External number representations are commonly used throughout the first years of instruction. The twenty-frame is a grid that contains two rows of 10 dots each, and within each row, dots are organized in two groups of five. The assumption is that children can make use of these structures for enumerating the dots, rather than relying on one-by-one…

This paper contributes to the theory and evidence that mathematical cognition is embodied. Drawing on the practices of primary teachers in South Africa engaged in a longitudinal research and development project -- Wits Maths Connect--Primary -- we report on aspects of lessons aimed at developing number sense through whole-class teacher-learner…

This study examines the use of cross-classified random effects models (CCrem) and cross-classified multiple membership random effects models (CCMMrem) to model rater bias and estimate teacher effectiveness. Effect estimates are compared using CTT versus item response theory (IRT) scaling methods and three models (i.e., conventional multilevel…

Value-added data have become an increasingly common evaluation tool for schools and teachers. Many school districts have begun to adopt these methods and have released results publicly. In this paper, we use the unique public release of value-added data in Los Angeles to identify how this measure of school quality is capitalized into housing…

The conventional fixed-effects (FE) and random-effects (RE) confidence intervals that are used to assess the average alpha reliability across multiple studies have serious limitations. The FE method, which is based on a constant coefficient model, assumes equal reliability coefficients across studies and breaks down under minor violations of this…

A detailed derivation of analytic solutions is presented for overlap, kinetic, nuclear attraction and electron repulsion integrals involving Cartesian Gaussian-type orbitals. It is demonstrated how s-type orbitals can be used to evaluate integrals with higher angular momentum via the properties of Hermite polynomials and differentiation with…

The transmission through a stack of identical slabs that are separated by gaps with random widths is usually treated by calculating the average of the logarithm of the transmission probability. We show how to calculate the average of the transmission probability itself with the aid of a recurrence relation and derive analytical upper and lower…

Generating, collecting, and analyzing data is an essential practice in the science classroom (NRC 2012). Taking this data and using it to craft an explanation that demonstrates understanding of content is another essential practice. But both practices can be challenging, and students often require teacher support to succeed. Accordingly, the…

The article provides an overview of goodness-of-fit assessment methods for item response theory (IRT) models. It is now possible to obtain accurate "p"-values of the overall fit of the model if bivariate information statistics are used. Several alternative approaches are described. As the validity of inferences drawn on the fitted model…

We present a synthesis of findings from constructivist teaching experiments regarding six schemes children construct for reasoning multiplicatively and tasks to promote them. We provide a task-generating platform game, depictions of each scheme, and supporting tasks. Tasks must be distinguished from children's thinking, and learning situations…

The present study has two main interests. First, some pending issues about the psychometric properties of the CTAC (an anxiety questionnaire for blind and visually-impaired people) are assessed using item response theory (IRT). Second, the linear model is compared to the graded response model (GRM) in terms of measurement precision, sensitivity…

This study examined whether counting and rapid automatized naming (RAN) could operate as significant predictors of both later arithmetic calculation and reading fluency. The authors also took an important step to clarify the cognitive mechanisms underlying these predictive relationships by controlling for the effect of phonological awareness and…

This article attempts to introduce the reader to computational thinking and solving problems involving randomness. The main technique being employed is the Monte Carlo method, using the freely available software "R for Statistical Computing." The author illustrates the computer simulation approach by focusing on several problems of…

Children's learning of early number bonds is a significant issue in South African schools because evidence shows that flexible and efficient (fluent and reasoned) knowledge of number bonds to 20 correlates with success at the end of primary schooling, yet the evidence is that many South African students are over-reliant on inefficient counting…

The purposes of this study were to investigate (a) undergraduate students' reasoning about the concepts of confidence intervals (b) undergraduate students' interactions with "well-designed" screencast videos on sampling distributions and confidence intervals, and (c) how screencast videos improve undergraduate students' reasoning ability…