Gender differences in mathematical achievement development: a family psychobiosocial model

This study proposes a family psychobiosocial model on gender differences in cognitive development. Specifically, the aim is to investigate how family biological, socioeconomic, and psychological factors predict child mathematics achievement (MAch) development. The data were obtained from the Millennium Cohort Study. Children’s pattern construction scores collected at ages 5 and 7 years worked as MAch (n = 18,497). The predictors were family data collected when the children were 9 months. The results of path analyses for all students indicate that all three factors in the family psychobiosocial model play some roles in children’s MAch development. Analyses for the female and male students separately reveal that girls’ positive MAch development was significantly predicted by four psychobiosocial factors (fewer mother in-pregnancy alcohol intakes, more family income, higher mother education levels, and more mother cognitive stimulation); boys’ MAch development is predicted by only one factor (higher mother education levels). The results support the psychobiosocial model as a whole. Family psychobiosocial factors, especially social factors, impact children’s cognitive development more for females than for males.

Various biological and cultural issues set the boundaries for individual development, with family factors playing key roles (McCulloch & Joshi, 2001).Children's cognitive developments may be informed by parental biological, social, and psychological factors in their early family experiences.
Mathematical achievement (MAch) or ability is one of the major indicators or representations of children's cognitive abilities.Child development in MAch may be properly understood by investigating the direct or proximal bio-ecological processes that impact children's early lives.With a large spectrum of contributing factors, influences of family's lives and parental behaviors start since children's birth (Bronfenbrenner, 1986(Bronfenbrenner, , 1994;;Bronfenbrenner & Ceci, 1994).
The relative importance of the diverse biological, social, and psychological factors of families for child cognitive or MAch development has rarely been addressed in previous academic research.Longitudinal data are especially valuable in examining this issue.The purpose of this study, therefore, is to use longitudinal data to identify relatively significant family biological, social, and psychological factors relating to child cognitive, MAch development.

Theoretical statement 1. Family psychobiosocial factors impact children's MAch along time
The psychobiosocial model (PM) explains why there are gender differences in cognitive abilities (Halpern et al., 2005).An individual has initial inputs from biological factors (e.g., genes), followed by psychological factors in the social context or environment, which, in turn, interact with biological factors (p.52).The PM emerges mainly in response to the two educational phenomena: (a) the persistent underrepresentation of females in science, technology, engineering, and mathematics (STEM) careers and (b) average (though unstable) lowered female mathematics achievement.For academic interests, the PM compensates the weakness of past research only studying puzzle pieces of elements that influence cognitive achievements or abilities.The PM also urges to place the pieces together in order to see the whole dynamic picture of the elements and their interactions and changes over time.
The PM focuses on children's personal factors in the biological, psychological, and social aspects, with little focus on family factors.For educational purposes, a family psychobiosocial model (FPM), posited in this study, can capture a fuller picture of family factors for children's development.The term "biopsychosocial" appears to be popularly used especially in medicine, psychiatry (Frazier, 2020), and gender issues (Leavitt et al., 2021), and a reasonable continuum from biology (more natural sciences), psychology, to sociology (more social sciences).This study starts its argument with the PM (Halpern et al., 2005), which aims at addressing the issue of gender differences in STEM achievements (including MAch), similar to the focus of this study.As such, this study uses the term "psychobiosocial," which is also used in research on sport (Filho, 2020) and on gender differences in clinical psychology (McCarthy et al., 2018).Both "biopsychosocial" and "psychobiosocial" mean the same for this study because the three factors are interwoven in each individual's lifespan though they may be with different degrees of relative significance in different phases.
To examine the FPM, three levels of questions should be raised: (1) Do family psychobiosocial factors impact children's mathematics ability?(2) Are there gender differences in the impact?(3) Because there are three major factors in the PM, the next question is: What is the relative importance of the psychobiosocial factors?
The answer to the first question should be positive because the FPM can be viewed as an extension from the PM.The PM assumes that gender differences in cognitive performance, especially mathematics achievement, can be explained by the three (psychobiosocial) factors.

Theoretical statement 2. Family psychobiosocial factors impact children's MAch development more for females than for males
A persistent phenomenon in gender difference is that boys have higher variances in achievement than girls, known as the greater male variability hypothesis (GMVH) (Chen et al., 2010;Hyde, 2014).The hypothesis has been evident in STEM and language fields in some studies (Baye & Monseur, 2016;Gray et al., 2019) but only in mathematics and not in languages in others (Pargulski & Reynolds, 2017).Most empirical studies support the hypothesis, and only a few studies do not (Chen, 2003;Hyde & Mertz, 2009).As evidenced by recent large-scale international studies, despite inter-cultural gender differences, a general phenomenon is that more boys than girls are top mathematics achievers (Organization for Economic Co-operation & Development, 2014, p. 72) or advanced mathematical problem solvers (Mullis et al., 2016).No cross-cultural studies to date have ever provided evidence to attribute social factors to the GMVH.
In summary, research on the GMVH implies that more diversity of boys' MAch than girls' is due to nature (i.e., sex or gender), not nurture.In terms of data analysis, the GMVH indicates that, compared to girls, boys have more variances in MAch.This means that boys' MAch is less likely to be accounted for by any certain factors other than gender.In this study, family biological and psychological factors cannot be completely separated from family social factors.It is because the three factors interact in the long-term socialized family environment and can be seen as a combined social factor, which is more nurture than nature.As such, the GMVH may successfully apply to the scenario of family psychobiosocial factors predicting MAch.In terms of statistical terms, Theoretical Statement 2 can be evidenced by higher variances or lower portions of the three family psychobiosocial factors combined in predicting MAch for boys than for girls.

Theoretical statement 3. Social factors account for more MAch for females than for males
The third question: What is the relative importance of the psychobiosocial factors?If addressed in statistical terms, that is: What are the proportions of the variances in mathematics achievement explained by the biological, socioeconomic, and psychological factors, respectively?This question is hard to answer because the three factors become interwoven along time of development.Speculations, however, may be inferred from related studies.
Research indicates that social support may reduce gender differences in MAch.Despite the small male advantages on MAch, recent meta-analysis and cross-cultural studies reveal that more gender-equal societies have fewer gender differences in MAch (Guiso et al., 2008), known as the gender stratification hypothesis (Else-Quest et al., 2010).The findings further support the gender similarities hypothesis (Hyde, 2005), which contends that gender differences in MAch are small and subject to social factors.In other words, social factors (e.g., gender-equal society and family socioeconomic status) account for gender differences in MAch.For educational practices, females need more positive social factors for MAch than males.
In summary, the gender stratification and similarities hypotheses give birth to Theoretical Statement 3: Relatively compared with family biological and psychological factors, family social factors serve as a more salient predictive factor for children's MAch in their later lives.The claim arises from past research evidence that gender-equal society and societal support would reduce gender differences in MAch.In statistical terms, regression and related methods (e.g., path analysis) can examine this claim because the standardized solutions for each factor indicate its capability in predicting outcome variables (MAch in this study), controlling for the other predictive factors.

Roles of family psychobiosocial factors in MAch
To examine the FPM, the best approach is to use a longitudinal dataset that includes valid, standardized, and repeatedly measured outcomes on child mathematical achievements and predictors of family biological, social, and psychological factors.The Millennium Cohort Study (MCS) provided high-quality and related measures to meet the need.For educational purposes, the author decided to focus on educationally meaningful, changeable family measures (e.g., maternal behaviors) that may fit the three theoretical statements of the FPM as suggested by the following literature reviewed.

Maternal biological factors
The family biological sub-factors used in this study focused on maternal ones, including maternal during-pregnancy and early-childhood alcohol intake and depression.Past literature recommends that the three sub-factors have a negative role in children's MAch.

During-pregnancy and early-childhood maternal alcohol intake
The timing of maternal alcohol intake is a factor in child development though with some uncertainty.Light drinking (more than one or two units of alcohol per week) during pregnancy does not relate to mathematical, spatial, and behavioral development for children at age 7 years (Kelly et al., 2013).A later study, however, finds that week-12 gestation and pre-pregnancy even with 2 units per week alcohol intake (an allowed intake suggested by the Department of Health guidance, England) both have detriment effects, leading to preterm birth or low-birth-weight babies (Nykjaer et al., 2014).A metaanalysis study comparing the effects of different quantities of prenatal alcohol intake on child cognitive development finds that binge alcohol intake has consistent negative effects on child cognition, and there is no known safe amount of alcohol intake during pregnancy (Flak et al., 2014).
The equivocal results about the effects of during-pregnancy maternal alcohol intake suggest a further investigation using a different model or methodology.There appears to be no study to date focusing on the effect of maternal alcohol intake during their children's early childhood.The MCS provides both during-pregnancy and early-childhood maternal alcohol intake data, which are reasonably effective predictors of children's cognitive abilities, including MAch, which is the outcome measure of this study.The juxtaposition of during-pregnancy and early-childhood maternal alcohol intake can facilitate our understanding of the relative importance of "biological" and "behavioral" alcohol intake in influencing children's MAch.

Maternal depression
Parental depression relates to increased physical discipline use, reduced parental warmth, lower child mathematics achievements, and undesirable approaches to learning (Bodovski & Youn, 2010).Mother prenatal depression may have a slightly negative effect on child IQ at age 8 years, but mother postnatal depression does not have an effect on child IQ (Evans et al., 2012).Maternal mental health when children at 0-30 months of age relate to the offspring's educational and occupational attainments at 30 years of age, although the relationship is mediated by the offspring's academic competence and mental health at 18 years of age (Slominski et al., 2011).Despite the diverse study contexts, previous research stably evidences the negative effect of maternal depression on children's cognitive development.

Family social factors
Family social or socioeconomic status (SES) has long been a concern of educators.SES disadvantages are stable factors for students' low cognitive achievements and high behavioral problems from early childhood to early adolescence (Rees, 2018).Three indicators of family SES selected in this study are family income, maternal education, and household minority language use, which are essential family SES factors for children's cognitive, mathematics achievement development, as indicated by related studies as follows.

Family income
Research on economics generally finds that family income is a stable (though generally weak) predictor of children's cognitive development.For example, Dickerson and Popli (2016) analyzed the MCS data and found that family poverty was a persistent, negative predictor of cognitive development for children of ages from 9 months to 7 years even after controlling for family background variables and parental investment (e.g., maternal education).Similar findings were found for US and Canadian children of ages 2 to 5 years (Duncan et al., 2011).

Maternal education
Maternal education is viewed as a major indicator of cultural capital or family investment.Even after controlling family income, maternal education remains a positive predictor of cognitive, mathematics achievement development for early-year children in the U.K. (Dickerson & Popli, 2016).Parental education plays a more significant role in the initial and growth of mathematics achievement of adolescents than family income in Taiwan (Chiu, 2016).It is, therefore, interesting to place family income and maternal education in a model to assess their relative impacts on children's MAch.

Household minority language use
Immigration or ethnic minority tends to be disadvantaged cultural capital for children's cognitive development.The main reason may be non-mainstream household language use, which detriments children's opportunity to learn, especially in the early years.A largescale cross-cultural study indicates that immigrant students at age 15-16 tend to have lower 1 3 mathematical and science problem-solving skills than non-immigrant students, which may be affected by multiple conditions related to immigration such as age, age of arrival, gender, language use, and family SES (Martin et al., 2012).Secondary students in Taiwanspeaking ethnic minority languages at home tend to have a low initial and negative growth in mathematics achievement (Chiu, 2016).However, research finds that the disadvantage of ethnic minorities may gradually diminish.For example, students with Bangladeshi heritage in the UK have higher progress from key stage 2 (ages 8-11) to KS4 (ages 14-16) than their counterparts (Sammons et al., 2014).

Maternal psychological factors
Parental psychological behaviors (e.g., playing shape games) impact their children's mathematical achievements (Chiu, 2018).There appear to be no studies to date focusing on parental psychological behaviors occurring during very early childhood (e.g., children at age 9 months), though when parental behaviors may not be able to focus on mathematics.This study focused on maternal psychological behaviors because mothers are the major caregivers in very early childhood in most societies.An exploratory factor analysis of the MCS data identified three potential maternal psychological behaviors (cognitive stimulation, positive affect, and secure attachment), which may impact children's cognitive developments, as suggested by the following literature.

Maternal cognitive stimulation or regulation
Maternal cognitive stimulation may positively impact children's later mathematics achievement development.It is because numeracy activities are a kind of cognitive stimulation, and research had indicated that parental numeracy activities (either formal or informal ones) have a positive impact on children's mathematical achievements (Dunst et al., 2017;Skwarchuk et al.'s (2014)).A meta-analysis indicates that the most effective parental involvement programs normally include some cognitive stimulation activities for their children (e.g., shared reading; Jeynes, 2012).

Maternal positive affect or emotion
As suggested by the positive-affect-to-success (PAS) hypothesis, the personal positive affect will generate successful outcomes (e.g., mathematical achievement; Lyubomirsky et al., 2005).Inferred from the personal PAS hypothesis to form a parental PAS (PPAS) hypothesis: positive parental affect may positively impact children's MAch development, which is supported by parenting studies.For example, parental support and democratic control positively relate to adolescent academic performance and negatively relate to the onset of smoking (Morin et al., 2012).Parental control behavior over child academic performance plays a negative role in 3rd and 4th graders' school achievements in German and mathematics in Germany (Su et al., 2015).
The underlying mechanism of the PPAS may be that parental adverse control behavior has a negative effect on child performance goals (aiming for high achievement), which may also be additionally mediated by child negative emotions such as anxiety, while parental involvement behavior plays a direct positive role in child mastery goals (aiming to learn and improve) (Duchesne & Ratelle, 2010).Furthermore, parental emotional sensitivity can 1 3 protect children from adverse experiences, especially for disadvantaged students (Oxford & Lee, 2011).Insensitive parenting may play a role in student social withdrawal in primary school (Booth-LaForce & Oxford, 2008).Dysfunctional parenting and negative family climate relate to child depression at age 7 (Castelao and Kröner-Herwig, 2013).All these studies suggest a positive effect of parental positive affective or emotional states on children's cognitive and behavioral development.

Maternal secure attachment
Secure attachment addresses the desirable quality of close relationships.There appears to be no study to date focusing on mothers' secure attachment reports.However, studies on performers' secure attachment suggest that maternal secure attachments will positively relate to children's MAch.For example, child secure attachment measured with the traditional attachment task (i.e., a standard strange situation) at ages 24 and 36 months (though not at 15 months) relates to child academic achievement and IQ test results in the 3rd and 4th grades (West et al., 2013).Child secure attachment responses to mother-child stories relate to some child coping behaviors reported by mothers for children at ages 10-11 (Brumariu et al., 2012).The mechanism as suggested by adult research may be that (employees') self-report secure attachment (with their supervisors) leads to positive performance through trust (Simmons et al., 2009).

The present study
The psychobiosocial model (PM) and family PM (FPM) propose family biological, socioeconomic, and psychological factors that predict children's cognitive development, especially in MAch.In addition, the prediction patterns are different between genders.We selected variables from the MCS with support from related literature to examine the three theoretical statements of the FPM (cf.theoretical basis).
In terms of statistical examinations, the research framework can be presented in Fig. 1.This study aims to examine the following three hypotheses for the three theoretical statements, respectively.

Hypothesis 1
Family biological, socioeconomic, and psychological factors at early childhood predict children's MAch development at later childhood.(Hypothesis 1 is for theoretical statement 1 excluding gender.)

Hypothesis 2
The prediction pattern described in hypothesis 1 as a whole fits data for girls better than for boys (theoretical statement 2; Fig. 1).

Hypothesis 3
In the prediction pattern described in hypothesis 1, socioeconomic factors predict mathematics achievement better for females than for males (theoretical statement 3; Fig. 1).
Hypotheses 2-3 examine whether gender plays a moderating role in the prediction pattern described in hypothesis 1.

Data source and sample
The present study used cohort data from the MCS compiled by the U.K. Data Service.The MCS is a longitudinal study collecting children's family and personal data, starting from their birth in 2000 to 2001.The first sweep of data (MCS1) was collected when children at age 9 months (family n = 18,552) in 2001, with additional families (n = 692) joining the second sweep data collection in 2004 (i.e., MCS2).The families and their children were continuously surveyed or followed up when the cohort members (children) were at ages 3 (MCS2), 5 (MCS3), 7 (MCS4), 11 (MCS5), 14 (MCS6), and 17 (MCS7) years old until 2018, when this paper was written.A common identifier ("MCSID") was used to merge five MCS datasets: (1) The longitudinal family dataset for sampling weights (further explained in the "Data analysis" section on sampling weight); (2) The parent interview dataset (for biological and psychological measures) collected at MC1; (3) The family-derived dataset (for social measures) collected at MCS1; (4) The child cognitive assessment dataset (as Outcome1) collected at MCS3; (5) The child cognitive assessment dataset (as Outcome2) collected at MCS4.
The two datasets of children's cognitive assessment (4 and 5) were used because "BAS Pattern construction" was the only MCS cognitive measure that could represent the construct of children's MAch and were collected using the same assessment tool, which could justify longitudinal development.The following steps merged the five datasets.
(1) Select only the families joining MCS1 in the longitudinal family dataset (n = 18,552).
The parent interview dataset and family-derived dataset collected in MCS1 have the same sample size (n = 18,552).The new families joining at MCS2 were not included because they did not have MCS1 parent interview data.
(2) Select only the data of the first child in one family from the two assessment datasets at MCS3 (n = 15,246) and MCS4 (n = 13,857), which let each child from a different family.The Step (1) family dataset was used as the base to merge all the other four datasets.
In other words, the sample size remained as 18,552 cases (observations, children, or families) until this step.
(3) Select only the cases that their natural mothers were interviewed.This procedure would reduce the bias produced by variations in respondents, though with a slightly reduced sample size (n = 18,497, including 8999 girls and 9498 boys) for the later analysis.

Measures
Two dependent variables were used to represent child MAch.The variables were children's spatial ability assessment results over two sweeps at ages 5 and 7 years (i.e., MCS3 and MCS4).Nine independent variables were selected to represent the constructs of family biological, socioeconomic, and psychological factors, each factor containing three variables, which were collected at MCS1 (i.e., children age 9 months).The supplementary file section A presents all the measures' names, labels, values, data preparation procedures, and located dataset names with their line numbers in the dataset.

Child mathematical achievement
The MCS used several standardized tests to examine children's cognitive abilities or achievement from MCS2 to MCS5.The only MAch-related tests administered more than once were the British Ability Scales (BAS) pattern construction (BAS-PC) at MCS3 (i.e., MAch5 in this study) and MCS4 (MAch7).We, therefore, used these two variables to represent MAch development.The BAS-PC test asked children to "put flat squares or solid cubes with black and yellow patterns on each side" to construct a geometrical design; the BAS-PC scores contained children's accuracy and speed in completing the tasks and represented children's achievement on spatial awareness (including dexterity and coordination) and traits of perseverance and determination (Hansen 2014 p. 64).For BAS-PC, the internal consistency reliability was 0.940-0.950,and the average test-retest reliability was 0.890; the current validity is evidenced by the moderate to high correlations (0.590-0.880) between the BAS (as a whole) and several intelligence tests (e.g., the WISC, p. 25; Willis et al., 2008).We chose to use the ability scores of BAS-PC, not raw scores or T-scores because this study focused on MAch development.The ability scores could represent the construct of longitudinal MAch development from early to later stages (i.e. from MCS3 to MCS4).The raw scores did not adjust for different item administration conditions and, thus, with little meaning; the T-scores were adjusted for children's age and would not contain the differences in variances at different ages, which, therefore, were not suitable for comparison between ages (Hansen 2014, pp. 66-68).

Family biological factors
Three biological factors were obtained from the MCS1 parental interview dataset.All the variables were mother self-report data, including (bio1) mother alcohol intake during pregnancy, (bio2) mother alcohol intake at MCS1, and (bio3) mother depression at MCS1.The bio3 was the mean score of mother reports on nine items such as "tired most of time" and "often miserable or depressed" on a 2-point scale (1 = yes; 2 = no).It was because exploratory factor analysis (EFA) results showed that the nine items could be viewed as one factor, with all the items' factor loading larger than 0.300; if two factors, there would be items with few differences between the primary and alternative factors (Howard, 2016).The 9-item bio3 also had acceptable internal consistency reliability (Cronbach's alpha = 0.731).

Family socioeconomic factors
Three social or socioeconomic-status (SES) factors were obtained from the MCS1-derived family dataset.The three factors were weekly net family income, maternal highest education level, and home foreign language use.

Family psychological factors
Psychological factors were obtained by using EFA on the 11 items regarding mother-baby interaction.The EFA resulted in three factors: (a) mother cognitive stimulation, including four items (psy1; Cronbach's alpha = 0.658): the importance of talking, cuddling, stimulation for development, and regular sleeping and eating, all on a 5-point Likert scale (1 = strongly agree to 5 = strongly disagree), (b) mother positive affect, including four items (psy2; Cronbach's alpha = 0.516): (1) feelings of annoyance or irritation (1 = almost all the time to 6 = never), (2) feeling when caring (1 = incompetent and a lack of confidence to 4 = very competent and confident), (3) patience (1 = very impatient to 4 = extremely patient), and (4) feeling like giving up due to baby (1 = resent a lot to 4 = don't resent at all), and (c) mother secure attachment, including two items (psy3; Cronbach's alpha = 0.583): thinking about the baby when apart (1 = all the time to 6 = never) and feeling when leaving baby (1 = sad; 5 = relieved).
The items were recorded to let higher scores represent higher degrees in the meaning of the factor names.For factors with items using different Likert scales (i.e., (b) and (c)), their items were scaled into standardized scores before the items of the factor were calculated to form mean scores.

Data analysis
Data analysis was performed using R version 3.5.1 (R Core Team, http:// www.R-proje ct.org/) and RStudio version 1.1.456(https:// www.rstud io.com/).The EFA was performed using the nFactors package to determine the number of factors to extract and using the psych and GPArotation packages to run the EFA with the oblimin rotation and principal factor method.To focus on examining the posited FPM, EFA was used as a data-reduction technique in this study (Watson, 2017), which may also reduce the possibility of multicollinearity in regression analysis.

Sampling weight and correlation
This study used the sample weight for the four countries of the UK (i.e., England, Wales, Scotland, and North Ireland) across waves ("WEIGHT2") provided in the longitudinal family dataset; the weight was to compensate for the sampling design of unequal selection probabilities (Hansen, 2014).Although the MCS user guide suggested using weights, taking account of attrition and non-response (e.g., avowt2 for MCS1 and dovwt2 for MCS4), this study used multiple waves of data altogether.Therefore, overall weight ("WEIGHT2") was used in combination with other sample design variables, including stratification ("PTTYPE2"), clustering ("SPTN00"), and finite population correction factor ("NH2") variables (Jones & Ketende, 2010, pp. 7-8).This procedure allowed for the analysis estimates to be generalized to or explained for the UK total population (e.g., the weight used on page 44 of Hansen (2014)).The major R syntax for setting the complex-sample plan and generating the correlation matrix is presented in the Supplementary File Section B.
Given the large sample size of this study, significant correlations were easily obtained even with small absolute correlation values.As such, this study used the criteria of the absolute correlation values smaller than 0.350 as low, between 0.360 and 0.670 as moderate, and larger than 0.680 as high relationships to assess the degrees of relationships between the measures (Taylor, 1990).

Path-model structural equation modeling (SEM) analysis
The hypothesis was examined by path analysis, the regression analysis part of SEM.The model set that the six factors predict MAch at ages 5 and 7 years (MAch5 and MAch7), respectively.In addition, MAch7 regressed on MAch5, which assumed that previous mathematics abilities would at least partially address MAch7.The sample weights also activated.The R syntax is presented in the Supplementary File section C.
Due to the use of path-model SEM analysis, the traditional model-fit criteria for SEM are not helpful in determining single-sample SEM analysis without additional constraints (e.g., path models 1 and 4-5 in Table 2).It is because the degree of freedom becomes zero with the numbers of parameter estimations equal to the numbers of measures' means, variances, and co-variances placed into the specified mode (i.e., a saturation model; Raykov et al., 2013).A saturation model is still trustworthy and robust (Raykov et al., 2017), and its model fit may be determined by local fit (e.g., significant path coefficient estimates; Chiu, 2022).

3
For multigroup SEM (including multigroup path-model SEM), the traditional criteria for determining model fit are still helpful.The following four indices with relatively unequivocal criteria are widely used to determine SEM model fit (Chiu, 2020;Hair et al., 2010;Ximénez et al., 2022).The root mean square error of approximation (RMSEA), a typically used criterion for SEM, should be below 0.050 or 0.080.Recently, both RMSEA and standardized root mean square residual (SRMR) have been used for comparing competing models with smaller RMSEA and SRMR, representing better model fit.The comparative fit index (CFI) and the Tucker-Lewis index (TLI) should be above 0.900 or 0.950.Larger CFI and TLI represented better model fit.Although a non-significant chi-square (χ 2 ) was a criterion for model fit, χ 2 value may easily become significant if there is a large sample size, as this study.As such, this study would not use χ 2 value as a criterion for judging model fit (Bollen & Long, 1993) but used RMSEA, SRMR, CFI, and TLI.

Results
The correlations between the factors are presented in Table 1.The largest absolute value of the correlations was a moderate one (0.417) between family income (ses1) and mother education (ses2).All the other correlations were small (i.e., below 0.360; Taylor, 1990).The results implied that the sub-factors were different constructs, and the results of the sub-factors could be compared.In addition, the regression analysis result would have few multicollinearity problems in regression analyses because all the correlations between the predictors were smaller than 0.900 (Hair et al., 2006).This section focuses on the results of examining the two hypotheses.

Hypothesis 1: model fit for all students
Hypothesis 1 was examined by using single-group path analysis; that is, all the students were viewed as a whole in the path analysis.The traditional SEM model-fit indices of model 1 (Table 2) cannot assess this path-model SEM result's model fit.Instead, the significance of path coefficients is used as a local fit index; for example, MAch7 can be significantly regressed on MAch5 (beta = 0.567; Table 3).
Two social factors (ses1 and ses2) positively predicted both MAch5 and MAch7 significantly (model 1 results in Table 3 and Fig. 2).Only one biological factor (bio1) negatively and one psychological factor (psy1) positively predicted MAch5.The nine factors accounted for only 6.5% of the total variance of MAch5.MAch5 moderately predicted MAch7 (0.567).The nine factors and MAch5 accounted for 37.6% of MAch7.The results implied that later MAch was moderately determined by previous MAch.
In summary, two social factors stably predicted both MAch5 and MAch7.One biological factor and one psychological factor predicted MAch5 only, not MAch7.The result concurred with the family psychobiosocial model (FPM)'s theoretical statement 1 (and support hypothesis 1) with four among nine family psychobiosocial factors impacting children's mathematics ability.

Table 1
Correlations between the predictors and outcomes for the total student sample The bold correlation coefficients are significant at p < 0.001, the bold, italic ones are significant at p < .050,and the underlined ones are not significant at p  The betas are standardized solutions.The bold betas are significant at p < 0.001; the bold, italic betas are significant at p < 0.050; and the underlined betas are not significant at p < 0.05.Bio1-bio3, ses1-ses3, and psy1-psy3, MAch5, and MAch7: Hypothesis 2: gender differences in the FPM's fit to data Firstly, hypothesis 2 was examined by using multigroup SEM to compare model parameter estimates between boys and girls.Model 2 set equal regression coefficients, which could be supported because the fit indices were all desirable (RMSEA = 0.038 < 0.050; SRMR = 0.027, the smaller, the better; CFI = 0.984, and TLI = 0.969 > 0.950; Table 2).Model 3, which set equal regressions, intercepts, and residuals across genders, however,  3)

B Female Students (Model 4 in Table 3)
C Male Students (Model 5 in Table 3) was not so desirable (RMSEA = 0.066; SRMR = 0.037; CFI = 0.943; and TLI = 0.905).The results implied that boys and girls may have different intercept and residual estimates.Secondly, single-group path-model SEM was used to examine Hypothesis 2 for boys and girls separately.Both female and male samples have significant path coefficients.Especially, MAch5 can stably predict MAch7, as a reasonable result and a sign of local fit.
The FPM's theoretical statement 2 was saliently supported by R 2 values.Girls' MAch5 (10.8%; model 4 in Table 3) was more explained by the nine factors than boys' (5.9%; model 5).Furthermore, girls' MAch7 (40.4%) was more explained by both the nine factors and MAch5 than boys' (37.6%).As indicated in the literature review, the FPM's theoretical statement 2 can be inferred by the greater male variability hypothesis (Hyde, 2014).

Hypothesis 3: gender differences in path parameter estimates
The examination of Hypothesis 3 also used single-group SEM for boys and girls separately.The path analysis results focused on comparing the path parameter estimates between model 4 and model 5 (Table 3; Fig. 2).Two social factors predicted both MAch5 and MAch5 (except for ses2 predicting MAch7).One psychological factor (psy1) predicted MAch5 only, not MAch7.
For boys, there was only one significant regression coefficient; that is, ses2 (mother education) positively predicted MAch5.The result generally supported the FPM's theoretical statement 3.

Discussion
The discussion section focuses on addressing the three theoretical statements of the FPM using the results obtained from examining hypotheses 1-2.

Theoretical statement 1
The results from a single-group SEM for the sample of all students mostly support the FPM's theoretical statement 1.The statistically significant predictors are ses1 (family income) and ses2 (mother education), predicting both MAch5 and MAch7.Bio1 (mother in-pregnancy alcohol intake) and psy1 (mother cognitive stimulation) only predict MAch5.

Positive factor: SES
SES plays the most role in children's mathematical ability development among the nine psychobiosocial indicators.However, only the SES indicators of family income and mother education play a significant role, not home foreign language.
Family income has persistent, positive effects on child MAch development.The result is consistent with Duncan et al.'s (2011) results for pre-school children.The role of family income, however, reduces after age 7, a result consistent with the finding that family 1 3 income could not substantially predict children's subjective well-being at 11 years old (Rees, 2018).
The finding of a non-significant role of home foreign language use at home is consistent with Sammons et al.'s (2014) finding.The result suggests that mixed-language, cultural backgrounds may not be a disadvantage for children's long-term academic and psychosocial development.Some longitudinal studies, however, indicate a negative role of immigration or ethnic minority status (Chiu, 2016;Martin et al., 2012).

Weak, positive factor: Mother cognitive stimulation
Mother cognitive stimulation during mother-baby interaction plays only a weak role on MAch at the age of 5 years.The results are consistent with research findings that programs focusing on parent-child cognitive interaction generate positive child learning outcomes (Jeynes, 2012).
Both mother positive affect and secure attachment fail to play a role.The results are not consistent with some related findings.Past findings indicate that sensitive, supportive parenting with a positive baby-parent emotional interaction relates to positive child cognitive and behavioral abilities (Castelao & Kröner-Herwig, 2013;Morin et al., 2012;Oxford & Lee, 2011).The inconsistent finding of this study compared with relevant past research may be due to the use of mother self-reported positive affect for children at 9 months of age as a measure.In this study, self-reporting of too much confidence, patience, and acceptance in caring for children may reveal that mothers actually lack sufficient involvement in children's learning (Duchesne & Ratelle, 2010) due to over-confidence and some biological and family reasons.Future research needs to consider the meanings or validity of the diverse positive affect measures.
The relationships between child abilities and secure attachment are unstable for different attachment measures (Brumariu et al., 2012).Another reason may be past research focused on earlier years (West et al., 2013), and this study focuses on later development in late primary school.

Weak, negative factor: In-pregnancy (prenatal) alcohol intake
The present findings emphasizing the negative longitudinal role of prenatal alcohol intake (bio1) are consistent with most related findings obtained using maternal alcohol intake "during pregnancy" as the factor (Flak et al., 2014).Nykjaer et al.'s (2014) study focuses on both pre-pregnancy and in-pregnancy alcohol intake but only on their roles in infant development.This finding adds to the literature that in-pregnancy alcohol intake has persistent roles in child development even at age 5, but not age 7 years.
The new findings, however, need to be examined further by future research and by considering related psychosocial factors.The results also have implications for policymakers to educate the public to reduce alcohol intake during pregnancy by emphasizing the prolonged negative role of in-pregnancy alcohol intake in children's mathematics achievement.
Mother depression (bio3) fails to play a role.Mother self-reported depression negatively relates to later child cognitive or psychosocial development.The result is consistent with the findings of most studies (Bodovski & Youn, 2010;Evans et al., 2012;Slominski et al., 2011).

Theoretical statement 2
Theoretical statement asserts that family psychobiosocial factors impact children's mathematics achievement development more for females than for males.This study supports Theoretical Statement 2 in terms of higher proportions of the outcome explained by the model (R squared, or lower variances) and more significant path coefficients for girls than boys (Table 3).
This study may be the first to attribute social factors to the greater male variability hypothesis.The family psychobiosocial model (FPM) actually combines three factors but places greater emphasis on social factors, as evidenced by more significant path coefficients for girls than boys.

Theoretical statement 3
Theoretical statement 3 states that social factors account for more mathematics achievement for females than for males.This study supports theoretical statement 3.It is because girls have both family income (ses1) and mother education (ses2), impacting mathematics achievements.However, boys only have mother education (ses2), impacting their achievements when they are 5 years old.
Theoretical statement 3 is based on the gender stratification hypothesis (Else-Quest et al., 2010) and gender similarities hypothesis (Hyde, 2005).This study extends the focus of the two hypotheses to social factors in determining gender differences in mathematics achievement.

Limitations of this study and suggestions for future research
This study focuses on examining the posited FPM and investigated the direct effects of early family psychobiosocial factors on later child MAch.Given the focus and multiple, diverse measures examined, this study did not examine the measurement models for the items of the latent measures (i.e., bio3, psy1, psy2, and psy3; Table 3).This leads to pathmodel (all measures being observed variables) SEM and cannot allow for measurement invariance tests between genders for this study.Future research can consider examining the FPM using full SEM, considering both path and measurement models, to increase the certainty, validity, and reliability of the structural relations and measures.
Before doing full SEM, latent measures need high validity and reliability to increase the possibility of SEM model fit.The acceptable-to-poor reliability of the latent measures in this study (Cronbach's alpha values from 0.516 to 0.731) led to poor model-fit indications, as have been examined using full SEM by the authors.
Although the present statistical tests found some significant correlations and path coefficients, their absolute values (i.e., effect sizes) are actually quite small.Furthermore, all the psychobiosocial factors used in this study account for only a small proportion of achievement.As such, readers need to be cautious about drawing conclusions from the results.Future research needs to use more relevant psychobiosocial predictors to examine the posited FPM.

3
Lastly, most predictors of this study were collected when children were months, which predicted children's mathematics achievements at the ages of 5 and 7 years.Between the times, things (e.g., family income, parenting strategies, and pre-and early-school programs) would change and may impact children's achievements.This study only focuses on some selected early psychobiosocial predictors at a one-time point, which, however, may change, and more influential factors may join over time.The complex datasets can be analyzed using diverse, suitable longitudinal data analysis methods (e.g., latent growth modeling).

Relevant publications:
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Fig. 1
Fig. 1 A research framework based on the family psychobiosocial model

Fig. 2
Fig. 2 Significant path coefficients.Note: Non-significant path coefficients are not presented A All Students (Model 1 in Table3)