From whole numbers to fractions to word problems: Hierarchical relations in mathematics knowledge for Chinese Grade 6 students.

It is well established in the literature that fraction knowledge is important for learning more advanced mathematics, but the hierarchical relations among whole number arithmetic, fraction knowledge, and mathematics word problem-solving are not well understood. In the current study, Chinese Grade 6 students (N = 1160; 465 girls; Mage = 12.1 years, SD = 0.6) completed whole number arithmetic (addition, subtraction, multiplication, and division), fraction (mapping, equivalence, comparison, and arithmetic), and mathematics word problem-solving assessments. They also completed two control measures: number writing speed and nonverbal intelligence. Structural equation modeling was used to investigate the hierarchical relations among these assessments. Among the four fraction tasks, the correlations were low to moderate, suggesting that each task may tap into a unique aspect of fraction understanding. In the model, whole number arithmetic was directly related to all four fraction tasks, but was only indirectly related to mathematics word problem-solving, through fraction arithmetic. Only fraction arithmetic, the most advanced fraction skill, directly predicted mathematics word problem-solving. These findings are consistent with the view that students need to build these associations into their mathematics hierarchy to advance their mathematical competence.

Longitudinal assessments of functional near-infrared spectroscopy background functional connectivity in low- and middle-income infants during a social cognition task.

Shortly after birth, human infants demonstrate behavioral selectivity to social stimuli. However, the neural underpinnings of this selectivity are largely unknown. Here, we examine patterns of functional connectivity to determine how regions of the brain interact while processing social stimuli and how these interactions change during the first 2 years of life. Using functional near-infrared spectroscopy (fNIRS), we measured functional connectivity at 6 (n = 147) and 24 (n = 111) months of age in infants from Bangladesh who were exposed to varying levels of environmental adversity (i.e., low- and middle-income cohorts). We employed a background functional connectivity approach that regresses out the effects of stimulus-specific univariate responses that are believed to affect functional connectivity. At 6 months, the two cohorts had similar fNIRS patterns, with moderate connectivity estimates for regions within and between hemispheres. At 24 months, the patterns diverged for the two cohorts. Global (brain-wide) connectivity estimates increased from 6 to 24 months for the low-income cohort and decreased for the middle-income (MI) cohort. In particular, connectivity estimates among regions of interest within the right hemisphere decreased for the MI cohort, providing evidence of neural specialization by 2 years of age. These findings provide insights into the impact of early environmental influences on functional brain development relevant to the processing of social stimuli. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

The hierarchical relations among mathematical competencies: From fundamental numeracy to complex mathematical skills.

Mathematical competencies can be conceptualized as layers of knowledge, with numeracy skills as the foundational core and more complex mathematical skills as the additional layers over the core. In this study, we tested an expanded hierarchical symbol integration (HSI) model by examining the hierarchical relations among mathematical skills. Undergraduate students (N = 236) completed order judgement, simple arithmetic, fraction arithmetic, algebra, and verbal working memory tasks. In a series of hierarchical multiple regressions, we found support for the hierarchical model: Additive skills (i.e., addition and subtraction) predicted unique variance in multiplicative skills (i.e., multiplication and division); multiplicative skills predicted unique variance in fraction arithmetic; and fraction skills predicted unique variance in algebra. These results support the framework of the HSI model in which mathematical competencies are related hierarchically, capturing the increasing complexity of symbolic mathematical skills. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Walking another pathway: The inclusion of patterning in the pathways to mathematics model.

According to the Pathways to Mathematics model [LeFevre et al. (2010), Child Development, Vol. 81, pp. 1753-1767], children's cognitive skills in three domains-linguistic, attentional, and quantitative-predict concurrent and future mathematics achievement. We extended this model to include an additional cognitive skill, patterning, as measured by a non-numeric repeating patterning task. Chilean children who attended schools of low or high socioeconomic status (N = 98; 54% girls) completed cognitive measures in kindergarten (Mage = 71 months) and numeracy and mathematics outcomes 1 year later in Grade 1. Patterning and the original three pathways were correlated with the outcomes. Using Bayesian regressions, after including the original pathways and mother's education, we found that patterning skills predicted additional variability in applied problem solving and arithmetic fluency, but not number ordering, in Grade 1. Similarly, patterning skills were included in the best model for applied problem solving and arithmetic fluency, but not for number ordering, in Grade 1. In accord with the hypotheses of the original Pathways to Mathematics model, patterning varied in its unique and relative contributions to later mathematical performance, depending on the demands of the tasks. We conclude that patterning is a useful addition to the Pathways to Mathematics model, providing further insights into the range of cognitive precursors that are related to children's mathematical development.

Paths to postsecondary education enrollment among adolescents with and without childhood attention-deficit/hyperactivity disorder (ADHD): A longitudinal analysis of symptom and academic trajectories.

We examined developmental trajectories of attention-deficit/hyperactivity disorder (ADHD) symptoms, standardized achievement, and school performance for adolescents with and without ADHD who did and did not enroll in postsecondary education (PSE; N = 749; 79% boys; 63% White, 17% non-Hispanic Black, 10% Hispanic, and 10% other ethnicities). In a multisite study (recruitment based in New York, North Carolina, Pennsylvania, California, and Quebec), participants were originally enrolled between 1994 and 1998 at ages 7 to 9.9 and followed up through 2012 (Mage = 25 at final follow-up). Adolescents who eventually enrolled in PSE had less severe symptoms, but differences were modest and trajectories were similar over time. For all adolescents, standardized achievement trajectories declined up to two thirds of a standard deviation from ages 9 to 17. By the end of high school, the average GPA of adolescents with ADHD was three quarters of a point higher for those who eventually enrolled in PSE compared to those who did not. Overall, school performance mattered more than academic achievement for understanding eventual enrollment of adolescents with ADHD.

Fraction mapping and fraction comparison skills among grade 4 Chinese students: An error analysis.

BACKGROUND: Mapping fraction symbols to magnitudes is easier for students to master than comparing fraction magnitudes. Fraction mapping assesses students' understanding of part-whole interpretations of fractions; fractions represent the parts of an object or set of objects. Fraction comparison assesses students' understanding of measurement interpretations of fractions; a fraction is a single numerical quantity, not a combination of two whole numbers. AIM: To examine and compare the types of errors made by emergent fraction learners on fraction mapping and comparison tasks. SAMPLE: Grade 4 Chinese students (N = 1036; 577 boys; Mage  = 9.9 years). METHOD: We examined performance and identified errors on fraction mapping and comparison tasks. For mapping, students converted pictorial representations into fraction notation. For comparison, they chose the larger of two symbolic fractions. RESULTS: Consistent with curriculum expectations, most students successfully mapped pictorial representations to fraction notation. In contrast, few students were able to accurately compare fraction magnitudes. Within each task, students' errors were consistent across trials, suggesting that they applied systematic but incorrect procedures. However, errors were not consistent between tasks and the correlation between mapping and comparison performance was weak. CONCLUSION: Emergent fraction learners can acquire part-whole knowledge of fractions without acquiring measurement interpretations of fractions. Moreover, misconceptions about different interpretations of fractions need not overlap. Awareness of the types of errors that students make can assist educators in identifying misconceptions early so that students do not build their fraction knowledge on erroneous beliefs.

Divide and conquer: Relations among arithmetic operations and emerging knowledge of fraction notation for Chinese students in Grade 4.

How do whole number arithmetic skills support students' understanding of fraction magnitude during the emerging stages of fraction learning? Chinese students in Grade 4 (N = 1038; Mage = 9.9 years; 55.6% boys) completed assessments of whole number arithmetic skills (i.e., addition, subtraction, multiplication, and division), fraction mapping (i.e., connecting visual fraction representations to fraction notations), and fraction comparison (i.e., comparing magnitudes of fraction symbols). We found that division skills uniquely differentiated students who had a basic understanding of fraction notation (mappers) from students with no understanding of fraction notation (non-mappers). Furthermore, we found that division mediated the relations between all three other arithmetic operations (i.e., addition, subtraction, and multiplication) and fraction mapping performance for the mappers. For fraction comparison, there was evidence of the whole number bias for the majority of students. The current results highlight the importance of the mastery of division skills and its dominance in predicting individual differences in fraction mapping for Chinese students in Grade 4.

Individual differences in the development of children's arithmetic fluency from grades 2 to 3.

In the present research, we provide empirical evidence for the process of symbolic integration of number associations, focusing on the development of simple addition (e.g., 5 + 3 = 8), subtraction (e.g., 5 - 3 = 2), and multiplication (e.g., 5 × 3 = 15). Canadian children were assessed twice, in Grade 2 and Grade 3 (N = 244; 55% girls). All families were English-speaking, and parent education levels ranged from high school to postgraduate, with a median of community college. In Grade 2, children completed general cognitive tasks (i.e., receptive vocabulary, working memory, nonverbal reasoning, and inhibitory control). In both grades, children completed single-digit addition and complementary subtraction problems. In Grade 3, they completed single-digit multiplication problems and measures of applied mathematics, specifically, word-problem solving, algebra, and measurement. We found that addition and subtraction were reciprocally related (controlling for cognitive skills). Subtraction fluency predicted multiplication in Grade 3, whereas addition fluency did not. In Grade 3, both subtraction and multiplication fluency were predictors of applied mathematics, with multiplication partially mediating the relation between subtraction and applied mathematics performance. These findings support the view that learning arithmetic associations is a hierarchical process. As students practice each new skill, individual differences reflect the integration of the novel component into the developing associative network. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

The subject matters: relations among types of anxiety, ADHD symptoms, math performance, and literacy performance.

In this study, we extended a model of the relations among ADHD symptoms, anxiety, and academic performance. Undergraduates (N = 515) completed self-report measures of anxiety (i.e. trait, test, math, and literacy) and ADHD symptoms. During the study, they completed math and literacy measures and reported their current (i.e. state) anxiety three times. There were significant correlations among ADHD symptoms and all measures of anxiety. However, neither ADHD symptoms nor trait anxiety were correlated with math or literacy performance. Test-anxious students reported increasing state anxiety as the study progressed. In contrast, math-anxious students reported increased state anxiety following the math tasks but not the literacy tasks and literacy-anxious students reported increased state anxiety following the literacy tasks but not the math tasks. With respect to performance, math-anxious students had worse math performance and literacy-anxious students had worse literacy performance whereas test anxiety was not directly related to performance on either math or literacy tasks. Patterns of relations did not differ for men and women, but, except for literacy anxiety, women reported higher levels of all types of anxiety. These results could help educators understand how affective responses are related to university students' academic performance.

Fixated in more familiar territory: Providing an explicit midpoint for typical and atypical number lines.

Does providing an explicit midpoint affect adults' performance differently for typical and atypical number line tasks? Participants (N = 29) estimated the location of target numbers on typical (i.e., 0-10,000) and atypical (i.e., 0-7,000) number lines with either an explicitly labelled midpoint or no midpoint. For the typical number line, estimation accuracy did not differ for the explicit- and implicit-midpoint conditions. For the atypical number line, participants in the explicit-midpoint condition were more accurate than those in the implicit-midpoint condition and their pattern of error was similar to that seen for typical number lines (i.e., M-shaped). In contrast, for participants in the implicit-midpoint condition, the pattern of error on the atypical line was tent-shaped, with less accurate estimates around the midpoint and quartiles than the endpoints. Eye-tracking data showed that, for all number lines, participants used the middle of the line to guide their estimates, but participants in the explicit-midpoint condition were more likely to make their first fixation around the true midpoint than those in the implicit-midpoint condition. We conclude that adults have difficulty in estimating on atypical number lines because they incorrectly calculate the numerical value of the midpoint.

Children's Home Numeracy Environment Predicts Growth of their Early Mathematical Skills in Kindergarten.

This study investigated the longitudinal associations between children's early mathematics and their home numeracy environment (HNE). Chilean children from families who varied widely in socioeconomic status were assessed at the beginning and end of prekindergarten in 2016 (N = 419, Mage  = 4:7 [years:months]), and at the end of kindergarten in 2017 (N = 368, Mage  = 5:10). Children whose parents provided frequent operational numeracy activities (e.g., learning simple sums) at prekindergarten showed better arithmetic performance and growth in nonsymbolic and symbolic number comparison at the end of kindergarten. Parents' knowledge of number-related games predicted children's arithmetic skills and growth in nonsymbolic number comparison. These findings underscore the persistent relations between the HNE and the development of children's mathematical skills.