If a curriculum developer's goal is to create a single linear sequence of tasks that lead to the development of some important mathematical concept, then some researchers have suggested that these sequences should follow progressions similar to stages of development that have been identified in Piaget-like research on the relevant concept(s).…

Using the problem of determining where a Flatland artist was standing, this article takes another look at perspective.

What may have been the birth of a new calculus problem took place when the author noticed that two coffee cups, one convex and one concave, fit nicely together, and he wondered which held more coffee. The fact that their volumes were about equal led to the topic of this article: complementary surfaces of revolution with equal volumes.

This article describes several creative solutions developed by calculus and modeling students to the classic optimization problem of testing in groups to find a small number of individuals who test positive in a large population.

Consider the series [image omitted] where the value of each a[subscript n] is determined by the flip of a coin: heads on the "n"th toss will mean that a[subscript n] =1 and tails that a[subscript n] = -1. Assuming that the coin is "fair," what is the probability that this "harmonic-like" series converges? After a moment's thought, many people…

This paper presents a visual representation of complex solutions of quadratic equations in the xy plane. Rather than moving to the complex plane, students are able to experience a geometric interpretation of the solutions in the xy plane. I am also working on these types of representations with higher order polynomials with some success.

Career and technical education is already ahead of the curve, since automotive technology, electrical technology and electronics students across the nation have been actively engaged in learning about hybrid, electric and even solar-powered cars. Students in the Environmental Technologies Club at Central High School in Phoenix, Arizona, have built…

Using the power series solution of a differential equation and the computation of a parametric integral, two elementary proofs are given for the power series expansion of (arcsin x)[squared], as well as some applications of this expansion.

A simple problem relating to birds chasing each other gives rise to a homogeneous differential equation. The solution draws on student skills in differential equations and basic co-ordinate geometry.

In order to facilitate the transition from learning from worked examples in earlier stages of skill acquisition to problem solving in later stages, it is effective to successively fade out worked solution steps--in comparison to the traditional method of employing example--problem pairs that is frequently used in cognitive-load research. In the…

As an elementary physical education specialist teaching 740 students in a diverse public school, the author is constantly looking for classroom management strategies that are age appropriate, build relationships, and allow students to take responsibility in solving their own problems beyond that of the author's basic behavior management plan.…

Several developments during the last decades have provoked spectacular changes in both Mathematics Teaching and the Education of Mathematics Teachers. From this perspective of renovation, we wish to define a new context in mathematics teacher education in which we consider that "pedagogical content knowledge" (Shulman, 1993, Mellado,…

Given three points in the plane, interest is in the locus of all points for which the sum of the distances to the given points is a prescribed constant. These curves turn out to be sixth degree polynominals in x and y , and thus are complicated. However, it turns out that often there is a point, within the triangle formed by the three given…

Given two circles C 1 and C 2 in a plane such that neither one of the two circles is contained in the other, there are either four common tangents when the circles do not intersect at all or the circles have three common tangents when they touch each other externally or only two common tangents when the circles intersect exactly at two points. The…

Technical problem solving as a model for instructional design (ID) has its strengths, and it has done much to provide designers strategies for their work, but it has substantial limitations as well. For one, it does not do a good job in describing how designers actually think. Slavish adherence to its methods can also be considered responsible for…