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Exploring effects of an early math intervention: The importance of parent–child interaction

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We explore whether training parents’ math skills or playing number games improves children’s mathematical skills. Participants were 162 parent–child dyads; 88.3% were white and children (79 female) were 4 years (M=46.88 months). Dyads were assigned to a number game, shape game, parent-only approximate number system training, parent-only general trivia, or a no-training control condition and asked to play twice weekly for eight weeks. Children in the number game condition gained 3 more points on an assessment of mathematical skill than did those in the no-training control. After eight additional weeks without training, effects diminished; however, children of parents in the ANS condition underperformed those in the no-treatment control. Iatrogenic effects were mediated by changes in the home numeracy environment.
Exploring effects of an early math intervention:
The importance of parent–child interaction
Andrew Ribner, Alex M. Silver, Leanne Elliott, & Melissa E. Libertus
University of Pittsburgh
Author Note
This work was supported by the National Science Foundation under grant DUE1534830 to ML,
by a James S. McDonnell Foundation Scholar Award to ML, by the National Institutes of Health
under grant T32GM081760 to AS, and by the National Institute of Child Health and Human
Development under grant F32HD102106-01 to AR. We would like to thank Abigail Haslinger,
Jamie Patronick, Chelsea MacNeil, Amy Veasey, Erinn Hanner, Lauren Krawczyk, Jessica
Ferraro, and the research assistants in the Kids’ Thinking Lab for help with data collection and
data entry. Finally, we especially thank the families who participated.
Correspondence concerning this article should be addressed to Andrew Ribner, Learning
Research and Development Center, University of Pittsburgh, 3240 Forbes Avenue #634,
Pittsburgh, PA 15213, USA. Email:
We explore whether training parents’ math skills or playing number games improves children’s
mathematical skills. Participants were 162 parent–child dyads; 88.3% were white and children
(79 female) were 4 years (M=46.88 months). Dyads were assigned to a number game, shape
game, parent-only approximate number system training, parent-only general trivia, or a no-
training control condition and asked to play twice weekly for eight weeks. Children in the
number game condition gained 3 more points on an assessment of mathematical skill than did
those in the no-training control. After eight additional weeks without training, effects
diminished; however, children of parents in the ANS condition underperformed those in the no-
treatment control. Iatrogenic effects were mediated by changes in the home numeracy
The mathematical skills of children entering kindergarten vary widely: Some children are
unable to recognize Arabic numerals or recite the count list, whereas others can already do
simple arithmetic (Jordan et al., 2009; Mazzocco & Thompson, 2005; Zill & West, 2001). This
variability has implications for later development and academic achievement, as individual
differences in mathematical skills demonstrate remarkable rank-order stability throughout
elementary school grades and beyond (Duncan et al., 2007; Jordan et al., 2009). That is, children
who enter school with lower levels of mathematical skills typically continue to underperform and
tend to take fewer high-level math courses compared to their peers (Davis-Kean et al., 2021). It
is therefore important to examine the origins of individual differences in mathematical skills
prior to school entry.
A large body of work has highlighted within-child factors that contribute to individual
differences including domain-general cognitive skills such as general intelligence (Hart et al.,
2009; Xenidou-Dervou et al., 2018), language (LeFevre et al., 2010; Slusser et al., 2019), and
executive function (Bull & Lee, 2014; Cragg & Gilmore, 2014; Ribner et al., 2018).
Additionally, domain-general basic number skills contribute in meaningful ways to the
development of children’s math skills (Halberda et al., 2008; Libertus et al., 2013; Slusser et al.,
2019; van Marle et al., 2014).
In addition to children’s domain-general and domain-specific cognitive skills, children’s
learning environments may have substantial impact on their math skill development. Prior to
entering formal schooling (around age 5 in the United States) much of the environmental
influence is a product of the home. Most notably, this home environment is shaped by parents
and caregivers and the learning opportunities they create for their children. Extant research has
demonstrated several ways in which parents and their daily practices may contribute to their
children’s mathematical skill development, including through their own mathematical skills (e.g.,
Brown et al., 2011) and through parenting practices including frequency of engagement with
activities that require mathematical thinking (e.g., LeFevre et al., 2009; Susperreguy et al.,
2020). In this study, we investigate whether experimentally manipulating aspects of parents’ own
skills or specific parenting practices with their children affects preschool-aged children’s
mathematical skill development.
The Role of Parents in Mathematical Skill Development
Math in the Home Environment. A broad research base has described relations
between facets of the home numeracy environment—which is often measured in terms of the
frequency with which children play with math-related toys and games—and math skills
(Daucourt et al., 2021; LeFevre et al., 2009; Segers et al., 2015; Susperreguy et al., 2020; but see
Elliott & Bachman, 2018 and Hornburg et al., 2021 for discussions of inconsistencies).
Researchers often use questionnaires about the frequency with which parents report child
engagement with and use of games, toys, and activities that involve mathematically relevant
information to ascertain the general home math environment. Individual differences in responses
to those questionnaires relate not only to children’s math skills concurrently, but also to the
development of math skills over time such that children who reportedly engage in more frequent
math-related activities such as board games that require counting and arithmetic tend to have
greater math skills (e.g., Niklas & Schneider, 2014; Susperreguy et al., 2020). Similarly, children
who hear their parents talk more about numbers and other math-related concepts during play and
other everyday interactions tend to also have better math skills (Levine et al., 2010; Ramani et
al., 2015; Susperreguy & Davis-Kean, 2016).
Importantly, the home numeracy environment appears to be malleable and at least some
interventions targeting the home numeracy environment yield positive effects on children’s math
skills. Simple, low-cost interventions to inform parents about the concept and importance of the
home numeracy environment have been shown to affect both the home numeracy environment
and children’s skills (Niklas et al., 2016), and interventions to make numerical information more
salient to adult–child dyads increase the amount of conversation around number concepts
(Braham et al., 2018; Hanner et al., 2019). For example, an intervention in a grocery exhibit in a
children’s museum which prompted parent-child dyads to engage with the concept of selecting a
meal given a limited budget yielded greater math talk than a condition in which dyads were
asked to create a healthy meal. Children in the budget condition were subsequently more likely
to spontaneously attend to number in an imitation task with a researcher (Braham et al., 2018).
Other targeted interventions have found similar results: Parent-child engagement with a
tablet-based intervention which provided scripted mathematical problem-solving opportunities
improved children’s mathematical skills compared to a non-mathematical story condition
(Berkowitz et al., 2015). Similarly, parent-child engagement with a picture book that drew
attention to quantity (as opposed to other salient characteristics such as color) improved
children’s understanding of number words and quantity (Gibson et al., 2020). However, not all
home-based interventions to increase conversation about and salience of numbers have yielded
effects. One intervention with preschool-aged children that successfully increased
mathematically relevant conversation during a cooking activity found no effects on children’s
math skills (Vandermaas-Peeler et al., 2012).
One intervention paradigm that has generally shown positive effects for children’s
mathematical skill development has involved the use of board games. Several studies have
shown engaging with board games which encourage children to count (typically dictated by the
use of a spinner to specify the number of spaces a player should move) improve mathematical
skill development (e.g., Elofsson et al., 2016; Ramani & Siegler, 2008, 2011; Siegler & Ramani,
2009; Whyte & Bull, 2008). Across various implementations and samples, participants
demonstrated increased mathematical skill after only four to six sessions (approximately 60
minutes of play time) over two to three weeks compared to control conditions (e.g., color board
games, other number activities). Though there is some evidence to suggest nuances of game
design might moderate the effectiveness of number board game play (e.g., circular versus linear,
Siegler & Ramani, 2009; Whyte & Bull, 2008; traditional versus digital, de Vries et al., 2021),
these positive effects of number board games appear to be robust to differences in country and
language (e.g., Cheung & McBride, 2017; Elofsson et al., 2016; Skillen et al., 2018; Whyte &
Bull, 2008), as well as individual versus group settings (e.g., Ramani et al., 2012). Numerical
card games administered by a trained research assistant have shown similar effects (Scalise et al.,
2017). However, effects might not be universal for all children: benefits of number board games
were stronger for children who had lower levels of initial knowledge and who were from lower-
income families (Ramani & Siegler, 2011).
Effective number game interventions have nearly exclusively taken place in classroom
settings and been administered by a trained game partner (i.e., a researcher or paraprofessional;
Ramani et al., 2012). More recent evidence has suggested that games played with parents at
home might be less efficacious than those played with a trained partner (Ramani & Scalise,
2020). Additional research is needed to test the efficacy of game-based interventions which
require little to no training for parent–child dyads. The current study seeks to fill this gap.
Parent Math Skills. While the mechanism of intergenerational transfer of math skills
remains unclear, there is substantial evidence that parents who are good at math tend to have
children who are good at math themselves (Borriello et al., 2020; Braham & Libertus, 2017;
Brown et al., 2011; Navarro et al., 2018). For example, parents’ math fluency, (i.e., their ability
to quickly retrieve basic arithmetic facts), is related to a range of their 5- to 8-year-old children’s
math skills including children’s math fluency as well as their ability to solve word problems and
written calculations (Braham & Libertus, 2017). One aspect of parents’ and children’s cognitive
functions that may be the basis of these intergenerational associations in math skills is parents’
and children’s non-symbolic number skills, or approximate number system (ANS) acuity. The
ANS is an evolutionarily preserved intuitive sense of number which is associated with and
underlies the acquisition of symbolic math skills (for reviews, see Chen & Li, 2014; Schneider et
al., 2017), and which produces imprecise estimates of quantity from input across sensory
modalities (e.g., sequences of tones, visually or tactilely presented objects, taps of a finger). The
imprecision of the numerical representations in the ANS increases with increasing number,
meaning that the accuracy of observers’ comparison between ANS representations follows
Weber’s Law (i.e., the discriminability of any two ANS representations is a function of the ratio
between them) (Buckley et al., 1974; Dehaene, 1996; Moyer & Landauer, 1967). Parents’ ANS
acuity is correlated with children’s ANS acuity (Braham & Libertus, 2017; Navarro et al., 2018)
and also predicts children’s overall math skills (Braham & Libertus, 2017). Furthermore, prior
research has demonstrated that the ANS is malleable: Adults who repeatedly engage in
increasingly difficult approximate number comparisons and operations demonstrate improved
ANS acuity (e.g., DeWind & Brannon, 2012), which in turn leads to improvements in symbolic
mathematical skills (Au et al., 2018; Bugden et al., 2016; Park & Brannon, 2013). As such, when
considering ways that improving parents’ math skills might improve children’s math ability, we
focus on acuity of the ANS as a target of intervention.
While there is suggestive evidence for a causal link between acuity of the ANS and
symbolic math skills (albeit with some ongoing debate, cf. Lindskog & Winman, 2016; Merkley
et al., 2017), it remains unclear whether improving adults’ ANS acuity also affects their
behaviors, which may in turn have effects on children’s math skills. Notably, a correlational
study found that individual differences in parents’ ANS acuity were associated with parents’ use
of number words when playing with their children (i.e., number talk; Elliott et al., 2017). Parents
with greater ANS acuity tended to use more numbers greater than ten, and this large number talk
was associated with children’s performance on a standardized math assessment. Thus, it is
possible that parents whose ANS acuity is improved as a result of training may also provide
greater opportunity for engagement with math for their children through the provision of more
number talk or math-relevant activities, yielding improvements in not only their own math skills,
but also the skills of their children. We will test this hypothesis in the current study.
Current Study
Prior research has suggested a range of influences on the development of children’s early
mathematical skills, including their parents’ number skills and the home numeracy environment;
however, many studies to date have focused on correlational associations among such skills.
Here, we assess causal associations in these two proposed pathways from parent factors to
children’s math skills by randomly assigning parents to training conditions designed to improve
either their own approximate number sense or to increase parents’ and children’s shared
engagement with mathematically-relevant information. For each of these proposed pathways
through parent number sense or parent-child interactions, we designed one training condition to
target number skills and another as an active control targeting a non-mathematical domain. As
such, we assigned parents of preschool-aged children to one of four training conditions, two of
which were focused on parents alone in the form of a computer task and two of which focused on
parent–child interactions in the form of a board game, or a no-training control condition.
We investigate the following questions: (1) Can “training” parents affect children’s
mathematical skills? That is, does training parents’ number skills and/or does encouraging
parents and children to play a board game in which numerical features are salient improve
children’s performance on a test of mathematical skills? (2) If so, do these training effects occur
through changes in the home numeracy environment—specifically, in children’s engagement
with math activities? We hypothesize that children of parents assigned to math conditions (i.e.,
math training for parents only and math training for parent–child dyads) will develop skills at a
faster rate. Given the leading hypothesis that parents who have better math skills likely facilitate
a more mathematically-friendly environment, we anticipate changes in child skills as a result of
assignment to treatment condition will occur through the home numeracy environment.
A total of 162 children (79 female) and their primary caregiver (9 fathers, 154 mothers)
from the [BLINDED] area were recruited through flyers, a central research participant database,
and mailing lists. Parents provided informed written consent prior to any data collection as
approved by the local Institutional Review Board. Children and their parent attended four in-lab
visits: Visit 1 (“Pre-Test”) occurred when children were approximately 47 months of age (MAge =
46.88 months, SD = 0.72), and the subsequent three visits occurred every two months following
Visit 1 (Visit 2 (“Assignment”): MAge = 48.95 months, SD = 0.90; Visit 3 (“Post-Test”): MAge =
51.05 months, SD = 1.15; Visit 4 (“Follow-Up”): MAgeV4 = 53.10 months, SD = 1.38). Of those
who participated in data collection at Pre-Test, 87.7% participated at Assignment (n = 143); of
those who participated in Assignment, 91.6% (n = 131) participated at Post-Test; 92.3% of those
who participated at Post-Test participated at Follow-Up (n = 121). Additionally, parents and
children were video-recorded during up to six 10-minute naturalistic free-play sessions in their
homes via video-conference between their first and third visit to the lab; however, those data are
not included as a part of the present investigation. Primary caregivers enrolled in the sample
were primarily white (88.3%), had a Bachelor’s Degree or higher (81.9%), and reported a family
income of $60,000 or more (69.9%).
Children received a small gift (e.g., a book or stuffed animal) as a reward for
participating in each in-lab data collection visit and received stickers to maintain attention and
motivation throughout data collection. Parents received $8 per hour for participating in each data
collection session.
Data are drawn from a longitudinal study that examined relations between parent and
child mathematical skills, as well as the role of parent-child dyadic interactions in the
transmission and development of those skills. Visit protocol followed a fixed order: During each
visit to the lab, children and their parents were first asked to play in a room filled with a standard
set of toys for ten minutes, after which the parent completed a set of questionnaires. Finally, both
parents and children completed a battery of cognitive tasks described in more detail below. Visits
took approximately 90 minutes, on average, and were conducted by two trained research
assistants. More details regarding the full study can be found in [AUTHORS], [AUTHORS],
Figure 1. Visual depiction of assigned training conditions.
Training Conditions
At the end of the Assignment visit (i.e., the second visit to the lab), parents and children
were randomly assigned to one of five training conditions: A number board game, a shape board
game, a computerized parent ANS training game, a computerized parent general knowledge
(trivia) training game, or a no-training control condition. All parents except for those in the no-
training control condition were given materials and instructions for their assigned condition as
well as a brief demonstration of how to use their assigned materials. Parents were asked to
complete their assigned training for 10 minutes at least 2 times per week for the intervening eight
weeks between the second and third lab visit for a total of 16 training sessions. Trained research
assistants were on call for technical support as needed, and check-ins were conducted during the
videoconference sessions between the second and third lab visit. Parents were asked to complete
a brief log with the date, time, and any issues that arose during each training session. Each of the
training conditions is described in detail below. A visual depiction of treatment conditions is
presented in Figure 1.
Number Board Game
In the number board game training, parents and children played a number board game
similar to the one used by Ramani and Siegler (2008). This game was designed using the
commercially available game “All Around the Playground” with modifications to include
number content. Specifically, families received a spinner with the numbers “1” through “6”
written on it and a board with a square labeled “Start”, followed by 64 colored shapes arranged
with the corresponding Arabic numerals written in each shape, followed by a square labeled
“Finish”. Arabic numerals were written on the board in permanent marker by research staff.
Parents and children were instructed to move their token from start to finish and advance it the
number of spaces indicated on the spinner. While they moved their token, they were instructed to
say the number words on the corresponding squares (e.g., if a child’s token was on “3” and they
spun a “2”, they said “4, 5”). If the child erred or could not name the numbers, the parent was
instructed to correctly name them and then ask the child to repeat the names while moving the
Shape Board Game
In the shape board game training condition, parents were instructed to play “All Around
the Playground” without the number-related modifications described above. Specifically, parents
and children were given the original game’s spinner, which contained six different shapes, and
the original board with 64 tiles that each contained a shape. Parents and children were instructed
to move their tokens to the next appropriate shape while saying the shape names of each shape
along the way. If the child erred, the parent was instructed to correctly name the shape and then
ask the child to repeat the names while moving the token. Labeling the shapes was not included
in the original instructions for this game but was included to parallel the number labeling aspect
of the Number Board Game condition.
Computerized ANS Training Game
To improve parents’ ANS acuity, parents received a laptop computer pre-loaded with a
non-symbolic arithmetic task similar to the one used by Park and Brannon (2013). On each trial
of this training task, parents viewed an animation of two dot arrays containing from 9 to 36 dots
sequentially moving behind an occluder too quickly to be counted. Trials included non-symbolic
addition (i.e., two sets of dots moving behind an occluder) and subtraction (i.e., one set moving
behind an occluder, and a smaller subset moving out from behind the occluder). On half of the
trials, parents were asked to indicate whether the sum of or difference between the dots in the
two arrays was more or less than the number of dots in a third array. In the other half of the
trials, they indicated which of two arrays contained a number of dots equivalent to the sum of or
difference between the number of dots in the two initially presented arrays. Task difficulty was
manipulated each session based on past performance by adjusting incorrect answers to be closer
or further from the correct response (i.e., making trials more or less difficult) to maintain
performance around 70-85% accuracy.
Computerized Trivia Training Game
Similar to the control condition used by Park and Brannon (2013), parents were trained to
solve general knowledge questions. Sample questions included “What does “pp” on a music
score mean? 1) Very Quiet, 2) Quiet, 3) Loud, 4) Very Loud 5) Repeat.” Or “What is a group of
toads called? 1) Club, 2) Knot, 3) Group, 4) Hub, 5) Pack.” After each question, parents were
told whether their selection was correct or not. When a question was answered incorrectly, the
correct answer was not given; instead, the question appeared again on later trials.
Parents and children each completed a series of standardized tasks in a quiet, one-on-one
setting at each lab visit.
Child Skills
Mathematical Skill. To measure children’s mathematical skill, the Test of Early Math
Ability-Version 3 (TEMA-3; Ginsburg & Baroody, 2003) was used. The TEMA-3 is a
standardized measure of children’s number skills, calculation skills, number comparison ability,
Arabic numeral literacy, and understanding of numerical concepts, and is normed for children
age 3 years 0 months to 8 years 11 months. Psychometric reports by the developers of the
TEMA3 demonstrate excellent internal consistency (with coefficients greater than 0.92) and
good testretest and alternative form reliability (with coefficients greater than 0.80; Ginsburg &
Baroody, 2003; Kline, 2000); internal consistency was also acceptable in the present sample ( =
.89-.93). Raw scores were used to adequately capture growth over time.
Parent Skills
Mathematical Skill. To measure parents’ mathematical skill, two subtests of the
Woodcock-Johnson Tests of Achievement III (Woodcock et al., 2001) were used. Parents first
completed the Math Calculation subtest, an untimed test in which they were asked to solve math
problems including arithmetic, algebra and calculus. Parents then completed the Math Fluency
subtest, a timed test in which they were asked to solve simple arithmetic problems as quickly and
as accurately as possible in three minutes. Scores on both subtests were then used to compute a
normed Math Calculation Skills Composite Score. The Math Calculation Skills Composite Score
has previously been shown to demonstrate excellent reliability with Cronbach’s alpha of 0.94
(Woodcock et al., 2001).
Approximate Number System Acuity. Parents completed a non-symbolic number
comparison task similar to that used by Halberda, Mazzocco, and Feigenson (2008). Parents
were shown sets of yellow and blue dots varying in size and asked to indicate as quickly and as
accurately as possible which color was more numerous by pressing one of two keys on a
keyboard labeled with yellow and blue stickers. Parents completed 4 practice trials with
preselected stimuli presented in a random order, followed by 150 test trials, including equal
number of trials for each of five ratios (1.33, 1.25, 1.2, 1.14, 1.11). Stimuli were displayed for
1500 ms on a 23-inch computer monitor, followed by a blank screen until parents responded.
The number of dots in each set ranged from 12 to 36. To avoid the use of perceptual cues instead
of number to solve the task, one-third of trials were Correlated (i.e., the side with the larger
number also had the larger cumulative area), one-third of trials were Anti-Correlated (i.e., the
side with the smaller number had the larger cumulative area but cumulative perimeter was equal)
and one-third of trials were Neutral (i.e., the arrays had equal cumulative areas). Performance
was quantified as the percentage of correct responses across all trials.
Home Activities
At each lab visit, parents completed a home activities questionnaire which was designed
to measure individual differences in the home numeracy environment (LeFevre et al., 2009);
however, only questionnaires from Pretest and Post-Test were used. Parents were asked to
indicate how frequently they participated in each of 40 activities (e.g., “Identifying names of
written numbers,” “Identifying sounds of alphabet letters”) with their children on a scale from 0
(did not occur) to 4 (almost daily) in the last month. Of the 40 total activities, 23 were related to
math and 3 were related to literacy. The remaining 14 items pertained to general activities (e.g.,
“Making collections”) or fine-motor skills (e.g., “Buttoning buttons”) and were not used in this
investigation. Responses to items pertaining to math were averaged to create a “frequency of
home numeracy activities” variable; responses pertaining to literacy were averaged to create a
“frequency of home literacy activities” variable. Internal consistency was acceptable for the
home math activities at Pre-Test (Math = .80) and Post-Test (Math = .84). Internal consistency
was below accepted norms for the home literacy activities (Pre-Test: Literacy = .41; Post-Test:
Literacy = .59); however, as literacy activities were included as a contrast for math activities and
were ancillary to the purposes of the investigation, the items were retained as written.
A series of covariates was included in all tested models. These included indicator
variables for whether or not the child was male and whether the primary caregiver had received a
Bachelor’s degree or higher. Additional covariates included child age and child vocabulary at
Pre-Test. Child vocabulary was assessed using the Developmental Vocabulary Assessment for
Parents (DVAP; Libertus et al., 2013), a parent-report measure of child vocabulary size for
children aged 2 to 7 based on the Peabody Picture Vocabulary Test (PPVT-4; Dunn & Dunn,
2007). The parent was given a list of 212 vocabulary words and was asked to indicate which of
those words they had heard their child say. Total number of words was used as a measure of
children’s vocabulary in analyses.
Child inhibitory control was assessed using a Day-Night Stroop task (Gerstadt et al.,
1994). In this assessment, children were told to say “night” when shown an image of a cartoon
sun and “day” when shown a cartoon image of a moon. This required children to inhibit their
prepotent association response. Children received 16 trials (8 “day” trials, 8 “night” trials).
Children were not given any feedback after responding to a trial. For each trial, children could
receive 0, 1 or 2 points for their response. Children received 2 points for a correct response, and
1 point for an incorrect response that they then self-corrected. Children received 0 points for
incorrect responses. Scores were averaged to create an aggregate measure ranging from 0 to 2.
Past work demonstrates that the Day-Night Stroop task is a reliable measure of young children’s
interference control that is highly correlated with other measures of inhibition, with correlation
coefficients as high as 0.79 (Montgomery & Koeltzow, 2010).
Descriptive Statistics
Descriptive statistics and correlations among study variables are reported in Table 1.
Table 1. Descriptive statistics and bivariate correlations among measures
1TEMA Score Pre-Test
2TEMA Score Post-Test 0.79***
3TEMA Score Follow-Up 0.78*** 0.88***
4Child Vocabulary 0.04 0.04 0.02
5Child Inhibitory Control 0.29** 0.22*0.21*0.07 —
6Child Age 0.01 0.13 0.08 0.11 -0.03 -0.04
N 152 123 129 140 133 162
Mean 10.23 16.23 16.99 104.42 1.31 3.91
SD 5.55 7.34 7.60 27.70 0.55 0.06
Range 0-27 1-43 2-43 44-182 0-2 3.75-4.04
Treatment Randomization
We first sought to test whether there were any baseline differences that existed despite
randomization to condition. One-way Analyses of Variance (ANOVAs) revealed that during
children’s first visit to the lab—two months prior to randomization to condition—there were no
systematic differences on the basis of the primary caregiver’s ANS acuity (F(4,152) = 0.66, p
= .619) or math skills (F(4,155) = 0.83, p = .508), nor were there differences on the basis of child
math skills (F(4,147) = 0.87, p = .483) by treatment condition. Finally, we found no significant
differences in assignment to treatment condition for planned covariates: Chi-squared tests
revealed there was no difference on the basis of whether or not the primary caregiver had
received a Bachelor’s degree (χ2(4) = 2.22, p = .695) or whether the child was male (χ2(4) = 3.59,
p = .424); one-way ANOVAs revealed no difference on the basis of child age (F(4,158) = 0.88,
p = .480), vocabulary (F(4,156) = 2.20, p = .071), or performance on the Day-Night Stroop task
(F(4,128) = 0.24, p = .915).
We were next interested in whether adherence to training—or fidelity—differed by
treatment condition. Each participating parent assigned to a treatment condition (that is, not
assigned to the no-treatment control condition) completed a log reporting frequency of
engagement in training. Number of trainings ranged from 0 to 18 (M = 8.04; SD = 6.19). There
was no difference by treatment condition (F(3,95) = 1.71, p = .170). As such, number of training
sessions parents reported to have completed was not considered as a covariate.
Treatment Effects
To test efficacy of assignment to training condition, we computed two linear regressions,
one each predicting TEMA scores at Post-Test (i.e., two months after Assignment) and TEMA
scores at Follow-Up (i.e., two months after Post-Test). TEMA scores at Post-Test and Follow-Up
were regressed on treatment condition (with “no training” as reference group) to test treatment
effects; TEMA scores at Pre-test were included as a covariate to estimate change in mathematical
skills due to training over and above stability. Additionally, outcomes were regressed on a set of
control covariates (i.e., indicator variables for whether or not the child was male and whether the
primary caregiver had received a Bachelor’s degree or higher; continuous variables for child age,
vocabulary, and inhibitory control). Data were determined to be missing completely at random
conditional upon included covariates, χ2(14) = 12.86, p = 0.538 (Li, 2013; Little, 1988). Models
were estimated in MPlus 8.0 (Muthén & Muthén, 2017) and missing data were accounted for
using Full Information Maximum Likelihood estimation (Enders, 2001). The analytic sample
included all dyads who participated in at least one wave of data collection (i.e., N = 162).
Treatment Effects on Child Math Skills at Post-Test
Results of the linear regression testing treatment effects at Post-Test are shown in
Column 1 of Table 2 and are shown visually in Figure 2. Children of parents randomized to the
parent–child math game developed math skills at a faster rate than did children in the no-training
control condition, b = 2.98, p = .021. That is, children who played the parent–child math game
answered, on average, three more questions correct on the TEMA (over 15% of a standard
deviation) than did children who were in the no-training control condition, controlling for scores
at pre-test. In contrast, children of parents in the other training conditions (parent ANS training,
parent trivia training, parent–child shape game) did not differ from those in the no-training
control condition in their TEMA scores (-1.21 < bs < 1.27, ps > .339).
Treatment Effects on Child Math Skills at Follow-Up
Results are shown in Column 2 of Table 2 and are shown visually in Figure 2. In contrast
to results at Post-Test, gains in TEMA scores for children in the parent–child math game did not
persist to Follow-Up. Compared to the no-training control condition, only children of parents in
the parent math condition differed; those children whose parents participated in the parent math
condition demonstrated lower math skills than did those in other conditions, b = -2.91, p = .024,
such that they answered, on average, three fewer questions correct on the TEMA than did
participants in other groups.
Figure 2. Child TEMA Scores in the four active training conditions at Post-Test and Follow-Up
relative to the no-training control condition
M ath A ssessmen t P oints C ha nge (R elave to n o-tr ain ing con trol)
NOTE: * denotes significant difference from no-training control, p < .05
To better understand this iatrogenic effect, we further examined effects of condition on
parents’ math abilities and ANS acuity (rather than children’s) at Follow-Up. We used the same
analytic approach as described above wherein the outcome measure from Follow-Up was
regressed on treatment condition (with “no training” as the reference group) and performance on
the same measure during Pre-Test to estimate change due to training over and above stability.
Only the covariate relevant to parents (i.e., completion of Bachelor’s degree or higher) was
included; those pertaining to child (i.e., age, sex, vocabulary, inhibitory control) were not.
Specifically, we were interested in whether being assigned to a parent training condition led to
change in parents’ cognitive skills, namely ANS acuity and mathematical skills. Results are
presented in Table 3. Indeed, we found a marginally significant effect of training condition on
parents’ ANS acuity such that being assigned to the parent ANS training condition resulted in
approximately a fifth of a standard deviation increase in performance on the ANS acuity task as
compared to the no-training control, b = 0.05, p = .051. There was no effect of condition on
parent symbolic math skills. To determine whether gains in ANS had any effect on child math
skills, we computed a latent change score to estimate the change in parent ANS acuity from Pre-
Test to Post-Test. There was no direct effect of change in parent ANS acuity on child math skills
(b = -0.93, p = .890). Tests of indirect effects with 1000 bootstraps revealed no indirect effect of
assignment to the parent ANS training condition to change in child TEMA scores via change in
parent ANS acuity (b = -0.002, 95% CI [-.59,.45]).
Table 2. Regression Results Testing Effects of Treatment Condition on Child Math Skills at
either Post-Test or Follow-Up
Post-Test Follow-Up
b SE p-value b SE p-value
TEMA Score Pre-Test 1.04 0.08 < .001 1.02 0.08 < .001
Parent–Child Math Game 2.98 1.27 .019 1.33 1.26 .290
Parent–Child Shape Game 0.30 1.22 .807 -0.06 1.25 .963
Parent ANS Game -1.21 1.34 .366 -2.91 1.29 .024
Parent Trivia Game 1.27 1.33 .339 -1.01 1.31 .443
Child Vocabulary 0.02 0.02 .324 0.02 0.02 .106
Child Inhibitory Control 0.64 0.77 .406 0.45 0.75 .549
Child Male 1.18 0.82 .150 1.02 0.81 .206
Child Age -3.20 4.61 .487 0.96 4.68 .838
Parent ≥ Bachelor's Degree -0.02 0.89 .980 -1.11 0.89 .214
NOTE: Bold—p < .05; ANS—Approximate Number System
As negative effects of parent training condition on child math were not attributable to
changes in parent skills, we then sought to investigate ways in which assignment to training
condition might have had unanticipated consequences for children’s home learning
environments. To test this, we used data from parent-reported home activities completed at each
visit. We computed two latent change scores to estimate change in home numeracy and home
literacy activities from Pre-Test to Post-Test and tested direct (change in activities on math
skills) and indirect effects (training effects on math skills through change in activities) with 1000
bootstraps. The amount of change in reports of both math and literacy activities did not appear to
differ by condition for either numeracy or literacy (ps > .240), although interestingly both
increased over time (ps < .001). In turn, change in parent-reported math activities was related to
child TEMA scores (b = 3.36, p = .002) such that a 1 SD increase in math activities corresponded
to over a 3-point gain in TEMA scores; change in parent-reported literacy activities was not
related to TEMA scores (b = 0.90, p = .127). Despite non-significant pathways from training to
change in math activities, a significant indirect effect of assignment to the parent ANS training
condition on child TEMA scores via changes in math activities emerged (b = -0.61, 95% CI [-
1.56,-.10]). Furthermore, when controlling for changes in home math activities, the negative
effect of assignment to parent training condition was reduced such that it was no longer different
from zero (b = -1.69, p = .098), suggesting changes in home math activities fully mediated the
negative effects of parent training condition. In contrast, there was no indirect effect through
changes in literacy activities (b = -0.29, 95% CI [-1.12,.02]).
Table 3. Regression Results Testing Effects of Treatment Condition on Parent Math Skills
Post-Test Follow-Up
b SE
value b SE p-value
Pretest Parent ANS 0.21 0.12 .067 0.29 0.08 < .001
Parent-Child Math Game -0.01 0.02 .579 0.07 0.11 .533
Parent-Child Shape Game -0.01 0.02 .556 -0.05 0.11 .667
Parent ANS Game 0.02 0.02 .321 0.20 0.10 .051
Parent Trivia Game 0.02 0.03 .448 0.19 0.11 .077
Parent ≥ Bachelor's Degree -0.02 0.02 .338 0.03 0.08 .752
Pretest Parent Math 0.95 0.04 < .001 1.03 0.05 < .001
Parent-Child Math Game -1.76 1.33 .186 0.45 1.72 .795
Parent-Child Shape Game -0.23 1.34 .867 1.32 1.72 .442
Parent ANS Game 0.04 1.34 .974 0.59 1.76 .736
Parent Trivia Game -1.10 1.40 .434 0.12 1.82 .947
Parent ≥ Bachelor's Degree -0.39 0.92 .671 -1.49 1.18 .209
NOTE: Bold—p < .05; Italicized—p < .07; ANS—Approximate Number System
The goal of this study was to assess whether environmental influences—namely parents’
math skills and the home numeracy environment—are causally related to the development of
young children’s mathematical skills. To test this, parent–child dyads were randomly assigned to
one of four training conditions (contrasted with a no-training control condition), two of which
targeted parent skills and two of which targeted parent–child interactions. Within each of the
parent skill and the parent–child interaction conditions, one condition was specifically designed
to improve age-appropriate mathematical skills (i.e., parents’ approximate number system acuity
and children’s counting abilities, respectively) and the other served as an active control. We
hypothesized that the two intervention conditions that targeted parents’ and children’s math skills
would improve children’s mathematical skills relative to the two active control conditions and
the no-training control. Specifically, we explored whether these improvements would occur via
changes in the home numeracy environment.
Our hypotheses were partially supported, though only for the parent-child interaction
condition. Immediately after training completion, children in the parent-child math game
condition outperformed those in the no-training control condition; however, positive effects of
condition faded out by Follow-Up two months later. In contrast and contrary to our hypotheses,
children of parents randomly assigned to the parent math training condition significantly
underperformed their peers in the no-training control condition by Follow-Up. This negative
association appears to be fully mediated by a change in parent-reported home numeracy
environment. Unsurprisingly, randomization to either of the non-math training conditions (i.e.,
the parent trivia condition and parent–child shape game condition) did not result in discernable
differences in child mathematical skill from the no-treatment control.
Dyadic Interaction around Math Supports Math Learning
Prior studies (e.g., Elofsson et al., 2016; Ramani & Siegler, 2008, 2011; Siegler &
Ramani, 2009; Whyte & Bull, 2008) have demonstrated the efficacy of board games as a tool to
improve children’s mathematical skills; however, extant studies have implemented game play in
classrooms and laboratory settings with researchers, paraprofessionals, and teachers as play
partners. This study marks an important transition to exploring the causal effects of board game
play in the home with parents as play partners. Consistent with prior studies we found that
playing a simple math game that prompted skills broadly associated with number sense (e.g.,
counting, understanding of cardinal values, symbol identification, one-to-one correspondence)
improved children’s mathematical skills to a greater extent than did other training conditions or
no training. That randomization to the parent–child math game condition corresponded to an
average three-point increase on a standardized test of mathematical skill suggests that sometimes
an easy, fun intervention can have meaningful effects.
Despite improved performance on a standardized test of mathematical skills as compared
to a no-training control condition, effects were non-significant eight weeks after training had
stopped. There are several reasons this might be the case. First, it is possible that some effects
were sustained but that the current study is underpowered to detect an effect of that size. A post-
hoc power analysis revealed that a simple analysis of group differences with repeated measures
given a sample of this size was approximately 0.60, far below commonly accepted thresholds.
Results of an analysis with a larger sample might have reached conventional levels of statistical
significance with regard to effects of parent–child math game being sustained through Follow-
Up (albeit attenuated). Either way, it is important to note that between Post-Test and Follow-Up,
parent–child dyads were no longer playing the assigned games; it is likely critical to sustain the
intervention in order to see sustained gains in mathematical skills.
The findings of short-term effects echo those of other successful home training studies
that have relied upon parent–child interaction. For example, Gibson and colleagues (2020)
randomly assigned children and parents to a picture book whose text specifically referenced
number and depicted a numerical match between text, objects, and Arabic numerals. Children in
this number book condition had improved understanding of number words as compared to
children in the control condition. It is likely that by receiving repeated practice engaging with
number words and set sizes (e.g., by seeing, counting, and discussing number words in the
context of a picture book or in the current study, by moving a token the number of spaces that
appear on a spinner), children might gain greater understanding of number symbols, counting,
and cardinality than by engaging with adjectives or colors. Similarly, Berkowitz and colleagues
(2015) found that randomization to a tablet-based instructional application to be used by parent–
child dyads which focused on mathematics (as opposed to reading) resulted in improvements in
children’s mathematical skills. While the tablet intervention was designed for older children
(first grade, or approximate age 6.5 years), effects might have been due to a similar mechanism
wherein there was simply increased attention to and practice with numerical information in the
home. Importantly, both studies by Gibson et al. (2020) and Berkowitz et al. (2015) found
moderations of treatment effects (Gibson et al. by children’s number understanding at the start of
the study; Berkowitz et al. by parents’ math anxiety), suggesting the main effects reported here
might be masking heterogeneity of treatment by other factors. Further investigation is needed to
better understand the role of children’s baseline knowledge in mathematical skill and parent
characteristics that might result in differential development from the same instructional material.
As a potential corollary to the positive effects of parent–child math games, we found a
negative effect of a parent ANS training condition (relative to a no-training control condition),
which was fully mediated by a change in the home numeracy environment. One possible
explanation for these unexpected findings lies in the indirect effects seen through parents’
engagement in math activities. However, this fails to address why parent ANS training might
have affected the home numeracy environment. On the one hand, parents may have found the
training games to be burdensome, particularly for the math condition, and this additional demand
may have decreased parents’ time to engage in math activities with their children. Alternatively,
the game was designed to be challenging and included feedback on performance, which may
have inadvertently shifted parents’ attitudes about math, math confidence or motivation to
engage in math activities and in turn affected their interactions with their children. Given that
these negative impacts were unique to the parent ANS intervention and not the parent general
world knowledge control condition, which required the same time of parents, we suspect the
latter explanation may be more likely. Surprisingly, although the parent-child math game
condition effects were observed at Post-Test but faded out by the Follow-Up assessment, the
opposite was true of these negative effects of parent ANS training, suggesting that the
mechanism underlying these effects may be more gradual. Alternatively, it is possible that—as
with the lack of sustained positive effects of the parent–child math training game—the present
study is simply underpowered to detect a small negative effect of training at Post-Test. Although
we are unable to determine the cause of these unexpected negative findings, we find them
concerning. Further study is needed to investigate potential sources of these negative effects,
including effects of training on attitudinal characteristics and on parents’ time.
Short-Term Parent ANS Training Does Not Relate to Child Math Learning
There has been some correlational evidence to suggest that parent math skills are
transmitted to their children. Several studies have found that—above and beyond factors
including sociodemographic characteristics—parents who are better at math tend to have
children who are better at math (e.g., Borriello et al., 2020; Braham & Libertus, 2016; Brown et
al., 2011; Navarro et al., 2018), though little is known about the mechanisms of transmission and
the extent to which it might be purely about shared environment. One recent investigation using
a behavioral genetics approach found that both parent skills and the home environment played a
role in the development of mathematical skills. Using a sample of children living with non-
relative adopted families since before the age of three months, Borriello and colleagues (2020)
found that both birth parents’ mathematical skills and adoptive fathers’ mathematical skills were
correlated with children’s skills, suggesting some support for both a genetic and environmental
component to individual differences in children’s skills.
Results from the present study complement these findings insofar as they suggest
intergenerational transmission of parent to child math skills does not operate at a rapid pace. In
other words, immediate increases in parents’ ANS acuity may not be sufficient to lead to changes
in parents’ behavior, both of which are likely rooted in habit and their dyadic routine. Although
we find that randomization to the parent ANS training condition was actually negatively related
to children’s math skills, we do not assume that these results indicate true negative
intergenerational associations. Despite the fact that we did see improvement in parent ANS
acuity as a result of randomization to the parent ANS training condition, changes in ANS did not
mediate associations between condition and children’s math skills, suggesting that the increases
in ANS acuity parents gained from training were not associated with children’s math skills. As
noted above, this effect was instead fully explained through differences in the home numeracy
environment. It remains possible the experimental manipulations of other aspects of parent math,
such as symbolic math skills or math attitudes, could positively relate to children’s math
outcomes, or that these causal pathways may be stronger at different developmental stages.
Alternatively, to the extent that these intergenerational links found in prior work reflect genetic
transmission, experimental manipulations in parents’ math skills would have no consequences
for children’s learning, as the process through which these skills are transmuted has already
occurred. In sum, although we did not find evidence that parents’ ANS acuity was causally
linked to children’s math skills, we cannot conclude that no causal pathways might exist between
parents’ and children’s math skills.
Despite several strengths of this study, there are a number of limitations that warrant
discussion. First, the study was underpowered to detect small effects that might result from
simple, low-cost interventions such as these. As such, results should be interpreted as
exploratory; it is possible that given a larger sample, other training conditions might have yielded
effects (positive or negative) on children’s math skills or that effects of the parent–child math
game might have been sustained through Follow-Up; indeed, a visual analysis of Figure 2
suggests the parent trivia training might also have had an iatrogenic effect and that effects of the
parent–child math game continue but are slightly attenuated. However, given that the direction
and magnitude of effects that resulted from randomization to the parent–child math game
condition are consistent with those from prior studies (e.g., Elofsson et al., 2016; Ramani &
Siegler, 2008, 2011; Siegler & Ramani, 2009; Whyte & Bull, 2008), we are cautiously optimistic
about this approach as an effective intervention for improving young children’s math skills.
Second, it is important to note the homogeneous group of parents and children enrolled in
the study. Nearly 90% of participating primary caregivers were white, and over 80% had a
Bachelor’s degree or higher. This is clearly not representative of a broader worldwide or US
population, nor even is it representative of the county in which most participants resided. While
we suggest there may be evidence for the efficacy of math games for young children as a way to
support mathematical skill development, further investigation is needed to better understand how
these effects might be accentuated or attenuated in other groups of participants.
Third and finally, we recognize that there are likely omitted variables that might have
contributed to results from this study. While treatment randomization minimized baseline
differences among groups on assessed characteristics, it is still possible that there were
unintended systematic differences among other unmeasured baseline characteristics.
Additionally—and inherent to low-touch interventions such as this—it is possible that treatment
fidelity differed across groups in ways beyond those we tested here. While the number of parent-
reported training sessions did not differ between groups, there may still have been differences in
the ways in which games were played (as compared to how they were designed or intended to be
played) or the amount of time dedicated to games within training sessions. Likewise, it is
possible that there were attitudinal differences that emerged over the course of training (e.g., if
participants in one condition found that game play was more fun or engaging than did those in
another). However, the limited control over fidelity to treatment is also one of the strengths of
this study. That we found children’s math skills improved as a result of randomization to the
parent–child math game condition suggests that even with limited oversight, this might be an
efficacious way to provide supports for young children’s math skill development.
In spite of these limitations, results from this study show the promise of low-touch, low-
cost interventions to improve young children’s mathematical skills. While further research is
needed to better understand generalizability and/or specificity of these findings, our results—
building upon the work of others who have found similar effects (e.g., Elofsson et al., 2016;
Ramani & Siegler, 2008, 2011; Siegler & Ramani, 2009; Whyte & Bull, 2008)—suggest that
introducing families to simple games that promote parent-child engagement around
mathematically meaningful play can promote math learning in fun, natural contexts. There is
now repeated evidence of positive findings of similar interventions, and the time may be ripe to
further sow the seeds of mathematical play. Programs such as “Reach Out and Read” have been
operating to improve access to literacy materials for over 30 years (Zuckerman, 2009); the time
has come to similarly improve access to research-based games that promote mathematical play.
Au, J., Jaeggi, S. M., & Buschkuehl, M. (2018). Effects of non-symbolic arithmetic training on
symbolic arithmetic and the approximate number system. Acta Psychologica, 185, 1–12.
Berkowitz, T., Schaeffer, M. W., Maloney, E. A., Peterson, L., Gregor, C., Levine, S. C., &
Beilock, S. L. (2015). Math at home adds up to achievement in school. Science,
350(6257), 196–198.
Borriello, G. A., Ramos, A. M., Natsuaki, M. N., Reiss, D., Shaw, D. S., Leve, L. D., &
Neiderhiser, J. M. (2020). The intergenerational transmission of mathematics
achievement in middle childhood: A prospective adoption design. Developmental
Science, 23(6), e12974.
Braham, E. J., & Libertus, M. E. (2017). Intergenerational associations in numerical
approximation and mathematical abilities. Developmental Science, 20(5), e12436. https://
Braham, E. J., Libertus, M. E., & McCrink, K. (2018). Children’s Spontaneous Focus on
Number before and after Guided Parent-Child Interactions in a Children’s Museum.
Developmental Psychology, 54(8), 1492–1498.
Brown, S., Mcintosh, S., & Taylor, K. (2011). Following in Your Parents’ Footsteps? Empirical
Analysis of Matched Parent–Offspring Test Scores*. Oxford Bulletin of Economics and
Statistics, 73(1), 40–58.
Buckley, P. B., Clifford, & Gillman, B. (1974). Comparisons of digits and dot patterns. Journal
of Experimental Psychology.
Bugden, S., DeWind, N. K., & Brannon, E. M. (2016). Using cognitive training studies to
unravel the mechanisms by which the approximate number system supports symbolic
math ability. Current Opinion in Behavioral Sciences, 10, 73–80.
Bull, R., & Lee, K. (2014). Executive Functioning and Mathematics Achievement. Child
Development Perspectives, 8(1), 36–41.
Chen, Q., & Li, J. (2014). Association between individual differences in non-symbolic number
acuity and math performance: A meta-analysis. Acta Psychologica, 148, 163–172. https://
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.
Cragg, L., & Gilmore, C. (2014). Skills underlying mathematics: The role of executive function
in the development of mathematics proficiency. Trends in Neuroscience and Education,
3(2), 63–68.
Davis-Kean, P. E., Domina, T., Kuhfeld, M., Ellis, A., & Gershoff, E. T. (2021). It matters how
you start: Early numeracy mastery predicts high school math course-taking and college
attendance. Infant and Child Development, e2281.
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),
Dearing, E., McCartney, K., & Taylor, B. A. (2009). Does Higher Quality Early Child Care
Promote Low-Income Children’s Math and Reading Achievement in Middle Childhood?
Child Development, 80(5), 1329–1349.
Dehaene, S. (1996). The Organization of Brain Activations in Number Comparison: Event-
Related Potentials and the Additive-Factors Method. Journal of Cognitive Neuroscience,
8(1), 47–68.
DeWind, N., & Brannon, E. (2012). Malleability of the approximate number system: Effects of
feedback and training. Frontiers in Human Neuroscience, 6.
Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P., Pagani,
L. S., Feinstein, L., Engel, M., Brooks-Gunn, J., Sexton, H., Duckworth, K., & Japel, C.
(2007). School readiness and later achievement. Developmental Psychology, 43(6),
Dunn, D.M., & Dunn, L.M. (2007). Peabody picture vocabulary test (4th edn.). Minneapolis,
MN: Pearson Assessments.
Elliott, L., & Bachman, H. J. (2018). How Do Parents Foster Young Children’s Math Skills?
Child Development Perspectives, 12(1), 16–21.
Elliott, L., Braham, E. J., & Libertus, M. E. (2017). Understanding sources of individual
variability in parents’ number talk with young children. Journal of Experimental Child
Psychology, 159, 1–15.
Elofsson, J., Gustafson, S., Samuelsson, J., & Träff, U. (2016). Playing number board games
supports 5-year-old children’s early mathematical development. The Journal of
Mathematical Behavior, 43, 134–147.
Enders, C. K. (2001). The Performance of the Full Information Maximum Likelihood Estimator
in Multiple Regression Models with Missing Data. Educational and Psychological
Measurement, 61(5), 713–740.
Gerstadt, C. L., Hong, Y. J., & Diamond, A. (1994). The relationship between cognition and
action: Performance of children 3 1/2-7 years old on a Stroop-like day-night test.
Cognition, 53(2), 129–153.
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.
Ginsburg, H. P., & Baroody, A. J. (2003). Test of early math ability. PRO-ED.
Halberda, J., Mazzocco, M. M. M., & Feigenson, L. (2008). Individual differences in non-verbal
number acuity correlate with maths achievement. Nature, 455(7213), 665–668.
Hanner, E., Braham, E. J., Elliott, L., & Libertus, M. E. (2019). Promoting Math Talk in Adult–
Child Interactions Through Grocery Store Signs. Mind, Brain, and Education, 13(2),
Hart, S. A., Petrill, S. A., Thompson, L. A., & Plomin, R. (2009). The ABCs of Math: A Genetic
Analysis of Mathematics and Its Links With Reading Ability and General Cognitive
Ability. Journal of Educational Psychology, 101(2), 388.
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),
Jordan, N. C., Kaplan, D., Ramineni, C., & Locuniak, M. N. (2009). Early math matters:
Kindergarten number competence and later mathematics outcomes. Developmental
Psychology, 45(3), 850–867.
LeFevre, J.-A., Fast, L., Skwarchuk, S.-L., Smith-Chant, B. L., Bisanz, J., Kamawar, D., &
Penner-Wilger, M. (2010). Pathways to Mathematics: Longitudinal Predictors of
Performance. Child Development, 81(6), 1753–1767.
LeFevre, J.-A., Skwarchuk, S.-L., Smith-Chant, B. L., 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(2), 55–66.
Levine, S. C., Suriyakham, L. W., Rowe, M. L., Huttenlocher, J., & Gunderson, E. A. (2010).
What counts in the development of young children’s number knowledge? Developmental
Psychology, 46(5), 1309–1319.
Libertus, M. E., Feigenson, L., & Halberda, J. (2013). Is approximate number precision a stable
predictor of math ability? Learning and Individual Differences, 25, 126–133.
Libertus, M., Odic, D., Feigenson, L., & Halberda, J. (2013). A Developmental Vocabulary
Assessment for Parents (DVAP): Validating Parental Report of Vocabulary Size in 2- to
7-Year-Old Children. Journal of Cognition and Development, 16, 141217110914002.
Lindskog, M., & Winman, A. (2016). No evidence of learning in non-symbolic numerical tasks –
A comment on Park and Brannon (2014). Cognition, 150, 243–247.
Mazzocco, M. M. M., & Thompson, R. E. (2005). Kindergarten Predictors of Math Learning
Disability. Learning Disabilities Research & Practice, 20(3), 142–155.
Merkley, R., Matejko, A. A., & Ansari, D. (2017). Strong causal claims require strong evidence:
A commentary on Wang and colleagues. Journal of Experimental Child Psychology, 153,
Montgomery, D. E., & Koeltzow, T. E. (2010). A review of the day–night task: The Stroop
paradigm and interference control in young children. Developmental Review, 30(3), 308–
Moyer, R. S., & Landauer, T. K. (1967). Time required for Judgements of Numerical Inequality.
Nature, 215(5109), 1519–1520.
Muthén, L. K., & Muthén, Bengt O. (2017). MPlus (Version 8) [Computer software].
Navarro, M. G., Braham, E. J., & Libertus, M. E. (2018). Intergenerational associations of the
approximate number system in toddlers and their parents. British Journal of
Developmental Psychology, 36(4), 521–539.
NICHD Early Child Care Research Network. (2005). Child Care and Child Development:
Results from the NICHD Study of Early Child Care and Youth Development. Guilford
NICHD Early Child Care Research Network. (2002). Early Child Care and Children’s
Development Prior to School Entry: Results from the NICHD Study of Early Child Care.
American Educational Research Journal, 39(1), 133–164.
Niklas, F., Cohrssen, C., & Tayler, C. (2016). Improving Preschoolers’ Numerical Abilities by
Enhancing the Home Numeracy Environment. Early Education and Development, 27(3),
Niklas, F., & Schneider, W. (2014). Casting the die before the die is cast: The importance of the
home numeracy environment for preschool children. European Journal of Psychology of
Education, 29(3), 327–345.
Park, J., & Brannon, E. M. (2013). Training the Approximate Number System Improves Math
Proficiency. Psychological Science, 24(10), 2013–2019.
Ramani, G. B., Rowe, M. L., Eason, S. H., & Leech, K. A. (2015). Math talk during informal
learning activities in Head Start families. Cognitive Development, 35, 15–33.
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,
Ramani, G. B., & Siegler, R. S. (2008). Promoting Broad and Stable Improvements in Low-
Income Children’s Numerical Knowledge Through Playing Number Board Games. Child
Development, 79(2), 375–394.
Ramani, G. B., & Siegler, R. S. (2011). Reducing the gap in numerical knowledge between low-
and middle-income preschoolers. Journal of Applied Developmental Psychology, 32(3),
Ramani, G. B., Siegler, R. S., & Hitti, A. (2012). Taking it to the classroom: Number board
games as a small group learning activity. Journal of Educational Psychology, 104(3),
Ribner, A., Moeller, K., Willougby, M., & Blair, C. (2018). Cognitive Abilities and
Mathematical Competencies at School Entry. Mind, Brain and Education : The Official
Journal of the International Mind, Brain, and Education Society, 12(4), 175–185. https://
Scalise, N. R., Daubert, E. N., & Ramani, G. B. (2017). Narrowing the Early Mathematics Gap:
A Play-Based Intervention to Promote Low-Income Preschoolers’ Number Skills.
Journal of Numerical Cognition, 3(3), 559–581.
Schneider, M., Beeres, K., Coban, L., Merz, S., Susan Schmidt, S., Stricker, J., & De Smedt, B.
(2017). Associations of non-symbolic and symbolic numerical magnitude processing with
mathematical competence: A meta-analysis. Developmental Science, 20(3), e12372.
Segers, E., Kleemans, T., & Verhoeven, L. (2015). Role of Parent Literacy and Numeracy
Expectations and Activities in Predicting Early Numeracy Skills. Mathematical Thinking
and Learning, 17(2–3), 219–236.
Siegler, R. S., & Ramani, G. B. (2009). Playing linear number board games—But not circular
ones—Improves low-income preschoolers’ numerical understanding. Journal of
Educational Psychology, 545–560.
Skillen, J., Berner, V.-D., & Seitz-Stein, K. (2018). The rule counts! Acquisition of mathematical
competencies with a number board game. The Journal of Educational Research, 111(5),
Slusser, E., Ribner, A., & Shusterman, A. (2019). Language counts: Early language mediates the
relationship between parent education and children’s math ability. Developmental
Science, 22(3), e12773.
Susperreguy, M. I., & Davis-Kean, P. E. (2016). Maternal Math Talk in the Home and Math
Skills in Preschool Children. Early Education and Development, 27(6), 841–857. https://
Susperreguy, M. I., Di Lonardo Burr, S., Xu, C., Douglas, H., & LeFevre, J.-A. (2020).
Children’s Home Numeracy Environment Predicts Growth of their Early Mathematical
Skills in Kindergarten. Child Development, 91(5), 1663–1680.
van Marle, K., Chu, F. W., Li, Y., & Geary, D. C. (2014). Acuity of the approximate number
system and preschoolers’ quantitative development. Developmental Science, 17(4), 492–
Vandermaas-Peeler, M., Boomgarden, E., Finn, L., & Pittard, C. (2012). Parental support of
numeracy during a cooking activity with four-year-olds. International Journal of Early
Years Education, 20(1), 78–93.
Whyte, J., & Bull, R. (2008). Number Games, Magnitude Representation, and Basic Number
Skills in Preschoolers. Developmental Psychology, 44, 588–596.
Xenidou-Dervou, I., Van Luit, J. E. H., Kroesbergen, E. H., Friso-van den Bos, I., Jonkman, L.
M., van der Schoot, M., & van Lieshout, E. C. D. M. (2018). Cognitive predictors of
children’s development in mathematics achievement: A latent growth modeling
approach. Developmental Science, 21(6), e12671.
Woodcock, R. W., McGrew, K. S., & Mather, N. (2001). Woodcock-Johnson Tests of
Achievement (Vol. 3). Riverside Publishing.
Zill, N., & West, J. (2001). Entering Kindergarten: A Portrait of American Children When They
Begin School. Findings from the Condition of Education, 2000. ED Pubs, P.
Zuckerman, B. (2009). Promoting Early Literacy in Pediatric Practice: Twenty Years of Reach
Out and Read. Pediatrics, 124(6), 1660–1665.
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