Supplementary Information files for: The association of basic numerical abilities and math achievement: The mediating role of visuospatial and arithmetical abilities

Supplementary Information files for: The association of basic numerical abilities and math achievement: The mediating role of visuospatial and arithmetical abilitiesBasic numerical abilities such as number line estimation have been observed repeatedly to be associated with mathematical achievement. Recently, it was argued that the association between basic numerical abilities and mathematical achievement is fully mediated by visuospatial abilities. However, arithmetical abilities have not yet been considered as influencing this association, even though solution strategies in number line estimation as well as mathematical achievement often involve arithmetical procedures. Therefore, we investigated the mediating role of arithmetical and visuospatial abilities on the association between number line estimation and mathematical achievement in a sample of n = 599 German elementary school students. Results indicated that arithmetical abilities as well as visuospatial abilities mediated the association between number line estimation and mathematical achievement. However, neither visuospatial nor arithmetical abilities fully mediated the association between number line estimation and mathematical achievement when considered in isolation. This substantiates the relevance of the intertwined development of visuospatial and arithemtical abilities as well as basic numerical abilities such as number line estimation (i.e. the combination of domain-specific numerical and domain-general abilities) driving mathematical achievement<br

The association of basic numerical abilities and math achievement: The mediating role of visuospatial and arithmetical abilities

Basic numerical abilities such as number line estimation have been observed repeatedly to be associated with mathematical achievement. Recently, it was argued that the association between basic numerical abilities and mathematical achievement is fully mediated by visuospatial abilities. However, arithmetical abilities have not yet been considered as influencing this association, even though solution strategies in number line estimation as well as mathematical achievement often involve arithmetical procedures. Therefore, we investigated the mediating role of arithmetical and visuospatial abilities on the association between number line estimation and mathematical achievement in a sample of n = 599 German elementary school students. Results indicated that arithmetical abilities as well as visuospatial abilities mediated the association between number line estimation and mathematical achievement. However, neither visuospatial nor arithmetical abilities fully mediated the association between number line estimation and mathematical achievement when considered in isolation. This substantiates the relevance of the intertwined development of visuospatial and arithemtical abilities as well as basic numerical abilities such as number line estimation (i.e. the combination of domain-specific numerical and domain-general abilities) driving mathematical achievement

Results of Study 1 (Adoption-Task).

<p><i>I</i>_1<i>t</i> and <i>I</i>_2<i>t</i> are Indicator or Dummy variables indicating the stimulus category (<i>I</i>_1<i>t</i>: 1 = African infants, 0 = Caucasian infants; <i>I</i>_2<i>t</i>: 1 = dog puppies, 0 = Caucasian infants). <i>S</i>_<i>t</i> represents participants’ sex (0 = male, 1 = female). <i>A</i> indicates participants’ age (0 = mean age of the sample).</p><p>Robust estimators were used for statistical inference with respect to fixed effects and variance components to account for possible violations of model assumptions, such as normality of Level-2 residuals. Degrees of freedom were computed based on the Satterthwaite’s Approximation to account for the moderate sample size at Level 2 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0121554#pone.0121554.ref046" target="_blank">46</a>]. Therefore, the degrees of freedom were not necessarily integers and could vary across tests independent of the number of parameters.</p><p>Results of Study 1 (Adoption-Task).</p

Descriptive statistics of Study 1: Adoption-Task.

<p>Relative frequencies are presented for cute and less cute faces being chosen for each stimulus category and separately for male and female participants.</p

Results of Study 2.

<p><i>I</i>_1<i>it</i> and <i>I</i>_2<i>it</i> are Indicator or Dummy variables indicating the stimulus category (<i>I</i>_1<i>it</i>: 1 = African infants, 0 = Caucasian infants; <i>I</i>_1<i>it</i>: 1 = dog puppies, 0 = Caucasian infants). <i>H</i>_<i>it</i> reflects the health state (0 = mean assessment of perceived health across all stimuli and participants, a positive value indicates perceived above-average illness frequency). <i>S</i>_<i>t</i> represents participants’ sex (0 = male, 1 = female). <i>A</i>_<i>t</i> indicates participants’ age (0 = mean age of the sample). For interpreting the coefficients all other predictor variables have to be held constant.</p><p>An unstructured covariance structure was used for the random part at Level 2. Hence, the variances and covariances of Level 2 residuals were estimated without any constraints. Robust estimators were used for statistical inference with respect to fixed effects and variance components to account for possible violations of model assumptions, such as normality of Level-2 residuals. Degrees of freedom were computed based on the Satterthwaite’s Approximation to account for the moderate sample size at Level 2 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0121554#pone.0121554.ref046" target="_blank">46</a>]. Therefore, the degrees of freedom were not necessarily integers and could vary across tests independent of the number of parameters.</p><p>Results of Study 2.</p

Descriptive statistics of Study 1: Toy-Task.

<p>Relative frequencies are presented for cute and less cute faces being chosen for each stimulus category and separately for male and female participants.</p

Results of Study 1 (Toy-Task).

<p><i>I</i>_1<i>t</i> and <i>I</i>_2<i>t</i> are Indicator or Dummy variables indicating the stimulus category (<i>I</i>_1<i>t</i>: 1 = African infants, 0 = Caucasian infants; <i>I</i>_2<i>t</i>: 1 = dog puppies, 0 = Caucasian infants). <i>S</i>_<i>t</i> represents participants’ sex (0 = male, 1 = female). <i>A</i> indicates participants’ age (0 = mean age of the sample).</p><p>Robust estimators were used for statistical inference with respect to fixed effects and variance components to account for possible violations of model assumptions, such as normality of Level-2 residuals. Degrees of freedom were computed based on the Satterthwaite’s Approximation to account for the moderate sample size at Level 2 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0121554#pone.0121554.ref046" target="_blank">46</a>]. Therefore, the degrees of freedom were not necessarily integers and could vary across tests independent of the number of parameters.</p><p>Results of Study 1 (Toy-Task).</p

Mean cuteness scores of the stimuli used in Study 1.

<p>Mean cuteness scores of the stimuli used in Study 1.</p

Results of the pre-study (Model 2).

<p><i>I</i>_1<i>it</i> and <i>I</i>_2<i>it</i> are Indicator or Dummy variables indicating the stimulus category (<i>I</i>_1<i>it</i>: 1 = African infants, 0 = Caucasian infants; <i>I</i>_2<i>it</i>: 1 = dog puppies, 0 = Caucasian infants). <i>C</i> represents cuteness category (<i>C</i>_<i>it</i>: 0 = less cute, 1 = cute). Robust estimators were used for statistical inference with respect to fixed effects and variance components to account for possible violations of model assumptions, such as normality of Level-2 residuals. Degrees of freedom were computed based on the Satterthwaite’s Approximation to account for the moderate sample size at Level 2 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0121554#pone.0121554.ref046" target="_blank">46</a>]. Therefore, the degrees of freedom were not necessarily integers and could vary across tests independent of the number of parameters.</p><p>Results of the pre-study (Model 2).</p

Getting fit for the Mathematical Olympiad: positive effects on achievement and motivation?

All around the world, there are numerous academic competitions (e.g., “Academic Olympiads”) and corresponding training courses to foster students’ competences and motivation. But do students’ competences and motivation really benefit from such courses? We developed and evaluated a course that was designed to prepare third and fourth graders to participate in the German Mathematical Olympiad. Its effectiveness was evaluated in a quasi-experimental pre- and posttest design (N = 201 students). Significant positive effects of the training were found for performance in the academic competition (for both third and fourth graders) as well as mathematical competences as measured with a curriculum-oriented test (for fourth graders only). Differential effects across grade levels (with more pronounced positive effects in fourth-grade students) were observed for students’ math self-concept and task-specific interest in mathematics, pointing to possible social comparison effects.</p