INVESTIGATING ASSOCIATIONS BETWEEN PARENT ENGAGEMENT AND TODDLERS’ MATHEMATICS PERFORMANCE

Abstract

Early mathematics skills relate to later mathematics achievement and educational attainment, which in turn predict career choice, income, health, and financial decision-making. Critically, large differences exist among children in early mathematics performance, with parental mathematics engagement being a key predictor. However, most prior work has examined mothers’ mathematics engagement with their preschool- and school-aged children. In this Registered Report, we tested concurrent associations between mothers’ and fathers’ engagement in mathematics activities with their two- to three-year-old toddlers and children’s mathematics performance. Mothers and fathers did not differ in their engagement in mathematics activities, and both parents’ mathematics engagement related to toddlers’ mathematics skills. Fathers’ mathematics engagement was associated with toddlers’ number and mathematics language skills, but not their spatial skills. Mothers’ mathematics engagement was only associated with toddlers’ mathematics language skills. Critically, associations may be domain-specific, as parents’ literacy engagement did not relate to measures of mathematics performance above their mathematics engagement. Mothers’ and fathers’ mathematics activities uniquely relate to toddlers’ developing mathematics skills, and future work on the nuances of these associations is needed.

Introduction

Children’s early mathematics skills relate to later mathematics achievement and educational attainment (Duncan et al., 2007; Jordan et al., 2009; Nguyen et al., 2016; Siegler et al., 2012), which in turn predict career choice, employment, income, health, and financial decision-making (Agarwal & Mazumder, 2013; Currie & Thomas, 2001; Reyna & Brainerd, 2007; Trusty et al., 2000). Notably, children vary substantially in emerging mathematics skills already by kindergarten (Jordan et al., 2006; Starkey & Klein, 1992). Thus, understanding the contributors to early individual differences may inform recommendations for parents, schools, and society.

Even before formal schooling, mathematics skills undergo critical developmental change. In infancy, children display informal skills such as distinguishing between quantities non-verbally (see Libertus et al., 2020). In toddlerhood, children begin transitioning from nonverbal to verbal skills that rely more heavily on language and symbolic understanding. These foundational mathematics skills include basic number knowledge (e.g., recognizing and identifying numbers, learning to count, understanding cardinality), spatial skills (e.g., recognizing and identifying shapes, learning spatial relations), and mathematics language skills (e.g., comprehending and producing words/phrases that refer to number and space), all of which support children’s later learning of mathematics skills in school. However, little is known about how mathematics concepts of number, space, and children’s use of mathematics language emerge in the toddler period as children transition from solely possessing nonverbal mathematics skills to acquiring symbolically mediated mathematics skills. Furthermore, minimal work has examined individual differences in foundational mathematics skills during the toddler period, a time when these important skills are first emerging; and thus, very little is known about the factors that may influence these processes. As such, we ask whether variability in the home mathematics environment—a significant correlate of children’s mathematics performance at later ages—relates to variability in toddlers’ mathematics performance.

The current investigation is rooted in sociocultural theory—the perspective that children’s skills and experiences are influenced by interactions with their environment, including parents, peers, and the broader community and culture (Callanan & Valle, 2008; Moll, 2013; Rogoff, 1998; Rogoff et al., 2018). We apply this framework to a specific focus on mathematics-related activities in the proximal home environment and the social process through which parents support their children’s learning of specific mathematics skills.

The Home Mathematics Environment and Children’s Mathematics Performance

Sources of variability in children’s mathematics performance range from genetics (Hart et al., 2009) to social and environmental influences (Jordan & Levine, 2009). Most notably, researchers have spotlighted the role of parents’ engagement in informal, mathematics-related activities with children in the home—such as measuring ingredients while cooking or playing board games that require counting and number knowledge. As documented for parents’ literacy engagement and children’s early literacy skills (e.g., Burgess et al., 2002; Huntsinger et al., 2016; Manolitsis et al., 2013; Senechal & LeFevre, 2014), parents’ mathematics-related activities and discussions at home support children’s growing mathematics skills (Blevins-Knabe & Musun-Miller, 1996; Elliott et al., 2017; Gunderson & Levine, 2011; Huntsinger et al., 2016; Kleemans et al., 2012; LeFevre et al., 2009; Levine et al., 2010; Mutaf Yildiz et al., 2018; Niklas & Schneider, 2013; Ramani et al., 2015; Silver et al., 2020; Susperreguy & Davis-Kean, 2016; Thompson et al., 2017).

However, studies have failed to replicate associations between home mathematics activities and children’s mathematics performance, perhaps due to the complexity of associations and methodological differences across studies (DeFlorio & Beliakoff, 2014; Missall et al., 2014; Skwarchuk, 2009). Notably, measures of the home environment, including the variety of activities that researchers examine, differ among studies and may not capture the full range of ways that families engage in mathematics and other academic activities (see Elliott & Bachman, 2018). Additionally, researchers use different measures of mathematics outcomes, which may yield divergent results. Finally, studies target children of different ages, largely preschool- and school-age children, leaving open questions about whether even young children may benefit from parents’ mathematics engagement (see Thompson et al., 2017). Here, we test how parent engagement relates to different specific mathematics skills in their young toddlers.

Extending the Home Mathematics Environment to Younger Children

Research on the home mathematics environment has primarily focused on preschool- and school-aged children, with little attention to children’s experiences in infancy and toddlerhood—a time when children are developing foundational mathematics skills (Jordan & Levine, 2009; Starr et al., 2013). The handful of studies on parents’ engagement in mathematics activities with toddlers suggest that parents who frequently discuss numbers and mathematics concepts, regardless of context, tend to have children who perform better in mathematics than those who hear less frequent number talk (Casey et al., 2018; Gibson et al., 2020; Gunderson & Levine, 2011; Levine et al., 2010). These results parallel work showing associations between parents’ engagement in literacy activities during toddlerhood and children’s literacy performance in preschool and elementary school (e.g., Adamson et al., 2021; Dodici et al., 2003). Notably, no work has investigated associations between toddlers’ engagement in mathematics activities at home and their early mathematics skills. The current study aims to address this gap.

Specificity in the Home Mathematics Environment

As described, parent mathematics engagement positively correlates with children’s mathematics performance. However, associations may depend on the specific types of mathematics skills that researchers target. Most studies report associations between parent numeracy activities and children’s numeracy skills (Blevins-Knabe & Musun-Miller, 1996; Huntsinger et al., 2016; Kleemans et al., 2012; LeFevre et al., 2009; Mutaf Yildiz et al., 2018; Niklas & Schneider, 2013; Ramani et al., 2015; Silver et al., 2020), with far less attention to other types of mathematics activities and other types of mathematics outcomes.

In addition to numeracy, children acquire spatial skills during toddlerhood, which are foundational to mathematics performance. Spatial skills involve the perception, retention, and manipulation of information on intrinsic (within-object; e.g., shape) and extrinsic (between-object; e.g., location) properties of people, objects, and space (Newcombe & Shipley, 2015). For example, differentiating between mirror images, mentally rotating an image, building a block construction, and navigating a playground all require children to invoke spatial skills. Children’s spatial skill performance correlates with their broader mathematics performance concurrently (e.g., Mix et al., 2016; Verdine et al., 2014) and longitudinally (Geer et al., 2019; Verdine et al., 2017). However, evidence on associations between parents’ engagement in spatial activities and children’s spatial skills are mixed. Some studies suggest that joint parent-child engagement with puzzles and blocks relate to children’s spatial skills (e.g., Jirout & Newcombe, 2015; Levine et al., 2012), whereas others do not (Dearing et al., 2012; Purpura et al., 2020; Zippert et al., 2021; Zippert & Rittle-Johnson, 2020). Here, we ask whether parents’ engagement across a broad set of mathematics activities relate to several measures of toddlers’ spatial skills.

Likewise, children’s mathematics language skills offer a window into their conceptual understanding of mathematics more generally. Beyond general language skills, mathematics language relates to young children’s mathematics performance in early childhood (Hornburg et al., 2018; Purpura & Logan, 2015; Purpura et al., 2016; Purpura & Reid, 2016; Toll & Van Luit, 2014). However, surprisingly little work has examined how parental engagement in home mathematics activities relates to children’s development of mathematics language. In one study, parents’ engagement in numeracy activities related to preschool-aged children’s mathematics performance and was mediated by children’s mathematics language skills (King & Purpura, 2021). Here we examine whether home activities involving broad mathematics concepts, including numeracy and spatial skills, extend downward to toddlers’ mathematics language.

Finally, studies of preschool- and school-aged children suggest cross-domain benefits of parent engagement, such that mathematics engagement may relate to literacy skills and vice versa (Huntsinger et al., 2016; Napoli & Purpura, 2018; Ribner et al., 2020). However, whether such cross-domain associations apply at younger ages, and whether parents’ engagement in literacy activities relates to specific mathematics skills remain unexamined. As such, we explore whether parental literacy engagement relates to toddlers’ mathematics performance domains of numeracy, spatial skills, and mathematics language.

Extending to Mothers and Fathers of Toddlers

Existing work on parents’ mathematics engagement with toddlers primarily focuses on mothers’ (Casey et al., 2018; Gibson et al., 2020; Gunderson & Levine, 2011; Levine et al., 2010). However, the importance of fathers’ contributions to children’s broader development has been widely acknowledged (see Cabrera, 2020; Taraban & Shaw, 2018), but not tested for the domain of mathematics. As such, we explore associations between both mothers’ and fathers’ mathematics engagement and toddlers’ mathematics performance.

The Current Study

We address several gaps on associations between parents’ engagement in mathematics activities and children’s mathematics skills through a secondary data analysis of a dataset consisting of measures of the home environment and mathematics performance from a diverse set of monolingual and bilingual families. First, we asked whether parents’ engagement in mathematics activities relates to the development of emerging mathematics skills in toddlerhood, thus extending to an understudied age group. Second, we tested specificity of parent-child associations by widening the lens to multiple mathematics skills—number knowledge, spatial skills, and mathematics language—and asking whether associations are specific to mathematics skills, or instead whether parents’ engagement in literacy activities also predicts toddlers’ mathematics performance. Finally, we moved beyond the dominant mother-centred focus by comparing mothers’ and fathers’ engagement in mathematics activities and associations to toddlers’ mathematics performance.

To this end, three series of hierarchical linear regression models tested associations to different measures of children’s mathematics performance (i.e., number skills, spatial skills, and mathematics language). For each mathematics skill, we examined whether mothers’ and fathers’ mathematics activities (e.g., counting, doing puzzles) related to toddlers’ performance, while covarying children’s gender (Chang et al., 2011; Jacobs & Bleeker, 2004; Thippana et al., 2020), children’s age (Thompson et al., 2017), language of task administration (Kung et al., 2020), children’s domain-general vocabulary skills (Purpura et al., 2011; Purpura & Logan, 2015; Toll & Van Luit, 2014), and mothers’ and fathers’ education (Schmitt et al., 2018), which have been shown to relate to parental mathematics activities and children’s mathematics skills.

We expected parents’ frequency of engaging in mathematics activities to be associated with toddlers’ mathematics performance across the three domains of mathematics knowledge. Furthermore, we expected associations to be robust (i.e., to maintain above controls as in previous work with older children, e.g., Anders et al., 2012; Silver et al., 2020; Toll & Van Luit, 2014). Next, we expected that parents’ frequency of literacy activities to be associated with children’s performance in the three mathematics domains, based on cross-domain findings in older children (e.g., Huntsinger et al., 2016; Napoli & Purpura, 2018; Ribner et al., 2020). However, associations between parental activities and skills may be highly specific at young ages, with mathematics activities relating to mathematics skills and literacy activities either not relating to mathematics performance or only with mathematics language. Finally, we expected variation in parent input (for both mothers and fathers) to relate to variation in toddlers’ emerging mathematics skills.

Methods

Participants

Participants were 220 toddlers and their biological mothers (n=206) and fathers (n=179). Children’s age was available for 215 toddlers (M age = 30.42 months, SD = 3.50 months, ranging from 23.80 months to 36.82 months of age). Because we used children’s age at the date of testing, we did not calculate age of the five children who did not complete any assessments. Children’s gender was reported by parents for 206 toddlers (53% female). All participants were reported by parents to be English-monolingual, English/Spanish bilingual, or Spanish-monolingual speakers, and were tested in their preferred language (80% English only, 15% English and Spanish, 5% Spanish only).

Requirements for inclusion in the broader study included family residence in the United States; family members speaking only English and/or Spanish in the home; identification of race and ethnicity for both parents as either White and non-Hispanic/Latino, White and Hispanic/Latino, or some other race and Hispanic/Latino; agreement of both the mother and father to participate; and parental age for both parents greater than 18 years. Additionally, children had to be between 24 and 36 months of age at the time of testing, born full term (i.e., 37 weeks of gestation or later, or between 35 and 37 weeks of gestation with a birth weight greater than or equal to 5.5 pounds), and have not experienced any diagnosed health or developmental problems since birth, including Autism Spectrum Disorder or other disabilities involving severe cognitive or motor issues that would prevent the child from completing the tasks. Due to experimenter error, one additional child participated at 21.21 months of age, and was excluded from analyses. An additional 15 families met inclusion criteria for the study and signed consent forms but did not complete any study procedures, and thus were not included in analyses.

Experimental Procedures

Data were collected as part of a collaborative study between three universities located in three metropolitan areas of the mid-Eastern United States. All data collection occurred remotely (January 2021 through October 2022), with families participating via Zoom from anywhere in the United States.

Administration of all materials occurred in English or Spanish as preferred by participants. Each parent completed an online questionnaire through Qualtrics that was emailed to them on their consent to participate. The questionnaire asked parents to report their frequency of engaging in academic activities with their child, their child’s expressive vocabulary including domain-general and math-specific vocabulary, and demographic information about the family. Additionally, each parent participated in a Zoom video call on a computer or tablet with their child, where they looked at and talked about pictures on the screen as if they were reading a book together, and parents completed a mathematics fluency assessment. The zoom interactions and mathematics fluency tasks are not relevant to the current paper, and thus not analysed further.

The order of parents’ video calls was counterbalanced across families, such that the father participated first for some of the families, and the mother participated first for the other families. During the first video call (regardless of which parent was participant), the experimenter administered all of the child assessments, testing children’s number skills and spatial skills. Children completed the Point-to-X task, the Point-to-Shape task, the Give-N task, and the Point-to-Spatial Relations task, always in that order. If children did not complete a task during the first video call, any items not administered were attempted at the video call with the second parent. During child assessments, parents were asked to hold up a second device (e.g., smartphone or tablet) that was also connected to the Zoom meeting. Parents were instructed to hold this device behind the child facing the family’s main screen to allow the experimenter to view the main screen and identify to which side of the screen children were pointing (see Figure 1 for example). All visits were video recorded to allow for offline coding. Parents were compensated $50 each, for a total of $100 per family. Families who completed all required study components were invited to select a toy (worth approximately $10) for their toddler as an extra thank-you gift.

Figure 1.

Example set up of Zoom study visit for pointing tasks

Measures

Children’s number skills

Scores from number tasks were averaged to create a composite score of children’s number skills.

Point-to-X.

Children viewed a series of paired images on the computer screen and for each, were asked to point to the one (left or right side) that showed the number of objects queried (Silver et al., 2021; Appendix A). To familiarize children with the task, children were given two practice trials with two different common objects and prompted to point to one image (e.g., “Which has a ball?”). Subsequently, in twelve number-word trials, each image showed two sets of identical stimuli differing only in number (e.g., four ducks and five ducks), and children were prompted to point to one of the images (e.g., “Which has four ducks?”). Number-word trials varied along three dimensions: (1) the numerical distance between the two sets (for “one-away” trials, the numbers differed by one; for “two-away” trials, the numbers differed by two; for “far-away” trials, the numbers differed by more than four); (2) the size of the target number (for eight trials the prompted number was small (1-4), and for four trials the prompted number was large (5-10)); and (3) the size of the response options (for five trials both numbers were small numbers (1-4), and for seven trials at least one of the numbers was large (5-10)). The side of the correct response was counterbalanced across trials.

Children received 1 point for each correct response. During administration, if children initially pointed to one image, then verbally indicated that they wanted to change their answer, the second point was counted as their response. In cases where children did not respond, the experimenter repeated the prompt once. If children still did not respond, the experimenter moved on to the next trial and children received 0 points for the trial. If children pointed to both images without clearly signalling their response, the experimenter prompted, “Remember, you can only choose one. Which has [number]?” After this prompt, if children continued to point to both images, they received 0 points for the trial. If children responded incorrectly to each of the first three number-word trials, the experimenter employed a stopping rule and ended the task. Fourteen children responded incorrectly to the first three number-word trials, however, due to experimenter error, 9 of these children were not stopped but should have been. In such cases, children’s scores would ordinarily have been imputed; however, because data were collected, we included children’s true scores in analyses. Children’s Point-to-X score was the percentage of trials for which they correctly pointed to the requested number.

Videos were coded by trained research assistants to identify which image children pointed to for each trial. A second researcher double-coded 20% of videos to assess inter-coder reliability. Inter-coder reliability was excellent; agreement for each trial ranged from 90.24% to 100% (κ’s from .81 to 1.00).

Give-N.

Children completed a Give-N task, where they were asked to create a set containing a number of items requested by the researcher (based on Wynn, 1990, 1992). Prior to the video visits, parents were asked to supply their child with a plate and six matching spoons. During the Give-N task, they were asked to place the plate in front of their child and put the spoons in a pile next to the plate.

Children were shown “Ellie the Elephant”, an elephant puppet held up to the webcam by the researcher. To introduce children to the game, children were shown the puppet and told that the “Ellie the elephant” loves to eat ice cream. They were asked to help “feed” Ellie by placing a set of spoons on the plate so the puppet could eat ice cream. The experimenter said, “Look, let’s feed Ellie!” and mimed placing a spoon on a plate that the experimenter held. Then the experimenter held the puppet up to the webcam and enacted the puppet “eating” ice cream off the spoon and saying, “Yum yum yum!”

Children were asked to produce sets of spoons in different numbers to “feed” Ellie scoops of ice cream. For each trial, children were asked, “Can you give Ellie [number] spoons?” and were instructed to put the set of spoons on the plate. After the child paused for more than 3 seconds or indicated that they were done creating the set, the experimenter prompted confirmation from the children, “Is that [number]?” If children said yes or nodded, the experimenter held the puppet up to the webcam and said, “Yum yum yum! Thank you!” If children said no or shook their head, they were given one chance to correct their response, and were instructed, “Ok, well, Ellie wants [number] spoons. Can you give Ellie [number] spoons on the plate?” Once children adjusted the number of spoons, or paused for more than 3 seconds, the experimenter held the puppet up to the webcam and said, “Yum yum yum! Thank you!” The spoons were then removed from the plate before the next trial. Children received 1 point for each correct response. If children did not respond to a trial, the experimenter repeated the prompt one time. If children still did not respond, the experimenter moved on to the next trial and children were considered to have responded incorrectly and received 0 points for that trial. All children were administered 8 total trials, always in the same order, where they were asked twice each for sets of 1 through 4 spoons (Appendix A presents the order of items). The percentage of trials where children produced the correct set size was used as their Give-N score.

Videos were coded by trained researchers to identify the number of spoons children produced for each trial. A second researcher coded 20% of videos to ensure reliability; and inter-coder reliability was excellent. Agreement for each trial ranged from 94.87% to 100% (κ’s from .72 to 1.00).

Children’s spatial skills

Scores from spatial skills tasks were averaged to create a composite score of children’s spatial skills.

Point-to-Shape.

Children viewed a series of images on the computer screen and were asked to point to the prompted shape (Appendix A). To familiarize children with the Point-to-Shape task, children were first given two practice trials with two different common shapes and were prompted to point to one image (e.g., “Where is the heart?”). Subsequently, in eight test trials, each trial showed two shapes on either side of the screen, and children were prompted to point to one of the images (e.g., “Where is the triangle?”). Children’s responses were handled and scored in the same way as the Point-to-X task. If children responded incorrectly to each of the first four test trials, the experimenter employed a stopping rule and did not continue to administer the remaining trials. Two children responded incorrectly to the first four test trials; however, due to experimenter error, both children were not stopped but should have been. In such cases, children’s scores would ordinarily have been imputed; however, because data were collected, we included children’s true scores in analyses. Children’s Point-to-Shape score was the percentage of trials for which they correctly pointed to the requested shape.

A second researcher coded 20% of videos to ensure reliability, and inter-coder reliability was excellent; agreement for each trial ranged from 92.54% to 98.53% (κ’s from .74 to .94).

Point-to-Spatial Relations.

Children viewed a series of images on the computer screen and were prompted to point to the image showing a particular spatial relation (Appendix A). In seven trials, two images on either side of the screen were presented, each showing a scene with a toy tiger and a cup. Children were prompted to point to one of the images (e.g., “Where is the tiger UNDER the cup?”). Children’s responses were handled and scored in the same way as the Point-to-X task. If children responded incorrectly to each of the first three trials, the experimenter employed a stopping rule and did not continue to administer the remaining trials. Seven children responded incorrectly to the first three trials, however, due to experimenter error, 3 of these children were not stopped but should have been. In such cases, children’s scores would ordinarily have been imputed; however, because data were collected, we included children’s true scores in analyses. Children’s Point-to-Spatial Relations score was the percentage of trials for which they correctly pointed to the image displaying the requested spatial relation between the tiger and the cup.

A second researcher coded 20% of videos to ensure reliability, and inter-coder reliability was excellent; agreement for each trial ranged from 93.85% to 100% (κ’s from .81 to 1.00).

Children’s language skills

Parents completed a vocabulary checklist reporting their child’s expressive vocabulary. Parents were instructed “Children understand many more words than they say. Right now, we are particularly interested in the words your child says. You're going to see a list of words and we ask that you select the words that you have heard your child say. Be sure to count words that your child says in any other language(s), as well as mispronunciations, like “sketti” for “spaghetti” or “raffe” for “giraffe”.”

Domain-General Vocabulary.

Parents reported their children’s expressive domain-general vocabulary by reading through a list of words (e.g., “train,” “dog,” “garden”). Domain-general vocabulary words were selected from the MacArthur-Bates Communicative Development Inventories Vocabulary Checklist (MCDI, (Fenson, 2007); see Appendix for items). To reduce the burden to parents, 37 words from the MCDI were chosen. Children’s domain-general vocabulary composite scores were calculated as the average of the number of words that mothers and fathers reported their children to have used. In cases where only one parent of a family completed the survey, the second parent’s score was imputed.

Mathematics Language.

Parents also reported their children’s expressive mathematics-specific vocabulary by checking which of 37 mathematics words (e.g., “three,” “triangle,” “more”) their child said based on a mathematics word inventory created for this study (see Appendix). Children’s mathematics language composite scores were calculated as the average of the number of mathematics words mothers and fathers reported their children to have used. Similar to the domain-general vocabulary score, if only one parent of a family completed the survey, the second parent’s score was imputed.

Parents’ home activities

Parents reported the frequency of home activities they engaged in with their children via questionnaire (see Appendix for items). The full Home Activities questionnaire contained 19 items, probing mathematics, literacy, motor, and music/art activities. Here we focused only on the mathematics and literacy items.

Parent mathematics activities.

Parents reported the frequency of home mathematics activities they engaged in with their children. Each parent was asked to indicate how often in the past month they had participated in listed mathematical activities (e.g., “Counting objects,”) with their child at home on a scale from 1 (“Did not occur”) to 5 (“Almost daily”). Responses for the 11 math-related items were averaged to create a mathematics activities score for each parent.

Parent literacy activities.

Parents reported the frequency of home literacy activities they engaged in with their children. Each parent was asked to indicate how often in the past month they had participated in listed literacy activities (e.g., “Identifying sounds of alphabet letters”) with their child at home on a scale from 1 (“Did not occur”) to 5 (“Almost daily”). Responses for the 4 literacy-related items were averaged to create a literacy activities score for each parent.

Demographic Information

Family demographics were measured via parent questionnaire. Parents reported their level of education (dichotomized as at least a college degree [1] or less than a college degree [0]), their child’s preferred language for task administration (dichotomized as English-only [1] or a combination of English and Spanish or Spanish-only [0]), their child’s gender (dummy coded to reflect female [1] or male [0]), and their child’s birthdate, which was used to calculate the child’s age in months on the date of testing.

Preregistered Data Analysis Pipeline

Analyses were preregistered on the Open Science Framework upon acceptance of the Stage 1 Registered Report (https://doi.org/10.17605/OSF.IO/NGX4E). As this was a secondary analysis of previously collected data, preregistration occurred after data collection but prior to any data inspection or analysis. All data, analysis scripts, and materials can be found on the Open Science Framework (https://osf.io/872yg/). Data for a particular trial of a task were excluded from analyses if the experimenter made an error during administration (e.g., using an incorrect prompt for a trial, failing to ask all questions necessary for administering the trial, giving the child feedback on their response), if the parent interfered (e.g., answering for the child, encouraging the child to use a particular strategy to respond to a trial, giving the child feedback on their response), or if the video recording was of insufficient quality to code the child’s response for that trial (i.e., it is impossible to see where the child is pointing in Point-to-X, Point-to-Shape, or Point-to-Spatial Relations, or it is impossible to determine how many items a child has produced in Give-N). Our preregistration plan had indicated that when a trial was excluded from analyses, children’s score for that trial would be imputed from available data on other trials, and that data for a particular measure would only be excluded if the entire video recording was of poor quality, preventing coding any of children’s responses, or if the child refused to participate in answering at least 80% of trials. However, deviating from our preregistered plan, for children who responded to at least 80% of trials but did not respond to every trial for a particular measure, we used their performance on the completed trials to calculate their score (rather than imputing at both trial and task-level), as this occurred in less than 2% of the cases (n=1 for Point-to-X, n=3 for Give-N, n=3 for Point-to-Shape, and n=4 for Point-to-Spatial Relations).

All data were entered at the item level into spreadsheets. Analyses were performed using Stata/SE 15.1 (StataCorp, 2017), using the mi, mim (Carlin et al., 2008) and milrtest (Medeiros, 2008) packages. Missing data were examined for missing at random, and then accounted for using multiple imputation (Muthén et al., 1987; Newman, 2003). Prior to composite variable creation and all analyses, the distribution of all variables was examined for normality and presence of outliers. If skewness or kurtosis was greater than +/− 1 and +/− 3, respectively, appropriate corrections were made (Cain et al., 2017).

The amount of missing data per measure ranged from 2%-6% for demographic variables (children’s age, children’s gender, children’s language), to 10%-20% for children’s direct assessments (Point-to-X, Give-N, Point-to-Shape and Point-to-Spatial Relations), with a high of 18%-21% for fathers’ report measures (fathers’ education, fathers’ reports of children’s domain-general and math-specific vocabulary, and fathers’ home activities). Little’s chi-squared test for the missing completely at random (MCAR) assumption indicated that data for the Point-to-X, Give-N, Point-to-Shape, Point-to-Spatial Relations, mothers’ report of children’s mathematics language, and fathers’ report of children’s mathematics language variables were not missing completely at random, χ²(85) = 111.84, p = .027, and including the covariates (children’s age, children’s gender, children’s language, mothers’ education, fathers’ education, mothers’ report of children’s domain-general vocabulary, fathers’ report of children’s domain-general vocabulary, mothers’ mathematics activities, fathers’ mathematics activities, mothers’ literacy activities and fathers’ literacy activities) suggested that there was no covariate-dependent missingness, χ²(516) = 188.97, p = 1.00. As our data were not missing completely at random, missing data were imputed in Stata to create 10 imputed data sets, and regression models were estimated on each of the 10 data sets that included complete data, in line with recommendations from Enders (2013).

All variables were evaluated for normality prior to analyses, and Mardia multivariate tests of normality suggested that the distributions of data significantly differed from normal, skewness = 6.57, χ²(165) = 2556.82, p < .001, and kurtosis = 99.97, χ²(1) = 2.75, p = .098. As such, we carried out all analyses using both Ordinary Least Squares (OLS) regression and robust regression (in line with recommendations from Cain et al., 2017), with the pattern of results remaining similar in both sets of models. Although robust regression addresses issues related to skew and kurtosis, it does not permit model comparison using milrtest. Therefore, we present the results of OLS regression on the imputed dataset with model comparison in the text below and the results of the robust regression analyses in Appendix A (Tables A1, A2, and A3).

First, we present preliminary descriptive statistics, including means, standard deviations, and ranges for all study variables, and bivariate correlations between them. We then run a series of hierarchical linear regression models, one for each of children’s mathematics abilities. We begin with a model testing associations between baseline covariates (children’s age, children’s gender, the language of task administration, mothers’ education level, fathers’ education level, and children’s domain-general vocabulary) and each mathematics measure.

In Step 2, we add mothers’ and fathers’ engagement in mathematics activities to evaluate whether adding parents’ mathematics activities explains significantly more variance than the baseline model of covariates by calculating the change in R². As we had no specific hypotheses about the role of mothers’ versus fathers’ mathematics engagement and we are most interested in how home mathematics activities are broadly associated with mathematics outcomes, adding mothers’ and fathers’ mathematics engagement in the same step allowed us to examine how the experiences that children have with mathematics activities at home generally relate to their mathematics abilities. Additionally, we explore whether mothers’ and fathers’ mathematics engagement each uniquely explains variance in children’s mathematics performance.

To examine the specificity of parental activity engagement, in Step 3 we added mothers’ and fathers’ literacy activities. By calculating the change in R² over the previous model, we test if home literacy engagement significantly relates to children’s mathematics abilities, beyond the effect of parents’ mathematics engagement.

Results

Preregistered Analyses

Descriptive statistics show that children’s number skills, spatial skills, and mathematics language skills (as reported by both mothers and fathers) were all significantly correlated (Table 1). Mothers and fathers reported engaging in mathematics activities with their toddlers on average between a few times a month and once per week. Both mothers and fathers reported engaging in literacy activities significantly more frequently than mathematics activities, t(204) = 15.20, p < .001, and t(174) = 11.54, p < .001, respectively. Parents’ mathematics activities and literacy activities were highly correlated. In addition, mothers’ mathematics activities correlated with children’s performance in Point-to-Shape and their own report of children’s mathematics language skills (albeit weakly). Fathers’ mathematics activities correlated with children’s performance in Point-to-X, Point-to-Shape, mothers’ reports of children’s mathematics language skills, and their own report of children’s mathematics language skills. We next examined whether mothers’ and fathers’ mathematics activities remained significantly related to children’s mathematical performance in regression models where we included important baseline covariates.

Table 1.

Descriptive statistics and bivariate correlations for study variables

	1	2	3	4	5	6	7	8	9	10	11	12	13
1 Point-to-X	-
2 Give-N	.51***	-
3 Point-to-Shape	.33***	.32***	-
4 Point-to-Spatial Relations	.29***	.36 ***	.28***	-
5 Mathematics vocabulary (Mother report)	.40***	.44***	.48***	.42***	-
6 Mathematics vocabulary (Father report)	.43***	.40***	.50***	.52***	.77***	-
7 General vocabulary (Mother report)	.40***	.43***	.44***	.45***	.91***	.75***	-
8 General vocabulary (Father report)	.38***	.36***	.44***	.46***	.73***	.89***	.80***	-
9 Child age months	.23**	.29***	.17*	.31***	.38***	.35***	.41***	.36***	-
10 Mother mathematics activities	.10	.09	.20**	.03	.20**	.13†	.10	.00	−.05	-
11 Father mathematics activities	.25**	.15†	.19*	.14†	.24**	.36***	.16*	.28***	−.05	.39***	-
12 Mother literacy activities	.07	.16*	.24***	.20**	.30***	.31***	.21**	.25***	.04	.60***	.34***	-
13 Father literacy activities	.25**	.24**	.26***	.10	.30***	.41***	.20**	.35***	.00	.29***	.75***	.44***	-
N	193	177	196	182	206	179	206	179	215	205	176	205	175
M	59.88	36.08	80.89	71.78	25.49	24.96	28.22	27.60	30.42	2.88	2.84	3.63	3.44
SD	19.84	27.37	21.60	23.38	9.83	9.81	9.87	9.78	3.50	0.61	0.75	0.89	1.05
Min	0	0	12.50	0	0	0	0	0	23.80	1.27	1	1.25	1
Max	100	100	100	100	37	37	37	37	36.82	4.64	4.73	5	5

^†p < .10

^*p < .05

^**p < .01

^***p < .001

Predicting Number Skills

We first predicted children’s performance in the number skills assessments in a series of regression models (Table 2). In the first model consisting only of baseline covariates, children’s age, children’s language, and children’s general vocabulary significantly predicted their number skills performance, such that children who were older, children who were administered tasks only in English, and children with larger domain-general vocabularies performed better in the number skills tasks.

Table 2.

OLS Regression models predicting children’s number skills, N = 220

	Step 1				Step 2				Step 3
	B (SE)	β	t	p	B (SE)	β	t	p	B (SE)	β	t	p
Children’s age	0.01 (0.00)	0.18	2.55	.013	0.01 (0.00)	0.22	3.03	.004	0.01 (0.00)	0.21	3.11	.003
Children’s gender	0.01 (0.03)	0.02	0.27	.786	0.01 (0.03)	0.03	0.43	.667	0.01 (0.03)	0.02	0.33	.745
Children’s language	0.16 (0.04)	0.30	3.80	< .001	0.18 (0.05)	0.33	3.85	.001	0.18 (0.05)	0.33	3.88	.001
Mothers’ education	−0.02 (0.05)	−0.03	−0.40	.691	−0.01 (0.05)	−0.02	−0.27	.787	−0.01 (0.05)	−0.01	−0.21	.836
Fathers’ education	0.06 (0.04)	0.11	1.47	.146	0.05 (0.04)	0.09	1.33	.190	0.05 (0.04)	0.09	1.24	.221
Children’s general vocabulary	0.01 (0.00)	0.30	3.54	.001	0.01 (0.00)	0.24	2.78	.008	0.01 (0.00)	0.24	2.70	.009
Mothers’ mathematics activities	-	-	-	-	0.02 (0.02)	0.06	0.94	.352	0.04 (0.03)	0.10	1.20	.233
Fathers’ mathematics activities	-	-	-	-	0.04 (0.02)	0.16	2.23	.029	0.01 (0.03)	0.06	0.50	.622
Mothers’ literacy activities	-	-	-	-	-	-	-	-	−0.01 (0.02)	−0.06	−0.65	.520
Fathers’ literacy activities	-	-	-	-	-	-	-	-	0.03 (0.02)	0.15	1.28	.209
Constant	−0.23 (0.14)	-	1.68	.099	−0.46 (0.17)	-	−2.72	.010	−0.47 (0.17)	-	−2.80	.008
Average R2	.36				.40				.41
Average Adj-R2	.34				.37				.38
Model F Test	F(6, 149.0) = 13.96***				F(8, 149.8) = 11.26***				F(10, 160.6) = 9.41***

^***p < .001

Note: Children’s age is in months, children’s gender is dummy coded with female = 1, children’s language is dummy coded with English = 1, mothers’ education and fathers’ education are dummy coded with college degree or more = 1.

In the second step, adding mothers’ and fathers’ mathematics activities to the model explained more variance in number skills, although this was not a significant increase from the baseline model, F(2, 61.42) = 2.94, p = .061, and parents’ math activities accounted for a small- to medium-sized effect on number skills (Cohen’s f² = .06). The pattern of results from the baseline model remained. Additionally, fathers’ mathematics activities were significantly associated with number skills performance. In contrast, mothers’ mathematics activities were not significantly associated with children’s number skills.

To examine domain-specificity, we added mothers’ and fathers’ literacy activities in a third step and found that above covariates and mathematics activities, parents’ literacy activities were unrelated to number skills performance, and the model did not explain a significant amount of additional variance in toddlers’ number skills compared to the previous model, F(2, 161.30) = 1.11, p = .331. However, with the addition of these stringent control variables, fathers’ mathematics activities no longer significantly related to children’s number skills. This model should be interpreted with caution, however, as the correlations between mothers’ and fathers’ mathematics and literacy activities were quite high (Variance Inflation Factor [VIF] for these four variables ranged from 1.81 to 2.78).

Predicting Spatial Skills

We next examined children’s performance in the spatial skills assessments (Table 3). In the first model of baseline covariates, children’s spatial skills performance was significantly associated with children’s language, fathers’ education, and children’s domain-general vocabulary, such that children administered tasks only in English, children whose fathers had obtained at least a college degree, and children with larger vocabularies performed significantly better in the spatial skills assessments.

Table 3.

OLS Regression models predicting children’s spatial skills, N = 220

	Step 1				Step 2				Step 3
	B (SE)	β	t	p	B (SE)	β	t	p	B (SE)	β	t	p
Children’s age	0.01 (0.00)	0.12	1.94	.054	0.01 (0.00)	0.14	2.34	.020	0.01 (0.00)	0.14	2.32	.022
Children’s gender	0.02 (0.02)	0.05	1.01	.313	0.02 (0.02)	0.06	1.11	.268	0.02 (0.02)	0.05	0.84	.403
Children’s language	0.09 (0.03)	0.18	2.96	.004	0.09 (0.03)	0.20	3.19	.002	0.10 (0.03)	0.20	3.27	.001
Mothers’ education	−0.03 (0.03)	−0.06	−0.92	.361	−0.02 (0.03)	−0.04	−0.70	.488	−0.03 (0.03)	−0.05	−0.74	.463
Fathers’ education	0.08 (0.03)	0.17	2.63	.010	0.08 (0.03)	0.16	2.50	.015	0.08 (0.03)	0.16	2.45	.017
Children’s general vocabulary	0.01 (0.00)	0.49	6.65	< .001	0.01 (0.00)	0.46	5.97	< .001	0.01 (0.00)	0.44	5.70	< .001
Mothers’ mathematics activities	-	-	-	-	0.03 (0.02)	0.11	1.51	.141	0.02 (0.02)	0.06	0.79	.430
Fathers’ mathematics activities	-	-	-	-	0.02 (0.02)	0.07	1.15	.256	0.01 (0.02)	0.06	0.68	.501
Mothers’ literacy activities	-	-	-	-	-	-	-	-	0.02 (0.02)	0.08	1.07	.287
Fathers’ literacy activities	-	-	-	-	-	-	-	-	0.00 (0.01)	0.01	0.06	.955
Constant	0.18 (0.10)	-	1.76	.081	0.01 (0.12)	-	0.07	.947	0.00 (0.13)	-	0.03	.975
Average R2	.46				.48				.48
Average Adj-R2	.44				.46				.46
Model F Test	F(6, 169.6) = 22.82***				F(8, 168.6) = 17.64***				F(10, 180.5) = 14.76***

^***p < .001

Note: Children’s age is in months, children’s gender is dummy coded with female = 1, children’s language is dummy coded with English = 1, mothers’ education and fathers’ education are dummy coded with college degree or more = 1.

In the next step we added mothers’ and fathers’ mathematics activities to the baseline model and found a similar pattern of covariate results, although children’s age became a significant predictor of performance. Furthermore, neither mothers’ nor fathers’ mathematics activities significantly related to children’s spatial skills performance, and adding mothers’ and fathers’ mathematics activities did not explain significantly more variance than the baseline model, F(2, 96.73) = 2.79, p = .067, and parents’ math activities accounted for a small-sized effect on spatial skills (Cohen’s f² = .04).

In the test of domain-specificity in step three, there was no difference in the pattern of results from the second step, and parents’ literacy activities were unrelated to toddlers’ spatial skills above and beyond the covariates and parents’ mathematics activities. Additionally, model comparison indicated that this model did not explain significantly more variance than the previous model, F(2, 1096.27) = 0.69, p = .504. As in the previous set of models, results from models including literacy activities should be interpreted with caution as parents’ mathematics and literacy activities were highly correlated.

Predicting Mathematics Language Skills

Finally, we tested associations to children’s mathematics language skills (Table 4). In the first model with baseline covariates, children whose fathers had obtained at least a college degree and children with larger domain-general vocabularies had higher mathematics language skills than children whose fathers had not obtained at least a college degree and children who scored lower in domain-general vocabulary.

Table 4.

OLS Regression models predicting children’s mathematics language, N = 220

	Step 1				Step 2				Step 3
	B (SE)	β	t	p	B (SE)	β	t	p	B (SE)	β	t	p
Children’s age	0.09 (0.08)	0.03	1.15	.253	0.16 (0.08)	0.06	2.05	.042	0.16 (0.08)	0.06	2.14	.034
Children’s gender	−0.69 (0.51)	−0.04	−1.35	.180	−0.58 (0.47)	−0.03	−1.23	.220	−0.79 (0.49)	−0.04	−1.63	.104
Children’s language	1.29 (0.75)	0.05	1.72	.087	1.70 (0.69)	0.07	2.45	.015	1.83 (0.70)	0.08	2.61	.010
Mothers’ education	−0.96 (0.86)	−0.03	−1.11	.270	−0.69 (0.82)	−0.02	−0.84	.403	−0.68 (0.80)	−0.02	−0.84	.401
Fathers’ education	1.78 (0.75)	0.07	2.36	.020	1.56 (0.74)	0.06	2.13	.037	1.43 (0.74)	0.06	1.94	.056
Children’s general vocabulary	0.87 (0.03)	0.89	26.44	< .001	0.83 (0.03)	0.85	26.25	< .001	0.82 (0.03)	0.83	25.21	< .001
Mothers’ mathematics activities	-	-	-	-	1.29 (0.42)	0.08	3.10	.002	1.187 (0.53)	0.08	2.22	.028
Fathers’ mathematics activities	-	-	-	-	1.15 (0.36)	0.09	3.21	.002	0.43 (0.56)	0.04	0.76	.450
Mothers’ literacy activities	-	-	-	-	-	-	-	-	0.22 (0.38)	0.02	0.57	.567
Fathers’ literacy activities	-	-	-	-	-	-	-	-	0.67 (0.40)	0.08	1.68	.097
Constant	−3.30 (2.47)	-	−1.33	.185	−11.47 (2.71)	-	−4.24	< .001	−11.88 (2.64)	-	−4.50	< .001
Average R2	.86				.88				.89
Average Adj-R2	.86				.88				.88
Model F Test	F(6, 184.2) = 184.31***				F(8, 188.2) = 162.82***				F(10, 187.2) = 130.00***

^***p < .001

Note: Children’s age is in months, children’s gender is dummy coded with female = 1, children’s language is dummy coded with English = 1, mothers’ education and fathers’ education are dummy coded with college degree or more = 1.

We next added mothers’ and fathers’ mathematics activities to this baseline model and found a similar pattern of covariate results, although children’s age and the language of administration both became significant, where older children and those administered the tasks in English had higher mathematics language skills than younger child and those administered tasks in Spanish. This model explained significantly more variance than the baseline model, F(2, 212.71) = 13.07, p < .001, and parents’ math activities accounted for a medium-sized effect on mathematics language skills (Cohen’s f² = .17). We also found that mothers’ and fathers’ mathematics activities both significantly related to children’s mathematics language skills.

To test domain-specificity, we added mothers’ and fathers’ literacy activities to the model in a third step. This model did not explain significantly additional variance than the prior model, F(2, 156.38) = 2.05, p = .133. Neither mothers’ nor fathers’ literacy activities were significantly related to children’s mathematics language skills, but with the addition of these stringent control variables fathers’ mathematics activities no longer predicted children’s mathematics language scores. In contrast, mothers’ mathematics activities remained a significant predictor of children’s mathematics language scores even with these stringent controls. As in the previous models, these domain-specificity results should be interpreted with caution due to the potential for multicollinearity.

Exploratory Analyses

Although mothers’ and fathers’ mathematics activities did not differ in their overall frequency, we explored whether they differed in the specific activities they engaged in with their toddlers. That is, although they engaged in similar frequencies of mathematics activities overall (about once per week), their activities might differ and lead to the different pattern of associations with children’s mathematics skills. Descriptive statistics and comparisons of mothers’ and fathers’ engagement in each mathematics activity are presented in Table 5. For families where both parents reported their mathematics engagement, we found that mothers more frequently engaged in counting objects and identifying the names of written numbers with their toddlers than did fathers. In contrast, fathers more frequently engaged in using number activity books (such as connect-the-dots) and playing board games with numbers than mothers.

Table 5.

Comparison of mothers’ and fathers’ mathematics activities

Mathematics Activity	Mothers’ M (SD)	Fathers’ M (SD)	Significant Difference?
Counting objects	4.27 (1.00)	4.01 (1.23)	t(168) = 2.65**
Sorting things by size, color, or shape	3.38 (1.19)	3.34 (1.32)	t(165) = 0.39
Counting down	2.51 (1.50)	2.43 (1.45)	t(168) = 0.69
Identifying names of written numbers	3.05 (1.53)	2.74 (1.51)	t(169) = 2.50*
Playing with number fridge magnets	2.12 (1.41)	2.16 (1.34)	t(169) = −0.34
Putting pegs in a board or shapes into holes	2.34 (1.27)	2.55 (1.25)	t(169) = −1.76
Playing with puzzles	3.33 (1.18)	3.28 (1.23)	t(170) = 0.53
Building with blocks or construction set	3.94 (1.04)	3.75 (1.14)	t(170) = 1.97
Using number activity books	1.51 (0.89)	1.74 (0.97)	t(168) = −2.74**
Playing board games with dice or spinner	1.48 (0.82)	1.68 (1.01)	t(167) = −2.23*
Reading books that teach simple shapes	3.49 (1.25)	3.42 (1.21)	t(171) = 0.54

^*p < .05

^**p < .01

Finally, we tested whether parents’ mathematics activities explain additional variance above parents’ literacy activities and baseline covariates, by adding parents’ literacy activities in step two and parents’ mathematics activities in step three. Results from these models are presented in Tables 6, 7, and 8. Parents’ mathematics activities did not explain significantly more variance in toddlers’ number skills or spatial skills above parents’ literacy activities and baseline covariates, F(2, 93.31) = 0.97, p = .382, and F(2,175.53) = 0.81, p = .447, respectively. However, adding parents’ mathematics activities explained significantly more variance in toddlers’ mathematics language skills when added to the model of parents’ literacy activities and baseline covariates, F(2, 137.65) = 3.22, p = .043.

Table 6.

Exploratory OLS Regression models predicting children’s number skills, N = 220

	Step 1				Step 2				Step 3
	B (SE)	β	t	p	B (SE)	β	t	p	B (SE)	β	t	p
Children’s age	0.01 (0.00)	0.18	2.55	.013	0.01 (0.00)	0.22	3.03	.004	0.01 (0.00)	0.21	3.11	.003
Children’s gender	0.01 (0.03)	0.02	0.27	.786	0.00 (0.03)	0.01	0.11	.910	0.01 (0.03)	0.02	0.33	.745
Children’s language	0.16 (0.04)	0.30	3.80	< .001	0.18 (0.05)	0.33	3.96	< .001	0.18 (0.05)	0.33	3.88	.001
Mothers’ education	−0.02 (0.05)	−0.03	−0.40	.691	−0.01 (0.05)	−0.02	−0.32	.746	−0.01 (0.05)	−0.01	−0.21	.836
Fathers’ education	0.06 (0.04)	0.11	1.47	.146	0.05 (0.04)	0.09	1.26	.212	0.05 (0.04)	0.09	1.24	.221
Children’s general vocabulary	0.01 (0.00)	0.30	3.54	.001	0.00 (0.00)	0.22	2.56	.013	0.01 (0.00)	0.24	2.70	.009
Mothers’ mathematics activities	-	-	-	-	-	-	-	-	0.04 (0.03)	0.10	1.20	.233
Fathers’ mathematics activities	-	-	-	-	-	-	-	-	0.01 (0.03)	0.06	0.50	.622
Mothers’ literacy activities	-	-	-	-	0.00 (0.02)	0.01	0.19	.849	−0.01 (0.02)	−0.06	−0.65	.520
Fathers’ literacy activities	-	-	-	-	0.04 (0.02)	0.19	2.38	.022	0.03 (0.02)	0.15	1.28	.209
Constant	−0.23 (0.14)	-	1.68	.099	−0.39 (0.15)	-	−2.54	.015	−0.47 (0.17)	-	−2.80	.008
Average R2	.36				.40				.41
Average Adj-R2	.34				.37				.38
Model F Test	F(6, 149.0) = 13.96***				F(8, 154.7) = 11.52***				F(10, 160.6) = 9.41***

^***p < .001

Note: Children’s age is in months, children’s gender is dummy coded with female = 1, children’s language is dummy coded with English = 1, mothers’ education and fathers’ education are dummy coded with college degree or more = 1.

Table 7.

Exploratory OLS Regression models predicting children’s spatial skills, N = 220

	Step 1				Step 2				Step 3
	B (SE)	β	t	p	B (SE)	β	t	p	B (SE)	β	t	p
Children’s age	0.01 (0.00)	0.12	1.94	.054	0.01 (0.00)	0.13	2.19	.030	0.01 (0.00)	0.14	2.32	.022
Children’s gender	0.02 (0.02)	0.05	1.01	.313	0.01 (0.02)	0.03	0.62	.536	0.02 (0.02)	0.05	0.84	.403
Children’s language	0.09 (0.03)	0.18	2.96	.004	0.09 (0.03)	0.20	3.28	.001	0.10 (0.03)	0.20	3.27	.001
Mothers’ education	−0.03 (0.03)	−0.06	−0.92	.361	−0.03 (0.03)	−0.05	−0.84	.405	−0.03 (0.03)	−0.05	−0.74	.463
Fathers’ education	0.08 (0.03)	0.17	2.63	.010	0.07 (0.03)	0.16	2.43	.018	0.08 (0.03)	0.16	2.45	.017
Children’s general vocabulary	0.01 (0.00)	0.49	6.65	< .001	0.01 (0.00)	0.44	5.67	< .001	0.01 (0.00)	0.44	5.70	< .001
Mothers’ mathematics activities	-	-	-	-	-	-	-	-	0.02 (0.02)	0.06	0.79	.430
Fathers’ mathematics activities	-	-	-	-	-	-	-	-	0.01 (0.02)	0.06	0.68	.501
Mothers’ literacy activities	-	-	-	-	0.03 (0.01)	0.12	1.77	.082	0.02 (0.02)	0.08	1.07	.287
Fathers’ literacy activities	-	-	-	-	0.01 (0.01)	0.05	0.76	.449	0.00 (0.01)	0.01	0.06	.955
Constant	0.18 (0.10)	-	1.76	.081	0.06 (0.11)	-	0.53	.599	0.00 (0.13)	-	0.03	.975
Average R2	.46				.48				.48
Average Adj-R2	.44				.46				.46
Model F Test	F(6, 169.6) = 22.82***				F(8, 176.0) = 18.18***				F(10, 180.5) = 14.76***

^***p < .001

Note: Children’s age is in months, children’s gender is dummy coded with female = 1, children’s language is dummy coded with English = 1, mothers’ education and fathers’ education are dummy coded with college degree or more = 1.

Table 8.

Exploratory OLS Regression models predicting children’s mathematics language, N = 220

	Step 1				Step 2				Step 3
	B (SE)	β	t	p	B (SE)	β	t	p	B (SE)	β	t	p
Children’s age	0.09 (0.08)	0.03	1.15	.253	0.15 (0.08)	0.05	1.93	.055	0.16 (0.08)	0.06	2.14	.034
Children’s gender	−0.69 (0.51)	−0.04	−1.35	.180	−0.98 (0.48)	−0.05	−2.04	.043	−0.79 (0.49)	−0.04	−1.63	.104
Children’s language	1.29 (0.75)	0.05	1.72	.087	1.82 (0.72)	0.08	2.52	.013	1.83 (0.70)	0.08	2.61	.010
Mothers’ education	−0.96 (0.86)	−0.03	−1.11	.270	−0.84 (0.81)	−0.03	−1.04	.301	−0.68 (0.80)	−0.02	−0.84	.401
Fathers’ education	1.78 (0.75)	0.07	2.36	.020	1.44 (0.72)	0.06	1.99	.050	1.43 (0.74)	0.06	1.94	.056
Children’s general vocabulary	0.87 (0.03)	0.89	26.44	< .001	0.8 (0.03)	0.83	24.22	< .001	0.82 (0.03)	0.83	25.21	< .001
Mothers’ mathematics activities	-	-	-	-	-	-	-	-	1.187 (0.53)	0.08	2.22	.028
Fathers’ mathematics activities	-	-	-	-	-	-	-	-	0.43 (0.56)	0.04	0.76	.450
Mothers’ literacy activities	-	-	-	-	0.75 (0.31)	0.07	2.46	.015	0.22 (0.38)	0.02	0.57	.567
Fathers’ literacy activities	-	-	-	-	0.88 (0.26)	0.10	3.37	.001	0.67 (0.40)	0.08	1.68	.097
Constant	−3.30 (2.47)	-	−1.33	.185	−9.17 (2.50)	-	−3.67	< .001	−11.88 (2.64)	-	−4.50	< .001
Average R2	.86				.88				.89
Average Adj-R2	.86				.88				.88
Model F Test	F(6, 184.2) = 184.31***				F(8, 188.5) = 160.15***				F(10, 187.2) = 130.00***

^***p < .001

Note: Children’s age is in months, children’s gender is dummy coded with female = 1, children’s language is dummy coded with English = 1, mothers’ education and fathers’ education are dummy coded with college degree or more = 1.

Discussion

Children develop important mathematical skills during the toddler years, including number, spatial, and mathematics language skills. Early mathematics performance shows remarkable stability, leading to questions of how best to support children’s developing skills. We extend previous research on home mathematics engagement – primarily with preschool-aged children and their English-speaking mothers – showing that mothers’ and fathers’ mathematics activities at home are associated with mathematics skills in children already in toddlerhood. Moreover, associations between parents’ engagement and toddlers’ skills may be somewhat domain-specific to mathematics, as mothers’ and fathers’ literacy activities did not explain additional variance in toddlers’ mathematics skills when accounting for their engagement in mathematics activities, whereas parents’ mathematics engagement continued to predict after controlling for their literacy activities.

Mothers’ and Fathers’ Mathematics Activities and Toddlers’ Mathematics Performance

We found partial support for our first hypothesis that parent engagement in mathematics activities relates to toddlers’ mathematics skills. Associations differed for the three mathematics skills we assessed: Home engagement in mathematics activities was associated with toddlers’ number and mathematics language skills, but we found no support for the hypothesis that parental mathematics activities are associated with toddlers’ spatial skills.

Differences across the three mathematics measures suggest that parental engagement in mathematics activities may be more relevant for toddlers’ mathematics language and number skills than for toddlers’ spatial skills. One possible explanation is that the mathematics activities assessed in this study were not broad enough to encompass the variety of ways that parents engage in mathematics with their toddlers, and we did not include activities that may be most relevant for spatial skills development. A more comprehensive list of mathematics activities might yield similar patterns across all mathematics skills (see Elliott & Bachman, 2018). Another explanation is that parents’ engagement with toddlers in mathematics activities support some spatial skills, just not those skills we measured here (e.g., shape and spatial relation comprehension). For instance, perhaps mathematics activities may relate to toddlers’ mental rotation skills or expressive spatial vocabulary (rather than their comprehension). Testing additional mathematics activities, potentially with a stronger focus on spatial activities and spatial skills would be helpful to understand the type of activities that support early-emerging spatial skills.

Parent mathematics activities (for both parents) related to toddlers’ mathematics language skills, whereas only fathers’ mathematics activities related to toddlers’ number skills above and beyond baseline covariates. Although our study was not designed to test the unique contributions of each parent, they tentatively point to unique relations between mothers’ and fathers’ activities and children’s developing skills. Findings with older children show distinct roles of mothers’ and fathers’ mathematics activities in children’s mathematics performance (del Río et al., 2017; Huang et al., 2022; Huang et al., 2017; Liu et al., 2019; Mutaf-Yildiz et al., 2020). For example, some work suggests an important role of fathers’ mathematics activities above mothers’ engagement in children’s mathematics development (Huang et al., 2022; Huang et al., 2017; Liu et al., 2019; but see del Rio et al., 2017). However, this work has focused on numeracy skills with preschool- and school-aged children and has studied children in China and Chile, leaving open questions about the role of culture in shaping associations between parent gender roles and mathematics engagement, and how these associations may differ for younger children across a broader variety of mathematics skills. Studies are needed to better understand mothers’ and fathers’ mathematics engagement with children of different ages, and in particular experimental methods can help identify causal effects.

Domain-Specificity in Associations of Parent Activities with Children’s Mathematics Skills

In addition to testing the associations between parents’ mathematics activities and children’s mathematics skills, we examined the domain-specificity of parents’ engagement. Parents’ literacy activities did not relate to any of toddlers’ mathematics skills above and beyond the effects of mathematics activities and other covariates. Such findings suggest that certain activities by parents may specifically support children’s math. These findings contrast with previous work that found that home literacy engagement is related to children’s mathematics performance in preschool- and school-aged children (e.g., Anders et al., 2012; LeFevre et al., 2010; Ribner et al., 2020; Soto-Calvo et al., 2019). It remains unclear whether inconsistencies reflect age-related differences in associations, methodological differences in measures of engagement, or demographic differences among samples. Given the high correlation between parents’ mathematics and literacy activities and the likelihood of collinearity between these variables, it is premature to draw strong conclusions and further work on cross-domain connections is warranted.

Limitations, Conclusions, and Future Directions

Although this study was well-powered to detect the tested effects, a few notable limitations warrant discussion. First, all data were collected during the COVID pandemic, which may have changed the ways that parents interacted with their children. In particular, parents may have taken on a much larger role in teaching their children with the closure of daycares and the additional time spent at home due to quarantine restrictions and lockdown measures. Additionally, parents may have spent more time with their children as they worked more from home, worked fewer hours, or did not work altogether. As such, some of the differences between our results and previous findings may reflect cohort effects. Unfortunately, it is not possible to test this directly in the current sample.

Next, although our sample was fairly diverse in socioeconomic background, due to the broader aims of the larger study from which these data were drawn, participants were only from White, non-Hispanic or Hispanic/Latino families. Future work should expand to samples from other ethnic/racial backgrounds to test the generalizability of results. Although we included language as a covariate, differences in children’s mathematics skills associated with children’s preferred language are difficult to interpret and should be treated with caution, due to the relatively large number of children administered tasks in English (and consequently much smaller number of children administered tasks in Spanish).

Additionally, although we included reports from both parents to attempt to gain as accurate and complete a picture as possible of children’s development and home environment, our measures of toddlers’ mathematics language skills, domain-general vocabulary, and parents’ activity engagement were drawn from parent questionnaires and may be subject to reporter bias (Paulhus & Vazire, 2007). Along these lines, although parents reported the frequency of their mathematics and literacy engagement, their reports do not take into account the quality of the interactions or how parents and children are actually doing the different activities, nor the duration or depth of their engagement. Finally, we queried parents only on their activity engagement over the previous month, which may not be the appropriate time scale for understanding associations with children’s mathematics skills. Frequent activities over an extended time period may yield more robust associations with children’s mathematics performance.

Nonetheless, both mothers’ and fathers’ mathematics activities relate to toddlers’ mathematics skills, albeit not to all skills and not always in the same way. Based on our exploratory analyses, mothers reported engaging more frequently than fathers in counting objects and identifying numbers, whereas fathers reported engaging more frequently than mothers in using number activity books and playing number board games. Future work identifying the mathematics activities that most strongly predict toddlers’ mathematics skills would help determine why associations exist between mothers’ and fathers’ mathematics activities and toddlers’ mathematics performance. For example, frequent numeracy activities may support development of number skills more strongly than spatial or mathematics language skills, and frequent spatial activities may predict spatial skills more strongly than other mathematics skills (e.g., Leyva et al., 2021). Furthermore, observations of parent-child interactions during mathematics activities would illuminate the nuances of parent-child associations.

Altogether, mothers’ and fathers’ mathematics activities already relate to toddlers’ developing mathematics skills, and future work should examine the nuances of associations. Examining the specificity in relations between activities and skills, identifying the independent contributions of each parent and the potential qualitative differences in their mathematics engagement, and investigating additional heterogeneity from age- and other demographic-related differences in these associations will provide further insight into home influences on early mathematics development.

Appendix A

Point-to-X Task

Let’s look at some pictures!

1. (Practice) Which has a tree?

2. (Practice) Which has a ball?

3. Which has 1 cookie?

4. Which has 2 fish?

5. Which has 4 ducks?

6. Which has 5 apples?

7. Which has 2 carrots?

8. Which has 3 ladybugs?

9. Which has 4 strawberries?

10. Which has 5 pears?

11. Which has 10 fish?

12. Which has 3 oranges?

13. Which has 7 blueberries?

14. Which has 1 turtle?

Give-N Task

This is my friend Ellie the Elephant. She loves to eat ice cream! Will you help me feed Ellie scoops of ice cream? Watch, we are going to give spoons to Ellie so she can eat ice cream, like this.

1. Can you give Ellie 1 spoon?

2. Can you give Ellie 4 spoons?

3. Can you give Ellie 2 spoons?

4. Can you give Ellie 3 spoons?

5. Can you give Ellie 4 spoons?

6. Can you give Ellie 3 spoons?

7. Can you give Ellie 1 spoon?

8. Can you give Ellie 2 spoons?

Point-to-Shape Task

Let’s look at these pictures. In this game, I’ll tell you the name of a shape and you show me which one it is.

1. (Practice) Where’s the heart?

2. (Practice) Where’s the star?

3. Where’s the diamond?

4. Where’s the triangle?

5. Where’s the rectangle?

6. Where’s the circle?

7. Where’s the triangle?

8. Where’s the square?

9. Where’s the oval?

10. Where’s the triangle?

Point-to-Spatial Relations Task

Look, my friend Tiger is hiding. I’m going to tell you where he is hiding, and I want you to show him to me.

1. Where’s tiger on top of the cup?

2. Where’s tiger under the cup?

3. Where’s tiger between the cups?

4. Where’s tiger in front of the cup?

5. Where’s tiger behind the cup?

6. Where’s tiger in the cup?

7. Where’s tiger next to the cup?

Home Activities

In the past month, how often did you and your child do the following things at home (not at daycare or elsewhere)?

1. Counting objects^M

2. Sorting things by size, color, or shape^M

3. Counting down (10, 9, 8, 7, …)^M

4. Identifying names of written numbers^M

5. Buttoning buttons

6. Movement songs (i.e., Itsy Bitsy Spider)

7. Coloring, painting, writing^L

8. Identifying names of written alphabet letters^L

9. Identifying sounds of alphabet letters^L

10. Making music

11. Playing with number fridge magnets^M

12. Putting pegs in a board or shapes into holes^M

13. Playing with puzzles^M

14. Building with blocks or construction set (Duplo, Megablocks, etc)^M

15. Playing with “Playdoh”, dough, or clay

16. Using number activity books (like connect-the-dots)^M

17. Playing board games with numbers^M

18. Reading books that teach simple shapes like squares, circles, and triangles^M

19. Recite nursery rhymes (such as “Mother Goose”) or read other rhyming books^L

Note: mathematics activities are denoted by a superscript M, and literacy activities are denoted by a superscript L

Vocabulary

Children understand many more words than they say. Right now, we are particularly interested in the words your child SAYS. You’re going to see a list of words and we ask that you select the words that you have heard your child SAY. Be sure to count words that your child says in any other language(s), as well as mispronunciations, like “sketti” for “spaghetti” or “raffe” for “giraffe.”

1. Dog

2. Bear

3. Lion

4. Airplane

5. Train

6. Candy

7. Apple

8. Cake

9. Strawberry

10. Cup

11. Egg

12. Doll

13. Sock

14. Dress

15. Purse

16. Head

17. Chair

18. Kitchen

19. Window

20. Soap

21. Clock / Watch

22. Plant

23. Flower

24. Moon

25. Garden

26. Cloud

27. Boy / Girl

28. Brother / Sister

29. Clown

30. Empty

31. Fast

32. Happy

33. Dry

34. Dirty

35. Good

36. Hot

37. Thank you

38. One

39. Two

40. Three

41. Four

42. Five

43. Six

44. Seven

45. Eight

46. Nine

47. Ten

48. Twenty

49. Fifty

50. Heart

51. Star

52. Diamond

53. Triangle

54. Rectangle

55. Circle

56. Oval

57. On top (of)

58. Under

59. Down

60. Between / In the middle

61. Behind

62. In front of

63. In / Inside

64. Next to / Beside / Close to

65. Up

66. Outside / Out

67. Upside down

68. Around

69. More

70. All

71. A lot / Many

72. Few

73. Here

74. There

Table A1.

Robust regression models predicting children’s number skills, N = 220

	Step 1				Step 2				Step 3
	B (SE)	β	t	p	B (SE)	β	t	p	B (SE)	β	t	p
Children’s age	0.01 (0.00)	0.18	2.60	.012	0.01 (0.00)	0.22	3.06	.003	0.01 (0.00)	0.21	3.13	.003
Children’s gender	0.01 (0.03)	0.02	0.27	.785	0.01 (0.03)	0.03	0.43	.670	0.01 (0.03)	0.02	0.33	.746
Children’s language	0.16 (0.04)	0.30	3.98	< .001	0.18 (0.04)	0.33	3.95	< .001	0.18 (0.04)	0.33	3.97	< .001
Mothers’ education	−0.02 (0.04)	−0.03	−0.41	.680	−0.01 (0.05)	−0.02	−0.28	.782	−0.01 (0.04)	−0.02	−0.21	.831
Fathers’ education	0.06 (0.04)	0.11	1.53	.133	0.05 (0.04)	0.09	1.39	.172	0.05 (0.04)	0.09	1.29	.202
Children’s general vocabulary	0.01 (0.00)	0.30	3.65	.001	0.01 (0.00)	0.24	2.82	.007	0.01 (0.00)	0.24	2.73	.008
Mothers’ mathematics activities	-	-	-	-	0.02 (0.02)	0.06	0.91	.364	0.04 (0.03)	0.10	1.19	.237
Fathers’ mathematics activities	-	-	-	-	0.04 (0.02)	0.16	2.25	.028	0.01 (0.03)	0.06	0.51	.613
Mothers’ literacy activities	-	-	-	-	-	-	-	-	−0.01 (0.02)	−0.06	−0.65	.519
Fathers’ literacy activities	-	-	-	-	-	-	-	-	0.03 (0.02)	0.15	1.28	.209
Constant	−0.23 (0.13)	-	−1.73	.090	−0.46 (0.17)	-	−2.76	.009	−0.47 (0.17)	-	−2.82	.007
Average R2	.36				.40				.41
Average Adj-R2	.34				.37				.38
Model F Test	F(6, 144.5) = 14.99***				F(8, 146.6) = 11.57***				F(10, 157.8) = 9.69***

^***p < .001

Note: Children’s age is in months, children’s gender is dummy coded with female = 1, children’s language is dummy coded with English = 1, mothers’ education and fathers’ education are dummy coded with college degree or more = 1.

Table A2.

Robust regression models predicting children’s spatial skills, N = 220

	Step 1				Step 2				Step 3
	B (SE)	β	t	p	B (SE)	β	t	p	B (SE)	β	t	p
Children’s age	0.01 (0.00)	0.12	1.91	.059	0.01 (0.00)	0.14	2.34	.021	0.01 (0.00)	0.14	2.31	.022
Children’s gender	0.02 (0.02)	0.05	1.01	.314	0.02 (0.02)	0.06	1.10	.272	0.02 (0.02)	0.05	0.83	.409
Children’s language	0.09 (0.03)	0.18	2.82	.005	0.09 (0.03)	0.20	3.00	.003	0.10 (0.03)	0.20	3.09	.002
Mothers’ education	−0.03 (0.03)	−0.06	−0.99	.325	−0.02 (0.03)	−0.04	−0.72	.472	−0.03 (0.03)	−0.05	−0.76	.451
Fathers’ education	0.08 (0.03)	0.17	2.82	.006	0.08 (0.03)	0.16	2.65	.010	0.08 (0.03)	0.16	2.58	.012
Children’s general vocabulary	0.01 (0.00)	0.49	6.57	< .001	0.01 (0.00)	0.46	5.81	< .001	0.01 (0.00)	0.44	5.50	< .001
Mothers’ mathematics activities	-	-	-	-	0.03 (0.02)	0.11	1.51	.141	0.02 (0.02)	0.06	0.83	.411
Fathers’ mathematics activities	-	-	-	-	0.02 (0.02)	0.07	1.13	.261	0.01 (0.02)	0.06	0.70	.485
Mothers’ literacy activities	-	-	-	-	-	-	-	-	0.02 (0.01)	0.08	1.19	.236
Fathers’ literacy activities	-	-	-	-	-	-	-	-	0.00 (0.01)	0.01	0.06	.953
Constant	0.18 (0.10)	-	1.79	.077	0.01 (0.13)	-	0.07	.947	0.00 (0.13)	-	0.03	.976
Average R2	.46				.48				.48
Average Adj-R2	.44				.46				.46
Model F Test	F(6, 166.2) = 23.57***				F(8, 167.8) = 18.81***				F(10, 179.2) = 16.65***

^***p < .001

Note: Children’s age is in months, children’s gender is dummy coded with female = 1, children’s language is dummy coded with English = 1, mothers’ education and fathers’ education are dummy coded with college degree or more = 1.

Table A3.

Robust regression models predicting children’s mathematics language, N = 220

	Step 1				Step 2				Step 3
	B (SE)	β	t	p	B (SE)	β	t	p	B (SE)	β	t	p
Children’s age	0.09 (0.08)	0.03	1.13	.261	0.16 (0.08)	0.06	1.95	.053	0.16 (0.08)	0.06	2.03	.044
Children’s gender	−0.69 (0.51)	−0.04	−1.34	.181	−0.58 (0.47)	−0.03	−1.24	.218	−0.79 (0.48)	−0.04	−1.65	.100
Children’s language	1.29 (0.82)	0.05	1.58	.117	1.70 (0.78)	0.07	2.19	.030	1.83 (0.77)	0.08	2.37	.019
Mothers’ education	−0.96 (0.82)	−0.03	−1.16	.248	−0.69 (0.81)	−0.02	−0.85	.395	−0.68 (0.79)	−0.02	−0.85	.395
Fathers’ education	1.78 (0.81)	0.07	2.18	.031	1.56 (0.78)	0.06	2.00	.049	1.43 (0.77)	0.06	1.85	.068
Children’s general vocabulary	0.87 (0.03)	0.89	29.44	< .001	0.83 (0.03)	0.85	27.13	< .001	0.82 (0.03)	0.83	25.64	< .001
Mothers’ mathematics activities	-	-	-	-	1.29 (0.41)	0.08	3.13	.002	1.17 (0.50)	0.08	2.35	.021
Fathers’ mathematics activities	-	-	-	-	1.15 (0.36)	0.09	3.22	.002	0.43 (0.55)	0.04	0.77	.442
Mothers’ literacy activities	-	-	-	-	-	-	-	-	0.22 (0.33)	0.02	0.66	.509
Fathers’ literacy activities	-	-	-	-	-	-	-	-	0.67 (0.40)	0.08	1.65	.102
Constant	−3.30 (2.62)	-	−1.26	.211	−11.47 (2.89)	-	−3.97	< .001	−11.88 (2.83)	-	−4.20	< .001
Average R2	.86				.88				.89
Average Adj-R2	.86				.88				.88
Model F Test	F(6, 180.3) = 268.10***				F(8, 187.7) = 207.31***				F(10, 185.0) = 165.04***

^***p < .001

Note: Children’s age is in months, children’s gender is dummy coded with female = 1, children’s language is dummy coded with English = 1, mothers’ education and fathers’ education are dummy coded with college degree or more = 1.

Data availability statement:

Following Stage 1 in principle acceptance, the authors registered the approved protocol on the Open Science Framework (https://doi.org/10.17605/OSF.IO/NGX4E) publicly. The raw data, digital study materials and analysis code are freely available in a public repository and have been deposited on the Open Science Framework (https://osf.io/872yg/).