This paper argues that design fixation, in part, entails fixation at the level of meta-representation, the representation of the relation between a representation and its reference. In this paper, we present a mathematical model that mimics the idea of how fixation can occur at the meta-representation level. In this model, new abstract concepts…

Functions provide powerful tools for describing change, but research has shown that students find difficulty in using functions to create and interpret models of changing phenomena. In this study, we drew on a models and modeling perspective to design an instructional approach to develop students' abilities to describe and interpret rates of…

A folded symplectic form on an even-dimensional manifold is a closed two-form that degenerates in a suitably controlled way along a smooth hypersurface. When a torus having half the dimension of the manifold acts in a way preserving the folded symplectic form and admitting a moment map, the manifold is called a folded symplectic toric manifold.…

Unemployment Insurance (UI) is the major social insurance program that protects against lost earnings resulting from involuntary unemployment. Existing literature finds that low-earning unemployed workers experience difficulty accessing UI benefits. The most prominent policy reform designed to increase rates of monetary eligibility, and thus UI…

A cognitive construct that is important when solving engineering design problems is executive control process, or metacognition. It is a central feature of human consciousness that enables one "to be aware of, monitor, and control mental processes." The framework for this study was conceptualized by integrating the model for creative design, which…

Learning to select optimal behavior in new and uncertain situations is a crucial aspect of living and requires the ability to quickly associate stimuli with actions that lead to rewarding outcomes. Mathematical models of reinforcement-based learning to select rewarding actions distinguish between (1) the formation of stimulus-action-reward…

The present study proposes and examines the multidimensional vector (MDV) model framework as a modeling schema for choice response times. MDV extends the Thurstonian model, as well as signal detection theory, to classification tasks by taking into account the influence of response properties on stimulus discrimination. It is capable of accounting…

The purpose of our modeling effort is to predict future outcomes. We assume the data collected are both accurate and relatively precise. For our oscillating data, we examined several mathematical modeling forms for predictions. We also examined both ignoring the oscillations as an important feature and including the oscillations as an important…

A quiet methodological revolution, a modeling revolution, has occurred over the past several decades, almost without discussion. In contrast, the 20th century ended with contentious argument over the utility of null hypothesis significance testing (NHST). The NHST controversy may have been at least partially irrelevant, because in certain ways the…

We represent the main stages of the process of mathematical modelling as fuzzy sets in the set of the linguistic labels of negligible, low intermediate, high and complete success by students in each of these stages and we use the total possibilistic uncertainty as a measure of students' modelling capacities. A classroom experiment is also…

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…

Part of the power of algebra is that it provides students with tools that they can use to model a variety of problems and applications. Such modeling requires them to understand patterns and choose from a variety of representations--numeric, graphical, symbolic--to construct a model that accurately reflects the relationships found in the original…

Various theoretical equations have been proposed to predict response rate as a function of the rate of reinforcement. If both the rate and probability of reinforcement are considered, a simple identity, defining equation, or "law" holds. This identity places algebraic constraints on the allowable forms of our mathematical models and can help…

This study focuses on assessing the contribution of ELL services to Spanish-speaking students' mathematics learning in elementary schools. ELL students tend to have lower average math achievement at school entry and throughout elementary school. The term "ELL services" encompasses English-as-a-second-language (ESL) programs, bilingual…

In Ireland, to encourage the study of STEM (science, technology, engineering and mathematics) subjects and particularly mathematics, the Mathematics Applications Consortium for Science and Industry (MACSI) and Clongowes Wood College (County Kildare, Ireland) organized a mathematical modelling workshop for senior cycle secondary school students.…