We present the findings of a study on prospective elementary teachers' proportional reasoning. After describing some of the teachers' performance in solving multiplicative structure problems that involve ratios and relations of direct proportionality between quantities, we were able to establish classifications of their answers according to…

The inter-derivability of the Pythagorean Theorem and Heron's area formula is explained by applying Al-Karkhi's factorization to Heron's formula.

Pompeiu's theorem states that if ABC is an "equilateral" triangle and M a point in its plane, then MA, MB, and MC form a new triangle. In this article, we have a new look at this theorem in the realm of arbitrary triangles. We discover what we call Pompeiu's Area Formula, a neat equality relating areas of triangles determined by the points A, B,…

Because mathematical formulae and problem solving are such prominent components of most introductory physics courses, many students consider these courses to be nothing more than courses in applied mathematics. As a result, students often do not develop an acceptable understanding of the relationship between mathematics and science and of the role…

This article discusses a variety of examples in errors in mathematical reasoning, the source of which is due to the tension between the syntax (form of mathematical expression) and semantics (underlying ideas or meaning). This article suggests that the heightened awareness of syntactic and semantic reasoning, and the consequent resolution of the…

We present a calculation involving a wide class of real integrals by means of integration in the complex plane. Some particular cases where Euler numbers and Bell polynomials appear are discussed and a generalisation of some previous results is also provided. (Contains 1 table and 1 figure.)

Several activities in which population dynamics can be modelled by tossing M&M's[R] candy are presented. Physical activities involving M&M's[R] can be modelled by difference equations and several population phenomena, including death and immigration, are studied. (Contains 1 note.)

The Lithuanian Informatics Olympiad is a problem solving contest for high school students. The work of each contestant is evaluated in terms of several criteria, where each criterion is measured according to its own scale (but the same scale for each contestant). Several jury members are involved in the evaluation. This paper analyses the problem…

Huntington's disease (HD), an autosomal-dominant genetic disorder, has historically been viewed as a degenerative movement disorder but it also includes psychiatric symptoms and progressive cognitive decline. There has been a lack of consensus in the literature about whether or not cognitive signs can be detected in carriers before clinical…

This study explored the mathematical problem-solving styles of middle school and high school deaf and hard-of-hearing students and the mathematical problem-solving styles of the mathematics teachers of middle school and high school deaf and hard-of-hearing students. The research involved 45 deaf and hard-of-hearing students and 19 teachers from a…

The purpose of this article is to discuss specific techniques for the computation of the volume of a tetrahedron. A few of them are taught in the undergraduate multivariable calculus courses. Few of them are found in text books on coordinate geometry and synthetic solid geometry. This article gathers many of these techniques so as to constitute a…

The aim of this paper is to investigate the role that auxiliary means (manipulatives such as cubes and representations such as number line) play for kindergartners in working out mathematical tasks. Our assumption was that manipulatives such as cubes would be used by kindergartners easily and successfully whereas the number line would be used by…

Many workshops and meetings with the US high school mathematics teachers revealed a lack of familiarity with the use of transformations in solving equations and problems related to the roots of polynomials. This note describes two transformational approaches to the derivation of the quadratic formula as well as transformational approaches to…

Invention and Productive Failure activities ask students to generate methods that capture the important properties of some given data (e.g., uncertainty) before being taught the expert solution. Invention and Productive Failure activities are a class of scientific inquiry activities in that students create, implement, and evaluate mathematical…

A. Lobb discovered an interesting generalization of Catalan's parenthesization problem, namely: Find the number L(n, m) of arrangements of n + m positive ones and n - m negative ones such that every partial sum is nonnegative, where 0 = m = n. This article uses Lobb's formula, L(n, m) = (2m + 1)/(n + m + 1) C(2n, n + m), where C is the usual…