A simple experiment is described to compare the descent time between two vertically separated points when an object slides down tracks of varying shape. A surprising result is that the descent time is shortest when it follows a circular track rather than a cycloidal track. Cycloidal tracks are usually predicted to result in the shortest descent…

Currently the design standards for single case experimental designs (SCEDs) are based on validity considerations as prescribed by the What Works Clearinghouse. However, there is a need for design considerations such as power based on statistical analyses. We compute and derive power using computations for (AB)[superscript k] designs with multiple…

Background/Context: Attrition has been described as "the Achilles Heel of the randomized experiment" (Shadish et al., 1998 p.3). Attrition looms as a threat because it can undermine group equivalence, eroding the methodological strength at the heart of a randomized evaluation. In particular, attrition could result in unobserved…

Counting is an intellectual activity that is usually identified with the assignment of positive whole numbers ("1, 2, 3, 4, ...") in one-to-one correspondence with a collection of items in one's attentional field. Taking into account a broader, "combinatorial" meaning of the term, counting extends far beyond this description.…

Once research questions are posed, researchers must answer many a priori questions regarding research design before analysis can be performed and any conclusions can be made, including sample selection criteria, data collection method, model specification, analysis and estimation technique. The choices made by researchers along this forking path…

The purpose of the study is to systematically review studies investigating the script fading procedure on individuals with autism and also determine the quality of the studies and the effect size related to the calculations of the percentage of nonoverlapping data (PND) and the percentage of data exceeding the median (PEM) in the studies. The…

Traditional physical chemistry conceptions of reaction mechanism are formulated in terms of stationary points of an Arrhenius-style "energy profile" that differs sharply (in purpose and form) from the corresponding Robinson-style "arrow-pushing" mechanistic conceptions of organic chemistry. We show here how these diverse…

In this article, we present a two-stage estimation approach applied to multilevel latent class analysis (LCA) with covariates. We separate the estimation of the measurement and structural model. This makes the extension of the structural model computationally efficient. We investigate the robustness against misspecifications of the proposed…

In this paper, we share the results of the piloting of national diagnostic assessments for strategic calculation with Grade 3 learners in South Africa. The diagnostic assessment pilot intervention was focused on promoting strategic use of calculation strategies and aimed to move learners on from the concrete one-to-one counting methods that…

In this paper, we first focus on the sum of powers of the first n positive odd integers, T[subscript k](n)=1[superscript k]+3[superscript k]+5[superscript k]+...+(2n-1)[superscript k], and derive in an elementary way a polynomial formula for T[subscript k](n) in terms of a specific type of generalized Stirling numbers. Then we consider the sum of…

Physical activity (PA) promotion is an ideal intervention target for public health because it has the potential to help individuals feel better, sleep better, and perform daily tasks more easily, in addition to providing disease prevention benefits. There is strong evidence that individual-level theory-based behavioral interventions are effective…

There is growing interest in mathematics learning progressions in early childhood education. Counting is a skill usually developed early in life. The application of the counting principles in early childhood typically entails counting objects. This poses challenges for learning about zero. Indeed, the word "zero" is seldom used in the…

This study examines the feasibility and potential advantages of using large language models, in particular GPT-4o, to perform partial credit grading of large numbers of student written responses to introductory level physics problems. Students were instructed to write down verbal explanations of their reasoning process when solving one conceptual…

Investigating students' thinking in classroom tasks, particularly in science and engineering, is essential for improving educational practices and advancing student learning. In this context, the notion of "Ways of Thinking" (WoT) has gained traction in STEM education, offering a framework to explore how students approach and solve…

The introduction of programming education in K-12 schools to promote computational thinking has attracted a great deal of attention from scholars and educators. Debugging code is a central skill for students, but is also a considerable challenge when learning to program. Learners at the K-12 level often lack confidence in programming debugging due…