This note shows a combinatorial approach to some identities for generalized Fibonacci numbers. While it is a straightforward task to prove these identities with induction, and also by arithmetical manipulations such as rearrangements, the approach used here is quite simple to follow and eventually reduces the proof to a counting problem. (Contains…

A stopper is called "universal" if it can be used to plug pipes whose cross-sections are a circle, a square, and an isosceles triangle, with the diameter of the circle, the side of the square, and the base and altitude of the triangle all equal. Echoing the well-known result for equal cubes that is attributed to Prince Rupert, we show that it is…

If you break a stick at two random places, the probability that the three pieces form a triangle is 1/4. How does this generalize? To answer this question, we give a method for finding the probability that n randomly chosen points in a given interval fall within a specified distance of one another. We use this method to provide solutions to…

Consider a circular segment (the smaller portion of a circle cut off by one of its chords) with chord length c and height h (the greatest distance from a point on the arc of the circle to the chord). Is there a simple formula involving c and h that can be used to closely approximate the area of this circular segment? Ancient Chinese and Egyptian…

In this article we propose building sets of problems rooted in the same concept, a process that allows students to develop understanding and take on increasingly complex problems. By beginning with "foundation stories" and using strategies similar to those for literary constructs, we have dramatically increased our students' mathematical…

An exercise on structure elucidation for advanced undergraduate students is described. To determine the structure of an unknown product, students are required to use spectra together with an organic chemistry mechanism. This exercise exemplifies the procedure commonly used in research, thus helping students develop problem-solving skills. In…

It is well known that the discoverer of radio waves, Heinrich Hertz, was the first man to apply Maxwell's electrodynamic theory to a problem in radio wave propagation. In this paper, we scrutinize his near-field lines of force using computers and his theory. In one of his four figures, a feature was found which was not to be obtained by…

The primary application of dimensional analysis (DA) is in problem solving. Typically, the problem description indicates that a physical quantity Y(the unknown) is a function f of other physical quantities A[subscript 1], ..., A[subscript n] (the data). We propose a qualitative problem-solving procedure which consists of a parallel decomposition…

Most differences that people see across identity groups are not essential. They're not hard coded into people's genes. They originate in communities and in cultures and are reinforced by daily practice. People act the ways they do because they're conforming to the people around them. In this article, the author introduces a new way of thinking…

The problem solving conversations of 92 adolescent romantic couples were analyzed using two innovative techniques for capturing the meaning-making processes in couples' interactions. Couples were coded using the narrative coding system developed by the Family Narrative Consortium [Fiese, B. H. & Sameroff, A. J. (Eds.), (1999). The stories that…

The classic experiment presented in this problem-solving test was designed to identify the template molecules of translation by analyzing the synthesis of phage proteins in "Escherichia coli" cells infected with bacteriophage T4. The work described in this test led to one of the most seminal discoveries of early molecular biology: it dealt a…

A study was conducted to promote higher order cognitive skills (HOCS) in a chemistry class using the GOAL (Gather, Organize, Analyze, and Learn) method. Students were assigned four qualitative problems specifically designed to be solved with the method over the course of the semester outside of normal homework and testing. The problems served as a…

In three experiments, children aged between 3 and 5 years (N = 38, 52, 94; mean ages 3-7 to 5-2) indicated their confidence in their knowledge of the identity of a hidden toy. With the exception of some 3-year-olds, children revealed working understanding of their knowledge source by showing high confidence when they had seen or felt the toy, and…

Though several researchers have thus far attempted to address the viable relationship between EI and burnout, it seems that few have approached the notion of burnout by considering the influence of teachers' coping resources such as EI especially in an EFL context. Thus, in an attempt to bridge this gap, the present study sought to explore the…

This paper describes the creation and evolution of an undergraduate dynamics and vibrations course for civil engineering students. Incorporating vibrations into the course allows students to see and study "real" civil engineering applications of the course content. This connection of academic principles to real life situations is in…