Getting students to think about the relationships between area and perimeter beyond the formulas for these measurements is never easy. An interesting, nonroutine, and accessible problem that will stimulate such thoughts is the Lattice Octagon problem. A "lattice polygon" is a polygon whose vertices are points of a regularly spaced array.…

The process of developing definitions is underemphasized in most mathematics instruction. Investing time in constructing meaning is well worth the return in terms of the knowledge it imparts. In this article, the authors present a third approach to "defining," called "constructive." It involves modifying students' previous understanding of a term…

We present a simple and pedagogical derivation of the quantum adiabatic theorem for two-level systems (a single qubit) based on geometrical structures of quantum mechanics developed by Anandan and Aharonov, among others. We have chosen to use only the minimum geometric structure needed for the understanding of the adiabatic theorem for this case.…

We present a sequence of classroom activities on Euclidean geometry, both plane and space geometry, used to make three dimensional geometry more catchy and simple. The activity consists of a guided research activity that leads the students to discover unexpected properties of two apparently distant geometrical entities, quadrilaterals and…

As calls are made for reasoning-and-proving to permeate school mathematics, several textbook analyses have been conducted to identify reasoning-and-proving opportunities outside of high school geometry. This study looked within geometry, examining six geometry textbooks and characterizing not only the justifications given and the…

Take a strip of paper and fold a crease intersecting the long edges, creating two angles. Choose one edge and consider the angle with the crease. Fold the opposite edge along the crease, creating a new crease that bisects the angle. Fold again, this time using the newly created crease and the initial edge, creating a new angle along the chosen…

Mathematics is full of surprises and beauty. In this article, the author discusses three favourite topics that he finds both amazing and amusing: (1) the problem of the possibility of subdividing a rectangle into a number of different squares; (2) the arrangement formed when several soap bubbles meet; and (3) circles and spheres.

Kinesthetic intelligence is one of the seven kinds of intelligence identified by Gardner's multiple intelligence theory (1983). The kinesthetic approach to teaching has numerous pedagogical advantages and can be adapted to the teaching of mathematics. This article describes a series of kinesthetic activities designed to explore the properties of…

We provide several constructions, both algebraic and geometric, for determining the ratio of the radii of two circles in an Apollonius-like packing problem. This problem was inspired by the art deco design in the transom window above the Shadek Fackenthal Library door on the Franklin & Marshall College campus.

Many illustrations and problems on the vector nature of forces have weights and forces in a vertical plane. One of the common devices for studying the vector nature of forces is a horizontal "force table," in which forces are produced by weights hanging vertically and transmitted to cords in a horizontal plane. Because some students have…

Educators often speak of teaching as "listening," yet they seldom encounter it in practice. What converts pedagogical space into authentic educational experience is a climate of respect that is born of serious, just, humble, and generous relationships in which both the authority of the teacher and the freedom of the students are ethically…

The authors wanted their students to consider mathematical notions of space and dimension, which are typically not part of the middle school curriculum. They developed a two-week lesson that was drawn from the curriculum unit "Exploring the Shape of Space" (Weeks 2001). They began by considering implications of living in a two-dimensional…

In this paper, we explore an approach to understanding how multimodality works in a community of practice. Using a social learning framework, we show how a community of practice, involving a pair of high school students, engaged in perceptual, bodily, and imaginary experiences while discussing about calculus concepts in a dynamic geometry…

The purpose of this study was to investigate the effect of technology on one student's solution methods when solving mathematical tasks that included algebra word problems. The student was presented with context tasks in both paper-pencil and computer environments. The results of this study revealed that the student's solution methods differed in…

Students with special needs often require additional assistance in order to learn at the university level. This article documents a professor's efforts in teaching a visually impaired prospective elementary school teacher geometry content knowledge. The goal of this article is to shed light upon the iterative process of accommodating for…