Mental representations of numerical magnitude are commonly thought to undergo discontinuous change over development in the form of a "representational shift." This idea stems from an apparent categorical shift from logarithmic to linear patterns of numerical estimation on tasks that involve translating between numerical magnitudes and spatial positions (such as number-line estimation). However, the observed patterns of performance are broadly consistent with a fundamentally different view, based on psychophysical modeling of proportion estimation, that explains the data without appealing to discontinuous change in mental representations of numerical magnitude. The present study assessed these 2 theories' abilities to account for the development of numerical estimation in 5- through 10-year-olds. The proportional account explained estimation patterns better than the logarithmic-to-linear-shift account for all age groups, at both group and individual levels. These findings contribute to our understanding of the nature and development of the mental representation of number and have more general implications for theories of cognitive developmental change.

Trust is an important dimension of parent educational involvement and parent-teacher relationships. Preliminary research suggests that parent trust in teachers and schools is associated with student learning and behavior. However, examinations of parent trust in children's education are limited. The present study investigated the influence of demographic variables on parent trust and relations among parent trust, student behavior, and parent involvement. Findings suggested that student eligibility for a free or reduced-price lunch predicted parent trust of teachers and schools, parent trust was correlated with dimensions of student behavior, and parent trust of teachers predicted parent involvement. Implications for parent involvement and parent-teacher relationships are discussed.

Family educational involvement and parent-teacher relationships are important for supporting student outcomes and have unique implications for families of children with autism spectrum disorder (ASD). However, little research has examined child and family characteristics among families of children with ASD as predictors of family involvement and parent-teacher relationships. The present study examined child and family variables that may affect family involvement and parent-teacher relationships for families of children with ASD. Findings suggested (a) parents of children with higher developmental risk reported less family involvement and poorer relationships with their child's teacher and (b) family histories accessing services predicted family involvement and parent-teacher relationships. Limitations of the current study and implications for science and practice are discussed.

Family-school partnerships are crucial for promoting positive outcomes and serving as a protective factor for children at-risk for poor school outcomes (Christenson & Sheridan, 2001). This may be particularly important for autistic children, who are at increased developmental risk . However, little research has examined variables related to dimensions of these partnerships for parents and teachers of autistic children. The present study examined family socioeconomic resources in relation to two dimensions of family-school partnerships (relationship quality and family involvement) among parents of autistic children, as well as dyadic perceptions of relationship quality among parents and teachers of autistic children. Data were collected across two time points (Time 1 N = 68 parents, child ages = 5-11; Time 2 parent-teacher dyad N = 22, child ages = 7-13). Results suggest that (a) higher appraisal of financial resources was associated with higher parent-reported family involvement after controlling for child autism spectrum disorder (ASD) characteristics and (b) higher parent-reported relationship quality and family involvement at one time were significantly associated with positive independent (parent and teacher) and congruent ratings of parent-teacher relationship quality 2 years later. Study limitations, future directions, and practical implications are discussed. Impact and ImplicationsThis study explored family-school partnerships among parents and teachers of autistic children. Findings suggest that (a) parents' perceptions of parent-teacher relationship quality at one time point were related to independent and congruent parent-teacher perceptions of their relationship about 2 years later and (b) parents' appraisals of lower family financial resources were associated with lower educational involvement. Findings highlight the importance of understanding families' perceptions of parent-teacher relationship quality and emphasizing these relationships in early years.

The purpose of this study was to examine parent-teacher congruent perceptions of middle school structures to support parent monitoring of children's behavior. We examined family educational involvement and school's communication and support to families as predictors of congruence, as well as congruence as a predictor of children following school rules and expectations. Participants were 415 teachers and 5003 parents across 40 public middle schools in the northwest region of the United States. Mixed-effects regression was used to examine the cross-sectional data. School size and student eligibility of free or reduced price school lunch were included as school-level covariates. Results suggested that higher levels of teacher-reported family involvement and school communication and support to families were significantly associated with greater parent-teacher congruence. In addition, results suggested that greater parent-teacher congruence was significantly associated with higher levels of teacher-report of children following school rules and expectations. Implications for research on parent-teacher congruence and family-school collaboration are discussed at the school system level in terms of parent monitoring of children's behavior. Practical applications are reviewed for public middle schools.