Mathematical modeling and applications are an important part of curriculum and considered to be important for students' current and future lives. In this contribution, we focus on mathematical modeling from a cognitive prospective. Following embedding the cognitive perspective within the discourse of mathematical modeling, we describe some of the…

This monograph provides in-depth mathematical logic as the foundational rationale for the novel and innovative online instructional methodology called the 4A Metric Algorithm. The 4A Metric has been designed to address and meet the meta-competency-based education challenges faced by 21st century students who must now adapt to and learn in a…

Mathematical modeling has taken on increasing curricular importance in the past decade due in no small measure to the Common Core State Standards in Mathematics (CCSSM) identifying modeling as one of the Standards for Mathematical Practice (SMP 4, CCSSI 2010, p. 7). Although researchers have worked on mathematical modeling (Lesh and Doerr 2003;…

The article first summarizes the assumptions of Luce (2004, 2008) for inherently binary (2-D) stimuli (e.g., the ears and eyes) that lead to a "p-additive," order-preserving psychophysical representation. Next, a somewhat parallel theory for unary (1-D) signals is developed for intensity attributes such as linear extent, vibration to finger, and…

Early elementary school students are expected to solve twelve distinct types of word problems. A math researcher and two teachers pose a structure for thinking about one problem type that has not been studied as closely as the other eleven. In this article, the authors share some of their discoveries with regard to the variety of…

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…

Thanks to technological progress the world becomes more and more complicated. People stand in front of new and difficult problems that need to be solved. These are problems, the solutions of which are not universal, and cannot be learned. Many solutions require specific data that cannot be learned, as new data is part of the ongoing generation of…

The white picket fence is an integral component of the iconic American townscape. But, for mathematics students, it can be a mathematical challenge. Picket fences in a variety of styles serve as excellent sources to model constant, step, absolute value, and sinusoidal functions. "Principles and Standards for School Mathematics" (NCTM 2000)…

Visual backward masking is a versatile tool for understanding principles and limitations of visual information processing in the human brain. However, the mechanisms underlying masking are still poorly understood. In the current contribution, the authors show that a structurally simple mathematical model can explain many spatial and temporal…

The utility of an "ecologically rational" recognition-based decision rule in multichoice decision problems is analyzed, varying the type of judgment required (greater or lesser). The maximum size and range of a counterintuitive advantage associated with recognition-based judgment (the "less-is-more effect") is identified for a range of cue…

Because higher level cognitive processes generally involve the use of world knowledge, computational models of these processes require the implementation of a knowledge base. This article identifies and discusses 4 strategies for dealing with world knowledge in computational models: disregarding world knowledge, "ad hoc" selection, extraction from…

Comments on articles by Botvinick et al. and Jacob et al. M. M. Botvinick, T. S. Braver, D. M. Barch, C. S. Carter, and J. D. Cohen (2001) implemented their conflict-monitoring hypothesis of cognitive control in a series of computational models. The authors of the current article first demonstrate that M. M. Botvinick et al.'s (2001)…

This paper reports on the implementation of a self-directed learning process in a circuit theory course, incorporating dynamic training elements which were designed on the basis of a cybernetic model of cognitive process management. These elements are centrally linked in a dynamic learning frame, created on the monitor screen, which displays the…

The model of the decision system in Murdock's two-stage memory- and-decision model for item recognition is developed and tested. The decision model is shown to be able to fit the accuracy and mean response latency data from four major recognition paradigms (Sternberg, study-test, continuous, and prememorized list). (Author/LMO)

This paper aims to describe a Bayesian approach to modeling and estimating cognitive models both in terms of statistical machinery and actual instrument development. Such a method taps the knowledge of experts to provide initial estimates for the probabilistic relationships among the variables in a multivariate latent variable model and refines…