The primary purpose of this study is to investigate the mathematical characteristics of the test reliability coefficient rho[subscript XX'] as a function of item response theory (IRT) parameters and present the lower and upper bounds of the coefficient. Another purpose is to examine relative performances of the IRT reliability statistics and two…

We describe a modelling activity for students in a course in which modelling with differential equations is appropriate. We have used this model in our coursework for years and have found that it enlightens students as to the model building process and parameter estimation for a linear, first-order, ordinary differential equation. The activity…

Regressions that control for confounding factors are the workhorse of evaluation research. When treatment effects are heterogeneous, however, the workhorse regression leads to estimated treatment effects that lack behavioral interpretations even when the selection on observables assumption holds. Regressions that use propensity scores as weights…

Based on the Lobatto quadrature, we develop several new closed form approximations to the mathematical constant "e." For validating effectiveness of our approximations, a comparison of our results to the existing approximations is also presented. Another objective of our work is to inspire students to formulate other better approximations by using…

Exploratory factor analysis (EFA) is commonly employed to evaluate the factor structure of measures with dichotomously scored items. Generally, only the estimated factor loadings are provided with no reference to significance tests, confidence intervals, and/or estimated factor loading standard errors. This simulation study assessed factor loading…

This study combined the kernel smoothing procedure and a nonparametric differential item functioning statistic--Cochran's Z--to statistically test the difference between the kernel-smoothed item response functions for reference and focal groups. Simulation studies were conducted to investigate the Type I error and power of the proposed…

Generalized structured component analysis (GSCA) is a component-based approach to structural equation modeling. In practice, researchers may often be interested in examining the interaction effects of latent variables. However, GSCA has been geared only for the specification and testing of the main effects of variables. Thus, an extension of GSCA…

This article describes a design activity that provides students with a solid understanding of the many issues involved with alternate energy system design. In this activity, students will be able to learn about electric vehicles and have the opportunity to design a way to recharge the batteries while the cars are parked in a commuter garage. The…

The Landau-Zener formula provides the probability of non-adiabatic transitions occurring when two energy levels are swept through an avoided crossing. The formula is derived here in a simple calculation that emphasizes the physics responsible for non-adiabatic population transfer. (Contains 2 figures.)

In the western half of the United States, fire bombers are not an uncommon sight. During the "fire season," which can extend from June through November, these specially modified aircraft are used to drop fire retardant chemicals or water on wildfires. It can be an entertaining and instructive classroom exercise to use pictures of these…

Connecting mathematical thinking to the natural world can be as simple as looking up to the sky. Volunteer bird watchers around the world help scientists gather data about bird populations. Counting flying birds might inspire new estimation methods, such as counting the number of birds per unit of time and then timing the whole flock's flight. In…

This document presents a review of evidence commissioned by the Education Endowment Foundation to inform the guidance document "Improving Mathematics in Key Stages Two and Three" (Education Endowment Foundation, 2017). The review draws on a substantial parallel study by the same research team, funded by the Nuffield Foundation, which…

This study explored the nature of mathematics anxiety in a sample of 106 ethnically and linguistically diverse first-grade students. Although much is known about mathematics anxiety in older children and adults, little is known about when mathematics anxiety first emerges or its characteristics in young children. Results from exploratory factor…

The results of an exploratory study into measurement of elementary mathematics ability are presented. The focus is on the abilities involved in solving standard computation problems on the one hand and problems presented in a realistic context on the other. The objectives were to assess to what extent these abilities are shared or distinct, and…

In a 9-month training experiment, 64 first graders with a risk factor were randomly assigned to computer-assisted structured discovery of the add-1 rule (e.g., the sum of 7 + 1 is the number after "seven" when we count), unstructured discovery learning of this regularity, or an active-control group. Planned contrasts revealed that the…