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Edited by: Korbinian Moeller, Knowledge Media Research Center, Germany

Reviewed by: Erin A. Harper, Miami University, USA; Silke Melanie Goebel, University of York, UK

*Correspondence: Silvia Benavides-Varela

This article was submitted to Developmental Psychology, a section of the journal Frontiers in Psychology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

It is currently accepted that certain activities within the family environment contribute to develop early numerical skills before schooling. However, it is unknown whether this early experience influences both the exact and the approximate representation of numbers, and if so, which is more important for numerical tasks. In the present study the mathematical performance of 110 children (mean age 5 years 11 months) was evaluated using a battery that included tests of approximate and exact numerical abilities, as well as everyday numerical problems. Moreover, children were assessed on their knowledge of number information learned at home. The parents of the participants provided information regarding daily activities of the children and socio-demographic characteristics of the family. The results showed that the amount of numerical information learned at home was a significant predictor of participants' performance on everyday numerical problems and exact number representations, even after taking account of age, memory span and socio-economic and educational status of the family. We also found that particular activities, such as board games, correlate with the children's counting skills, which are foundational for arithmetic. Crucially, tests relying on approximate representations were not predicted by the numerical knowledge acquired at home. The present research supports claims about the importance and nature of home experiences in the child's acquisition of mathematics.

At the time children enter school education they already show great individual differences in their numerical performance (e.g., Aunola et al.,

Parents usually report using literacy activities (e.g., sharing book reading) more frequently than numeracy activities with their children at home (Blevins-Knabe et al.,

Parents' expectations about numeracy play a significant role in the basic calculation skills of their children. One longitudinal study showed that children's attitudes toward mathematics were influenced more by their parents' beliefs about their child's abilities than by the child's own results in previous mathematical assessments (Parsons et al.,

The socioeconomic background of the family also influences arithmetical development (Melhuish et al.,

One potential causal factor in the above studies is that socially more advantaged parents with higher numeracy expectations engaged in more numeracy-related practices, which in turn is associated with children's higher mathematics achievements (LeFevre et al.,

The above studies consistently show a relation between direct numerical instruction at home and children's math performance. With respect to the relations between numeracy knowledge and other daily activities that indirectly facilitate acquisition, the literature is much less developed. LeFevre et al. (

In agreement with the study of LeFevre and colleagues, Ramani and Siegler (

Whilst the ability to roughly approximate the numerosities of sets is present in humans even from birth (Izard et al.,

In the present work, we thus aimed at studying the effects of indirect numerical exposure further, focusing on how activities within the family environment and the numerical information learned at home relate to each of the representational systems underlying mathematical performance. This information should contribute to understand the permeability of the exact and the approximate systems in early childhood and should consequently be useful to elaborate refined educational programs adapted to the needs of the learners who begin the transition from the family to the pre-school/school learning environments. The tasks implemented in the current study are well known in the field. They were adopted from established groups that have evaluated approximate representations in magnitude comparison (Halberda et al.,

The current work is different from the previous studies in several respects. First and foremost: the abilities under investigation. The most influential study that has so far focused on the effects of indirect exposure to numbers in the family environment measured arithmetical performance (LeFevre et al.,

A hundred and ten pre-school children (59 females; 100 right-handers; mean age: 5.95 years-old; age range: 5.46–6.43 years-old) participated in the study. Participants were enrolled in the last year of kindergarten in five different urban and extra-urban schools of the province of Padua, Italy. The schools and teachers involved in the research were contacted in a meeting of Educational Psychology organized by the University. All children had normal or corrected-to-normal eyesight and their parents reported no history of health problems during pregnancy, infancy, or childhood that would compromise their perceptual or intellectual abilities. Direct measures of intellectual abilities were not obtained because the Italian policies allow them only for clinical purposes. The range of socio-economic status (SES) of the families was very wide as indexed by parental education and occupation. Parental education was coded on a six-point scale based on the level of education completed by each parent; zero = no schooling; one = primary school; two = middle-school; three = high-school or professional school; four = university degree; five = post-graduate studies. Occupation was coded on a scale of zero to two; zero = no occupation or not paid job; one = manual work; two = service sector or intellectual work. The level of education and occupation of the mother and the father were added into a single index of SES that ranged from 0 to 14. Families distributed according to the SES index in the following way: 1% were of high SES, 97.2% of middle SES, 1.8% of low SES. All parents provided informed consent for the tests and interviews. The data of all participants were collected in accordance with the Helsinki Declaration II and the Institutional Ethics Committee of the Psychology Department at Padova University.

Children completed a battery of numerical tasks, including tasks relying on approximate representations: magnitude comparison (Halberda et al.,

Additionally, we obtained data about family activities from two questionnaires -one for the child and one for the parents. The child's questionnaire (see Supplementary Material) was designed to directly assess the children's knowledge of numerical facts acquired in the family environment such as birthdates, number of siblings, phone numbers. Each correct response scored one point. The total scoring of the children's questionnaire corresponds hereafter to “numerical information learned at home.”

In order to assess congruency between parents' and children's accounts of the information children knew, we also interviewed the parents about number-related information, asking whether they thought their children knew this information. Each affirmative response scored one point. We also asked parents about the family constitution and background. Moreover, parents indicated the frequency on a weekly basis of number-related and non-related activities of the child (e.g., playing board games, videogames, etc.) and those involving the family (e.g., shopping, reading, tv watching, etc.). A five-point incremental scale was used for scoring frequency: does not carry out this activity, does it for about 1, 2, 3, 4 h, or more per week.

The criteria for including specific daily activities in the questionnaire derives from studies indicating that such activities facilitate academic performance and cognitive control more generally (i.e., reading –LeFevre et al.,

The tests were administered individually in a silent room. No feedback was given, only general praise and encouragement. A brief description of each test follows.

Counting tests assessed the child's mastery of the number word. The children were requested to count forwards (from 1 to 10; from 4 to 10) and backwards (from 6 to 1; from 10 to 1). Children were instructed to count as fast as possible. One point was given for each correct sequence.

One-to-one correspondence task was adapted from the Early Numeracy Test (ENT-part A, Van Luit et al.,

Magnitude comparison: children were presented (maximum 2 s) with two panels –one on the left and one on the right– that contained randomly arranged sets of squares of varying sizes. In each trial participants were asked to point as fast as possible to the panel that contained more squares, and were prevented for counting. There were six test trials randomized across participants (6:9, 5:6, 6:8, 6:9, 2:5, 8:9). One additional trial (1:2) was used for practicing before starting the test. Each correct response scored one point. This task was based on previous studies assessing approximate skills in children (Halberda et al.,

Number line task

Everyday numerical problems consisted of seven tests meant to assess the children's ability to use numbers in situations of the daily living. The problems involved the application of basic arithmetic operations (e.g., could you show me two types of fruits or vegetables that together will make 7 pieces?), magnitude comparison (e.g., are there fewer lemons or fewer strawberries?), or money usage [e.g., “each one of these vegetables costs 1 euro. Could you bring me the box that contains the number of vegetables that will make you spend all this money (5 euros)?]. Each correct response scored one point.

We first performed Pearson's correlation tests to determine the association between the number-related tasks and the child and family informal activities. To make sure that the relationships between those tasks were not mediated by a general cognitive or demographic factor, we computed partial correlations between the number-related tasks and the child and family activities, in which the impact of children's age, short-term verbal memory, and the SES of the family were controlled for.

Additionally, we used a stepwise regression procedure to determine which of the environmental variables predicted variation on each of the math cognition tasks. We included the numerical tests as the dependent variables and the knowledge of number information learned at home, short-term verbal memory, age, and SES of parents as potential predictors. Before running the regression analysis we checked for collinearity among predictors. Because there was a high collinearity (

The mean percentage of errors was 14% (range 0–75%) in the counting task, 22% (range 0–80%) in the one-to-one correspondence task, 13% (range 0–67%) in the magnitude comparison task, and 25% (range 0–75%) in everyday numerical problems. For the number line task mean percent absolute error was 21 (range 11–37%). It is worth noting that the accuracy of estimation in the number line task is similar to the one reported in comparable age groups (interval 1–10, studied by Berteletti et al.,

The results of the Pearson's correlation analysis showed that certain early numerical abilities were intercorrelated (see Table

1 | Counting | |||||||||||||||||||

2 | Magnitude comparison | 0.22 | ||||||||||||||||||

3 | Everyday numerical problems | 0.21 | ||||||||||||||||||

4 | Number line | –0.09 | –0.23 | |||||||||||||||||

5 | One–one correspondence | 0.18 | –0.01 | |||||||||||||||||

6 | Child's answers | 0.12 | 0.06 | |||||||||||||||||

7 | Parent's answers | 0.16 | 0.06 | 0.02 | 0.09 | 0.19 | ||||||||||||||

8 | Videogames | –0.11 | –0.11 | –0.14 | 0.01 | –0.07 | 0.02 | –0.05 | ||||||||||||

9 | Reading | –0.09 | –0.10 | –0.25 | 0.08 | –0.01 | 0.02 | –0.02 | ||||||||||||

10 | TV watching | –0.02 | –0.15 | –0.19 | 0.06 | –0.14 | –0.07 | 0.14 | 0.15 | |||||||||||

11 | Sports | –0.03 | –0.06 | –0.03 | –0.05 | –0.18 | 0.06 | 0.13 | 0.07 | 0.15 | –0.05 | |||||||||

12 | Music | 0.02 | –0.07 | 0.06 | 0.09 | –0.03 | 0.11 | 0.08 | –0.10 | –0.04 | –0.13 | 0.02 | ||||||||

13 | Other activities | 0.20 | 0.14 | 0.02 | –0.19 | 0.18 | –0.02 | 0.13 | –0.13 | 0.27 | 0.08 | 0.24 | –0.11 | |||||||

14 | Shopping | –0.23 | –0.02 | –0.18 | 0.16 | 0.05 | 0.03 | 0.21 | 0.11 | 0.13 | –0.08 | –0.02 | –0.07 | 0.00 | ||||||

15 | TV watching | –0.08 | –0.08 | –0.20 | 0.10 | –0.22 | –0.13 | 0.03 | 0.27 | –0.07 | 0.10 | –0.21 | 0.01 | 0.17 | ||||||

16 | Reading | 0.06 | 0.13 | 0.00 | 0.02 | –0.01 | –0.01 | 0.13 | –0.13 | 0.23 | –0.17 | 0.16 | 0.04 | 0.14 | 0.02 | 0.10 | ||||

17 | Sports | 0.03 | –0.09 | 0.05 | –0.06 | 0.08 | –0.01 | 0.08 | 0.02 | –0.10 | 0.22 | 0.05 | –0.02 | –0.06 | 0.15 | |||||

18 | Board games | 0.04 | 0.18 | 0.09 | 0.19 | 0.12 | 0.11 | 0.02 | 0.17 | 0.00 | 0.01 | 0.03 | –0.03 | 0.13 | –0.01 | |||||

19 | Videogames | –0.12 | –0.12 | –0.15 | 0.00 | –0.09 | 0.01 | –0.06 | –0.03 | 0.06 | –0.11 | –0.15 | 0.11 | 0.28 | –0.14 | 0.07 | 0.11 |

The children's responses to the questionnaire inquiring into the knowledge of number related information, and the responses of the parents to the same questions were positively correlated (

Considering specific activities within the family environment, we found a positive correlation between counting and the frequency with which children played board games at home (

The stepwise regression with the counting ability as the dependent variable settled on a final model that included only numerical information learned at home as the actual predictor [β = 0.397; ^{2} = 0.157, _{(1, 109)} = 20.154; ^{2} = 0.091, _{(1, 109)} = 11.885;

Similarly, numerical information learned at home predicted children's performance on one-to-one correspondence tasks [β = 0.287, ^{2} = 0.083, _{(1, 109)} = 9.728, ^{2} = 0.188, _{(1, 109)} = 12.364, ^{2} = 0.033, _{(1, 109)} = 3.676; ^{2} = 0.064, _{(1, 109)} = 7.690; ^{2} = 0.009, _{(1, 109)} = 0.938;

In the current study, we explored whether activities within the family environment and numerical information learned at home relate to preschoolers' performance on different numerical tasks. The main question was whether numerical instruction embedded in real-life settings influences tasks that depend on approximate representations, on exact representations, or both.

The findings indicate that the early acquisition of numerical information within the family environment significantly predicts the children's ability to solve numerical problems in everyday situations, counting abilities, and the skills for identifying one-to-one correspondences between sets. Crucially, not every numerical skill was related to the acquisition of numerical information at home. Children's performance on number line or magnitude comparison tasks, that essentially tap the approximate system, were not predicted by the amount of numerical information learned in the family environment.

These findings are in agreement with previous studies showing a relation between numerical instruction at home and children's math performance (e.g., LeFevre et al.,

The numerical information learned at home had a large effect on the counting ability of the children, and this appears fundamental to arithmetic learning. For example, Geary and colleagues showed that individual differences in first graders' counting abilities correlated positively with differences in their arithmetic proficiency (Geary et al.,

The fact that parents' answers regarding children's knowledge of information did not correlate well with their children's performance on the test is not at all surprising. It supports previous views suggesting that children's accounts might be more informative than parental ones. It has been argued that, particularly in the mathematical domain, parents may miss a lot of activities, those when they are busy with other things, or when the child is at some distance from the parent (Tudge and Doucet,

The present data also show a positive correlation between the frequency with which children performed specific activities within the family environment and their numerical abilities. In particular the results showed a correlation between the frequency with which children practiced sports and their performance in the number line task, which might be due to the common involvement of spatial abilities in both activities. Moreover, there was a correlation between the frequency with which children played board games at home, their knowledge of number related information and their counting abilities. These results add to the body of literature on early numerical cognition, providing evidence that playing board games correlates with the development of numerical skills in children (Ramani and Siegler,

The current results showing differential correlations between numerical knowledge acquired at home and each of the two numerical systems, questions the possibility that the exact and the approximate representations are interconnected, at least in this particular group. If the exact representations linked to the approximate system, one would expect parallel effects on both approximate representations and exact representations. Instead, we found that children who knew more number-related information only showed better exact representations; the same association was not apparent or transferred to the approximate system. Results from previous studies are in agreement with these findings. Using highly controlled training programs for children on either exact or approximate skills, Obersteiner and colleagues showed that participants improved only the skill trained with no crossover effects (Obersteiner et al.,

The question arises at to the limits of the information learned at home and activities within the family environment as potential modulators of the children's numerical skills. Are younger children, who generally spend more time at home, more susceptible to the numerical activities within the family environment than older children? Is there a critical period after which these activities do not impact significantly the children's representations of numbers? While the current study focused on 5- to 6-year-old children's numerical abilities, future studies with children at different ages should be able to establish whether, for instance, earlier numerical experiences can provide cascading advantages to the young math learner. A comprehensive exploration of the developmental trajectories of different numerical abilities (e.g., counting, estimation, magnitude comparison) and their interaction with everyday exposure to numbers should contribute to further clarify these issues. Such information will be also crucial for identifying temporal windows in which interventions for children at risk might be more effective.

To summarize, the present study sheds light on the way numerical information acquired at home links to numerical representations in 5–6 years-old children. It is pointed out that the development of certain basic numerical concepts is associated with the amount of numerical facts acquired at home and the frequency with which children carry out specific activities within the family environment. In particular, the data suggests that early mathematical concepts associated with exact—but not approximate—representations can be enhanced when learning numerical information takes place in real–life activities at home.

At this point of our research it is not possible to identify the directionality of the correlations between numerical information learned at home and children's performance on certain numerical tests. One possibility is that children with higher intrinsic mathematical abilities demand more numerical information from the family environment. However, these results could also suggest that the frequent use of numbers in familiar interactions enhances children's numerical knowledge. Although it seems likely that a regular exposure to numerical information boosts basic numerical understanding in young children, more research is needed to determine the directionality of the influences that occur between intrinsic numerical abilities and family factors.

Another limitation of the work regards the extent to which the numerical information learned at home interacts with the information acquired in other institutions. Although current pre-school programs in Italy are not concerned with teaching birthdates, age, phone numbers of the family members, number of brothers, etc. we do not exclude the possibility that other number related everyday knowledge might be consolidated in this environment.

Finally, we obtained information about the specific board games the children played at home. All Italian parents who reported their children playing board games mentioned numerical games such as “tombola,” “uno,” “carte,” but also non-numerical games such as memory or puzzles. The time invested in each of these types of games was not assessed, however this information might be important to determine the nature of the games that significantly influence numerical understanding.

SB, BB, CS conceived and designed the work. SB, FB, and GA acquired the data and performed the analysis. SB, BB, CS, and DL interpreted the final results. All the authors contributed writing and revising the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We wish to thank the teachers and the principals who collaborated with us during the present research, as well as the parents and children who participated in the study. We also thank Anna Tosin, Luis Diego Benavides, and Vanessa Vargas for helping acquiring and processing the data. The present study was supported by the Italian Ministry of Health (F-2009-1530973) and by “Progetto strategico NEURAT” from the University of Padua to CS.

The Supplementary Material for this article can be found online at: