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Prior research has shown that the home learning environment (HLE) is critical in the development of spatial skills and that various parental beliefs influence the HLE. However, a comprehensive analysis of the impact of different parental beliefs on the spatial HLE remains lacking, leaving unanswered questions about which specific parental beliefs are most influential and whether inducing a growth mindset can enhance the spatial HLE. To address these gaps, we conducted an online study with parents of 3- to 5-year-olds. We found that parents’ growth mindset about their children’s ability strongly predicted the spatial HLE after controlling for parents’ motivational beliefs about their children, beliefs about their own ability, children’s age, children’s gender, and family SES. Further, reading an article about growth mindset led parents to choose more challenging spatial learning activities for their children. These findings highlight the critical role of parents’ growth mindset in the spatial HLE. Crucially, these findings demonstrate that general growth mindset messages without specific suggestions for parental practices can influence parental behavior intentions. Further, these effects were also observed in the control domain of literacy, underscoring the broad relevance of the growth mindset in the HLE. 

Verbal labels for math concepts influence multiple aspects of math learning. In this study, we examined the influence of point labels (e.g., .42 as “point four two”), decomposed labels (e.g., “four tenths and two hundredths”), and common-unit labels (e.g., “forty-two hundredths”) on children’s processing and representation of decimal magnitudes. We randomly assigned 162 5th- and 6th-graders to briefly learn decomposed, common-unit, or point labels. Children then completed measures of decimal magnitude processing and representation. We found that the place-value labels (i.e., decomposed and common-unit labels) each showed unique advantages in reducing the whole-number bias, and common-unit labels also reduced componential processing. No difference was found in the ratio effect – which served as an index of the precision of decimal magnitude representation - among children from the three conditions. These findings add to our understanding of the role of verbal labels in math learning and have important implications for instructional practices. 

In this paper, we develop parameter-robust numerical algorithms for Biot model and apply the algorithms in brain edema simulations. By introducing an intermediate variable, we derive a multiphysics reformulation of the Biot model. Based on the reformulation, the Biot model is viewed as a generalized Stokes subproblem combining with a reaction–diffusion subproblem. Solving the two subproblems together or separately leads to a coupled or a decoupled algorithm. We conduct extensive numerical experiments to show that the two algorithms are robust with respect to the key physical parameters. The algorithms are applied to study the brain swelling caused by abnormal accumulation of cerebrospinal fluid in injured areas. The effects of the key physical parameters on brain swelling are carefully investigated. It is observed that the permeability has the biggest influence on intracranial pressure (ICP) and tissue deformation; the Young’s modulus and the Poisson ratio do not affect the maximum value of ICP too much but have big influence on the tissue deformation and the developing speed of brain swelling. 

Children's failure to reason often leads to their mathematical performance being shaped by spurious associations from problem input and overgeneralization of inapplicable procedures rather than by whether answers and procedures make sense. In particular, imbalanced distributions of problems, particularly in textbooks, lead children to create spurious associations between arithmetic operations and the numbers they combine; when conceptual knowledge is absent, these spurious associations contribute to the implausible answers, flawed strategies, and violations of principles characteristic of children's mathematics in many areas. To illustrate mechanisms that create flawed strategies in some areas but not others, we contrast computer simulations of fraction and whole number arithmetic. Most of their mechanisms are similar, but the model of fraction arithmetic lacks conceptual knowledge that precludes strategies that violate basic mathematical principles. Presentingbalanced problem distributions and inculcating conceptual knowledge for distinguishing flawed from legitimate strategies are promising means for improving children's learning.