Introduction

Mathematics and gender have already been analysed in many places. In Austria, in contrast to other countries, there are greater gender differences in mathematical performance already at the primary school level (Kelz, 2020). The aim of many educational systems and reforms is to overcome this gender gap. This is hindered by the fact that findings regarding mathematical performance at primary level are controversial and not sufficiently analysed compared to secondary level. It is therefore our goal to understand which factors play a role in the development of mathematical competence at the beginning of children’s school career by contributing to modelling the development of mathematics achievement.

Modelling the mathematics achievement of primary school students has already been done by Niklas (2015; cf. Fig. 1) and Luttenberger et al. (2018; cf. Fig. 2). Niklas (2015) examined structural traits of origin such as socio-economic status, migration background, the educational level, and home learning environment, consisting of home literacy environment and home numeracy environment. According to him, these constructs affect mathematics achievement via precursor skills. Luttenberger et al. (2018) include additional interacting variables in their model. These comprise self-constructs, prior knowledge, motivation, interest and mathematics anxiety. These two models can already explain some of the variance in mathematics achievement. Unfortunately, these models have only been investigated separately. In the course of this study, we aim to combine the merits of both models and empirically test the factors effecting mathematical achievement. The reason for the integration or further development of the two models is that antecedent competencies and school competencies cannot be explained solely by structural traits of origin and the home learning environment (Graß et al., 2018; Kelz, 2017, 2020). Vice versa, among the antecedents or interacting variables (self-concept, self-efficacy, prior knowledge, motivation, interest, mathematics anxiety), the essential influencing factor family with its division into home literacy and home numeracy environment is missing (LeFevre et al., 2009).

Fig. 1
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Model of the home learning environment according to Niklas (2015)

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Fig. 2
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A framework for understanding mathematics anxiety according to Luttenberger et al. (2018)

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In order to motivate this study, the theoretical part deals with modelling of math performance and its influencing factors. In the empirical part, a longitudinal study over 4 years with 203 pupils from 13 primary schools in Styria, Austria.

Theoretical framework.

Children's early mathematics development

Precursor abilities and their importance for later math achievement

According to Schuler (2008), precursor skills are skills that are acquired at kindergarten age and are a prerequisite for learning in primary school. Hasemann (2001), for example, examined number knowledge, digit knowledge, the ability to grasp quantities and arithmetic skills in school beginners. Grassman (1995) looked at geometric knowledge at the beginning of the first grade. In addition to these mathematics-specific precursor skills, which can be assigned to the areas of arithmetic and spatial imagination, the general cognitive precursor skills are also of interest in mathematics education (Prenzel et al., 2008). In general, several studies have already provided evidence that precursor skills, such as prior knowledge of numbers and quantities, have a significant influence on later mathematical performance (Kolkman et al., 2013; Krajewski & Schneider 2006; Weißhaupt et al. 2006). The range of prediction of precursor skills was measured by Watts et al. (2018) in a longitudinal study up to age 15 and by Niklas and Schneider (2012) up to age 23.

Depending on the underlying model, different abilities are defined as precursor skills. Nevertheless, all precursor skills are based on quantity-number precognition, which is the foundation for further finer definitions. The precursor skills defined by Moraske et al. (2018), digit knowledge, digit identification, counting skills, quantity recognition, and quantity decomposition, provide an example of the finer definitions of precursor skills based on quantity-number precognition. Moraske et al. (2018) demonstrated a predictor effect for these precursor skills. Specifically relevant to this work is the prediction of precursor skills by the end of the elementary school years.

In the study by Gallit et al. (2018), precursor skills were able to explain 42% of the mathematical performance assessed later. This is consistent with the studies presented previously, which measured predictive power between 30 and 50%. Moreover, Gallit et al. (2018) confirm math-specific prior knowledge as the most predictive predictor, while general cognitive precursor skills act indirectly through number-set knowledge. Conclusively, they summarize that mathematical precursor skills substantially evolve and are closely related to each other.

Later math performance

Mathematical performance in the context of this study can be described as a collection of mathematical competences that can be taken from the areas of cognition, arithmetic and spatial conception for the school entry phase and from the areas of building natural numbers, arithmetic operations, quantities and geometry for the rest of primary school.

Math performance is related to different variables. For example, it has been shown that performance is reciprocally related to mathematics anxiety and self-concept. Precursor skills also predict math performance in primary school (Kelz, 2020). Another study shows that the home learning environment, controlling for intelligence, predicts both precursor skills and achievement at the end of first grade (Niklas & Schneider, 2014).

Influencing family factors

Socio-economic status (SES)

Major international education studies, such as PISA, make it clear that there is a close connection between the socio-economic background of students and their academic performance, whereby this connection is particularly strong in the German-speaking countries compared to other countries (Bos et al., 2007a). Along this line, studies have shown that students coming from a high socioeconomic status (SES) outperform their peers with low SES by over one standard deviation (Baird, 2012). The math gap is also present early in child development (e.g., Reardon & Portilla, 2016), and it is widespread (Karakus et al., 2023).

Migration background

In addition to SES, migration background is another important structural trait of origin, which is mostly measured by the country of birth of the parents or by the family language. Various studies have shown that children with a migration background often have lower linguistic and written language as well as mathematical skills (Niklas et al., 2012; Oppermann & Lazarides, 2023).

Educational level of the mother/father

The educational level of the family is usually operationalised as part of SES. As already mentioned, the educational level of parents is significant in terms of possible influences on school competences (Baumert & Maaz, 2006). Structural characteristics of a family are significant for the educational development of the children living in it (Niklas, 2015).

Home learning environment

For young children, the most important learning environment for various educational processes is the family context (Melhuish et al., 2008). Brunner and Noack (2010) describe the family as an important context both for healthy emotional development of children and adolescents and as a prerequisite for optimal intellectual development. Lehrl et al. (2020) also emphasise the important role of the home learning environment for academic achievement. Although family influences are relevant, there is no clear, generally applicable classification of the various characteristics of a home learning environment. A differentiation according to content-related aspects makes sense, since the development of different child competence areas is often domain-specific (Wellman & Gelman, 1998). For example, the subdivision of HLE into a so-called home literacy environment, which refers to the written language domain, and a so-called home numeracy environment (HNE), which refers to the mathematical domain, has emerged (LeFevre et al., 2009). Whereas aspects of HLE predict linguistic competencies, HNE is more predictive of mathematical competencies (e.g. Zippert & Ramani, 2017). A relationship between home number experiences and number skills would be considered a within-domain relationship. However, an additional role for home literacy experiences supporting early number skills has been discussed (see Soto-Calvo et al., 2019).

Home Numeracy Environment (HNE)

Niklas and Schneider (2014) were able to show that, under control of age, gender and intelligence, the HNE plays a decisive moderating role in the development of mathematical competences at kindergarten age and at the beginning of primary school. HNE consists of cultural practice and cultural capital, Implicit Learning, Teaching by Parents, Parental Attitudes and Expectations and Parental Support. These parts of the HNE have already been supported with evidence (Baumert & Maaz, 2006; LeFevre et al., 2009; McElvany et al., 2009; Niklas et al., 2013; Sénéchal & LeFevre, 2002). The exact definitions of the sub-items are explained by Niklas (2015). This division of the HNE is an attempt to operationalise and capture the home learning environment. In this article, the HNE is operationalised alongside the home literacy environment as an independent construct and is seen and understood as part of the home learning environment.

Model of the home learning environment according to Niklas (2015)

Bronfenbrenner’s (1979) ecological theory distinguishes several factors influencing child development at different levels. Besides effects such as society, factors such as family, kindergarten and school are directly decisive. This motivated Niklas (2015) to develop his model, wherein he combines proximal process features (such as parent–child interactions) to the home learning environment and distal background characteristics (such as socioeconomic status) to structural traits of origin. According to Niklas (2015), the home learning environment is understood as those aspects that provide the child with opportunities and support within the family framework to acquire and practice specific precursor skills and additional skills in written language and mathematics. Besides Niklas (2015), there are different operationalisations (e.g. Skwarchuk et al., 2014). The model of Niklas (2015) is not specifically developed to model mathematics achievement in primary school. It is developed to operationalise the HLE and to stress the association to children´s early precursor abilities and later mathematics achievement. The model is used in this study to include possible factors for math performance in our own statistical modelling.

Influencing child factors of mathematics development

Self-concept

According to Moschner and Dickhäuser (2006), the general self-concept corresponds to the ideas about one’s own personal abilities and characteristics, while the mathematical self-concept is specified as the totality of the cognitive representations of one’s own abilities in the subject of mathematics (Schöne & Stiensmeier-Pelster, 2011). Self-concept emerged as the most significant predictor of achievement among 14 psychosocial ones in the 2000 PISA study (Marsh and O’Mara, 2008). In a study, Feng et al., (2018) found that among the two constructs self-concept and self-efficacy, self-concept predicts school grades significantly better statistically.

The relationship between self-concept and school performance has already been established by the relevant literature (Lohbeck et al., 2016). In particular, there is a close correlation between subject-specific self-concepts and performance in the same subject (Ehm et al., 2014). There are also some findings in this regard specifically for mathematics (Arens et al., 2011). For primary education, however, the results regarding self-concept and achievement are contradictory and scarce (Ehm et al., 2019).

Enjoyment of the subject

First and foremost, enjoyment of mathematics refers to the positive emotions that children bring to mathematics education (Golob, 2017). In addition, this concept also includes the factors of motivation and interest mentioned by Luttenberger et al. (2018). In more detail, enjoyment of the subject thus refers to both the aspect of the action itself, including its cause (interest) and its effect (joy of learning), as well as the aspect of goal orientation (motivation). However, this term is not used to describe cognitive-evaluative aspects (self-concept) or emotions as a result of one’s own performance (performance emotions or mathematics anxiety) (Golob, 2017).

On the one hand, Walther et al. (2008), referring to the results of TIMSS 2007, report no systematic correlations between the enjoyment of mathematics as a subject and the mathematics achievement of fourth graders. Students with higher motivation invest more time and effort in learning and use better learning strategies (Macher et al., 2013). While motivation helps to increase achievement, mathematics anxiety decreases task readiness (Luttenberger et al., 2018). However, Luttenberger et al., (2018) found that very few studies investigated the interaction between motivation, mathematics anxiety and achievement. The relationship between mathematics interest and mathematics achievement has traditionally been attributed to the fact that interest exerts a positive effect on subsequent achievement.

Math anxiety

Luttenberger et al. (2018) were able to show that poorly developed precursor skills promote mathematics anxiety and vice versa. Mathematics anxiety is defined as “feelings of tension and anxiety that interfere with the manipulation of numbers and the solving of mathematical problems in a wide variety of ordinary life and academic situations” (Richardson & Suinn, 1972, p. 551). Mathematics anxiety affects all students in different countries and at all school levels (Foley et al., 2017). Cross-country comparisons across several periods of PISA studies show that mathematics anxiety is negatively correlated with mathematics achievement (Skaalvik, 2018).

For the primary level, meta-analyses found correlations between mathematics anxiety and achievement that were relatively high and negative (r = -0.19 to r = -0.49), similar to those found for the secondary level (Ma et al., 2004). Wang et al. (2015) were able to show that “basic numerical skills” or precursor skills correlate with mathematics anxiety. Mathematics anxiety in early grades influences not only math performance in the same grade, but also in more advanced mathematical skills (Skaalvik, 2018). In addition to the importance of mathematics anxiety for performance in precursor skills and academic competencies, it is related to self-concept in the model as an interacting variable (Luttenberger et al., 2018).

A framework for understanding mathematics anxiety according to Luttenberger et al. (2018)

Luttenberger et al. (2018) describe their framework for understanding math anxiety in three steps. Antecedents consist of environment, culture and genetics and affect interacting variables. These are self-concept, self-efficacy, precursor skills, motivation, interest and maths anxiety, which in their model influence each other directly or reciprocally. The outcomes of these interactions include achievement, grades, academic decisions and learning behaviour (Luttenberger et al., 2018).

Gender of the child

In general, the gender aspect of the model is most evident in mathematics anxiety and self-concept. Girls have higher mathematics anxiety (Ashcraft, 2002; Todor, 2014) and a lower self-concept (OECD, 2015). The evidence in primary education largely confirms these gender differences for girls and boys. Rodríguez et al. (2019) also emphasise that boys are more motivated and interested than girls.

Mathematical self-concept is more positively related to achievement for boys than for girls, whereas the effect of interest and enjoyment of the Subject on achievement is more significant for girls than for boys (Ganley & Lubienski, 2016). Mathematics anxiety appears to have a greater effect on achievement for boys (Goetz et al., 2013). With regard to math performance in the early precursor skills, Kelz (2017) could not identify any gender differences in the areas of cognition, arithmetic and spatial imagination in his study of 397 school beginners. The effect size of the overall performance (d = -0.133) indicates a small advantage for girls, and in the 16 precursor skills, girls showed significantly better performance in four (seriality, quantity comparison, phonological awareness of quantities and number comparison).

The development of the construct gender in the tension of the model is highly worth investigating. Ideas about how gender interacts with origin characteristics, the home learning environment and interacting variables can provide explanatory approaches that get to the bottom of mathematical gender disparities. The decisive question is to what extent the construct of gender gains or loses significance in the course of school enrolment until the end of primary school. Gender disparities are found in later math performance, mostly in favour of boys (Niklas & Schneider, 2012). However, according to the current state of knowledge, it is not possible to say unequivocally when and why gender differences in mathematics break out. Indeed, it is not clear whether gender differences exist in the school entry phase or in primary education (Kelz, 2020).

Connections of family, child and mathematics

The question of what influences a child's maths performance is fundamental to mathematics education and also to this study. Not only the factors themselves but also the connections between them are highly worthy of investigation. In this study, the family is characterised in a first step by the structural traits of origin (SES, migration background, educational level) proposed by Niklas (2015). The child in its family itself, however, has measurable factors such as self-concept, enjoyment of the subject and maths anxiety. The high relevance of self-concept is demonstrated by the empirically well-established correlations with various individual learning characteristics, such as subject-related interests, high educational aspirations, positive learning emotions, specific learning behaviour characteristics, such as the willingness to make an effort, as well as the school performance of pupils (Lohbeck et al., 2016). Büch et al. (2015) explain that pupils affected by maths anxiety have a negative self-concept, are socially isolated and quite unwilling to work and attribute learning failures to themselves due to their alleged talent deficits. The gender of the child is also worth investigating with regard to maths performance. A study within this project shows that there are no gender differences at the start of school in Styria, Austria (Kelz, 2017), but that boys outperform girls over the course of their school career (Kelz, 2017). This is explained by the higher self-concept of boys (Kelz, 2020). The aim of our study is to empirically test the described factors and their connections in a model for the primary school level.

Our Study: Empirical testing of the influencing factors

In our study, we set out to test in a longitudinal study how the variables described above – SES, migration background, educational level of the mother/father, home learning environment, precursor skills, math anxiety, self-concept, enjoyment of the subject – are related to 1) each other and to 2) math performance. Furthermore, we set out to 3) test the role of gender on these relations. To this end, we study primary school children in Austria beginning from their school entrance phase until four years later (in Austria, primary school ends after four years). While research question 1 generally analyses the direction of effect and the reciprocal influences of the structural traits of origin, variables and performance in the sense of a structural equation model, research question 2 will include gender in the effect structure.

  • Research question 1: Which statistically significant factors influence mathematical performance?

  • Research question 2: What role does the construct of gender play in this structure of effects?

  • Hypothesis 1: Hypothesis 1 assumes that the following research results are confirmed. According to Baird (2012), higher SES is associated with better math performance. According to McElvany (2008) and Niklas et al. (2012), students with a migrant background perform worse in mathematics. According to Baumert and Maaz (2006), the educational level of parents has a significant and positive influence on school performance. According to Helmke and Schrader (2010), this also applies to home learning environment. With regard to the interacting variables, precursor skills have a positive influence on school competencies (Claessens et al., 2009; Krajewski & Schneider, 2009; Seitz & Weinert, 2022). Mathematics anxiety has a negative effect on mathematics achievement in primary school (Skaalvik, 2018). Self-concept and mathematics achievement are positively related (Lohbeck et al., 2016). According to Walther et al. (2008), enjoyment of the subject and mathematics achievement do not correlate with each other.

  • Hypothesis 2: Gender has no effect on the structural traits of origin. Similarly, according to Kelz (2017), no gender disparities in precursor skills should be measured. With regard to the other interacting variables, girls have greater mathematics anxiety (Ashcraft, 2002; Todor, 2014), lower self-concept (OECD, 2015) and lower enjoyment of the subject (Golob, 2017).

Method

Sample

In total, 549 children were surveyed in Styria, Austria. We first excluded children of which we did not have data for an entire block of variables (i.e. no information on their social background, no information on their home literacy/numeracy environment, no information at t1, no information on their math performance from t2 to t4, or no information on their math anxiety from t2 to t4). These exclusion criteria reduced the sample to 239 children. We then also removed those children of which we did not have longitudinal data at t3 and t4. We thereby allowed one missing per children in the longitudinal data. This resulted in the final sample of n = 203.

Among these children, there were 104 girls and 99 boys (51.23% vs 48.77%; in the initial n = 549: 50.27% vs 49.73%). Their mean HISEI was 58.82 (SD = 15.53; in the initial n = 549: M = 57.18, SD = 14.87; t[226.35] = -0.86, p = 0.39, d = -0.07). Their average migration background was 0.42 (SD = 0.84; in the initial n = 549: M = 0.56, SD = 0.97; t[466.73] = 1.88, p = 0.06, d = 0.15). Regarding the education of their parents, the mean education of their mother was 3.81 (SD = 2.13; in the initial n = 549: M = 3.74, SD = 2.18; t[430.06] = -0.37, p = 0.71, d = -0.03) and of their father was 3.83 (SD = 2.23; in the initial n = 549: M = 3.61, SD = 2.22; t[409.12] = -1.03, p = 0.30, d = -0.08). No significant differences were found in regard to our study variables between the original sample and the final sample.

Materials

Early precursor abilities were assessed with the ERT0 + (Lenart et al., 2014) and the math performance with the DEMAT (2 + /3 + /4 +), which is available for the different school levels. Self-concept and enjoyment of the subject were assessed using a questionnaire developed by Golob (2017). Mathematics anxiety was assessed with the “Fragebogen für Rechenangst” (FRA) (Questionnaire for Mathematics Anxiety), which can be used in German-speaking countries for children aged six to nine (Krinzinger et al., 2007).

The structural traits of origin and the home learning environment were assessed with the “Elternfragebogen Standardüberprüfung Mathematik 4. Schulstufe 2013” (Parents’ Questionnaire Standard Examination Mathematics 4th Grade 2013) according to Austrian Ministry of Education’s subdepartment responsible for standardized testing (Bifie, 2013). The SES of the respective pupil is determined with the help of the ISEI value (International Socio-Economic Index). The migration background is defined analogously to the IGLU study by at least one parent born foreign (Bos et al., 2007b). The educational level of the parents is operationalised through the parents’ educational qualifications.

The following overview gives an insight into the measurement schedule of the study (Fig. 3).

Fig. 3
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Overview of the research design denoting which data were collected at each measurement point

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Early precursor abilities

The ERT 0 + is designed to assess precursor skills at the school entry phase with 71 items. The test takes about two hours to complete and can only be administered by a trained test administrator. Despite the more precise differentiation in the lower achievement range (it can also be used as a pre-diagnostic for dyscalculia), the ERT 0 + allows the assessment of mathematical precursor skills at school entry. The ERT0 + was chosen over the DEMAT 1 + because it is specially designed for the school entry phase. It tests over 15 precursor skills from the areas of cognition, arithmetic and spatial reasoning with a Cronbach's alpha of 0.89.

Mathematics achievement

We assessed math performance with the DEMAT (e.g. Krajewski et al., 2020). The DEMAT is a German math test with version tailored to the different age groups, i.e. we used DEMAT 2 + (Krajewski et al., 2020) in 2nd grad, DEMAT 3 + (Roick et al., 2018) in 3rd grade and DEMAT 4 + (Gölitz et al., 2006) in 4th grade. The DEMAT 2 + tests the categories number properties, length comparison, addition and subtraction, doubling and halving, division, calculating with money, factual tasks and geometry with a Cronbach's alpha of 0.92. The DEMAT 3 + (Cronbach's alpha of 0.83) and DEMAT 4 + (Cronbach's alpha of 0.85) test the three areas arithmetic, factual arithmetic and geometry. The DEMAT 3 + is suitable for economically recording the maths performance of an entire school class. The area of arithmetic is covered by the four task types of number lines, additions, subtractions and multiplications. The area of factual arithmetic is covered by the two task types of factual calculations and converting lengths. Geometry performance is measured with the three task types mirror drawings, laying out shapes and estimating lengths. The DEMAT 4 + is divided into three areas (arithmetic, factual arithmetic and geometry) and nine task types. The area of arithmetic is covered by the task types of number lines, additions, subtractions, multiplications and divisions. The area of factual arithmetic is covered by the task types of size comparisons and factual calculations. Geometry performance is measured with the task types positional relationships and mirror drawings.

Self-concept

In this study, only the children's self-concept was surveyed. The self-concept was assessed by means of a questionnaire according to Golob (2017). An item assessing self-concept is for example "For me, arithmetic is... difficult/easy.". Golob (2017) draws these items from the literature, so that the assessment of current mathematical skills (items 1–4), mathematical ability (items 1–4), future assessment (item 5) and the dimensional comparison (item 6) are assessed. Responses range from "worst" to "best" on a four-step scale phrased for comparison with classmates. The questionnaires were filled out by the test leaders in the course of individual interviews with the children.

Enjoyment of the subject

Enjoyment of the Subject was assessed by means of a questionnaire according to Golob (2017). An item assessing self-concept is for example "I would like to have more arithmetic at school." Following Hagenauer (2011), the degree of agreement with the statements was visualised with the help of a smiley scale on a 4-step scale. Test leaders conducted interviews with the children and recorded results.

Math anxiety

We assess math anxiety with the FRA (“Fragebogen für Rechenangst”: Krinzinger et al., 2007), the German version of the MAQ (Mathematics Anxiety Questionnaire: Krinzinger et al., 2007) for primary school children. High reliability, standard scores corrected for gender, and economic handling make it an instrument well suited for use in clinical settings (e.g., dyscalculia diagnostics and intervention) and research. This self-report questionnaire measures four emotional components (self-perceived performance, attitudes, unhappiness with poor-performance, and anxiety) with the help of 5-step, child-friendly response scales. The four emotional components can be aggregated to form a total score of math anxiety. Each of the four scales corresponds to a specific type of question, which is asked in relation to two practice situations and seven mathematics-related situations from the primary school material or lessons. The FRA was applied three times in a group setting.

Parents questionnaire

The parents questionnaire collected information on the SES, migration background, educational level of mother/father and home learning environment and children’s gender.

Structural traits of origin

The SES of students is determined by the ISEI (International Socio-Economic Index) score. ISEI is an internationally standardised measure of students' SES. In the parent questionnaire, parents answer questions about occupation. These occupational details are classified on the basis of the International Standard Classification of Occupations (ISCO) from 1988 (ISCO-88). The ISEI values are calculated by coding the occupational data into the so-called ISCO codes. Migration background is defined analogously to the IGLU study (Bos et al., 2007) by at least one parent born abroad. This was also collected by the parent questionnaire. The educational level of mother/father was assessed by items The educational level of the parents was queried by asking for all possible education in Austria.

Home learning environment

Home literacy environment was assessed by the literacy resources and the literacy activities. Resources were assessed via the number of books and children’s books on a 5-point scale, which, for example, scales the number of books in 5 steps (0–10; 11–25; 26–100; 101–200; more than 200 books). Activities were assessed via the frequency of literacy related activities (library visits, museum/art exhibition visits, theatre/musical/ballet/classic concert visits, cinema visits, discussing shared events) on a 3-point scale. We computed scores for resources and activities separately, and then due to the different scales, standardized both to form the total home literacy score based on the mean of the two z-scores.

HNE was assessed through a three-step scale regarding frequency of mathematical activities. These activities include saying number rhymes, playing number games, counting things, playing games with shapes, playing with building blocks and playing board or card games. We calculated a total score for HNE through 6 activities. The more often an activity was conducted, the higher the score (1–3). The higher the score of the individual activities, the higher the total score of the home numeracy environment.

The home learning environment and the HNE were surveyed along Niklas (2015) using the parent questionnaire.

Statistical analyses

We conducted path analyses to examine the relations between the variables in question. Figure 4 gives an overview of the final model we tested, and the precursor models used to build the final model.

Fig. 4
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Overview of the path analyses estimated. Note. Gender is dummy coded. Within model 1, all three sets of variables were allowed to intercorrelate freely. SES = SES, math perform. = math performance

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First, model #1 and model #2 were estimated separately. In model #1 we tested how the variables at t1 were related to each other. In accordance with the hypotheses, the SES, migration background and the education of the mother/father should predict the home literacy/numeracy environment, which in turn should predict precursor skills, self-concept and enjoyment of the subject. The variables within each block were allowed to correlate freely. In model #2 we tested how math performance and math anxiety (co-)developed. The respective path model is a cross-lagged panel design with three measurement points. We did not allow for the paths from t2 to t4.

Second, we integrated model #1 and model #2 into a joined model, i.e. model #3. In model #3, we analysed the influences on mathematical performance, not yet considering the gender effects. More specifically, we examined with model #3 how precursor skills, self-concept and enjoyment of the subject predicted math performance and math anxiety at t2.

Lastly, we estimated model #4 to test the effects of gender. To do so, we augmented model #3 by the predictor gender. Therein, gender had a direct effect on every variable within model #3. We gauged the gender effects by comparing how accounting for gender (model #4) and not accounting for gender (model #3) caused differences in outcomes.

Within all models, we first estimated a model with all the free paths described above. Next, we iteratively constrained non-significant (p ≥ 0.05) parameters to zero until only significant (p < 0.05) relations remained. This was done to achieve the most parsimonious model, and in doing so to counteract overfitting and underestimating regressions weights due to multicollinearity.

All models were estimated in R (R Core Team, 2022) with lavaan (Rosseel, 2012). We used a full-information maximum likelihood estimator with robust standard errors. We used full-information, as each child only had a marginal amount of missings (see exclusion criteria). We used maximum likelihood as all but one variable were measured on a metric scale. We used Yuan-Bentler corrected standard errors, as not all variables were distributed normally (skewness and kurtosis exceeded |1|). We accepted a model as fitting with RMSEA < 0.06, SRMR < 0.08, and CFI ≥ 0.95 (Hu & Bentler, 1999; Marsh et al., 2004). Should a model fit not be acceptable, we inspected the modification indices to examine if relations were missed.

Results

The descriptive statistics and intercorrelations of all variables are presented in Table 1. As can be seen, gender was distributed very equally. It can be noted positively that children scored rather high on precursor skills, self-concept, and enjoyment of the subject. Furthermore, math performance was on an average level at t1, enabling ample room for improvement. Along this line, math performance steadily increased to t2 and t3 on the descriptive level. Math anxiety, on the other hand, was already present at t1, with a slight descriptive tendency to sink to t2 and t3.

Table 1 Descriptive statistics and intercorrelations (bold correlations are significant with α = 5%) of the variables entered into the path analyses
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All model fits of the path analyses are reported in Table 2.

Table 2 Model fits of all path analyses
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Relations of social background, home environment, and precursor skills, self-concept and enjoyment of the subject (model #1).

Model #1 did not fit the data. To achieve model fit, the migration also needed to predict the precursor skills. Adding this path to the model yielded good model fit (cf. Model #1.1). In the next step, we iteratively constrained the non-significant regression weights to zero. This resulted in constraining 10 paths to zero. The resulting model fit the data well (cf. Model #1.2). The left half of Fig. 5 shows the remaining paths.

Fig. 5
figure 5

The final path analyses. All parameters are standardized path coefficients. Non-significant gender effects are grey. Within model 3, the parameters on the left side are from model #3 showing the effects without considering gender; the parameters on the right side are from model #4 showing the effects while considering gender. Gender is coded with higher values denoting boys. Note. Within model 1, all three sets of variables were allowed to intercorrelate freely. SES = SES, math perform. = math performance

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(Co-)Development of math performance and math anxiety (model #2).

Model #2 did not fit the data). Inspection of the modification indices revealed that over and above the lagged co-development of math performance and math anxiety, math performance at t4 was also predicted by math performance at t2. Including this path led to good model fit (cf. Model #2.1). In the next step, we iteratively constrained the non-significant paths to zero. This resulted in constraining six regressions weights and three covariances to zero. The resulting model fit the data well (cf. Model #2.2). The right half of Fig. 5 shows the remaining paths.

Influences on mathematical performance (model #3).

Model #3 did not fit the data well). According to the modification indices, one more paths from the socio-economies status to the precursor skills was missing. Adding this path yielded good model fit (cf. Model #3.1). In the next step, we iteratively constrained the non-significant paths to zero. This resulted in constraining four regressions weights to zero. The resulting model fit the data well (cf. Model #3.2). Figure 5 shows the remaining paths, where the regression weights given on the left side are from this model.

The result showed that the social background was related to the home learning environment. More specifically, children with a higher SES, higher education of the mother and higher education of the father had a more favourable home literacy environment. A more favourable home numeracy environment, on the other hand, was related only to a higher SES.

Regarding precursor skills, self-concept, and enjoyment of the subject, only the precursor skills were related to other variables of model #1. To be precise, children with higher SES and less migration background had higher precursor skills.

Our results also showed that math anxiety was not related to any other variable, not even previous math anxiety. These results suggest that math anxiety was an independent construct in each year of schooling in our sample.

Math performance, on the other hand, could be explained by other variables. Having a higher math performance at t2 was related to a higher math performance at t3 and t4, and higher performance at t3 went hand in hand with higher performance at t4. Math performance was also related to precursor skills and self-concept. Both precursor skills and self-concept positively predicted math performance at t2. Comparing the results to the zero-order correlations in Table 1 shows that the precursor skills mediate the effects of socio-economies status and migration on math performance (Table 1: socio-economies is positively related to math performance, while migration is negatively related to math performance).

Gender effects (model #4).

In the last step, we examined the effects of gender on all variables and how the relations changed when accounting for gender. This model #4 fit the data well) and is depicted in Fig. 4. Gender directly affected the HNE (boys having a more favourable environment than girls) and math performance (t2: in favour of boys; t4: in favour of girls). However, taking effects of gender into account did not alter the relations otherwise found (compare left vs right parameters in Fig. 5).

Gender only had an effect on the math performance at t2 and at t4. Moreover, these two effects were of opposite direction: while boys outperformed girls at t2, girls outperformed boys at t4. Comparing these gender differences with the zero-order correlations in Table 1 shows that the gender effect at t4 is a suppression effect: gender is not related to math performance at t4 (cf. Table 1: r = 0.040), however when controlling for previous math performances a gender effect arises (cf. Figure 5: β = -0.154, p = 0.030). This suppression effect implies that there was a gender difference in math performance only at t2, not at t3 and t4. However, when controlling for previous math performance at t4, girls outperformed boys. In other words, only if boys and girls were equal in their math performance previously, girls outperformed boys at t4.

Discussion

Influences on mathematical performance

We examined in our first research question the factors influencing mathematical performance. Results showed that above all SES is decisive in the effect structure. SES affects the home learning environment (home numeracy and home literacy environment) and precursor skills. The better the SES, the better the home learning environment and the better the precursor skills. This is thus the most decisive factor in the structural traits of origin. Therefore, the first part of hypothesis 1 regarding the SES can be accepted and our results are in line with previous findings (cf. Baird, 2012). In Austria, education is inherited at an above-average rate (see PISA). This could be an explanation for the decisive factor of SES. The migration background has a negative correlation with the precursor skills. This goes hand in hand with hypothesis 1, which can therefore still be assumed and is in line with previous studies (cf. McElvany, 2008; Niklas et al., 2012).

The educational level of the parents in the study does not have a positive influence on performance, calling into question the findings of Baumert and Maaz (2006). Please note that while the educational level in itself is related to performance (cf. Table 1), its predictive utility decreased due to multicollinearity when socioeconomic status was factored in. The educational level of the parents can be seen as part of the definition of socioeconomic status, but unlike socioeconomic status, it is not relevant in its own right for precursor skills or later mathematics achievement. As a result, parental occupational status and household or family income appear more relevant than educational attainment. One reason for this could be that the differences in educational levels only gains relevance in the course of secondary or tertiary education, while social and cultural resources already play a role in primary education. Otherwise, future studies may also expand on this finding by scrutinizing if and how socioeconomic status and parents’ educational level may have not additive effects on attainment, but may actually interact. Research along this line may guide us in mitigating harmful effects on the lower bounds such as, for example, that low parents’ educational level may be even more detrimental when combined with low socioeconomic status being. However, future studies would first need to look into these possible interactions.

The home learning environment is influenced by SES and the educational level of the parents. Contrary to hypothesis 1, however, there is no significant influence of the home learning environment on math performance. Therefore, we should treat carefully when making home learning environment responsible for the fact that education is passed on. One explanation for this could be that other test instruments were used than those of Niklas, Skwarchuck or LeFevre. In general, such comparisons are very difficult, as different operationalizations can be found in the literature. Considering the structural traits of origin, only SES and migration background are decisive. Parents' educational level and the home learning environment play less of a role regarding math performance.

The interacting variables precursor skills and self-concept have a positive influence on math performance. This is consistent with hypothesis 1 and studies on precursor skills (Claessens et al., 2009; Krajewski & Schneider, 2009; Seitz & Weinert, 2022) and self-concept (Lohbeck et al., 2016). An explanation for the importance of the self-concept can also be found in the HATTIE study, in which the self-concept was counted among the most important factors for the influence of math performance after a comprehensive meta-study (Hattie, 2009). Precursor skills may be the missing link between the structural traits of origin and math performance. Precursor skills predict mathematics achievement in the 2nd grade. These results emphasise the importance of precursor skills for the acquisition of mathematical competences.

In addition to self-concept, it is mainly mathematics anxiety that is used as a construct to explain differences in performance in mathematics. Most studies assume a negative correlation, which could not be found in this study. This contradicts Luttenberger et al. (2018) and hypothesis 1 (Skaalvik, 2018). Mathematics anxiety appears as an isolated construct every year. This means that students worry about mathematics every year a new. One assumption for this isolation could be that mathematics anxiety begins to develop in primary school, but does not affect mathematics achievement until secondary school. Mathematics has the reputation of being particularly complicated and difficult. Children feel this already at primary school age and may be prone to adopt mathematics phobia from their parents and older siblings—before they even know what it implies. These beginnings of mathematics anxiety are based on several factors like society, but it should be emphasized that according to this study no influence on performance could be found.

According to Walther et al. (2008), the last part of the hypothesis assumes no connection between enjoyment of the subject and math performance. This was confirmed in this study. The emotion-related enjoyment of the subject is thus negligible in a modelling compared to the other constructs. In summary, hypothesis 1 can be partially refuted.

Role of gender

The second research question investigated whether the variable gender has an influence on the effect structure. Our results confirm hypothesis 2: gender has no influence on the structural traits of origin. Similarly, no gender disparities were found in the home learning environment. Therefore, our results suggest that gender differences in math performance cannot be closed by promoting a better home learning environment. In line with Kelz (2017), no effect was found in the precursor skills. Likewise, we found no gender differences in the constructs mathematics anxiety, self-concept and enjoyment of the subject.

Hypothesis 2, which posits that girls have greater mathematics anxiety (Ashcraft, 2002; Todor, 2014), lower self-concept (OECD, 2015) and lower enjoyment of the subject (Golob, 2017), needs to be rejected. These constructs do not seem to show gender disparities, at least in primary school in the school entry phase. This reinforces the assumption that girls and boys begin their school careers with the same performance (precursor skills) and the same affective characteristics (mathematics anxiety, self-concept, enjoyment of the subject).

In summary, the gender aspect appears to be very low. Neither in the performance at the beginning of school nor in the structural traits of origin nor in the home learning environment nor in the interacting variables were gender disparities measured. It is only the math performance at the time of the 2nd and 4th grade that has to be analysed with regard to gender.

Math performance

Math performance depends, not surprisingly, on prior math performance. Precursor skills could predict a substantial part of later math performance (up to 2nd grade). This is consistent with studies by Claessens et al. (2009) and Krajewski and Schneider (2009). In addition, self-concept predicted math performance at the 2nd measurement point (Arens et al., 2011). Thus, precursor skills and self-concept seem to be essential for predicting performance, whereas enjoyment of the subject and math anxiety are not not decisive for the pattern of effects.

Regarding math performance and gender, our results give a nuanced picture that may help to consolidate some of the controversial findings scholars have reported for studying mathematic performance in primary education (e.g., Kelz, 2020). Our results showed varying gender differences: first, boys outperform girls (cf. second measurement point), then we can find no differences (cf. third measurement point) and final girls outperform boys (cf. fourth measurement point). At first glance, these results show a gradual shift in the gender differences that lastly is in the girls’ favour. Taking a closer look (cf. suppression effect described for Fig. 5), however, reveals that these varying gender differences need to be interpreted with utmost care. As such, when viewing each measurement point in isolation (cf. Table 1) the only notable gender differences is at the second measurement point in favour of boys. This finding is in line with the longitudinal analyses. In the longitudinal analyses, however, prior mathematic performance is also factored in for the gender differences at the third and fourth measurement point. Accordingly, the results show that girls outperform boys at the last measurement point only when controlling for previous mathematic performance. These results imply that a cross-sectional study at the end of primary education will show other gender differences than a longitudinal study with the last measurement at the end of primary education. Thereby our findings stress that studies comparability needs to be scrutinized when comparing gender effects in mathematic performance (i.e., was it a gender effect observed in isolation or when controlling for previous mathematic performance?) and contradictory findings may actually be less controversial then previously suggested.

Finally, the theoretical models considered for this study of Niklas (2015) and Luttenberger et al. (2018) raises some questions in light of this study. While the structural traits of origin influence the home learning environment, the home learning environment, the central construct in the modelling of Niklas (2015), does not seem to be relevant for precursor skills and later math performance. Similarly, among the interacting variables of Luttenberger et al. (2018), only self-concept and not mathematics anxiety is relevant for performance. As can be seen in Fig. 5, only socioeconomic status and migration background are relevant for precursor skills. School competencies are only influenced by precursor skills and self-concept. Based on the review of all the factors discussed in this study, the educational level of the parents, the home learning environment, enjoyment of the subject and math anxiety would be dropped from a new modelling of math performance.

Implications

The precursor skills are crucial for math performance in primary school. One implication is that the acquisition of these skills should be promoted at kindergarten age. One result of the study indicates that this is predominantly promoted in kindergarten and not at home or through the home learning environment. One problem with mathematics anxiety is that children are stressed differently and newly every school year. Therefore, one implication would be to address this in education and training programs. The study also emphasizes that no gender differences can be assumed at the beginning of school. Gender differences in math performance exist independently of precursors skills. The constructs enjoyment of the subject, self-concept and mathematics anxiety are also not affected by gender differences. This means that these constructs are not primarily explanations for possible gender differences. However, SES and migration seems to condition a heterogeneous classroom environment. Nevertheless, this study gives the impression that girls and boys can start school equally, regardless of gender.

Limitations

The biggest limitation is the size of the sample (N = 203) and its regionality limited to the surrounding area of Graz, Styria, Austria. Therefore, the sample is only representative for Styrian primary schools in the vicinity of Graz. Another limitation is the ERT0 + (Eggenberger Rechentest 0 +) is a prediagnostic for dyscalculia, so that it differentiates only weakly in the upper third of performance. The third limitation is that not all constructs were collected at t2, t3 and t4 in the measurement schedule. Moreover, mathematics anxiety was not collected at the first measurement point. Another limitation arises from the use of longitudinal data, as a lack of individual data at measurement time t2, t3 or t4 was tolerated.

Conclusion

In this study, a longitudinal section with 203 pupils was conducted over four measurement points across the entire primary level. For the first time, structural traits of origin such as socioeconomic status, migration background and the educational level of the parents, the home learning environment and the interacting variables precursor skills, self-concept, enjoyment of the subject and mathematics anxiety were discussed theoretically for the primary level and tested empirically with regard to math performance. In addition, the aspect or construct of gender was included in the effect structure. In view of the theoretically supported modelling of Niklas (2015) and Luttenberger et al. (2018), it can be stated that, in contradiction to Niklas (2015), the home learning environment does not play a role for mathematics achievement and, in contradiction to Luttenberger et al. (2018), the interacting variables enjoyment of the subject and mathematics anxiety do not correlate with mathematics achievement. This is significant in the sense that a modelling of mathematics achievement in primary education based on this study would only consider the influencing factors socioeconomic status, migration background, self-concept, and precursor skills. Moreover, gender differences have only been measured in performance and not in structural traits of origin, home learning environment or interacting variables. One indication for this lies in the gender similarity hypothesis. This hypothesis states that girls and boys do not differ in most psychological variables (Hyde, 2005). The gender differences in math performance could not be completely explained by the influencing factors we used. Nevertheless, our study provides results that would facilitate the drawing of a holistic picture of relevant factors influencing mathematics achievement in primary education.