A test method is described for determining the divisibility of non-negative integers by a prime number. The test uses an integer multiplying factor that is defined for each prime, designated as [beta], to reduce the non-negative integer that is being tested by an order of magnitude in each of a sequence of steps to obtain a series of new numbers.…

We review the history and previous literature on radical equations and present the rigorous solution theory for radical equations of depth 2, continuing a previous study of radical equations of depth 1. Radical equations of depth 2 are equations where the unknown variable appears under at least one square root and where two steps are needed to…

The cosine theorem is used in solving triangulation problems and in physics when solving problems of addition of unidirectional oscillations. However, this theorem is used only for the analytical calculation of triangles or when solving problems of adding two oscillations. Here we propose a generalization of the cosine theorem for the case of…

In this regular "American Educator" column, findings from the field of cognitive science that are strong and clear enough to merit classroom application are considered. Individuals vary in their views of what students should be taught, but there is little disagreement on the importance of critical thinking skills. In free societies, the…

The distant magnetic field of a magnetic dipole is usually derived via the magnetic vector potential and substantial vector calculus. This paper presents an alternate proof that is less mathematically intensive, and that ties together various problem-solving tricks (the principle of virtual work, observation that only instantaneous quantities…

The purpose of this note is to describe several challenging problems in Euclidean geometry. The note also contains author's solution sketches to the two problems.

In a recent submission to "The Physics Teacher," we related how trigonometric identities can be used to find the extremes of several functions in order to solve some standard physics problems that would usually be considered to require calculus. In this work, the functions to be examined are polynomials, which suggests the utilization of…

There are many problems whose solution requires proof that a quadrilateral is cyclic. The main reason for writing this paper is to offer a number of new tools for proving that a particular quadrilateral is cyclic, thus expanding the present knowledge base and ensuring that investigators in mathematics and teachers of mathematics have at their…

The importance of mathematical problem posing has been acknowledged by many researchers. In this theoretical paper, we want to capture different meanings and aspects of problem posing by approaching it from three different levels: (1) by comparing definitions, (2) by relating it to other constructs, and (3) by referring to research and teaching…

Educators recognize that parental participation is a critical factor in the success of children within a school setting. This is particularly true for parents who have children with disabilities or who are from a culturally or linguistically diverse background. However, reaching out to these families can be a difficult task even for the most…

For years there has been an acknowledged interest in having students assess the rationality of their solutions to physics problems. In fact, many textbooks now routinely include end-of-problem assessments as part of the authors' detailed solutions to examples. Over the past two decades, I have experimented with various forms of end-of-problem…

Computer Programming Learners usually fail to get approved in introductory courses because solving problems using computers is a complex task. The most important reason for that failure is concerned with motivation; motivation strongly impacts on the learning process. In this paper we discuss how techniques like program animation, and automatic…

This article discusses the role of the three components of executive functions (EF) in geometric understanding. Discussing several examples of geometry problems, this article shows how EF are actively employed to solve geometry problems. Inhibition as the first component of EF helps the individual to suppress contextually irrelevant information.…

In this article, the authors define numberless math stories and highlight three strategies for supporting equitable problem solving: (1) The Turn-and-Tell Strategy; (2) Storyboard and Story Mat; and (3) The Act-It-Out Strategy. These three strategies open up problem solving and give every student access to a challenging mathematics curriculum…

This article aims to share five practical approaches for how teachers can identify, adapt, and create their own writing prompts they assign to students: (1) Promote Students' Solution Paths; (2) Go beyond Asking Students to Simply "Explain"; (3) Prompt Students to Share Their Reasoning; (4) Have Students Consider the Validity of a Given…