Typically, secondary and college algebra students attempt to utilize either completing the square or the quadratic formula as techniques to solve a quadratic equation only after frustration with factoring has arisen. While both completing the square and the quadratic formula are techniques which can determine solutions for all quadratic equations,…

Response to Intervention (RTI) models have attracted great attention as an alternative to traditional methods used for identifying students with learning disabilities. A major feature of this approach is the implementation of academic interventions in general education and measuring the student's response to those interventions. A common question…

Purpose--To illuminate the importance of the communal deliberative process, as a form of collective thinking, in overcoming the growing complexities of schoolwork in uncertain and turbulent environments. Design/methodology/approach-- Introduces the notion of deliberation as evolved from Dewey's moral theory, the essential phases and elements of…

Despite significant progress over the past six decades, a number of obstacles still exist as far as the understanding and application of inventive ideation techniques is concerned. This paper describes the development of a model aimed at simplifying the selection and application of these techniques. Analysis of a diverse range of creative thinking…

In this article, the author talks about Destination ImagiNation (DI), the world's largest creative problem solving (CPS) program, and describes how this program helps students. In many schools and communities, DI supplements or even serves as the talented and gifted program. CPS is also popular and appropriate for homeschool families because it…

We present and discuss three examples of misapplication of the notion of conditional probability. In each example, we present the problem along with a published and/or well-known incorrect--but seemingly plausible--solution. We then give a careful treatment of the correct solution, in large part to show how careful application of basic probability…

It was shown by Costa and Levine that the homogeneous differential equation (1-x[superscript N])y([superscript N]) + A[subscript N-1]x[superscript N-1)y([superscript N-1]) + A[subscript N-2]x[superscript N-2])y([superscript N-2]) + ... + A[subscript 1]xy[prime] + A[subscript 0]y = 0 has a finite polynomial solution if and only if [for…

There are four Euclidean centres of a triangle--the circumcentre, the centroid, the incentre and the orthocentre. In this article, the authors prove the following: if the centre is the incentre (resp. orthocentre) then there exists a triangle with given distances of its vertices from its incentre (resp. orthocentre). They also consider uniqueness…

People are inclined to desire proof of theories if they have developed a certain philosophical style when they are quite young. It is a style that questions the authority for things, so that they can hold fast to what is good. Regarding mathematical proof, this author argues that it is only those who are prepared to take their own authority for…

The author has always been fascinated by the title identity. It's charming and simple, as well as easy to believe after pressing a few calculator keys. Several fine proofs have appeared in the literature, including several proofs without words. His own earlier proof is trigonometric, and he has often been dissatisfied with not being able to…

Students analyze a photograph to solve mathematical questions related to the images captured in the photograph.

Technology is a powerful tool in assisting students in problem solving by allowing for multiple representations. The vignette offered in this article provides insight into ways to solve open-ended problems using multiple technologies.

Building on their knowledge of the three possible outcomes of solving 2x2 systems of equations, students use three-dimensional geometric figures to investigate the eight possible outcomes for solving 3x3 systems of equations.

This article describes Comer's Method, a student's discovery of an alternative approach to solving the determinant of a 3x3 matrix.

The time of flight, range and the angle which maximizes the range of a projectile in a linear resisting medium are expressed in analytic form in terms of the recently defined Lambert W function. From the closed-form solutions a number of results characteristic to the motion of the projectile in a linear resisting medium are analytically confirmed,…