Drew H. Bailey

Fractions: the new frontier for theories of numerical development

Recent research on fractions has broadened and deepened theories of numerical development. Learning about fractions requires children to recognize that many properties of whole numbers are not true of numbers in general and also to recognize that the one property that unites all real numbers is that they possess magnitudes that can be ordered on number lines. The difficulty of attaining this understanding makes the acquisition of knowledge about fractions an important issue educationally, as well as theoretically. This article examines the neural underpinnings of fraction understanding, developmental and individual differences in that understanding, and interventions that improve the understanding. Accurate representation of fraction magnitudes emerges as crucial both to conceptual understanding of fractions and to fraction arithmetic.

Adolescents' functional numeracy is predicted by their school entry number system knowledge.

One in five adults in the United States is functionally innumerate; they do not possess the mathematical competencies needed for many modern jobs. We administered functional numeracy measures used in studies of young adults’ employability and wages to 180 thirteen-year-olds. The adolescents began the study in kindergarten and participated in multiple assessments of intelligence, working memory, mathematical cognition, achievement, and in-class attentive behavior. Their number system knowledge at the beginning of first grade was defined by measures that assessed knowledge of the systematic relations among Arabic numerals and skill at using this knowledge to solve arithmetic problems. Early number system knowledge predicted functional numeracy more than six years later (ß = 0.195, p = .0014) controlling for intelligence, working memory, in-class attentive behavior, mathematical achievement, demographic and other factors, but skill at using counting procedures to solve arithmetic problems did not. In all, we identified specific beginning of schooling numerical knowledge that contributes to individual differences in adolescents’ functional numeracy and demonstrated that performance on mathematical achievement tests underestimates the importance of this early knowledge.

Competence with fractions predicts gains in mathematics achievement.

Competence with fractions predicts later mathematics achievement, but the codevelopmental pattern between fractions knowledge and mathematics achievement is not well understood. We assessed this codevelopment through examination of the cross-lagged relation between a measure of conceptual knowledge of fractions and mathematics achievement in sixth and seventh grades (N=212). The cross-lagged effects indicated that performance on the sixth grade fractions concepts measure predicted 1-year gains in mathematics achievement (ß=.14, p<.01), controlling for the central executive component of working memory and intelligence, but sixth grade mathematics achievement did not predict gains on the fractions concepts measure (ß=.03, p>.50). In a follow-up assessment, we demonstrated that measures of fluency with computational fractions significantly predicted seventh grade mathematics achievement above and beyond the influence of fluency in computational whole number arithmetic, performance on number fluency and number line tasks, central executive span, and intelligence. Results provide empirical support for the hypothesis that competence with fractions underlies, in part, subsequent gains in mathematics achievement.

Fact Retrieval Deficits in Low Achieving Children and Children With Mathematical Learning Disability

Using 4 years of mathematics achievement scores, groups of typically achieving children (n = 101) and low achieving children with mild (LA–mild fact retrieval; n = 97) and severe (LA–severe fact retrieval; n = 18) fact retrieval deficits and mathematically learning disabled children (MLD; n = 15) were identified. Multilevel models contrasted developing retrieval competence from second to fourth grade with developing competence in executing arithmetic procedures, in fluency of processing quantities represented by Arabic numerals and sets of objects, and in representing quantity on a number line. The retrieval deficits of LA–severe fact retrieval children were at least as debilitating as those of the children with MLD and showed less across-grade improvement. The deficits were characterized by the retrieval of counting string associates while attempting to remember addition facts, suggesting poor inhibition of irrelevant information during the retrieval process. This suggests a very specific form of working memory deficit, one that is not captured by many typically used working memory tasks. Moreover, these deficits were not related to procedural competence or performance on the other mathematical tasks, nor were they related to verbal or nonverbal intelligence, reading ability, or speed of processing, nor would they be identifiable with standard untimed mathematics achievement tests.

Women's attractiveness changes with estradiol and progesterone across the ovulatory cycle

In many species, females are more sexually attractive to males near ovulation. Some evidence suggests a similar pattern in humans, but methodological limitations prohibit firm conclusions at present, and information on physiological mechanisms underlying any such pattern is lacking. In 202 normally-cycling women, we explored whether womens attractiveness changed over the cycle as a function of two likely candidates for mediating these changes: estradiol and progesterone. We scheduled women to attend one session during the late follicular phase and another during the mid-luteal phase. At each session, facial photographs, voice recordings and saliva samples were collected. All photographs and voice recordings were subsequently rated by men for attractiveness and by women for flirtatiousness and attractiveness to men. Saliva samples were assayed for estradiol and progesterone. We found that progesterone and its interaction with estradiol negatively predicted vocal attractiveness and overall (facial plus vocal) attractiveness to men. Progesterone also negatively predicted womens facial attractiveness to men and female-rated facial attractiveness, facial flirtatiousness and vocal attractiveness, but not female-rated vocal flirtatiousness. These results strongly suggest a pattern of increased attractiveness during peak fertility in the menstrual cycle and implicate estradiol and progesterone in driving these changes.

Salivary testosterone does not predict mental rotation performance in men or women

Multiple studies report relationships between circulating androgens and performance on sexually differentiated spatial cognitive tasks in human adults, yet other studies find no such relationships. Relatively small sample sizes are a likely source of some of these discrepancies. The present study thus tests for activational effects of testosterone (T) using a within-participants design by examining relationships between diurnal fluctuations in salivary T and performance on a male-biased spatial cognitive task (Mental Rotation Task) in the largest sample yet collected: 160 women and 177 men. T concentrations were unrelated to within-sex variation in mental rotation performance in both sexes. Further, between-session learning-related changes in performance were unrelated to T levels, and circadian changes in T were unrelated to changes in spatial performance in either sex. These results suggest that circulating T does not contribute substantially to sex differences in spatial ability in young men and women. By elimination, the contribution of androgens to sex differences in human performance on these tasks may be limited to earlier, organizational periods.

Persistence and Fadeout in the Impacts of Child and Adolescent Interventions

ABSTRACT Many interventions targeting cognitive skills or socioemotional skills and behaviors demonstrate initially promising but then quickly disappearing impacts. Our article seeks to identify the key features of interventions, as well as the characteristics and environments of the children and adolescents who participate in them, that can be expected to sustain persistently beneficial program impacts. We describe three such processes: skill-building, foot-in-the-door and sustaining environments. We argue that skill-building interventions should target “trifecta” skills—ones that are malleable, fundamental, and would not have developed eventually in the absence of the intervention. Successful foot-in-the-door interventions equip a child with the right skills or capacities at the right time to avoid imminent risks (e.g., grade failure or teen drinking) or seize emerging opportunities (e.g., entry into honors classes). The sustaining environments perspective views high quality of environments subsequent to the completion of the intervention as crucial for sustaining earlier skill gains. These three perspectives generate both complementary and competing hypotheses regarding the nature, timing, and targeting of interventions that generate enduring impacts.

Early predictors of middle school fraction knowledge

Recent findings that earlier fraction knowledge predicts later mathematics achievement raise the question of what predicts later fraction knowledge. Analyses of longitudinal data indicated that whole number magnitude knowledge in first grade predicted knowledge of fraction magnitudes in middle school, controlling for whole number arithmetic proficiency, domain general cognitive abilities, parental income and education, race, and gender. Similarly, knowledge of whole number arithmetic in first grade predicted knowledge of fraction arithmetic in middle school, controlling for whole number magnitude knowledge in first grade and the other control variables. In contrast, neither type of early whole number knowledge uniquely predicted middle school reading achievement. We discuss the implications of these findings for theories of numerical development and for improving mathematics learning.

State and Trait Effects on Individual Differences in Children’s Mathematical Development

Substantial longitudinal relations between children’s early mathematics achievement and their much later mathematics achievement are firmly established. These findings are seemingly at odds with studies showing that early educational interventions have diminishing effects on children’s mathematics achievement across time. We hypothesized that individual differences in children’s later mathematical knowledge are more an indicator of stable, underlying characteristics related to mathematics learning throughout development than of direct effects of early mathematical competency on later mathematical competency. We tested this hypothesis in two longitudinal data sets, by simultaneously modeling effects of latent traits (stable characteristics that influence learning across time) and states (e.g., prior knowledge) on children’s mathematics achievement over time. Latent trait effects on children’s mathematical development were substantially larger than state effects. Approximately 60% of the variance in trait mathematics achievement was accounted for by commonly used control variables, such as working memory, but residual trait effects remained larger than state effects. Implications for research and practice are discussed.

The codevelopment of skill at and preference for use of retrieval-based processes for solving addition problems: individual and sex differences from first to sixth grades.

The ability to retrieve basic arithmetic facts from long-term memory contributes to individual and perhaps sex differences in mathematics achievement. The current study tracked the codevelopment of preference for using retrieval over other strategies to solve single-digit addition problems, independent of accuracy, and skilled use of retrieval (i.e., accuracy and reaction time [RT]) from first to sixth grades inclusive (N=311). Accurate retrieval in first grade was related to working memory capacity and intelligence, and it predicted a preference for retrieval in second grade. In later grades, the relation between skill and preference changed such that preference in one grade predicted accuracy and RT in the next grade as RT and accuracy continued to predict future gains in preference. In comparison with girls, boys had a consistent preference for retrieval over other strategies and had faster retrieval speeds, but the sex difference in retrieval accuracy varied across grades. Results indicate that ability influences early skilled retrieval, but both practice and skill influence each other in a feedback loop later in development and provide insights into the source of the sex difference in problem-solving approaches.