The latent growth model (LGM) is a popular tool in the social and behavioral sciences to study development processes of continuous and discrete outcome variables. A special case are frequency measurements of behaviors or events, such as doctor visits per month or crimes committed per year. Probability distributions for such outcomes include the…

In Bayesian statistics, the most widely used criteria of Bayesian model assessment and comparison are Deviance Information Criterion (DIC) and Watanabe-Akaike Information Criterion (WAIC). A multilevel mediation model is used as an illustrative example to compare different types of DIC and WAIC. More specifically, the study compares the…

Research shows the salient place of mathematical teaching talk, including the mathematical-linguistic practices of naming and explaining, in the enactment of students' mathematical talk and learning with understanding in the classroom. Our study was developed to examine the noticing of two groups of secondary-school mathematics teachers in one-day…

In a randomized controlled trial (RCT), some subjects assigned to the treatment condition may not fully comply. Often there is interest in the effect of the treatment within the "principal stratum" of subjects who would comply if assigned to treatment. However, it is unknown which control subjects would have complied if treated and which…

We consider here two questions posed by first-year undergraduate students in a statistics tutorial session. These questions are related, and each concerns an aspect of independence in probability. The notion of independence does tend to be a little less intuitive than some of the other elementary probabilistic concepts, thereby providing students…

Teaching quantum physics (QP) to high school (HS) students is gaining momentum, necessitating the exploration of various effective methods. Specifically, the research on quantitative teaching methods is still in its early stages. Understanding the power of Dirac notation (DN) in teaching is crucial for grasping the complexities of QP, as it…

We consider two-level models where a continuous response R and continuous covariates C are assumed missing at random. Inferences based on maximum likelihood or Bayes are routinely made by estimating their joint normal distribution from observed data R[subscript obs] and C[subscript obs]. However, if the model for R given C includes random…

Techniques for applying mathematical concepts in the real world: six rarely taught but crucial tools for analysis, research, and problem-solving. Many young graduates leave school with a solid knowledge of mathematical concepts but struggle to apply these concepts in practice. Real scientific and engineering problems are different from those found…

Several recent studies have employed item response theory (IRT) to rank incorrect responses to commonly used research-based multiple-choice assessments. These studies use Bock's nominal response model (NRM) for applying IRT to categorical (nondichotomous) data, but the response rankings only utilize half of the parameters estimated by the model.…

Most likely a large number of students have heard of the Northern Lights (Aurora Borealis), and know that they are a phenomenon in the sky. Although they may not have seen the Northern Lights, most have probably seen other phenomena in the sky, such as a rainbow, shooting star, or super moon. These experiences provide an opportunity for students…

Some IRT models can be equivalently modeled in alternative frameworks such as logistic regression. Logistic regression can also model time-to-event data, which concerns the probability of an event occurring over time. Using the relation between time-to-event models and logistic regression and the relation between logistic regression and IRT, this…

W scores are used in a number of commercially available tests. Due to their complex nature, it can be hard for applied researchers and practitioners to understand them or even acquire information about them beyond what is provided in technical manuals. In this article, we provide information regarding the background and derivation of W scores that…

The mathematics education literature shows that encouraging students to develop multiple solutions for given problems has a positive effect on students' understanding and creativity. In this paper, we present an example of multiple-solution problems in statistics involving a set of non-traditional dice. In particular, we consider the exact…

We tested a method for solving Bayesian reasoning problems in terms of spatial relations as opposed to mathematical equations. Participants completed Bayesian problems in which they were given a prior probability and two conditional probabilities and were asked to report the posterior odds. After a pretraining phase in which participants completed…

Data from published studies is used to recover information about the sample mean self-reported number of times cheated by college students. The sample means were estimated by fitting distributions to the reported data. The few estimated sample means thus recovered were roughly 2 or less.