Solution of problems in mathematics, and in particular in the field of Euclidean geometry is in many senses a form of artisanship that can be developed so that in certain cases brief and unexpected solutions may be obtained, which would bring out aesthetically pleasing mathematical traits. We present four geometric tasks for which different proofs…

Generalizing is a foundational mathematical practice for the algebra classroom. It entails an act of abstraction and forms the core of algebraic thinking. Kinach (2014) describes two kinds of generalization--by analogy and by extension. This article illustrates how exploration of fractals provides ample opportunity for generalizations of both…

One goal of a mathematics education is that students make significant connections among different branches of mathematics. Connections--such as those between arithmetic and algebra, between two-dimensional and three-dimensional geometry, between compass-and-straight-edge constructions and transformations, and between calculus and analytic…

The desire to persuade students to avoid strictly memorizing formulas is a recurring theme throughout discussions of curriculum and problem solving. In combinatorics, a branch of discrete mathematics, problems can be easy to write--identify a few categories, add a few restrictions, specify an outcome--yet extremely challenging to solve. A lesson…

Financial analysts generally create static formulas for the computation of NPV. When they do so, however, it is not readily apparent how sensitive the value of NPV is to changes in multiple interdependent and interrelated variables. It is the aim of this paper to analyze this variability by employing a dynamic, visually graphic presentation using…

The leap into the wonderful world of differential calculus can be daunting for many students, and hence it is important to ensure that the landing is as gentle as possible. When the product rule, for example, is met in the "Australian Curriculum: Mathematics", sound pedagogy would suggest developing and presenting the result in a form…

Light refracts as it travels from one medium to another. The angle of incidence "i" and the angle of refraction "r" are related by Snell's law, sin"i" ÷ sin"r"="k," where "k" is a constant. The diagram in Fig. 1 shows a geometric representation of the formula for light passing from…

Two points determine a line. Three noncollinear points determine a quadratic function. Four points that do not lie on a lower-degree polynomial curve determine a cubic function. In general, n + 1 points uniquely determine a polynomial of degree n, presuming that they do not fall onto a polynomial of lower degree. The process of finding such a…

Quantifier Elimination is a procedure that allows simplification of logical formulas that contain quantifiers. Many mathematical concepts are defined in terms of quantifiers and especially in calculus their use has been identified as an obstacle in the learning process. The automatic deduction provided by quantifier elimination thus allows…

A simple mathematical formulation of Mach's principle is given based on a century of investigation into inertia, and used to check the results of Newton's famous bucket experiment.

In this paper we consider the main properties of cylindrical lenses and propose several demonstrational experiments that can be performed with them. Specifically we use simple glasses full of water to demonstrate some basic geometrical optics principles and phenomena. We also present some less standard experiments that can be performed with such…

Two practical activities are described, which aim to support critical thinking about statistics as they concern multiple outcomes testing. Formulae are presented in Microsoft Excel spreadsheets, which are used to calculate the inflation of error associated with the quantity of tests performed. This is followed by a decision-making exercise, where…

The gamma function, which has the property to interpolate the factorial whenever the argument is an integer, is a special case (the case "g"?=?2) of the general term of the sequence factorial of "g"-gonal numbers. In relation to this special case, a formula for calculating the general term of the sequence factorial of any…

Mr. Perez and Mrs. Peterson co-teach a ninth-grade algebra class. Perez and Peterson's class includes four students with individualized education programs (IEPs). In response to legislation, such as the No Child Left Behind (NCLB) Act (2001) and the Individuals with Disabilities Education Improvement Act (2006), an increasing number of students…

Equality is such an important concept for children to develop. In this article it is argued that a precise definition is needed to ensure that students are provided with a consistent "picture" of what it is that equality really means.