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Objectives: A large body of research exists on families and their impact on children's developmental outcomes. Fewer studies examine family processes within different family structures and their impact on adolescents. This study extends prior work by evaluating associations between family structure (two biological parents, single-parent—mother only, and blended families with a step parent), family functioning (communication, cohesion and flexibility) and adolescents' logic and reasoning skills and math performance. Methods: A convenience sample of 269 African American adolescents was recruited from a university-based summer enrichment program. The sample was mostly female (63%), with a mean age of 16.34 years. Data were derived from the baseline assessment during which participants completed the Family Adaptability and Cohesion Evaluation Scales (FACES IV), a computer-administered test of cognitive skills, and a paper-based math performance test. Results: Families with different structures varied in their family functioning (Wilks' ʌ = 0.942, F(6, 496) = 2.486, p = 0.022, η2 = 0.835). Families differed on the dimensions of cohesion (F(2, 250) = 5.80, p = 0.003, η2 = 0.044), flexibility (F(2, 250) = 5.48, p = 0.005, η2 = 0.042) and communication (F(2, 250) = 5.80, p = 0.003, η2 = 0.047). Different dimensions of family functioning significantly predicted both logic and reasoning skills (F(7, 244) = 2.14, p = 0.040, R2 = 0.06) and math outcomes (F(7, 142) = 3.16, p = 0.004, R2 = 0.14). Conclusions: Families with different household structures differ in their functioning, including in their levels of cohesion, flexibility and communication. Family functioning is more important than family structure in predicting African American adolescents' reasoning skills and math performance. [ABSTRACT FROM AUTHOR]

In a recent paper Engle et al. (Commun Math Phys 354:231–246, 2017) showed that there is a unique state on the reduced holonomy–flux ∗ -algebra of homogeneous isotropic loop quantum cosmology, that is invariant under residual diffeomorphims. This result has been claimed to be true both for the Ashtekar–Bojowald–Lewandowski framework and for that introduced by the present author. Unfortunately, the uniqueness proof relies on an incorrect argument which spoils the second case. In our short note, we are going to patch this issue, this way keeping the nice uniqueness result in both cases. Moreover, we will even extend the underlying operator algebraic statements as this might help later for studying higher-dimensional models. [ABSTRACT FROM AUTHOR]

This paper offers for discussion and critique a conceptual framework that applies a situative perspective on learning to the study of learning to teach mathematics. From this perspective, such learning occurs in many different situations -- mathematics and teacher preparation courses, pre-service field experiences, and schools of employment. By participating over time in these varied contexts, mathematics teachers refine their conceptions about their craft -- the big ideas of mathematics, mathematics-specific pedagogy, and sense of self as a mathematics teacher. This framework guides a research project that traces the learning trajectories of teachers from two reform-based teacher preparation programs into their early teaching careers. We provide two examples from this research to illustrate how this framework has helped us understand the process of learning to teach. [ABSTRACT FROM AUTHOR]