Complexity of parental number talk predicts preschoolers’ gains in cardinal knowledge

Abstract

The study sought to identify the parental, child, and classroom predictors of gains in children’s (n = 86) cardinal knowledge of number words and numerals (child M_age = 3.83 years; parent M_age = 35.17 years). Children’s cardinal and related knowledge was assessed at the beginning of preschool, and their cardinal knowledge assessed again five months later. Children’s executive functions, working memory, and intelligence were also assessed. Parents (n = 86) reported on their math anxiety, attitudes, beliefs, and home numeracy activities, and their academic achievement and intelligence were assessed. A parent-child number talk task assessed the content and complexity of quantitative talk in the home. Classroom measures included children’s attentive behavior, and teacher-reported mathematics content presented in these classrooms. Children showed substantive gains in cardinal knowledge of number words (d = 0.57) and numerals (d = 0.65). A series of Bayesian and standard regression analyses revealed that complexity of parental number talk predicted gains in children’s understanding of the cardinal values of number words (β = 0.27) and numerals (β = 0.25). Gains for number words were also predicted by children’s executive functions (β = 0.30), whereas gains for numerals were predicted by recognition of numerals (β = 0.24). The results provide insights into the most critical contributors to children’s emerging conceptual understanding of number.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-025-27405-y.

Introduction

Even before kindergarten, young children vary widely in their number skills, such as knowing the count list (i.e. counting “one, two, three, …”) and their understanding of number concepts (e.g. that “three” refers to three and only three things or events)^1,2. These early differences predict later mathematics achievement in elementary³, middle⁴, and high⁵ school, including risk for learning disabilities (LD)⁶. These are compounding deficits that follow from the hierarchical nature of mathematics^7,8, whereby early deficits compromise the learning of more complex material. By adulthood, these individuals face difficulties in the labor market⁹ and in navigating the quantitative demands of daily life¹⁰.

These well-established patterns indicate that the identification of and interventions with children at risk for long-term difficulties with mathematics should begin before kindergarten. There are a few such interventions¹¹, but these are all child-centered and often suffer from intervention fade out¹². In fact, fade out is the norm for child-centered interventions in mathematics, that is, interventions focused on teaching specific skills and knowledge to children result in only short-term relative gains¹³. A similar risk of fade out is found with individual therapy for juvenile offenders, but sustained outcomes can be achieved with multi-systemic approaches that involve coordinated individual-, home-, and school-based interventions¹⁴.

The current study is part of a larger project focused on identifying the child, home, and classroom factors that contribute to individual differences in the development of preschoolers’ number skills and conceptual knowledge. The overall goal of the project is to provide the foundation for the development of multi-systemic preschool interventions for mathematics. The specific goal here is to identify the child, home, and classroom predictors of gains in children’s conceptual understanding of number words and numerals during the first year of preschool (i.e. three years before the start of 1st grade).

Early number development

The foundation for formal mathematics learning in elementary school is laid during the preschool years. A longitudinal study from preschool to the end of 1st grade provided a comprehensive attempt to identify the most critical of these early competencies^2,15. A broad range of counting, number, and arithmetic skills were assessed through 2 years of preschool, kindergarten, and 1st grade, as was standardized mathematics achievement. The first key finding was that children who started preschool with strong knowledge of the count list, recognized many numerals, understood ordinal relations (i.e. more, less) and, most critically, had some knowledge of the cardinal value of number words—understanding that each number word refers to a specific quantity (e.g. that ‘three’ = three objects or events of any kind)—had higher mathematics achievement at the end of preschool, controlling other factors (e.g. intelligence). The second finding was that gains in numeral recognition and cardinal knowledge during the preschool years predicted later readiness for mathematics learning at the beginning of 1st grade¹⁵.

These results and others suggest that children’s early learning of the cardinal knowledge of number words and later numerals might be the key to their preparation for later mathematics learning. Cardinal knowledge is children’s first conceptual understanding of formal mathematics and emerges slowly during the preschool years^16–21. The give-a-number task illustrates children’s progression to this conceptual insight^21,22. Here, children are asked to provide x number of objects from a pile of objects. One-knowers provide one object when asked to do so but random numbers of objects for other number words. Two- to four-knowers successively learn the relation between these number words and the quantities they represent over the course of about a year. Soon after children become four-knowers, they begin to generalize to larger familiar numbers and are then considered cardinal-principle knowers (CPKs)¹⁸, although their knowledge of cardinality continues to grow for several more years²³.

The protracted development of cardinal principle knowledge contributes to individual differences in early mathematics achievement such that children who become CPKs in the first of two years of preschool have a substantive advantage over children who become CPKs in the second year of preschool or in kindergarten. This is because the age of achieving CPK appears to be the critical first step in preschoolers’ early understanding of the relations between numerals (e.g. 3 > 2), and predicts more complex number and arithmetic knowledge in 1st grade^15,24. By 1st grade, children who have developed a network of number system knowledge—representations of the relations among numerals including in the context of arithmetic (e.g. 7 = 6 + 1 = 5 + 2)—have higher mathematics achievement in subsequent grades relative to children who start elementary school with sparse knowledge of the relations among numbers^4,25,26. In other words, the conceptual understanding of the magnitude of number words and numerals appears to be a gatekeeper for the development of the number system knowledge that is foundational for subsequent mathematics learning²⁷.

Precursors to the development of cardinal knowledge include knowing the count list (i.e. “one, two, …”) and becoming skilled with counting one-to-one (i.e. pointing at successive objects and counting them, or enumerating)^19,28–30. Other potentially important knowledge includes children’s understanding of the relative quantity of groups of items which is undergirded by the evolved approximate number system³¹. The latter might contribute to learning the cardinal value of the first few number words²⁸. Children’s spontaneous focus on number is another potential contributor to the development of cardinal knowledge³². Spontaneous focus involves encoding the number of objects or events in an episode even when this information is not explicitly relevant to task goals. Children who do this perform better on numerical tasks³³, including cardinality measures³⁴, than their peers who do not spontaneously encode number.

Individual differences in early number development and mathematics achievement more generally are also related to domain-general abilities, including working memory, executive functions more broadly, and intelligence^7,8,35,36. Geary et al.²⁸ found that intelligence predicted individual differences in the age at which children become CPKs, and that children with poor executive functions were at high risk of not becoming CPKs by the end of preschool.

Home and parental correlates of children’s early mathematics outcomes

It is also likely that parental traits and home experiences contribute to individual differences in children’s early cardinal knowledge and related competencies, given these differences are found before children begin preschool^2,37–41. These traits and experiences include parental attitudes and beliefs about the importance of mathematics, their engagement in simple (e.g. rote counting) and more complex (e.g. focus on cardinal value) numeracy activities with their children, the complexity of their number talk (e.g. labeling a collection of toys with its cardinal value), as well as parent math anxiety. However, research on the relation between these factors and children’s early mathematics outcomes is inconclusive^42–45. As an example, Skwarchuk⁴⁰ asked parents to rate how often their preschoolers engaged in various math-related activities. Parents who reported their children frequently engaged in more complex math activities (e.g. add objects) had children with higher math achievement scores, but children who frequently engaged in simple math activities (e.g. recite numerals) had lower scores.

These and related findings suggest a nuanced relation between engagement in math activities at home and children’s mathematical development⁴⁶. Indeed, Daucourt et al.’s⁴⁷ extensive and well-done meta-analysis of various aspects of the home math environment revealed a positive, but modest relation to children’s mathematics outcomes (r = .13), with larger effects for preschool and kindergarten (r = .15) than school-age (r = .06) children. The estimated effects were largest for frequent engagement in direct and indirect math activities (r = .20) and high expectations for math achievement (r = .22). Their extensive analysis is consistent with a robust but modest link between the home math environment and children’s mathematics outcomes, but much remains to be learned about which specific activities influence which specific mathematics outcomes⁴⁸. For instance, a smaller-scale meta-analysis suggested that informal (e.g. games that involved numbers) rather than direct math activities were more strongly related to young children’s mathematics outcomes⁴⁹, in contrast to Daucourt et al.’s finding that both are important.

Parent-child number talk is another component of the home math environment. Daucourt et al.’s⁴⁷ results suggested that it is less important for young children’s mathematics outcomes than other types of math activities, but that its importance may emerge later in development. In contrast, a meta-analysis focused on parent-child number-talk found that it was related to mathematics outcomes, but only in 3- to 5-year-olds³⁹. Detailed studies of specific aspects of number talk suggest a nuanced relation, even when one is found. Levine et al.⁵⁰ found that frequent parental engagement in number talk was associated with preschoolers’ advanced understanding of the cardinal value of number words. However, Ramani et al.⁵¹ did not find a relation between parental (or primary caregiver) number talk and preschoolers’ cardinal knowledge but complex number talk (e.g. comparing quantities) was related to their understanding of quantitative magnitude more broadly (e.g. as related to the number line). The results of related studies suggest that number talk involving larger quantities—those greater than 10—or labeling sets of objects with number words is more consistently related to children’s early mathematics outcomes than is talk focused on smaller quantities^52–55, but a consensus has yet to be achieved.

It is well documented that students with relatively high levels of mathematics anxiety generally have lower mathematics achievement, but the cause-and-effect relation remains debated^56–58. There is some evidence that parental mathematics anxiety might contribute to their children’s mathematics anxiety and outcomes^59–61. Maloney et al.⁶¹ found smaller academic-year mathematics achievement gains for 1st - and 2nd -graders whose parents were math anxious and who frequently helped them with homework. Becker et al.⁵⁹ found that young children with math anxious parents showed smaller preschool gains in mathematics achievement than their peers with less anxious parents (see also Tomasetto et al.⁶²). Simmons et al.⁶³ found that preschoolers with math-anxious parents had lower mathematics achievement a year later, but this relation was not related to home numeracy activities. Although these studies suggest a relation between parental math anxiety and young children’s mathematics outcomes, the mechanism (e.g. numeracy activities) through which parental anxiety might influence these outcomes is not clear⁶⁴.

Classroom behavior and mathematics activities

For kindergarten and older students, poor attentive behavior in classroom settings is consistently related to lower concurrent and later academic achievement, controlling for cognitive abilities, family background, and prior achievement^3,65. These relations are found in observational studies⁶⁶, and using teacher reports of children’s attentive behavior^4,67. Most of these studies have been conducted with elementary school children, although there is modest, but inconclusive evidence that this relation extends to preschool settings⁶⁸.

In-class attentive behavior effectively influences the opportunity to learn in classroom settings⁶⁹, but this opportunity depends on the actual mathematics content presented in these settings^70,71. Wang⁷², for instance, analyzed outcomes from the National Childhood Longitudinal Study (U.S.) and found that some teacher-reported opportunities to learn mathematics (e.g. estimating quantities) predicted kindergarteners’ mathematics achievement at the end of the academic year. The importance of broad mathematics exposure in preschool classrooms is also acknowledged and has revealed considerable class-to-class variation in the overall exposure to mathematics^73,74. At the same time, the relation between variation in preschool exposure to mathematics in the classroom and children’s mathematics outcomes has not been as systematically explored.

Current study

The current study provides the broadest multi-system longitudinal assessment of young children’s emerging understanding of the cardinal values of number words and numerals conducted to date. The focus on cardinal knowledge is based on its importance for the development of the number system knowledge that undergirds long-term mathematical development^4,15,26. The broad-based approach follows from the as of yet unresolved and sometimes mixed relation between parental attributes and beliefs, specific home math activities, and children’s mathematics outcomes⁴⁷. Accordingly, the home measures included parental math anxiety and attitudes and beliefs about the importance of mathematics, engagement in home numeracy activities, and features of parent-child number talk. Parents’ mathematics and reading achievement and intelligence were also assessed to explore whether parental achievement and ability influenced children’s mathematical development, perhaps through complexity of the home math-related environment (e.g. Do more mathematically knowledgeable parents engage in more numeracy activities with their children? )⁷⁵. Several studies of older children have revealed that parental mathematics achievement or spatial abilities predict children’s mathematics outcomes^76–78, but these relations have not been systematically explored for younger children. Children’s prerequisite counting and number knowledge were also assessed, as these appear to contribute to the emergence of cardinal knowledge. Children’s behavior in preschool classrooms was assessed through direct observation and teacher reports, and their exposure to mathematics content in the classroom through teacher report.

Methods

Participants

All students in the first year of a Title I preschool program within the public school system in Columbia, MO were invited to participate in a two-year longitudinal study, along with their self-identified primary parent or caregiver (P1) (i.e. spends the most time with the study child). Ninety-three of the 200 students and their parents agreed to participate, but 7 moved or dropped out of the study before completing many (or any) assessments. The final sample included 86 children (43 boys) and their primary parent or caregiver. At the first quantitative assessment, the children were 3.83 years (SD = 0.24), and at the first home assessment their primary caregiver was 35.17 years (SD = 5.13). There was a secondary caregiver in 74 of the 86 homes (M = 36.75 years, SD = 7.63). In 73 of the 86 homes, the primary caregiver was a female and typically the biological mother; in 13 homes the primary caregiver was male and typically the biological father.

Children’s demographic information was obtained through a parental survey. 14% of the children were Hispanic or Latino, ethnically, 79% were not Hispanic or Latino, and 7% were Unknown or Not Reported. The children’s racial composition was 44% White, 27% Black or African American, 8% Asian, 8% Middle Eastern, 12% More than One Race, and the remaining Unknown or Not Reported. 12% of the primary parents were Hispanic or Latino, 76% were not Hispanic or Latino, and 13% were Unknown or Not Reported (the total is > than 100 due to rounding). The parent’s (P1) racial composition was 53% White, 28% Black or African American, 8% Asian, 7% Middle Eastern, 2% More than One Race, and the remaining Unknown or Not Reported. In all, the sample was more diverse than the city of Columbia, Missouri (about 71% non-Hispanic White, 12% Black or African American, 6% Asian, 5% More than One Race, and 5% Hispanic).

47% of the primary parents had a college degree, as did 38% of the secondary caregivers. 38% of the primary parents completed high school or had some college (but not a four-year degree), as did 39% of the secondary caregivers. 6% of the primary parents and 9% of the secondary caregivers did not complete high school; the remaining caregivers did not report their educational level. Self-reported annual household income was as follows: $0-$14,999 (9%); $15,000-$24,999 (10%); $25,000-$34,999 (14%); $35,000-$44,999 (9%); $45,000-$54,999 (12%); $55,000-$74,999 (10%); $75,000-$99,999 (5%); and $100,000 or more (22%); the remaining parents did not respond to this question. As a comparison, 56% of adults in Columbia have at least a 4-year college degree, 23% have some college, 17% have only a high school diploma, and 5% with no diploma. The median family income in Columbia is $102,000 per year.

English was the primary or only language spoken at home for 65 of the 86 families and was spoken at home for 13 of the other families. Spanish was the primary language for 7 families and spoken at home for 6 of the remaining families. For 6 families, the number talk assessment (below) was conducted in Spanish, and the transcripts were translated by a native Spanish speaker who was fluent in English. We were unable to code the parent-child number talk assessments for the 8 remaining families (e.g. spoke Arabic, Russian, ASL, unknown language, or technology failure), although these primary parents were able to complete the parent assessments in English. Data for the number talks assessment variables were imputed (below), and the core results were the same whether or not we included these families.

Written informed consent was obtained from parents, and verbal assent was obtained from children for each assessment session. This study was approved by the University of Missouri Institutional Review Board (IRB; Approval # 2091602), and all assessments adhered to the principles outlined in the Declaration of Helsinki.

Procedure

Children were administered all quantitative and cognitive measures in a quiet location at their school site at the ages shown in Table 1. The child assessments were generally kept to 15–30 min, and the research assistant would end the session if the child had difficulties attending and later conduct a follow-up assessment to complete the associated tasks. One or two (for reliability) research assistants observed and coded each child’s behavior during a typical instructional period in their classroom. The parent-child number talk assessment occurred in the fall to mid-year and the parental assessments in the spring, both in the child’s home. Two research assistants were at each home assessment, one who conducted the parental surveys or assessments and the other who engaged with the child(ren) (i.e. siblings). The home visits were about 60–90 min, and parents received $75 at the completion of each visit.

Table 1.

Child measures and age of administration.

Measure	Mean age years (SD)	Min (Months)	Max (Months)
Fall year 1
Give-a-Number: Words	3.83 (0.24)	40	51
Give-a-Number: Numerals	3.83 (0.24)	40	51
Ordinal choice	3.83 (0.24)	40	51
Verbal counting	3.83 (0.24)	40	51
Numeral recognition	3.83 (0.24)	40	51
Enumeration	3.83 (0.24)	40	51
Spontaneous focus on number	3.83 (0.24)	40	51
Classroom behavioral observation	3.91 (0.24)	41	51
Parent-child number talk	3.91 (0.26)	41	54
Spring year 1
Executive functions	4.01 (0.27)	42	54
Give-a-Number: Words	4.25 (0.23)	45	56
Give-a-Number: Numerals	4.25 (0.23)	45	56
Year 2
WPPSI-IV: Intelligence	5.17 (0.24)	56	67
WPPSI-IV: Working memory	5.17 (0.24)	56	67

All the quantitative assessments and the classroom observations were conducted in the fall of the first year of preschool. The follow-up cardinal knowledge measures used here were administered in the spring semester at the child’s school. Children’s executive functions were assessed mid-year for the first year (late fall to early spring semester) and other cognitive assessments were in the spring of the second year (time constraints precluded a first-year assessment).

Materials

Child quantitative assessments

Cardinal knowledge of number words was assessed using the give-a-number task^21,22. Here, children were asked to give the experimenter exactly 1, 2, 3, 4, 5, and 6 objects from a pile of small identical toys (i.e. fish). Children began at set size 1 and advanced to the next set size after a correct response; if they were incorrect, they went down one set size. Following the procedure used by Geary and vanMarle², the highest number of objects they correctly gave the experimenter on at least 2 of 3 attempts was taken as the highest set size for which the child understood cardinality. The same procedure was used to assess cardinal knowledge of numerals except the experimenter now held up a card showing (in order) the Arabic numerals 1, 2, 3, 4, 5, or 6 and said, “Put this many fish in the water (i.e. round blue plate)”^17,27. We focused on gains in knower level across the year, following Geary et al.¹⁵. Given these are core variables, we calculated the fall-to-spring correlations which provides an estimate of test-retest reliability, although not optimal given rapid gains for many children from fall to spring. In any case, the fall-spring correlation for numbers words was r = 0.68 (p < .001) and r = 0.70 (p < .001) for numerals, suggesting adequate reliability.

For the enumeration task, children were shown an array of 20 stickers and asked to count them, touching each one as they counted. The score was the highest number counted before committing an error. The verbal counting task involved the child reciting the count list, starting from “one” and counting as high as they could without an error, or until they reached 100. Numeral recognition was assessed by showing children the Arabic numerals 1 to 15 one-at-a-time and in a random order and asking them to name each one. The score was the total number of numerals correctly named.

The ordinal choice task assesses children’s sensitivity to more and less. Geary and vanMarle² found that performance on the task was predicted by a measure of the sensitivity of children’s approximate number system and not counting, suggesting it assesses an implicit judgment of more versus less. The task involved the child watching an experimenter sequentially hide two different numbers of objects (e.g. small toy bears) in two opaque cups²; items were dropped into the cups one at a time². The task was to choose the cup that contained more objects. There were 6 different comparisons (i.e. 1v2, 2v3, 3v4, 4v5, 5v6, and 6v7). In order to successfully identify the larger quantity, children had to estimate the items in each set and then mentally compare the two sets. The comparisons varied in difficulty—ratios varied from 0.5 to 0.86—and thus the score was weighted for the difficulty of each comparison. This was done by first multiplying each trial’s score (incorrect = 0, correct = 1) by the ratio of the comparison (e.g. 2v3 = 0.67) and then summing the products across trials (maximum score = 4.41).

The spontaneous focus on number (SFON) task involved the experimenter placing a paper with a dinosaur outline in front of themselves and another in front of the child. The experimenter then stamped dots on the dinosaur and asked the child to make their dinosaur look the same. There were four trials, with the experimenter placing 2, 4, 5, or 3 dots on the dinosaur across successive trials. The score was the number of trials on which the number of dots produced by the child matched the number produced by the experimenter^32,79.

Child cognitive assessments

Children were administered the WPPSI-IV⁸⁰ over two sessions to keep assessments under 30 min and to reduce fatigue: Day 1 included Block Design, Information, Matrix Reasoning, and Bug Search subtests; Day 2 included Picture Memory, Similarities, and Zoo Locations subtests. Using standard procedures, subtest scores were used to calculate intelligence and working memory scores.

Children’s executive functions (EF) were assessed using the standardized Minnesota Executive Function Scale⁸¹. The scale is based on the Conflict EF measure developed for the assessment of preschool children⁸², although it has been extended to older individuals. The measure captures attentional and inhibitory functions and predicts preschoolers’ mathematics achievement and their cardinal knowledge of number words^2,28. The measure is administered via a proprietary Reflection Sciences MEFS app on Apple iPads and provides an age-based standard score (M = 100, SD = 15). All administrators of the assessment completed the training provided by the test developers.

In-class observation

The in-class observation protocol was based on Jacobs et al.’s⁸³ Revised Edition of the School Observation Coding System (REDSOCS). Observers were trained on the measure through collaborative discussions of fifteen practice videos of preschooler behavior in classroom settings, followed by live practice with feedback and debriefing. Minimum training time was eight hours over 3–4 weeks. Reliability was estimated through the Intraclass Correlation Coefficient (ICC) program in the psych package in R⁸⁴, and the ICC was over 85% for live practice.

In-class behaviors during structured learning time were documented live by research assistants who were discretely in the child’s classroom. Following Jacobs et al.⁸³, 30 consecutive 10-second observations were scored on Command Compliance, Appropriateness of Behavior, and whether the student was On- or Off-task using the criteria shown in Supplementary Information Table S1.

In the classroom setting, at the onset of each observation a black mask obscured the iPad screen for 10 s while the researcher observed the target child. After 10 s, the mask cleared, and the researcher scored the child’s behavior. After 18 s a “submit” button appeared and after pressing it the user moved to the next 10 s assessment. Average time spent on the behavioral observations was about 11 min per child. To assess reliability of the behavioral observations dual observers were used for a minimum of 25% of all observations. Reliability was calculated using ICC, and was 0.83 for Command Compliance, 0.85 for Appropriateness of Behavior, and 0.93 for On- or Off-Task.

Teacher-reported in-class behavior and opportunity to learn

Inattention and hyperactive behavior were assessed using The Strength and Weaknesses of ADHD-Symptoms and Normal-Behavior (SWAN-P) measure⁸⁵. The 18 items assess attentional deficits and hyperactivity relative to the behavior of a typical student. For school-age children, the inattention subscale is more sensitive to variation in classroom behavior than are diagnostic screeners for ADHD⁶⁵, and this subscale is more strongly correlated with academic outcomes than the hyperactivity subscale⁶⁸. Nevertheless, we included both subscales due to the young age of our sample. The 9-item inattention subscale (e.g. “Gives close attention to detail and avoids careless mistakes”, α = 0.95) and the 9-item hyperactivity subscale (e.g. “Stays seated when required by rules or social conventions”, α = 0.95) were administered to the students’ teachers using a Qualtrics survey in the spring semester.

Teachers also completed a 25-item, Opportunity to Learn (OTL) survey on the frequency with which they presented or engaged in number, counting, arithmetic, and related activities (e.g. “Count using fingers”, “Write numbers”) in the classroom³⁸ using a Qualtrics survey in the fall semester. They rated each activity on a 0 to 4 scale; (0) Not at all; (1) 1–2 times per week; (2) 3–5 times per week; (3) about once a day; (4) more than once a day. A principle components analysis with oblique rotation was used for data reduction and identified seven components with Eigen values > 1⁸⁶. The associated standardized loadings > |0.40| are shown in Supplementary Information Table S2. One component was defined by a single item and was not considered further. The remaining were Compare (8 items, α = 0.91), Count (3 items, α = 0.79), Order (4 items, α = 0.77), Relation (3 items, α = 0.79), Number (3 items, α = 0.66), and Numeral (4 items, α = 0.74).

Parent math anxiety, attitudes, and beliefs assessments

The parental assessments included measures of mathematics attitudes, beliefs, and anxiety that have been found to correlate with their engagement in math activities with their children or correlate with their children’s math outcomes^38,45,54,61. Mathematics anxiety was assessed with a single item using on a slider on an iPad. The slider had anchors at 1 (not anxious) and 10 (very anxious) and the parent was asked, “How anxious are you about math?”⁸⁷. The parent moved the slider to the location that best represented their level of math anxiety. Scores on this measure are correlated (0.49 to 0.85) with scores on multi-item math anxiety measures and correlated with mathematics achievement^56,87.

Mathematics attitudes and beliefs were assessed using 13 items from Missall et al.³⁸. The items (e.g. “When I was growing up my family valued math.”) ask parents to rate “how you think and feel about mathematics” on a one (strongly disagree) to four (strongly agree) scale. Although Missall et al. did not find significant correlations between a composite based on a subset of these items and preschoolers’ math achievement, they were included here to ensure a broad assessment of parental beliefs and attitudes about mathematics. Principle components analysis with oblique rotation identified four components with Eigen values > 1. The associated standardized loadings > |0.40| are shown in Supplementary Information Table S3. The first component was Math Teaching where parents reported teaching math at home. The second (Math Importance) was defined by items related to the importance of math in daily life and in school. The third (Math Value) taps the valuation of mathematics, with a negative factor loading for one item which means lower scores indicate more home engagement in math than literacy activities (this item was reverse coded for analyses). The final component (Math Help) indicates seeking help with mathematics as a child. The Math Teach (α = 0.84) and Math Importance (α = 0.75) items were internally consistent and thus the means of the respective items were used to create two composite variables. The items for the two other factors were not internally consistent (αs < 0.41) and thus were dropped from any further analyses.

Eight items from a subjective numeracy scale (SNS) asked parents to rate their math competencies (e.g. “How good are you at working with fractions?”) on a one (not at all) to six (extremely) scale for five items⁸⁸. The remaining items focused on their preferences, for instance, whether or not they preferred quantitative or non-quantitative information (e.g. “When reading the newspaper, how helpful do you find tables and graphs that are parts of a story?”). A principle components analysis confirmed the SNS items loaded on a single component except for one reverse coded item that was dropped. The score was the mean of the seven remaining items (α = 0.88). Following Ramani et al.⁵¹, parents reported on 16 home numeracy activity items. A principle components analysis revealed five components (Eigen values > 1). The items that defined these respective components (see Supplementary Information S4) focused on Counting (three items, α = 0.85), Measurement (four items, α = 0.79), Games (four items, α = 0.72), Add (three items, α = 0.71), and Computer Usage (two items, α = 0.77). The score was the mean of the respective items.

Parents were also administered an abbreviated version of the Home Observation for Measurement of the Environment Inventory, Early Childhood (HOME) measure⁸⁹, including Sections I (LM Learning Materials, 10 items), II (LS Language Stimulation, six items) and V (AS Academic Stimulation, six items). The HOME is administered through naturalistic observation and a semi-structured interview. The LM, LS and AS items were submitted to independent principle components analyses; see Supplementary Information Table S5, S6, and S7. Only two of the six associated composites showed adequate internal consistency (α > 0.60). These were retained for the analyses. The first (Toys) was defined by two learning materials items (“Child has toys which teach colors, sizes, and shapes.”; “Child has toys or games which help teach numbers.”; α = 0.60). The second (Learn) was defined by three academic stimulation items (“Child is encouraged to learn shapes.”; “Child is encouraged to learn colors.”; “Child is encouraged to learn numbers.”; α = 0.83). The score was the mean of the respective items.

Parent cognitive and achievement assessments

Parents’ mathematics and reading achievement was assessed using the Numerical Operations and Word Reading subtests of the Wechsler Individual Achievement Test⁹⁰. Their intelligence (IQ) was estimated using the Vocabulary and Matrix Reasoning subtests of the Wechsler Abbreviated Scale of Intelligence (WASI)⁹¹. The IQ scores were highly correlated with both standardized Vocabulary (r = .90, p < .0001) and Matrix Reasoning (r = .86, p < .0001) scores.

Number talk assessment

Measures of parental number talk were obtained from a structured parent-child interaction task (Number Talk) that was developed for this study. For 15 min, the primary parent (P1) and child (C) planned and acted out the child’s next birthday party. To elicit number talk, carefully selected homogeneous (e.g. ten each of identical plates, cups, forks) and heterogeneous stimuli (e.g. twelve colorful goodie bags and twelve sets of three goodie bag toys) were presented to the dyads (see Fig. 1) and instructions for the task referenced boundedness based on the number of invitees, specifically no more than ten people including P1 and C⁹². Interactions were video-recorded, and transcripts were prepared verbatim, using Otter.ai and human transcribing, at the level of an utterance or a “continuous unit[s] of speech beginning and ending with a clear pause”⁹³.

Fig. 1.

Number talk task stimuli.

Each transcript was coded at two levels: (a) number talk instances and (b) interaction summaries. Number talk instances were defined as an inclusive unit of analysis that captures a string of sequential numbers, a comparison of numbers, an effort to add or subtract numbers, or the use of a single number word to communicate a cardinal value or to answer a question, adapted from Gunderson and Levine⁵⁵. Each number talk instance was coded for: (1) speaker; (2) highest magnitude; (3) context (adapted from Levine et al.⁵⁰ and Lu et al.⁹²); and (4) presence or absence of objects. Our adaptations to coding the context of number talk were made because Levine et al.’s number talk data were gathered when children were 14 and 30 months of age; to account for the older children in the current sample, our coding system included a wider range of number talk context codes (i.e. more advanced forms of number talk between parent and child). For instance, their coding system captured only instances of counting and use of cardinal values whereas our system included number talk about both absolute and relative magnitudes.

Summary coding included: (1) duration of interaction in seconds; (2) total P1 utterances; (3) total child utterances; (4) frequency of saying, “count” and “how many” by speaker; and (5) presence of written numerals on the party paper (yes/no) and the highest number written. Number talk variables were created to represent the speaker, set size (small = 1–3; medium = 4–6; large = > 6), context, and presence or absence of objects.

For the present study, the following P1 number talk variables were created for analysis (see Supplementary Information Table S8 for detailed descriptions and examples): (a) frequency of P1 saying, “count”; (b) frequency of P1 saying, “how many?”; (c) summaries of absolute magnitude number talk associated with small sets, medium sets, and large sets of present and non-present objects separately and, (d) frequency of number talk referring to relative magnitudes. Although the target length for all parent-child interactions was 15 min (900 s), the actual duration varied slightly across dyads. The mean session length was 902.44 s (SD = 54.63; range = 618 to 977 s). To account for this variation in session length, all number talk variables were pro-rated to reflect the amount of number talk that would be expected if the session had lasted exactly 900 s, or 15 min (i.e. multiplied by 900/actual session length in seconds).

Coding was conducted by trained research staff, and some transcripts were coded by team consensus. 36% of (English) transcripts (n = 28) were coded by two trained coders and inter-coder reliability was calculated using percent agreement for identification of number talk instances (average = 89%) and present versus not present objects (average = 95%) and using intra-class correlations for the number talk summary variables listed in Supplementary Information Table S8.

The seven core parent number talk variables were submitted to a principle components analyses with oblique rotation (SAS⁹⁴. The analysis identified two components (Eigen values > 1) and the associated standardized loadings > 0.40 are shown in Table 2. The first component was defined by variables that index more simple number talk (e.g. counting) and the second by variables that index more complex number talk (e.g. magnitude comparisons). These variables were standardized (M = 0, SD = 1). A simple number talk composite was created based on the mean of the four associated variables (α = 0.65), and a complex number talk composite by the mean of the three associated variables (α = 0.76). Frequent engagement in simple number talk was associated with frequent engagement in complex number talk (r = .41, p = .0008).

Table 2.

Standardized loadings (> |0.40|) for parental number talk variables.

Variable	Component
	Number talk: Simple	Number talk: Complex
How many?	0.597
Count	0.541
Small magnitude	0.611
Medium magnitude	0.875
Large magnitude: Objects		0.781
Large magnitude: No objects		0.814
Relative magnitude		0.803

Statistical methods

The 4.0% of missing data were estimated using the mean of five multiple imputations using the proc mi procedure of SAS⁹⁴. We used Bayes factors and standard regression techniques to make decisions about the best predictors of fall-to-spring gains in cardinal knowledge for number words and numerals. The Bayes factors have the advantage of being less sensitive to sample size than standard methods and thus we began with them to reduce the number of potential predictors before moving to the standard regressions. Moreover, rather than using the Bayesian analyses to simply predict the core outcomes of interest, that is, the spring cardinality scores, we included the fall measure of the same cardinality variable in all models. Controlling fall cardinality scores should reduce the chance of identifying spurious relations, that is, false positives, with spring cardinality scores, although this might increase the chance of false negatives. With a sample of 86, a multiple regression with five core predictors has a power = 0.80 with a medium effect size (f² = 0.16) and α = 0.05.

The Bayes factors were computed for regression models⁹⁵ using the BayesFactor package (version 0.9.2+) for R⁹⁶. Following Geary and vanMarle², we ran the Bayes models for similar sets of variables (i.e. child variables, classroom variables, parental cognitive and number talk variables, and parental attitudes, beliefs, and anxiety) to predict spring performance on the cardinal number word and numeral variables, controlling for fall performance on the same variable. The models included all potential combinations of similar variables and provided evidence for the identified set of variables in terms of odds ratios relative to chance (intercept in these models). Generally, ratios greater than 3 are considered suggestive evidence for the set of variables and those greater than 10 are considered strong evidence⁹⁷.

The top variables identified in each analysis were then included in a combined Bayes regression analysis along with three control variables. The first was whether English was used the most at home (i.e. -1 = English was the primary or only language spoken at home, 1 = English was spoken at home but not the primary language, 0 = English was not often spoken at home). The two others were parental education level for the primary parent and the other parent (4-year college degree = 1, no degree = 0). Although the Bayes approach provides some advantages over standard regressions, it is still susceptible to Type 1 errors and thus for the final model, we dropped each predictor one at a time and evaluated change in model probability. Variables that were less than 33% as probable without them in the model were retained. For instance, if a model without variable A is 20% as probable as the model with it, then evidence for the importance of variable A is 5 to 1. The predictors that survived the Bayesian analyses were then used in a standard regression analysis, after standardizing the variables (M = 0, SD = 1), along with the educational and home language controls.

Results

The mean scores for the key variables used in the analyses are shown in Table 3, and correlations among them are in Supplementary Information Table S9. Dependent t-tests confirmed that children’s knowledge of the cardinal value of number words (M = 1.23, SD = 1.69, t = -6.76, df = 85, p = 1.672e-09, d = 0.57) and numerals (M = 1.57, SD = 1.84, t = -7.89, df = 85, p = 9.257e-12, d = 0.65) improved across the 5 months between the fall to spring assessments.

Table 3.

Variables used in analyses (note standardized achievement scores for all variables are presented but Raw scores were used in analyses, because there is more variability in Raw scores and thus, they are more sensitive to individual differences than standardized scores.)

Variable	Mean (SD)	Minimum	Maximum
Child quantitative
Cardinal knowledge words (fall)	3.17 (2.18)	0	6
Cardinal knowledge numbers (fall)	2.24 (2.40)	0	6
Verbal counting	10.60 (12.23)	0	100
Enumeration	11.55 (4.82)	0	20
Spontaneous focus on number	1.79 (1.19)	0	4
Numeral recognition	5.70 (4.71)	0	15
Ordinal choice	2.49 (1.02)	0.5	4.41
Cardinal knowledge words (spring)	4.41 (2.04)	0	6
Cardinal knowledge numbers (spring)	3.81 (2.36)	0	6
Child cognitive
Executive functions	95.68 (10.97)	61	114
Intelligence	98.49 (13.48)	66	126
Working memory	99.13 (13.97)	64	134
Child classroom
Command compliance	11.99 (7.87)	0	36
Appropriate behavior	58.72 (2.76)	40	60
On/off task	54.97 (11.28)	2	60
In-class attention	4.08 (0.93)	1.89	6
In-class hyperactivity	4.11 (0.93)	1.44	6
Opportunity to learn in classroom
Count	5.23 (2.60)	1	11
Compare	11.48 (6.76)	3	29
Order	4.30 (2.39)	0	13
Relation	7.23 (2.53)	3	12
Number	7.93 (2.39)	4	12
Numeral	4.69 (3.12)	0	14
Parent attitudes and home environment
Mathematics anxiety	3.78 (2.72)	1	10
Math teach	3.34 (0.48)	2.2	4
Math importance	3.67 (0.37)	2.75	4
Subjective numeracy	4.17 (1.63)	1.43	6
Numeracy activities counting	3.86 (1.05)	1	5
Numeracy activities measurement	1.72 (1.25)	0	4.5
Numeracy activities games	1.75 (1.03)	0	5
Numeracy activities add	2.08 (1.34)	0	5
Numeracy activities computer	1.71 (1.53)	0	5
HOME learn	0.58 (0.42)	0	1
HOME toys	0.77 (0.36)	0	1
Parent number talk and cognitive assessments
Number talk simple	5.99 (3.74)	0.25	17.55
Number talk complex	1.80 (1.87)	0	7.67
Intelligence	92.73 (13.95)	59	120
Mathematics achievement	92.62 (17.08)	50	137
Reading achievement	91.42 (14.76)	49	114

Bayes regressions

The best set of child predictors of spring performance on the cardinal knowledge of number words measure included fall cardinal knowledge of number words, enumeration, and executive functions; for this analysis and all others, the top five models are shown in Supplementary Information. The associated BF was 1.74534e + 11, meaning that the likelihood of this set of predictors relative to a null model with no predictors is 1.74534e + 11 to 1. For the classroom predictors, the best model was fall cardinal knowledge of number words with no other identified variables. This result indicates that none of the specific classroom variables related to the child’s behavior or teacher-reported mathematics activities added to the prediction of spring cardinal knowledge of number words. Among the parental cognitive and number talk variables, fall cardinal knowledge of number words, word reading achievement, and parental complex number talk were the best set of predictors of spring cardinal knowledge of number words (BF = 28,942,087,149). For the parental attitudes and beliefs predictors, the best model was fall cardinal knowledge of number words with no other identified variables.

The combined analysis included fall cardinal knowledge of number words, enumeration, executive functions, parental word reading achievement and parental complex number talk. The best model included fall cardinal knowledge, enumeration, executive functions, and parental complex number talk (BF = 2.058549e + 12 ); word reading was no longer included in the top model. These four predictors were included in a follow-up model that added the parental education and home language controls. The corresponding top model included fall cardinal knowledge, executive functions, parental complex number talk, and the education contrast for P1 (BF = 3.637201e + 12). These were then dropped one at a time and changes in model fit were evaluated. The model without the parental education contrast was 31.5% as likely as the model with it. The corresponding values for parental complex number talk, executive functions, and fall cardinal knowledge were 6.6%, 1.2%, and < 1%, respectively. In other words, the BF for retaining parental complex number talk was 15.15 (1/0.066), and 83.33 and > 1,000 for executive functions and fall cardinal knowledge, respectively.

The best set of child predictors of spring performance on the cardinal knowledge of numerals measure included fall cardinal knowledge of numerals and number words and numeral recognition (BF = 245,592,707,959). Again, none of the classroom or parental attitudes predictors emerged beyond the importance of fall cardinal knowledge of numerals. Among the parental cognitive and number talk variables, fall cardinal knowledge of numerals, word reading achievement, and parental complex number talk were the best set of predictors of spring cardinal knowledge of numerals (BF = 626,954,789,453).

The combined analysis confirmed that fall cardinal knowledge of numerals and number words, numeral recognition, and parental complex number talk predicted spring cardinal knowledge of numerals (BF = 1.681668e + 12), but word reading was no longer included in the top model. These same four predictors emerged as the top model with inclusion of the parental education and home language controls. Dropping parental complex number talk resulted in a model that was 16.6% less probable than the model with it; in other words, the BF to retain it was 6.02 (1/0.166). The corresponding BFs for retaining numeral recognition, fall numeral cardinal knowledge, and fall number word cardinal knowledge were, respectively, 3.18, 76.92, and 1.58. The latter value suggests that cardinal knowledge of number words does not substantively add to the prediction of spring cardinal knowledge of numerals and thus was dropped.

Standard regressions

The standard regression results for predicting spring performance on the cardinal knowledge of number words measure are shown in the top section of Table 4, adjusted R² = 0.57, p = 4.865e-14. As can be seen in Table 4, controlling other variables in the equation, a 1 SD advantage in fall cardinal knowledge of number words was associated with a 0.48 SD advantage in spring cardinal knowledge of number words, whereas 1 SD advantages in executive functions and parental complex number talk were associated with 0.30 and 0.27 SD advantages in spring cardinal knowledge of number words, respectively. The same core findings emerged after removing the 8 families with imputed number talk data (i.e. they did not do the task in English); fall cardinal knowledge of number words (β = 0.52, t = 5.61, p < .0001), children’s executive functions (β = 0.32, t = 3.58, p = .0006), and parental complex number talk (β = 0.24, t = 2.70, p < .0087). Similar findings were found for the 65 participants who only or primarily spoke English at home; fall cardinal knowledge of number words (β = 0.49, t = 4.69, p < .0001), children’s executive functions (β = 0.24, t = 2.22, p = .0301), and parental complex number talk (β = 0.23, t = 2.47, p < .0165) (see Supplementary Information).

Table 4.

Standardized regression results predicting spring performance on Cardinal knowledge measures.

Variable	Coefficients		Statistics
	β	SE	t	p
Predicting spring cardinal knowledge of number words
Intercept	-0.17	0.11	-1.62	0.109
Parent 1 education control	0.35	0.15	2.32	0.023
Parent 2 education control	-0.04	0.18	-0.22	0.824
Home language control	-0.10	0.10	-1.01	0.316
Fall cardinal knowledge of number words	0.48	0.09	5.58	0.000
Executive functions	0.30	0.09	3.55	0.001
Parental complex number talk	0.27	0.09	3.09	0.003
Predicting spring cardinal knowledge of numerals
Intercept	-0.18	0.11	-1.67	0.099
Parent 1 education control	0.15	0.16	0.94	0.349
Parent 2 education control	0.25	0.18	1.34	0.183
Home language control	-0.09	0.10	-0.91	0.366
Fall cardinal knowledge of numerals	0.56	0.09	6.29	0.000
Numeral recognition	0.24	0.09	2.62	0.010
Parental complex number talk	0.25	0.09	2.80	0.006

The results for predicting spring performance on the cardinal knowledge of numerals measure are shown in the bottom section of Table 4, adjusted R² = 0.57, p = 6.842e-14. As can be seen in Table 4, controlling other variables in the equation, a 1 SD advantage in fall cardinal knowledge of numerals was associated with a 0.56 SD advantage in spring cardinal knowledge of numerals, whereas a 1 SD advantage in numeral recognition and parental complex number talk were associated with 0.24 to 0.25 SD advantages in spring cardinal knowledge of numerals, respectively. The same core findings emerged after removing the 8 families with imputed number talk data; fall cardinal knowledge of numerals (β = 0.58, t = 6.21, p < .0001), numeral recognition (β = 0.30, t = 3.14, p = .0024), and parental complex number talk (β = 0.25, t = 2.72, p < .0083). The same pattern emerged for the 65 participants that only or primarily spoke English at home; fall cardinal knowledge of numerals (β = 0.53, t = 4.76, p < .0001), numeral recognition (β = 0.33, t = 2.95, p = .0046), and parental complex number talk (β = 0.23, t = 2.33, p < .0230) (see Supplementary Information).

Discussion

The study provides the most comprehensive (to our knowledge) multi-system, longitudinal assessment of the child, home, and classroom factors associated with children’s early mathematical skills and knowledge, with a focus on gains in children’s conceptual understanding —cardinal knowledge—of the quantities associated with number words and numerals. Children’s development of cardinal knowledge is theoretically important because it is their first conceptual understanding of formal mathematics^16,18,19,21, and practically important because individual differences in the rate of developing this knowledge predict readiness for mathematics learning at the beginning of 1st grade¹⁵. The core finding was that gains in cardinal knowledge were related to individual differences in the complexity of parental number talk, with executive functions also contributing to gains in cardinal knowledge of number words and numeral recognition contributing to gains in cardinal knowledge of numerals.

Child factors

The assessed child factors included domain-general cognitive abilities, such as executive functions, that have emerged in prior studies as predictors of cardinality performance²⁸ or mathematics achievement more broadly^7,35,36. The importance of the executive functions measure in predicting gains in cardinal knowledge of number words supports prior findings for cardinality²⁸ and for early mathematics achievement generally^35,36. The result is consistent with the importance of cognitive control networks that support attentional focus and the efficient coordination of brain regions and cognitive processes to meet task goals for aspects of early number learning⁹⁸.

The child factors also included domain-specific counting and number competencies that have been implicated as potential prerequisites to children’s development of cardinal knowledge^28,29,32,99. In the Bayesian analyses, enumeration emerged as the key domain-specific predictor of gains in cardinal knowledge of number words, replicating an earlier finding²⁸ and consistent with the relation between counting a set of objects and the last word of this count representing the cardinal value of the set^21,22,29,100. The learning of this relation might initially occur as children associate small sets (< 4) with the associated number word, even when they do not have to count^18–21. Mix et al.’s¹⁰¹ experimental study suggested that labeling the cardinal value of a set—done by an experimenter—and then asking the child to count the set contributed to children’s gains in cardinal knowledge, specifically realizing that the last number word in a count represents the cardinal value of the set. However, enumeration did not survive the inclusion of parental education and home language controls, suggesting overlap between children’s competence with enumerating and parental factors–indeed, enumeration scores showed a stronger correlation with parental math achievement (r = .27) than other child fall quantitative scores (rs < 0.12) (Supplementary Information Table S9). In all, these results are in keeping with prior studies showing that skill at enumerating is a precursor to developmental gains in learning the cardinal value of number words.

The predictors of gains in children’s understanding of the cardinal value of numerals are consistent with Knudsen et al.’s¹⁷ results, that is, these gains are influenced by children’s recognition of numerals and perhaps understanding of the cardinal value of number words. The latter is more tentative because the influence of the number word predictor did not survive the inclusion of parental education and home language controls. Our results also suggest that enumeration—verbally labeling objects with number words as they are counted —does not directly facilitate gains in children’s cardinal knowledge of numerals, but may be uniquely important for gains in knowledge of number words, in keeping with Spelke’s²⁰ proposal about the importance of language for early number learning. Moreover, the finding that executive functions predicted gains in number word, but not numeral knowledge suggests that generalizing from the cardinal values of number words to the corresponding numerals might not be as cognitively demanding as the initial learning of the cardinal values of number words.

Classroom factors

Studies of kindergarten and older children have consistently shown that in-class attentive behavior predicts various mathematics outcomes, controlling other factors (e.g. domain-general abilities, prior mathematics achievement)^3,4,65,67. These relations have not been as consistently studied in preschool children, and our results suggest it is less important than it is for older children. However, the children in our study were in the first year of preschool and it could be that classroom behavior becomes more important in the second year of preschool as academic material becomes more difficult and teacher expectations of children’s attentive behavior increase, but this remains to be determined. Given in -class attentive behavior predicts older students’ mathematics achievement, it would be premature to abandon in-class assessments of preschoolers’ attentive behavior⁶⁸.

Studies of the opportunity to learn (OTL) mathematics in classroom settings indicate that frequently presented content is learned better than less frequently presented content in the classrooms of kindergarten⁷² and elementary school⁷⁰ students. The main analyses did not reveal this pattern for early preschool children. The math content in our OTL measure was reported by teachers in the fall semester and thus it is possible that there was not enough content exposure by this point to significantly influence children’s understanding of cardinality. Additional OTL assessments will be needed (and are planned as part of the larger project) to fully evaluate the relative importance of math-content exposure in preschool classrooms for students’ early mathematical development.

Parent and home factors

There are many studies that have focused on the relation between parental and home factors and young children’s mathematics outcomes^37,38,40,41, as well as theoretical models of how these factors might influence child outcomes^102–104. Commonly hypothesized and assessed attributes include parental math anxiety and their attitudes and beliefs about the importance of mathematics, as well as their engagement in home numeracy activities and number talk with their children^49,61. Overall, there is a modest relation between these parental and home factors and children’s mathematics outcomes, especially for younger children⁴⁷. However, there is no consensus on which of these factors or combinations of them are the most important for facilitating young children’s early mathematical development⁴⁵. The current study provided a broad assessment of all these factors and sought to identify the most plausible contributors to preschoolers’ gains in cardinal knowledge.

The complexity of parental number talk emerged as an important predictor of gains in preschoolers’ cardinal knowledge of both number words and numerals and emerged in both the Bayesian and standard regressions. In keeping with prior studies^{39,50,53–55}, it is not necessarily number talk per se, but rather talk that involves larger quantities (> 6 in this study) and comparing, manipulating, and highlighting the relations between sets of numbers (e.g. “You would be upset if [sibling] got two and you only got one.”). The frequency of this type of number talk does not always predict performance on cardinality measures but does appear to contribute to children’s understanding of quantitative magnitude⁵¹. Our results suggest that at least aspects of parents’ complex number talk contribute to young children’s learning of cardinality. Carrazza and Levine’s⁵² recent intervention study focused on parent-child number talk and indicated that having the parent state the cardinal value of a set of objects and then asking the child to count the set facilitated children’s learning that the last number word in a count represents the cardinal value of the set, as found in Mix et al.’s¹⁰¹ experimental study.

The parental number talk captured by our task elicited these types of parent-child interactions, but not as systematically as Carrazza and Levine’s intervention. Moreover, it is possible that a different type of number task, such as one that prompted parents to teach specific skills, would elicit different parental number talk than our task, and more explicitly address issues related to the relative importance of direct (e.g. teaching the count string) or indirect (e.g. number games) activities for early quantitative development.

No other parent or home measure predicted children’s gains in cardinal knowledge, which was unexpected given prior studies showing that higher parental math anxiety, for instance, might be related to lower math outcomes for preschool children^62,63, as well as Daucourt et al.’s⁴⁷ findings of a robust but modest relation between home mathematics activities, parental expectations, and children’s mathematics outcomes. It is possible that we did not find these relations because of our narrow, but theoretically driven focus on gains in cardinal knowledge. In other words, relations between parental math anxiety and home math activities and expectations might be more robust for broader measures of math outcomes. Moreover, our home measures were exploratory and not aligned with theoretical debates regarding the relative contributions of direct home activities (e.g. use of flashcards) as contrasted with less direct activities (e.g. games that involve numbers) to children’s mathematics outcomes. Future studies that are better aligned with this theoretical debate might reveal stronger relations than found here.

Limitations and conclusion

The biggest limitation is the correlational nature of the data. Although focusing on gains in cardinal knowledge (i.e. spring performance, controlling fall performance) allows for stronger inferences about potential causal relations than correlations, these inferences are not as strong as would be possible with a randomized intervention study. Due to the labor-intensive nature of data collection and coding in this area, many studies have relatively small sample sizes, including the current one. This could result in false negatives or spuriously large effects that will not replicate, which is why our initial analyses were Bayesian and thus less sensitive to sample size than standard (frequentist) statistics⁹⁷. Nevertheless, the large number of potential predictors of gains in cardinal knowledge creates a potential for false positives. The Bayesian analyses were used to identify the most plausible predictors of cardinality gains and provided a first-step in reducing the risk of false positives, but this is not a guarantee that we eliminated all of them. Accordingly, our main findings regarding the importance of the complexity of parental number talk for early gains in children’s cardinal knowledge should be considered provisional until replicated, preferably with an intervention study focused on modifying parental number talk. Finally, we attempted to assess a broad range of child, home, and school factors that are correlated with young children’s mathematics outcomes, but this does not mean that we captured all these potential factors.

Despite these limitations, the study provided a very broad multi-system assessment of preschoolers’ emerging understanding of the cardinal value of number words and numerals. Understanding the factors that contribute to these gains is important, because acquiring cardinal knowledge appears to be a critical step in forming an integrated system of number knowledge that is important for mathematics learning in elementary school^15,24–26. The results also suggest that early preschool interventions should focus on improving children’s ability to enumerate and label set sizes and their recognition of numerals, along with a home component that facilitates complex parent-child number talk focused on counting and cardinality such as that done by Carrazza and Levine⁵².

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

D. C. G., E.S., S.G., & M.K. H. drafted the manuscript. E.S., S. G., and J.A.B developed the number talk task and scoring protocol. D.C.G. conducted the analyses. N.L. & M.K.H. managed data collection, task preparation, and drafted task manuals. All authors reviewed the manuscript.

Funding

The project was supported by grant P20 HD109951 from the Eunice Kennedy Shriver National Institute of Child Health and Human Development of the National Institutes of Health.

Data availability

Data, analyses code, and task manuals are available on Open Science Framework, [https://osf.io/2ew7d/](https:/osf.io/2ew7d).

Declarations

Competing interests

The authors declare no competing interests.