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Results from a 2-year longitudinal study of 181 children from 4th through 5th grade are reported. Levels of growth in children's computation, word problem, and estimation skills by means of common fractions were predicted by working memory, attentive classroom behavior, conceptual knowledge about fractions, and simple arithmetic fluency. Comparisons of 55 participants identified as having mathematical difficulties to those without mathematical difficulties revealed that group differences in emerging fraction skills were consistently mediated by attentive classroom behavior and conceptual knowledge about fractions. Neither working memory nor arithmetic fluency mediated group differences in growth in fraction skills. It was also found that the development of basic fraction skills and conceptual knowledge are bidirectional in that conceptual knowledge exerted strong influences on all 3 types of basic fraction skills, and basic fraction skills exerted a more modest influence on subsequent conceptual knowledge. Results are discussed with reference to how the identification of potentially malleable student characteristics that contribute to the difficulties that some students have with fractions informs interventions. Also, results will contribute to a future theoretical account concerning how domain-general and domain-specific factors influence the development of basic fraction skills. Keywords: fraction skills, mathematical learning disabilities, mathematical learning difficulties, mathematical development DOI: 10.1037/a0019824

The mathematical performance of 182 third and fourth graders in 8 different areas of mathematics was examined. The children belonged to 4 achievement groups: children with mathematic difficulties (MD only), children with both mathematic and reading difficulties (MD-RD), children with reading difficulties (RD only), and normally achieving children (control group). Both MD groups performed worse than the normally achieving children in all but 1 area, place value knowledge. The MD-only and the MD-RD children performed equally in all areas of mathematics. The RD-only group performed at the same level as the control group on all areas of mathematics. The study provides further evidence that fact retrieval deficits are a cardinal characteristic of children with MD. The MD children's substantial difficulties with mathematic word problem solving can be attributed to several processes involved in problem solving. Besides poor skills in multidigit calculation, arithmetic fact retrieval, and poor understanding of calculation principles, children with MD might have deficits related to specific problem-solving processes such as establishing a problem representation and developing a solution plan. Keywords: mathematical difficulties, reading difficulties, calculation, mathematic problem solving, telling time

The purpose of this study was to explore patterns of difficulty in 2 domains of mathematical cognition: computation and problem solving. Third graders (n = 924; 47.3% male) were representatively sampled from 89 classrooms; assessed on computation and problem solving; classified as having difficulty with computation, problem solving, both domains, or neither domain; and measured on 9 cognitive dimensions. Difficulty occurred across domains with the same prevalence as difficulty with a single domain; specific difficulty was distributed similarly across domains. Multivariate profile analysis on cognitive dimensions and chi-square tests on demographics showed that specific computational difficulty was associated with strength in language and weaknesses in attentive behavior and processing speed; problem-solving difficulty was associated with deficient language as well as race and poverty. Implications for understanding mathematics competence and for the identification and treatment of mathematics difficulties are discussed. Keywords: calculations, word problems, cognitive predictors, mathematics

The number skills of groups of 7- to 9-year-old children with specific language impairment (SLI) attending mainstream or special schools were compared with an age and nonverbal reasoning matched group (age control [AC]) and with a younger group matched on oral language comprehension. The SLI groups performed below the AC group on every skill. They also showed lower working memory functioning and had received lower levels of instruction. Nonverbal reasoning, working memory functioning, language comprehension, and instruction accounted for individual variation in number skills to differing extents depending on the skill. These factors did not explain the differences between SLI and AC groups on most skills. Keywords: number development, working memory, specific language impairment, instruction

This study examined the efficacy of preventive 1st-grade tutoring in mathematics, estimated the prevalence and severity of mathematics disability, and explored pretreatment cognitive characteristics associated with mathematics development. Participants were 564 first graders, 127 of whom were designated at risk (AR) for mathematics difficulty and randomly assigned to tutoring or control conditions. Before treatment, all participants were assessed on cognitive and academic measures. Tutoring occurred 3 times weekly for 16 weeks; treatment fidelity was documented; and math outcomes were assessed. Tutoring efficacy was supported on computation and concepts/applications, but not on fact fluency. Tutoring decreased the prevalence of math disability, with prevalence and severity varying as a function of identification method and math domain. Attention accounted for unique variance in predicting each aspect of end-of-year math performance. Other predictors, depending on the aspect of math performance, were nonverbal problem solving, working memory, and phonological processing. Keywords: mathematics, learning disability, prevention, first grade, intervention

This longitudinal study tracked 1,110 adolescents identified as mathematically precocious at Age 13 (top 1%) with plans for a math-science undergraduate major. Participants' high school educational experiences, abilities, and interests predicted whether their attained undergraduate degrees were within math-science or nonmath-nonscience areas. More women than men eventually completed undergraduate degrees outside math-science, but many individuals who completed nonmath-nonscience degrees ultimately chose math-science occupations (and vice versa). At Age 33, the 2 degree groups reported commensurate and uniformly high levels of career satisfaction, success, and life satisfaction. Assessing individual differences is critical for modeling talent development and life satisfaction; it reveals that equal male-female representation across disciplines may not be as simple to accomplish as many policy discussions imply.

When do students detect algebra word problems with missing information, and how do they do it? To determine whether such detection operates automatically or requires conscious effort and attention, this study manipulated whether a hint was provided that problems might be unsolvable. A signal detection analysis revealed that the hint resulted in a large improvement in the detection of unsolvable problems, implying that many individuals possess the means to detect missing information but that conscious effort is required for those means to be deployed. Participants with moderate mathematical ability had good success at detecting missing information when given the hint, but only when the problems had familiar cover stories. High math ability individuals were successful regardless of the cover story's familiarity. In addition, participants were more likely to conclude that a solvable problem was unsolvable when the hint was provided, when the problems' cover story was unfamiliar, and with the passage of time. As a result, problem solving of solvable problems was sometimes abandoned prematurely.

Three studies were conducted to examine the effects of individual differences and language differences on children's understanding of teen quantities (11 [less than or equal to] n [less than or equal to] 19) as counted cardinal tens and ones (embedded-ten cardinal understanding). At age 4, most Chinese children, using named-ten number words (e.g., 12 is said as 'ten two'), did not show such understanding on a task in which y quantifies were added to 10 quantities. At age 5, half the children of average or above intelligence who had high rote-counting sequences (M = 90) did show such understanding; those with lower rote-counting sequences did not. English-speaking 5-year-old children in England and in the United States, whose teen words obfuscate the tens and ones, showed no evidence of understanding teen quantities as cardinal tens and ones.

The unique associations between 3 families of mathematical knowledge and individual differences in 7th and 8th graders' fraction skills were examined. Procedural knowledge regarding the steps to solve fraction computation problems and conceptual knowledge of part-whole and measurement interpretations of fractions were measured. Math fact knowledge was indexed as the speed of solving single-digit addition and multiplication problems. Conceptual knowledge and procedural knowledge uniquely explained variability in fraction computation solving and fraction word problem set-up accuracy. Only conceptual knowledge uniquely explained individual differences in fraction estimation skills. The findings are discussed within a framework that describes the 3 necessary steps toward an information-processing account of individual differences in fraction skills. Four important limitations to this study were described.

For this study of the organization of cognitive abilities and gender differences in young children advanced in mathematical reasoning, parents identified 778 preschoolers and kindergartners. After screening with 2 arithmetic subtests of standard intelligence tests, 310 high scorers (55% boys) were given 15 additional measures. Mean performance of these high scorers on all standardized measures was 1 to 2 standard deviations above the mean of the norming samples. Boys scored higher on 8 of 11 quantitative measures, 0 of 3 verbal measures, and 1 of 3 spatial measures, including quantitative and spatial working memory span. Three factors (quantitative, verbal, and spatial) were modeled using confirmatory factor analysis; patterns of relationships were similar for older and younger groups and for girls and boys. Spatial and quantitative factors were highly correlated; the verbal factor correlated weakly with the others but showed a stronger relationship with the spatial factor for boys than girls.

This study examined relations between children's conceptual understanding of mathematical equivalence and their procedures for solving equivalence problems (e.g., 3 + 4 + 5 = 3 + ___). Students in 4th and 5th grades completed assessments of their conceptual and procedural knowledge of equivalence, both before and after a brief lesson. The instruction focused either on the concept of equivalence or on a correct procedure for solving equivalence problems. Conceptual instruction led to increased conceptual understanding and to generation and transfer of a correct procedure. Procedural instruction led to increased conceptual understanding and to adoption, but only limited transfer, of the instructed procedure. These findings highlight the causal relations between conceptual and procedural knowledge and suggest that conceptual knowledge may have a greater influence on procedural knowledge than the reverse.