This article discusses how mathematicians and mathematics educators worked together on creating the "new math"--a short-lived effort to reform mathematics and to bring mathematics education in K-12 more in step with the mathematics being taught in the universities. A number of large international research studies have helped the mathematics…

Teachers of mathematics face a particular tension, which the authors call the "planning paradox". If teachers plan from objectives, the tasks they set are likely to be unrewarding for the pupils and mathematically impoverished. Planning from tasks may increase pupils' engagement but their activity is likely to be unfocused and learning difficult…

This work extends the theory of the fixed charge transportation problem (FCTP), currently based mostly on a forty-year-old publication by Hirsch and Danzig. This paper presents novel properties that need to be considered by those using existing, or those developing new methods for optimizing FCTP. It also defines the problem in an easier way,…

In this paper, we distinguish between two ways that an individual can construct a formal proof. We define a syntactic proof production to occur when the prover draws inferences by manipulating symbolic formulae in a logically permissible way. We define a semantic proof production to occur when the prover uses instantiations of mathematical…

I. Klugkist, O. Laudy, and H. Hoijtink (2005) presented a Bayesian approach to analysis of variance models with inequality constraints. Constraints may play 2 distinct roles in data analysis. They may represent prior information that allows more precise inferences regarding parameter values, or they may describe a theory to be judged against the…

This article is a discussion of and reaction to two collections of papers on research on Reasoning about Variation: Five papers appeared in November 2004 in a Special Issue 3(2) of the Statistics Education Research Journal (by Hammerman and Rubin, Ben-Zvi, Bakker, Reading, and Gould), and three papers appear in a Special Section on the same topic…

Researchers have been aware for some time of the differences between extensive and intensive quantities but the significance of these differences has not been recognized in mathematics curricula. In England children are provided with many opportunities in their first few years in school to manipulate, measure and reason about extensive quantities…

Assignments were altered to determine if interspersing additional briefer and easier mathematics problems enhanced students' accuracy on longer, more difficult target problems across two types of mathematics assignments. Students (N = 52) from three fifth-grade classes completed six math assignments incorporating two task demands and three…

The card game SET has attracted the attention of math and game enthusiasts alike. In this article, I present a first semester Abstract Algebra project that guides the students through an algebraic formulation of the game. There are many interesting mathematical questions that one can ask about the game, and I illustrate how the project can be used…

Developing students' abilities to communicate technical information is increasingly becoming a higher priority. A calculus course is an appropriate setting for the beginning of this development. In particular, this article relates the details behind a letter-writing assignment that the author has created to address this development.

A survey was administered to calculus students who had previously been exposed to a course on integral calculus. The purpose of the survey was to explore students' understanding of the definition of a definite integral, their abilities to evaluate definite integrals, and their graphical interpretations of definite integrals. The analysis of…

Most beginning calculus courses spend little or no time on a technical definition of the limit concept. In most of the remaining courses, the definition presented is the traditional epsilon-delta definition. An alternative approach that bases the definition on infinite sequences has occasionally appeared in commercial textbooks but has not yet…

A new trigonometric identity derived from factorizations and partial fractions is given. This identity is used to evaluate the Poisson integral via Riemann sum and to establish some trigonometric summation identities.

This note provides a simple method to extend the usual Leibniz rule for higher derivatives of the product of two functions to several functions, which is within the reach of freshman calculus students.

This paper discusses a result of Li and Shen which proves the existence of a unique periodic solution for the differential equation x[dots above] + kx[dot above] + g(x,t) = [epsilon](t) where k is a constant; g is continuous, continuously differentiable with respect to x , and is periodic of period P in the variable t; [epsilon](t) is continuous…