The idea of using mathematical research in the classroom to collect real data has been spoken about within education for many years. The question is, why should teachers bother with real data from their students' worlds and how do they actually put it into practice in the classroom? When using the mathematical research strategy for collecting,…

The purpose of this study is to determine the latent structure of the California Verbal Learning Test-Second Edition (CVLT-II; Delis, Kramer, Kaplan, & Ober, 2000) at three different age levels, using the standardization sample. Maximum likelihood confirmatory factor analyses are performed to test four competing hypothetical models for fit and…

Math-education reformers encourage the incorporation of mathematical modeling activities into K-12 curricula. Many of the purported educational benefits derive from the authenticity of the activities-how well they reflect the everyday and occupational mathematical practices of adults. But a paucity in the literature of observational descriptions…

This paper deals with a prey-predator model incorporating a prey refuge and harvesting of the predator species. A mathematical analysis shows that prey refuge plays a crucial role for the survival of the species and that the harvesting effort on the predator may be used as a control to prevent the cyclic behaviour of the system. The optimal…

The existing vector formalism method for thermodynamic relationship maintains tractability and uses accessible mathematics, which can be seen as a diverting and entertaining step into the mathematical formalism of thermodynamics and as an elementary application of matrix algebra. The method is based on ideas and operations apt to improve the…

This paper explores the public awareness that there presently exists a crisis in mathematics education and a "dumbing down" of the curriculum, examines the nature of this crisis and argues that there has been a lowering of cultural, pedagogical and cognitive expectations with respect to most learners. The notion of cognitive development in…

The goal of this paper is to demonstrate the infrastructural nature of many modern conceptual technologies. The focus of this paper is on conceptual tools associated with elementary types of data modeling. We intend to show a variety of ways in which these conceptual tools not only express thinking, but also mold and shape thinking. And those ways…

In current curriculum materials for middle school students in the US, data and chance are considered as separate topics. They are then ideally brought together in the minds of high school or university students when they learn about statistical inference. In recent studies we have been attempting to build connections between data and chance in the…

This paper provides a microanalysis of one Algebra I teacher's instruction to explore the advantages that are afforded us by coordinating two perspectives to document and account for the teacher's mathematical understandings. We use constructs associated with Stein, Grover and Henningsen's domain of mathematical didactics and Realistic Mathematics…

The ability to apply existing knowledge in new situations and settings is clearly a vital skill that all students need to develop. Nowhere is this truer than in the rapidly developing world of Web-based learning, which is characterized by non-sequential courses and the absence of an effective cross-subject guidance system. As a result, questions…

The partial credit model (PCM) (G. N. Masters, 1982) can be viewed as a generalization of the Rasch model for dichotomous items to the case of polytomous items. In many cases, the PCM is too restrictive to fit the data. Several generalizations of the PCM have been proposed. In this paper, a generalization of the PCM (GPCM), a further…

This discussion of new methods for calibrating item response theory (IRT) models looks into new optimization procedures, such as the Genetic Algorithm (GA) to improve on the use of the Newton-Raphson procedure. The advantages of using a global optimization procedure like GA is that this kind of procedure is not easily affected by local optima and…

In the technique developed by K. G. Joreskog to solve the problem for oblique rotation to a specified simple structure, the basic concept is that the simple structure solution itself is determined only by the zero coefficients of the reference-structure matrix and not by the coefficients of non-zero magnitude. Following this, prior information…

Cross-validation, an economical method for assessing whether sample results will generalize, is discussed in this paper. Cross-validation is an invariance technique that uses two subsets of the data sample to derive discriminant function coefficients. The two sets of coefficients are then used with each data subset to derive discriminant function…

Conditional observed score (COS) and latent trait (LT) definitions of differential item functioning (DIF) are explored to determine when they are equivalent. COS methods rely solely on observed measurements, and LT methods model the response to an item as a function of an unobserved hypothetical latent ability or trait. For the case of dichotomous…