According to Poincare, a geometry cannot be an object of experience since any geometrical experiment must be realized with physical objects, such as rulers and light rays, and it is only their properties that can be tested. This position was apparently refuted by general relativity and the successful confirmation of its predictions by astronomical…

North Korea is one of the most closed nations in the world. It is difficult to access information about their education system. North Korea's economic system was reorganized on July 1, 2002, as well as the education system. This change has impacted North Korea's mathematics education. With limited information on North Korean mathematics textbooks,…

Under current learning trajectories for length and area, researchers examine how students coordinate and systematize measurement using a given external iterating unit. This study investigates the possible cognitive behaviors missing in that approach by giving students the larger task of comparing areas to see how they construct internal units…

This article describes one classroom activity in which the author simulates the Newtonian gravity, and employs the Euclidean Geometry with the use of new technologies (NT). The prerequisites for this activity were some knowledge of the formulae for a particle free fall in Physics and most certainly, a good understanding of the notion of similarity…

In this article, we demonstrate that prospective teachers' content knowledge related to defining mathematical concepts is dependent on content area. We use the example of generation (a research tool we developed in a previous study) to investigate prospective teachers' knowledge. We asked prospective secondary mathematics teachers to provide…

Introduction: The present paper shows the importance of a joint use of pencil and paper activities and of technology so that students may develop a complete understanding of ratio and proportion. A previous experience with strategy use when solving ratio and proportion problems provided background. Prompted by a recognition of the cognitive…

For the forty-third time, the Association for Educational Communications and Technology (AECT) is sponsoring the publication of these Proceedings. Papers published in this volume were presented online during the annual AECT Convention. Volume 1 contains 37 papers dealing primarily with research and development topics. Papers dealing with the…

This article presents a mathematical function for measuring the congruence among the strand representation in the enacted curriculum, state assessments, and district pacing guides. Using the Manhattan distance as its basis, this method allows researchers to incorporate the strand coverage of the enacted curriculum (i.e., the curriculum actually…

This paper examines three historical geometric constructions for handcrafting stringed instruments. Using elementary geometry--in particular, triangles--these methods can provide quite good rational approximations to the irrationals that arise from tuning instruments in equal temperament. Interestingly, continued fractions help explain the…

The seventeen-year-old Christiaan Huygens was the first to prove that a hanging chain did not take the form of the parabola, as was commonly thought in the early seventeenth century. We will examine Huygen's geometrical proof, and we will investigate the later history of the catenary.

A stopper is called "universal" if it can be used to plug pipes whose cross-sections are a circle, a square, and an isosceles triangle, with the diameter of the circle, the side of the square, and the base and altitude of the triangle all equal. Echoing the well-known result for equal cubes that is attributed to Prince Rupert, we show that it is…

Consider a circular segment (the smaller portion of a circle cut off by one of its chords) with chord length c and height h (the greatest distance from a point on the arc of the circle to the chord). Is there a simple formula involving c and h that can be used to closely approximate the area of this circular segment? Ancient Chinese and Egyptian…

Using the elementary tools of matrix theory, we show that the product of two rotations in the three-dimensional Euclidean space is a rotation again. For this purpose, three types of rotation matrices are identified which are of simple structure. One of them is the identity matrix, and each of the other two types can be uniquely characterized by…

A first-year teacher thrust mid-term into a mixed-grade geometry class in a tough high school struggles to sort out what works and does not work to engage the 120 students in meaningful learning. The students are largely from low-income urban families and demonstrate a broad range of abilities and motivation. The purpose of this study is to…

The Mathematics Performance Expectations (MPE) define the content and level of achievement students must reach to be academically prepared for success in entry-level, credit-bearing mathematics courses in college or career training. They were developed through a process that involved faculty from Virginia's two- and four-year colleges and…