Meta-analytic structural equation modeling was used to estimate the relative contributions of general cognitive ability or g (defined by executive functions, short-term memory, and intelligence) and basic domain-specific mathematical abilities to performance in more complex mathematics domains. The domain-specific abilities included mathematics fluency (e.g., speed of retrieving basic facts), computational skills (i.e., accuracy at solving multi-step arithmetic, algebra, or geometry problems), and word problems (i.e., mathematics problems presented in narrative form). The core analysis included 448 independent samples and 431,344 participants and revealed that g predicted performance in all three mathematics domains. Mathematics fluency contributed to the prediction of computational skills, and both mathematics fluency and computational skills predicted word problem performance, controlling g. The relative contribution of g was consistently larger than basic domain-specific abilities, although the latter may be underestimated. The patterns were similar across younger and older individuals, individuals with and without a disability (e.g., learning disability), concurrent and longitudinal assessments, and family socioeconomic status, and have implications for fostering mathematical development.

Adolescents' (n = 342, 169 boys) general algebra and algebra word problems performance were assessed in 9th grade as were intelligence, academic achievement, working memory, and spatial abilities in prior grades. The adolescents reported on their academic attitudes and anxiety and their teachers reported on their in-class attentive behavior in 7th to 9th grade. There were no sex differences on the general algebra measure or for mathematics achievement, but boys had an advantage on the algebra word problems measure (d = .51) and for spatial abilities (ds = .29 to .58). Boys had higher mathematics self-efficacy (d = .24 to .33), lower mathematics anxiety (ds = -.31 to -.53) and were less attentive in classrooms (ds = -.28 to -.37). A series of structural equation models revealed the sex difference for algebra word problems was mediated by spatial abilities and mathematics anxiety, controlling myriad confounds. Public Significance Statement Sex differences in mathematics are typically small but larger for word problems. The latter assess the ability to use mathematics in problem-solving situations and are often included on high-stakes tests. Boys' higher spatial abilities appeared to provide them with an advantage and girls' mathematics anxiety a disadvantage in solving algebra word problems.

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.

There are consistent correlations between mathematics achievement, attitudes, and anxiety, but the longitudinal relations among these constructs are not well understood nor are sex differences in these relations. To address this gap, mathematics achievement, attitudes, and anxiety were longitudinally assessed for 342 (169 boys) adolescents from seventh to ninth grade, inclusive, and latent growth curve models (LGCM) were used to assess the relations among these traits and developmental change in them. Spatial abilities (seventh and eighth grades) and trait anxiety (eighth and ninth grades) were also assessed and used to control for sex differences in these traits. Overall, boys had stronger spatial abilities and more positive mathematics attitudes and were less anxious than girls, but there were no sex differences in mathematics achievement. Across grades, mathematics achievement improved, attitudes became less positive, and anxiety increased for both boys and girls. Higher than average cross-grade growth in mathematics achievement mitigated boys' developmental declines in mathematics attitudes and increases in anxiety. Girls with strong spatial abilities had lower mathematics anxiety, but girls overall maintained higher mathematics anxiety and less positive mathematics attitudes relative to boys, even when they showed strong cross-grade gains in mathematics achievement. The study demonstrated that longitudinal gains in mathematics are associated with cross-grade changes in attitudes and anxiety but with several different developmental patterns for boys and girls.

In this article, the authors describe the two interpretations of mathematical expressions, as a process and as a product. They begin with an initial discussion of Tall's (2013) interpretations of expressions and connect this to state standards. Next, they describe the different interpretations of algebraic expressions based on their work with middle and high school students. They conclude with an example of how they provided instruction that promotes a deep understanding of the interpretations of expressions.

The articles in this special issue provide state-of-the-art reviews of the brain and cognitive systems that are engaged during some aspects of mathematical learning, as well as the self-beliefs, anxiety, and social factors that influence engagement with mathematics, along with discussion of any associated sex differences. These issues are integrated into an evolutionary perspective that includes discussion of how evolved brain and cognitive systems might be co-opted for learning in the evolutionarily novel domain of mathematics. Attitudes and beliefs about mathematics are considered in the context of the evolution of self-awareness that in turn explains why many students do not value mathematics, despites its importance in the modern world, as highly as many other personal traits, such as their physical appearance. The overall argument is that reflecting on academic learning and attitudes from an evolutionary perspective provides insights into student learning and self-beliefs about learning that might otherwise elude explanation.

Educational Impact and Implications Statement Given the frequent co-occurrence of difficulties in learning mathematics and reading, identifying the processes and skills that support success in both domains is important. The study showed that several domain-general abilities, such as working memory and attentive behavior, contributed to earlier and later calculation skill, word-problem solving, and word-reading fluency. More critically, the study showed that one bridge between the codevelopment of reading and mathematics achievement is the fluency of processing basic numerical relationships. The numerical measure in turn likely indexes the ease with which students form symbol-concept associations, and this cognitive system might be contributing to the co-occurrence of difficulties in mathematics and reading. We examined longitudinal relations between 1st-grade cognitive predictors (early nonverbal reasoning, processing speed, listening comprehension, working memory, calculation skill, word-problem solving, word-reading fluency, attentive behavior, and numerical cognition) and 2nd-grade academic outcomes (calculations, word-problem solving, and word reading) in 370 children (M-age = 6.55 years, SDage = 0.33 years at the start of the study) who were identified as at-risk or not-at-risk for mathematics disability. Path analysis mediation models revealed that numerical cognition, assessed at an intermediary timepoint, mediated the effects of processing speed, working memory, calculation skill, word-problem solving, and attentive behavior on all 3 outcomes. Findings indicate that multiple early domain-general cognitive abilities are related to later mathematics and reading outcomes and that numerical cognition processes, which may track ease of forming symbol-concept associations, predict later performance across both academic domains.

We investigated the role of working memory in symbolic and spatial algebra and related tasks across five experiments. Each experiment combined a processing task (expression evaluation, arithmetic, coordinate plane, geometry, or mental rotation) with verbal and spatial memory loads in a dual-task design. Spatial memory was compromised in the presence of more difficult processing tasks, and verbal memory was only compromised in the presence of algebraic tasks. The latter was related to the demands of retaining quantities associated with variables in verbal memory. We suggest that both verbal and spatial working memory retention engage domain-general attention, but that their maintenance mechanisms differ. Verbal memory has attention-based and rehearsal-based mechanisms, and thus sustaining verbal information over a short period is less attention-demanding than holding spatial information. We suggest that effects of a memory load on processing (e.g., x = 6) depend on whether use of maintenance strategies are possible for the specific memory load while carrying out processing. In all, our results indicate that algebraic tasks use domain-general attention and include verbal processing of algebraic variables (i.e., information conveyed in x, y). We discuss the implications for algebra learning and working memory theories.

Identifying meaningful cognitive and noncognitive predictors of mathematical competence is critical for developing targeted interventions for students struggling with mathematics. Here, 317 students' short-term verbal memory. verbal and visuospatial working memory, complex spatial abilities, intelligence, and mathematics attitudes and anxiety were assessed. and their teachers reported on their attentive behavior in 7th-grade mathematics classrooms. Bayesian regression models revealed that complex spatial abilities and in-class attention were the most plausible predictors of 7th-grade mathematics, but not word reading achievement, controlling for prior mathematics achievement. These results were confirmed with multilevel models that revealed interactions between these factors and prior achievement. The largest gains were among students with strong mathematical competencies in 6th grade, and average or better in-class attention in 7th grade as well as above average spatial abilities. High mathematics anxiety was associated with lower attention and through this indirectly influenced achievement gains. These results have implications for how to best target interventions for students at risk for long-term difficulties with mathematics.