This article demonstrates the potential of using hierarchical Bayesian methods to relate models and data in the cognitive sciences. This is done using a worked example that considers an existing model of category representation, the Varying Abstraction Model (VAM), which attempts to infer the representations people use from their behavior in…

A primary criterion on which models of cognition are evaluated is their ability to fit empirical data. To understand the reason why a model yields a good or poor fit, it is necessary to determine the data-fitting potential (i.e., flexibility) of the model. In the first part of this article, methods for comparing models and studying their…

We use the Implicit Function Theorem to establish a result of non-existence of limit to a certain class of functions of several variables. We consider functions given by quotients such that both the numerator and denominator functions are null at the limit point. We show that the non-existence of the limit of such function is related with the…

In this third of a series of four articles, the author deconstructs the primary national strategy's approach to written multiplication. The approach to multiplication, as set out on pages 12 to 15 of the primary national strategy's "Guidance paper" "Calculation" (DfES, 2007), is divided into six stages: (1) mental…

Recent studies conclude that teachers are important for student learning but it remains uncertain what actually determines effective teaching. This study directly peers into the black box of educational production by investigating the relationship between lecture style teaching and student achievement. Based on matched student-teacher data for the…

"Culture and the Interaction of Student Ethnicity with Reward Structure in Group Learning" examined the effects of different reward systems used in group learning situations on the math skills of African-American and white students. The study analyzed data on 75 African-American and 57 white fourth- and fifth-grade students from urban…

Drawing upon research, theory, classroom and personal experiences, this paper focuses on the development of primary-aged children's computational fluency. It emphasises the critical links between number sense and a child's ability to perform mental and written computation. The case of multi-digit multiplication is used to illustrate these…

We begin by answering the question, "Which natural numbers are sums of consecutive integers?" We then go on to explore the set of lengths (numbers of summands) in the decompositions of an integer as such sums.

The well-known "hats" problem, in which a number of people enter a restaurant and check their hats, and then receive them back at random, is often used to illustrate the concept of derangements, that is, permutations with no fixed points. In this paper, the problem is extended to multiple items of clothing, and a general solution to the problem of…

When a dog (in this case, Tim Pennings' dog Elvis) is in the water and a ball is thrown downshore, it must choose to swim directly to the ball or first swim to shore. The mathematical analysis of this problem leads to the computation of bifurcation points at which the optimal strategy changes.

This article describes the use of Fermi questions as a problem-solving tool.

This article engages students in a simulation of the Coat Check problem, in which four women check their coats only to have them returned at random. Students examine the experimental and theoretical probability of at least one woman getting her own coat back.

Principal components analysis (PCA) is used to explore the structure of data sets containing linearly related numeric variables. Alternatively, nonlinear PCA can handle possibly nonlinearly related numeric as well as nonnumeric variables. For linear PCA, the stability of its solution can be established under the assumption of multivariate…

This paper provides surface area and volume formulas for surfaces of revolution in R[superscript n]. In addition the authors illustrate how to obtain the formulas for volume and surface areas of revolution about the x- or y-axis in two different ways: a "heuristic" argument and a rigorous calculation using "cylindrical" coordinates. In the last…

We have developed a simple, resilient experiment that illustrates the Nernst equation and Beer-Lambert law for our second-semester general chemistry students. In the experiment, the students monitor the reduction of ferricyanide ion, [Fe(CN)[subscript 6]][superscript 3-], to ferrocyanide, [Fe(CN)[subscript 6]][superscript 4-], both…