We propose a new limited-information goodness of fit test statistic C[subscript 2] for ordinal IRT models. The construction of the new statistic lies formally between the M[subscript 2] statistic of Maydeu-Olivares and Joe (2006), which utilizes first and second order marginal probabilities, and the M*[subscript 2] statistic of Cai and Hansen…

We show that human observers strongly underestimate a linear or circular trajectory that a luminous spot follows in the dark. At slow speeds, observers are relatively accurate, but, as the speed increases, the size of the path is progressively underestimated, by up to 35%. The underestimation imposes little memory load and does not require…

Learning about statistics is a lot like learning about science: the learning is more meaningful if you can actively explore. This third installment of "Explorations in Statistics" investigates confidence intervals. A confidence interval is a range that we expect, with some level of confidence, to include the true value of a population parameter…

A question we posed in a recent final examination has uncovered a fundamental difficulty for students in understanding destructive interference. The problem stated that glass of index n[subscript 3] was coated with a thin film of a substance with index n[subscript 2]. The question then asked the student to calculate (a) the minimum coating…

A cubication procedure of the nonlinear differential equation for conservative nonlinear oscillators is analysed and discussed. This scheme is based on the Chebyshev series expansion of the restoring force, and this allows us to approximate the original nonlinear differential equation by a Duffing equation in which the coefficients for the linear…

Seven independent groups estimated the location of North American cities using both spatial and numeric response modes and a variety of perceptual and memory supports. These supports included having location markers for each city color coded by nation and identified by name, giving participants the opportunity to see and update all their estimates…

In the Basic Local Independence Model (BLIM) of Doignon and Falmagne ("Knowledge Spaces," Springer, Berlin, 1999), the probabilistic relationship between the latent knowledge states and the observable response patterns is established by the introduction of a pair of parameters for each of the problems: a lucky guess probability and a careless…

A method is presented for estimating reliability using structural equation modeling (SEM) that allows for nonlinearity between factors and item scores. Assuming the focus is on consistency of summed item scores, this method for estimating reliability is preferred to those based on linear SEM models and to the most commonly reported estimate of…

When first introduced to the analysis of algorithms, students are taught how to assess the best and worst cases, whereas the mean and amortized costs are considered advanced topics, usually saved for graduates. When presenting the latter, aggregate analysis is explained first because it is the most intuitive kind of amortized analysis, often…

Estimation of parameters of random effects models from samples collected via complex multistage designs is considered. One way to reduce estimation bias due to unequal probabilities of selection is to incorporate sampling weights. Many researchers have been proposed various weighting methods (Korn, & Graubard, 2003; Pfeffermann, Skinner,…

"Classroom Discussions: Seeing Math Discourse in Action, Grades K-6" provides preservice and inservice instructors, coaches and facilitators with real, classroom-based video examples that illustrate the principles and practices covered in the authors' best-selling book, "Classroom Discussions: Using Math Talk to Help Students Learn, Grade K-6,…

In 2008, the Computer and Information Science and Engineering Directorate of the National Science Foundation asked the National Research Council (NRC) to conduct two workshops to explore the nature of computational thinking and its cognitive and educational implications. The first workshop focused on the scope and nature of computational thinking…

This report introduces a short, hands-on activity that addresses a key challenge in teaching quantitative methods to students who lack confidence or experience with statistical analysis. Used near the beginning of the course, this activity helps students develop an intuitive insight regarding a number of abstract concepts which are key to…

The presented strategy of chemistry calculation is based on mole-concept, but it uses only one fundamental relationship of the amounts of substance as a basic panel. The name of LEGO-method comes from the famous toy of LEGO[R] because solving equations by grouping formulas is similar to that. The relations of mole and the molar amounts, as small…

Although it can be visualized fairly easily and its symmetry group is easy to calculate, the permutahedron is a somewhat neglected combinatorial object. We propose it as a useful case study in abstract algebra. It supplies concrete examples of group actions, the difference between right and left actions, and how geometry and algebra can work…