The purpose of this paper is to propose a revision of the well-known Campbellian system for causal research. The revised system, termed the COPS model, applies to both applied and basic research. Five validities are included, where two validities are adopted from the Campbellian system, and the validities are partly hierarchically ordered.…

This article presents the consensus of an National Council on Measurement in Education Presidential Task Force on Foundational Competencies in Educational Measurement. Foundational competencies are those that support future development of additional professional and disciplinary competencies. The authors develop a framework for foundational…

Researchers need to know what is an appropriate sample size for interview work, but how does one decide upon an acceptable number of people to interview? This question is not relevant to case study work where one would typically interview every member of a case, or in situations where it is both desirable and feasible to interview all target…

Cross-validation is a common statistical procedure applied to problems that are otherwise computationally intractable. It is often employed to assess the effectiveness of prediction procedures. In this report, cross-validation is discussed in terms of "U"-statistics. This approach permits consideration of the statistical properties of…

This paper proposes the replication of Bird's (2010) and Serrano's (2011) studies, which examined distribution of practice effects in second language acquisition (SLA). These studies, which took place in authentic classroom contexts, produced conflicting results regarding the degree to which the learning of a second language (L2) benefited from…

In this note, we propose a generalization of the famous Bernoulli differential equation by introducing a class of nonlinear first-order ordinary differential equations (ODEs). We provide a family of solutions for this introduced class of ODEs and also we present some examples in order to illustrate the applications of our result.

We perform dynamic investigation of two surprising geometrical properties, each of which involves additional properties. In the first task the property belongs to two regular polygons with the same number of sides and with one common vertex. It is found that all the straight lines that connect corresponding vertices of the two polygons intersect…

This paper discusses an interesting, classic problem that provides a nice classroom investigation for dynamic geometry, and which can easily be explained (proved) with transformation geometry. The deductive explanation (proof) provides insight into why it is true, leading to an immediate generalization, thus illustrating the discovery function of…

We present a model for describing the growth of students' understandings when reading a proof. The model is composed of two main paths. One is focused on becoming aware of the deductive structure of the proof, in other words, understanding the proof at a semantic level. Generalization, abstraction, and formalization are the most important…

Despite the call for an argument-based approach to validity over 25 years ago, few examples exist in the published literature. One possible explanation for this outcome is that the complexity of the argument-based approach makes implementation difficult. To counter this claim, we propose that the Assessment Triangle can serve as the overarching…

Experimental studies are often employed to test the effectiveness of teaching innovations such as new pedagogy, curriculum, or learning resources. This article offers guidance on good practice in developing research designs, and in drawing conclusions from published reports. Random control trials potentially support the use of statistical…

This document introduces, describes and exemplifies the technical features of some recently implemented automated reasoning tools in the dynamic mathematics software GeoGebra. The new tools are based on symbolic computation algorithms, allowing the automatic and rigorous proving and discovery of theorems on constructed geometric figures. Some…

A proof is a connected sequence of assertions that includes a set of accepted statements, forms of reasoning and modes of representing arguments. Assuming reasoning to be central to proving and aiming to develop knowledge about how teacher actions may promote students' mathematical reasoning, we conduct design research where whole-class…

A well-known result of Feinberg and Shannon states that the tribonacci sequence can be detected by the so-called "Pascal's pyramid." Here we will show that any tribonacci-like sequence can be obtained by the diagonals of the "Feinberg's triangle" associated to a suitable "generalized Pascal's pyramid."…

In empirical applied linguistics research it is essential that the key variables are operationalized in a valid and reliable way, and that the scores are treated appropriately, allowing for a proper testing of the hypotheses under investigation. The current article addresses several theoretical and practical issues regarding the use of measurement…