lerkkanen, M.-K. (2020). Home Literacy Activities and Children's Reading Skills, Independent Reading, and Interest in Literacy Activities From Kindergarten to Grade 2. *Frontiers in Psychology, 11*.
https://doi.org/10.3389/fpsyg.2020.01508

Silver, A. M., Elliott, L., Ribner, A. D., & Libertus, M. E. (2024). The benefits of math activities depend on the skills children bring to the table. *Developmental Psychology, 60*(2), 376-388. https://doi.org/10.1037/dev0001637

Skwarchuk, S.-L., Sowinski, C., & LeFevre, J.-A. (2014). Formal and informal home learning activities in relation to children’s early numeracy and literacy skills : The development of a home numeracy model. *Journal of Experimental Child Psychology, 121*, 63-84.
https://doi.org/10.1016/j.jecp.2013.11.006

Sonnenschein, S., Metzger, S. R., Dowling, R., Gay, B., & Simons, C. L. (2016). Extending an Effective Classroom-Based Math Board Game Intervention to Preschoolers' Homes. *Journal of Applied Research on Children, 7*(2). https://doi.org/10.58464/2155-5834.1304

Susperreguy, M. I., Douglas, H., Xu, C., Molina-Rojas, N., & LeFevre, J.-A. (2020).

Expanding the Home Numeracy Model to Chilean children : Relations among parental expectations, attitudes, activities, and children's mathematical outcomes. *Early Childhood Research Quarterly, 50*, 16-28. https://doi.org/10.1016/j.ecresq.2018.06.010

Turan, E., & De Smedt, B. (2023). Understanding mathematical development in preschool children : The association between mathematical language and mathematical abilities. *Cognitive Development, 66*, 101318. https://doi.org/10.1016/j.cogdev.2023.101318

Vandecruys, F., Vandermosten, M., & De Smedt, B. (2024). The inferior fronto-occipital fasciculus correlates with early precursors of mathematics and reading before the start of formal schooling. *Cortex, 174*, 149-163. https://doi.org/10.1016/j.cortex.2024.02.014

Xu, C., & LeFevre, J.-A. (2021). Children's Knowledge of Symbolic Number in Grades 1 and 2 : Integration of Associations. *Child Development, 92*(3), 1099-1117.
https://doi.org/10.1111/cdev.13473