This article could be of interest to teachers of applied mathematics as well as to people who are interested in applications of linear algebra. We give a comprehensive study of linear systems from an application point of view. Specifically, we give an overview of linear systems and problems that can occur with the computed solution when the…

Just as students need plenty of time to practice skills such as solving fraction problems, they also need time to practice the skills of discourse to become better communicators and stronger mathematicians. Embedded within discourse strategies are specific ways to maximize communication. When repeatedly practiced, students learn to listen to one…

This note explains how Emil Artin's proof that row rank equals column rank for a matrix with entries in a field leads naturally to the formula for the nullity of a matrix and also to an algorithm for solving any system of linear equations in any number of variables. This material could be used in any course on matrix theory or linear algebra.

This article describes a grading approach that encourages students to employ particular problem-solving skills. Some strengths of this method, called "process-based grading," are that it is easy to implement, requires minimal time to grade, and can be used in conjunction with either an online homework delivery system or paper-based homework.

In this paper we present an innovative instructional sequence for an introductory linear algebra course that supports students' reinvention of the concepts of span, linear dependence, and linear independence. Referred to as the Magic Carpet Ride sequence, the problems begin with an imaginary scenario that allows students to build rich imagery and…

The problem of charge motion governed by image force attraction near a plane metal surface is considered and solved self-consistently. The temporal dispersion of metal dielectric permittivity makes the image forces dynamic and, hence, finite, contrary to the results of the conventional approach. Therefore, the maximal attainable velocity turns out…

Literacy in STEM is critical to individual and national success in the twenty-first century, the century of science. Gifted students are in a position to make especially profound contributions to solving the world's major challenges and creating new economic opportunity. But modern American culture and the way STEM education is approached and…

In computing real-valued functions, it is ordinarily assumed that the input to the function is known, and it is the output that we need to approximate. In this work, we take the opposite approach: we show how to compute the values of some transcendental functions by approximating the input to these functions, and obtaining exact answers for their…

Based on Steiner's fascinating theorem for trapezium, seven geometrical constructions using straight-edge alone are described. These constructions provide an excellent base for teaching theorems and the properties of geometrical shapes, as well as challenging thought and inspiring deeper insight into the world of geometry. In particular, this…

This article describes "modeling" as the act of creating an "idealized version of a real-life situation" that has been "translated into mathematical terms." By exploring an open-ended investigation involving proportional reasoning, sixth grade students were able to walk through both problem solving and modelling. The…

Debates in education are often polarised by those who want students to be firmly told what they should take on board, and those who insist individuals learn best if they were liberated from all forms of collective arrangement (such as an education authority). Some politicians even fluctuate between the two sets of views without any underlying…

Teachers are natural problem solvers, and they should be using this quality to their advantage when it comes to solving the systemic issues that plague Philadelphia's education system. Many of the articles in this issue have already gone into great detail about what is happening in Philadelphia. Torch Lytle has provided a summary of the recent…

Students' success in solving problems involving proportional reasoning is an indication that they have moved beyond additive thinking to multiplicative thinking. However, classroom work indicates that many students do not reason proportionally in many practical contexts. The authors discuss a particular task that reveals students'…

Designing solutions to social problems requires some degree of interpretive accountability to the sociocultural systems in which design solutions must live. Our case studies show how ethnography of communication research generates distinctive resources for design. (Contains 5 notes.)

Einstein was blessed with a rare genius. He also understood the intellectual weight of a flight of fancy. He turned over the idea in his mind for a decade before concluding that the light beam next to him would appear to be at rest even though it was traveling at the speed of light. While it may be tempting to focus on Einstein's cognitive…