The circle of Apollonius is named after the ancient geometrician Apollonius of Perga. This beautiful geometric construct can be helpful when solving some general problems of geometry and mathematical physics, optics, and electricity. Here we discuss two of its applications: localizing an object in space and calculating electric fields. First, we…

Different technological artefacts may offer distinct opportunities for students to develop resources and strategies to formulate, comprehend and solve mathematical problems. In particular, the use of dynamic software becomes relevant to assemble geometric configurations that may help students reconstruct and examine mathematical relationships. In…

The electrochemical synthesis of transition-metal acetylacetonates described here can form the basis of assisting in the transformation of an entry-level laboratory course into a research-like environment where all members of a class are working on the same problem, but where each member has a personal responsibility for the synthesis and…

Organic chemistry students typically struggle with the retrosynthetic approach to solving synthesis problems because most textbooks present the chemistry grouped by "reactions of the functional group". In contrast, the retrosynthetic approach requires the student to envision "reactions that yield the functional group". A second challenge is the…

Mutational inactivation of a specific gene is the most powerful technique to analyze the biological function of the gene. This approach has been used for a long time in viruses, bacteria, yeast, and fruit fly, but looked quite hopeless in more complex organisms. Targeted inactivation of specific genes (also known as knock-out mutation) in mice is…

Reversing is the notion of thinking or working in reverse. Computer science textbooks and tutors recognize it primarily in the form of recursion. However, recursion is only one form of reversing. Reversing appears in the computer science curriculum in many other forms, at various intellectual levels, in a variety of fundamental courses. As such,…

This note generalizes the formula for the triangular number of the sum and product of two natural numbers to similar results for the triangular number of the sum and product of "r" natural numbers. The formula is applied to derive formula for the sum of an odd and an even number of consecutive triangular numbers.

Recently in many Canadian schools, escalated violence presents an evident threat. Rates of severe youth violence are considerably lower than sensational media coverage would have people believe, but at the same time, many young people, as well as adults, do not feel safe, respected, or constructively engaged as they should be. While schools cannot…

We solve exactly the problem of calculating the electromagnetic fields produced by a finite wire with a constant current, by using two methods: retarded potentials and Jefimenko's formalism. One result in this particular case is that the usual Biot-Savart law of magnetostatics gives the correct magnetic field of the problem. We also show…

A methodology is described that teaches science process by combining informal logic and a heuristic for rating factual reliability. This system facilitates student hypothesis formation, testing, and evaluation of results. After problem solving with this scheme, students are asked to examine and evaluate arguments for the underlying principles of…

Excel, Microsoft's spreadsheet program, offers several tools which have proven useful in solving some optimization problems that arise in operations research. We will look at two such tools, the Excel modules called Solver and Goal Seek--this after deriving an equation, called the "cash accumulation equation", to be used in conjunction with them.

This article describes investigative calculus projects in which students explore a question or problem of their own construction. Three exemplary pieces of student work are showcased. Investigative calculus projects are an excellent way to foster student understanding and interest in calculus. (Contains 4 figures.)

This article provides a review of research in cognitive strategy instruction for improving mathematical problem solving for students with learning disabilities (LD). The particular focus is on one of the salient components of this instructional approach--self-regulation. Seven studies utilizing this approach for teaching problem solving to…

Our aim is to describe open-ended case studies for learning real-life problem solving skills, and relate this approach to conventional, closed-ended decision case studies. Teaching methods are open-ended cases in agroecology, an alternative to traditional strategies that lead students through prepared materials and structured discussions to…

One way to merge imagination with problem-posing and problem-solving in the English classroom is by asking students to respond to "cultural and political vignettes" (CPVs). CPVs are cultural and political situations that are presented to students so that they can practice the creative and essential decision-making skills that they will need to use…