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Sixty (35 girls and 25 boys) 9th-grade students’ conceptual understanding of the number line was qualitatively assessed through verbal explanations and visual representations. The assessment included an open-ended question focused on students’ number line descriptions and the explanations coalesced around six features: sequential ordering (i.e., numbers are sequentially represented), positivity-negativity of numbers (i.e., the number line contains positive and negative numbers), non-centrality (i.e., zero does not have to be in the center), infinity, increment flexibility (i.e., number line increments can vary), and continuity (i.e., numbers can be placed anywhere between minus infinity and plus infinity without breaks). The students’ explanations show that these six features emerge in five successive stages in the conceptual understanding of the number line. These stages are (1) no knowledge, (2) sequential ordering and positivity-negativity, (3) infinity and non-centrality, (4) incremental flexibility, and (5) continuity. The last two stages were not found in most descriptions. The results suggest that students’ understanding of the number line is incomplete and may be overestimated by commonly used quantitative assessments of number line knowledge.

Sixty (35 girls) ninth graders were assessed on measures of algebraic reasoning and usage of visual and symbolic representations (with a prompt for visual use) to solve equations and inequalities. The study grouped visual representations into two categories: arithmetic-visual, which entailed the use of real-world objects to represent specific values of variables, and algebraic-visual, which involved formal representations like the number line and the coordinate plane. Symbolic representations, on the other hand, encompassed the use of standard algorithms to solve equations, such as changing the place of terms in an equation. The results reveal that the use of algebraic visuals, as opposed to arithmetic visuals, was associated with enhanced algebraic reasoning. Further, although the students initially relied on standard algorithms to explain equations and inequalities, they could produce accurate algebraic-visual representations when prompted. These findings suggest that students have multiple representations of equations and inequalities but only express visual representations when asked to do so. In keeping with the general relationship between visuospatial abilities and mathematics, self-generated algebraic-visual representations partially mediated the relation between overall mathematics achievement and algebraic reasoning.

This study describes the development and initial validation of a mathematics-specific spatial vocabulary measure for upper elementary school students. Reviews of spatial vocabulary items, mathematics textbooks, and Mathematics Common Core State Standards identified 720 mathematical terms, 148 of which had spatial content (e.g., edge). In total, 29 of these items were appropriate for elementary students, and a pilot study (59 fourth graders) indicated that nine of them were too difficult (< 50% correct) or too easy (> 95% correct). The remaining 20 items were retained as a spatial vocabulary measure and administered to 181 (75 girls, mean age = 119.73 months, SD =4.01) fourth graders, along with measures of geometry, arithmetic, spatial abilities, verbal memory span, and mathematics attitudes and anxiety. A Rasch model indicated that all 20 items assessed an underlying spatial vocabulary latent construct. The convergent and discriminant validity of the vocabulary measure was supported by stronger correlations with theoretically related (i.e., geometry) than with more distantly related (i.e., arithmetic) mathematics content and stronger relations with spatial abilities than with verbal memory span or mathematics attitudes and anxiety. Simultaneous regression analyses and structural equation models, including all measures, confirmed this pattern, whereby spatial vocabulary was predicted by geometry knowledge and spatial abilities but not by verbal memory span, mathematics attitudes and anxiety. Thus, the measure developed in this study helps in assessing upper elementary students' mathematics-specific spatial vocabulary.

Understanding the magnitudes represented by numerals is a core component of early mathematical development and is often assessed by accuracy in situating numerals and fractions on a number line. Performance on these measures is consistently related to performance in other mathematics domains, but the strength of these relations may be overestimated because general cognitive ability has not been fully controlled in prior studies. The first of two meta-analyses (162 studies, 33,101 participants) confirmed a relation between performance on whole number (r = 0.33) and fractions number (r = 0.41) lines and overall mathematics performance. These relations were generally consistent across content domains (e.g., algebra and computation) and other moderators. The second (71 studies, 14,543 participants) used meta-analytic structural equation modeling to confirm these relations while controlling general cognitive ability (defined by IQ and working memory measures) and, in one analysis, general mathematics competence. The relation between number line performance and general mathematics competence remained significant but reduced ([beta] = 0.13). Controlling general cognitive ability, whole number line performance consistently predicted competence with fractions but not performance on numeracy or computations measures. The results suggest an understanding of the magnitudes represented by whole numbers might be particularly important for students' fractions learning.

The goal was to identify the domain-general cognitive abilities and academic attitudes that are common and unique to reading and mathematics learning difficulties that in turn will have implications for intervention development. Across seventh and eighth grade, 315 (155 boys) adolescents (M age = 12.75 years) were administered intelligence, verbal short-term and working memory, and visuospatial memory, attention, and ability measures, along with measures of English and mathematics attitudes and mathematics anxiety. Teachers reported on students’ in-class attentive behavior. A combination of Bayesian and multi-level models revealed that intelligence and in-class attentive behavior were common predictors of reading accuracy, reading fluency, and mathematics achievement. Verbal short-term memory was more critical for reading accuracy and fluency, whereas spatial ability and mathematics self-efficacy were more critical for mathematics achievement. The combination of intelligence and in-class attentive behavior discriminated typically achieving students from students with comorbid (D = 2.44) or mathematics (D = 1.59) learning difficulties, whereas intelligence, visuospatial attention, and verbal short-term memory discriminated typically achieving students from students with reading disability (D = 1.08). The combination of in-class attentive behavior, verbal short-term memory, and mathematics self-efficacy discriminated students with mathematics difficulties from their peers with reading difficulties (D = 1.16). Given the consistent importance of in-class attentive behavior, we conducted post hoc follow-up analyses. The results suggested that students with poor in-class attentive behavior were disengaging from academic learning which in turn contributed to their risk of learning difficulties.