Dodgson's method of computing determinants is attractive, but fails if an interior entry of an intermediate matrix is zero. This paper reviews Dodgson's method and introduces a generalization, the double-crossing method, that provides a workaround for many interesting cases.

In Pre-Calculus courses, students are taught the composition and combination of functions to model physical applications. However, when combining two or more functions into a single more complicated one, students may lose sight of the physical picture which they are attempting to model. A block diagram, or flow chart, in which each block…

Generations have been inspired by Edwin A. Abbott's profound tour of the dimensions in his novella "Flatland: A Romance of Many Dimensions" (1884). This well-known satire is the story of a flat land inhabited by geometric shapes trying to navigate the subtleties of their geometric, social, and political positions. In this article, the authors…

Fay and Sam go for a walk. Sam walks along the left side of the street while Fay, who walks faster, starts with Sam but walks to a point on the right side of the street and then returns to meet Sam to complete one segment of their journey. We determine Fay's optimal path minimizing segment length, and thus maximizing the number of times they meet…

We give an alternative to the standard method of reduction or order, in which one uses one solution of a homogeneous, linear, second order differential equation to find a second, linearly independent solution. Our method, based on Abel's Theorem, is shorter, less complex and extends to higher order equations.

This article describes an investigation of Buffon's coin problem and related problems with the aid of an applet. The problems are accessible at a variety of grade levels and facilitate making connections between geometry and probability.

Children are naturally curious and want to make sense of their world. To implement mathematical tasks that nurture children's desire to reason, it is valuable for teachers to have experienced for themselves comparable tasks and learning environments (Ball and Bass 2000). In this article, the author describes three strategies to facilitate…

The emphasis on problem solving and problem solving courses has become a standard staple of most teacher preparation programs in mathematics. While opinions differ on whether problem solving should be integrated throughout the preparatory courses or teachers should also have a dedicated course for it, everybody agrees that problem solving skills…

We generalize a standard example from precalculus and calculus texts to give a simple description in polar coordinates of any circle that passes through the origin. We discuss an occurrence of this formula in the context of medical imaging. (Contains 1 figure.)

Providing student choice and opportunities for independent study are recognized as viable differentiation techniques. Daily homework sets that contain more demanding problems even though not required allow the teacher to provide challenge without incurring undue stress. The modest incentive of some homework bonus points is enough to whet the…

This article outlines some of the factors leading to challenges in succeeding in college environments for individuals with autism spectrum disorders (ASD). Individuals with the intellectual ability to pursue postsecondary education still need individualized and ongoing supports from their families and others to ensure success. Social challenges…

It has been argued that to increase societal impact behavioral researchers must do more to address problems of obvious practical importance. The basic science wing of behavior analysis has been described as especially detached from this goal, but is it really necessary that basic science demonstrate social relevance? If so, why hasn't this…

In this article, the author describes the Launch-Explore-Summarize teaching model which provides a way to structure how students participated in tasks. Shifting the structure of lessons helped the author foster her desired mathematical environment. This approach to teaching was different for her in that it recommended giving students very little…

In this article, the author wishes to emphasize two fundamental points related to theory that were significantly underplayed in Simon et al.'s "A Developing Approach to Studying Students' Learning through Their Mathematical Activity" (2010). The author believes these points are central to any study of children's mathematical learning. The first…

In this article, the author describes an art project wherein students make fanciful connections between art and medicine. This project challenges students to interpret "ear idioms" (e.g. "blow it out your ear," "in one ear and out the other") by relying almost entirely on realistic ear drawings, the placement of them, marks, and values. In that…