This article is essentially devoted to a brief historical introduction to Euler's formula for polyhedra, topology, theory of graphs and networks with many examples from the real-world. Celebrated Konigsberg seven-bridge problem and some of the basic properties of graphs and networks for some understanding of the macroscopic behaviour of real…

This note proposes a unique solutions to find out the value of x, y and z which satisfies the equation x[superscript 2] + y[superscript 2] = z[superscript 2]. The uniqueness of the proposed formulae is to find the total number of y's and z's at a given value of x. The value of y and z can be calculated by factoring x[superscript 2] or…

A base for linear recursive sequences, such as the sequence of Fibonacci numbers, is defined within the framework of the sum of the digits of a number. Examples of bases of a number of such sequences are then outlined, and a Mobius strip is also used to illustrate the effects diagrammatically.

The purpose of our article is to show two simpler and clearer methods of proving Stirling's formula than the traditional and conventional ones. The distinction of our method is to use the simple trapezoidal formula.

At the end of primary school all children more of less know what a percentage is, but yet they often struggle with percentage problems. This article describes a study in which students of 13 and 14 years old were given a written test with percentage problems and a week later were interviewed about the way they solved some of these problems. In a…

The standard for measuring fuel efficiency in the U.S. has been miles per gallon (mpg). However, the Environmental Protection Agency's (EPA) switch in rating fuel efficiency from miles per gallon to gallons per hundred miles with the 2013 model-year cars leads to interesting and relevant mathematics with real-world connections. By modeling…

This article presents the author's response to Yan Ping Xin and Dake Zhang's recent critical evaluation of her and colleagues' work in "Exploring a Conceptual Model-Based Approach to Teaching Situated Word Problems," published in "The Journal of Educational Research" in 2009 (Vol. 102, No. 6). Most critiques of prior research are written in a fair…

An outbox of a quadrilateral is a rectangle such that each vertex of the given quadrilateral lies on one side of the rectangle and different vertices lie on different sides. We first investigate those quadrilaterals whose every outbox is a square. Next, we consider the maximal outboxes of rectangles and those quadrilaterals with perpendicular…

The teaching and learning of mathematics cannot be understood without considering the resolution of word problems. These kinds of problems not only connect mathematical concepts with language (and therefore with reality) but also promote the learning related to other scientific areas. In primary school, problems are solved by using basic…

Trends in the curriculum reform propose that algebra should be taught throughout the grades, starting in elementary school. The aim should be to decrease the discontinuity between the arithmetic in elementary school and the algebra in upper grades. This study was conducted to investigate and characterise upper elementary school students…

When the author asked her preservice elementary teachers what "order of operations" is and why it is important for children to learn, they recited "PEMDAS" or "Please Excuse My Dear Aunt Sally" but typically became silent when questioned about the importance of the order of operations. Reys and Fennell once suggested that the mathematics component…

This study investigated children's understanding of area measurement, including the concept of area and the area formula of a rectangle, as well as their strategic knowledge for solving area measurement problems. Twenty-two fourth-graders from three classes of a public elementary school in Taipei, Taiwan, participated in a one-on-one interview.…

It is essential to retain a focus on building students' mathematical reasoning and comprehension rather than merely developing superficial understanding through procedural learning. All too often this approach "takes a back seat" because of examination and assessment pressure, where the importance of "How?" supersedes that of…

The results of a simulation conducted by a research team involving undergraduate and high school students indicate that when data is symmetric the student's "t" confidence interval for a mean is superior to the studied non-parametric bootstrap confidence intervals. When data is skewed and for sample sizes n greater than or equal to 10,…

It is a routine matter for undergraduates to find eigenvalues and eigenvectors of a given matrix. But the converse problem of finding a matrix with prescribed eigenvalues and eigenvectors is rarely discussed in elementary texts on linear algebra. This problem is related to the "spectral" decomposition of a matrix and has important technical…