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Children's ability to relate number to a continuous quantity abstraction visualized as a number line is widely accepted to be predictive of mathematics achievement. However, a debate has emerged with respect to how children's placements are distributed on this number line across development. In the current study, different models were applied to children's longitudinal number placement data to get more insight into the development of number line representations in kindergarten and early primary school years. In addition, longitudinal developmental relations between number line placements and mathematical achievement, measured with a national test of mathematics, were investigated using cross-lagged panel modeling. A group of 442 children participated in a 3-year longitudinal study (ages 5-8 years) in which they completed a number-to-position task every 6 months. Individual number line placements were fitted to various models, of which a one-anchor power model provided the best fit for many of the placements at a younger age (5 or 6 years) and a two-anchor power model provided better fit for many of the children at an older age (7 or 8 years). The number of children who made linear placements also grew with age. Cross-lagged panel analyses indicated that the best fit was provided with a model in which number line acuity and mathematics performance were mutually predictive of each other rather than models in which one ability predicted the other in a non-reciprocal way. This indicates that number line acuity should not be seen as a predictor of math but that both skills influence each other during the developmental process.

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Symbolic (i.e., with Arabic numerals) approximate arithmetic with large numerosities is an important predictor of mathematics. It was previously evidenced to onset before formal schooling at the kindergarten age (Gilmore et al., 2007) and was assumed to map onto pre-existing nonsymbolic (i.e., abstract magnitudes) representations. With a longitudinal study (Experiment 1), we show, for the first time, that nonsymbolic and symbolic arithmetic demonstrate different developmental trajectories. In contrast to Gilmore et al.’s (2007) findings, Experiment 1 showed that symbolic arithmetic onsets in grade 1, with the start of formal schooling, not earlier. Gilmore et al. (2007) had examined English-speaking children, whereas we assessed a large Dutch-speaking sample. The Dutch language for numbers can be cognitively more demanding, for example, due to the inversion property in numbers above 20. Thus, for instance, the number 48 is named in Dutch “achtenveertig” (eight and forty) instead of “forty eight.” To examine the effect of the language of numbers, we conducted a cross-cultural study with English- and Dutch-speaking children that had similar SES and math achievement skills (Experiment 2). Results demonstrated that Dutch-speaking kindergarteners lagged behind English-speaking children in symbolic arithmetic, not nonsymbolic and demonstrated a working memory overload in symbolic arithmetic, not nonsymbolic. Also, we show for the first time that the ability to name two-digit numbers highly correlates with symbolic approximate arithmetic not nonsymbolic. Our experiments empirically demonstrate that the symbolic number system is modulated more by development and education than the nonsymbolic system. Also, in contrast to the nonsymbolic system, the symbolic system is modulated by language.

Research has identified various domain-general and domain-specific cognitive abilities as predictors of children's individual differences in mathematics achievement. However, research into the predictors of children's individual growth rates, namely between-person differences in within-person change in mathematics achievement is scarce. We assessed 334 children's domain-general and mathematics-specific early cognitive abilities and their general mathematics achievement longitudinally across four time-points within the first and second grades of primary school. As expected, a constellation of multiple cognitive abilities contributed to the children's starting level of mathematical success. Specifically, latent growth modeling revealed that WM abilities, IQ, counting skills, nonsymbolic and symbolic approximate arithmetic and comparison skills explained individual differences in the children's initial status on a curriculum-based general mathematics achievement test. Surprisingly, however, only one out of all the assessed cognitive abilities was a unique predictor of the children's individual growth rates in mathematics achievement: their performance in the symbolic approximate addition task. In this task, children were asked to estimate the sum of two large numbers and decide if this estimated sum was smaller or larger compared to a third number. Our findings demonstrate the importance of multiple domain-general and mathematics-specific cognitive skills for identifying children at risk of struggling with mathematics and highlight the significance of early approximate arithmetic skills for the development of one's mathematical success. We argue the need for more research focus on explaining children's individual growth rates in mathematics achievement.

Preschool children have been proven to possess nonsymbolic approximate arithmetic skills before learning how to manipulate symbolic math and thus before any formal math instruction. It has been assumed that nonsymbolic approximate math tasks necessitate the allocation of Working Memory (WM) resources. WM has been consistently shown to be an important predictor of children's math development and achievement. The aim of our study was to uncover the specific role of WM in nonsymbolic approximate math. For this purpose, we conducted a dual-task study with preschoolers with active phonological, visual, spatial, and central executive interference during the completion of a nonsymbolic approximate addition dot task. With regard to the role of WM, we found a clear performance breakdown in the central executive interference condition. Our findings provide insight into the underlying cognitive processes involved in storing and manipulating nonsymbolic approximate numerosities during early arithmetic.

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Despite the widely accepted view that low self-efficacy beliefs negatively influence students' intention to opt for a STEM field oriented study or career path, it remains unclear how to effectively stimulate these beliefs in students who do seem to have the ability and motivation to opt for a STEM career. A suggestion from previous literature is that students' implicit beliefs about the malleability of their learning ability can have a major impact on their self-efficacy beliefs, and, importantly, that these implicit beliefs are malleable themselves. Even though this relation between implicit beliefs, self-efficacy, and STEM field aspirations has been suggested multiple times, there is no empirical evidence to support this claim. The goal of the current study was to examine whether implicit beliefs about the malleability of STEM ability are associated with secondary school students' intention to opt for a STEM field bachelor's degree, using a Structural Equation Modelling approach. Furthermore, we examined the mediating role of STEM-oriented self-efficacy beliefs on the relationship between implicit ability beliefs and STEM intention. We used a Likert-type questionnaire, consisting of subscales to measure ability beliefs, self-efficacy, and intention to opt for a STEM degree of secondary school students in their fifth grade (n = 483). Results showed that there is a positive relation between implicit STEM ability beliefs and the intention to opt for a STEM field bachelor degree, and that this relation is partly mediated by self-efficacy beliefs. Incremental STEM ability beliefs predicted positive selfefficacy beliefs and increased STEM intention. These findings provide a foundation for a novel approach to stimulate