The use of the Mira, a geometric device for studying reflections, is discussed. Proved are that all Euclidean constructions can be made with a Mira and that the Mira can be used to trisect an angle. (DT)

A course for high school students who have completed one semester of geometry explores neutral geometry (no parallel postulate), Lobachevskian, and Riemannian geometries. The teachers believe that students who study several postulational systems gain a better understanding of Euclidean geometry. (SD)

This clinical, exploratory study describes processes used by 8 ninth-graders learning to solve non-routine geometry problems and changes in those processes as instruction in heuristic methods was given. Directions for future research are indicated, and several hypotheses to be investigated are suggested. (DT)

Using Taxi-Cab Geometry (a non-Euclidean geometry program) as the starting point, 14 mathematically gifted British secondary students (ages 12-14) were asked to consider the differences between Euclidean and Non-Euclidean geometries, then to construct their own geometry and to consider the non-Euclidean elements within it. The positive effects of…

Illustrates how mathematicians work and do mathematical research through the use of a puzzle. Demonstrates how general rules, then theorems develop from special cases. This approach may be used as a research project in high school classrooms or math club settings with the teacher helping to formulate questions, set goals, and avoid becoming…

This study investigated the understanding of and reasoning about geometry of 120 mathematically talented students in the sixth through eight grades prior to taking a course in geometry. Results found that, although the students were able to deduce meaning from context, they lacked understanding of basic definitions, concepts, and properties.…

Presents new proofs of the Pythagorean theorem while exploring examination questions. Briefly reviews the work of Elisha Scott Loomis, a mathematician who amassed 320 different proofs of the theorem. (DDR)

Shows how an empirical approach to geometry using computer-based dynamic geometry software can create didactic situations in which students require proofs. Reports classroom experiences that show where students felt the need for proof in order to explain phenomena or to convince themselves of counterintuitive results. (Author/MKR)

Describes three activities in discrete mathematics that involve coloring geometric objects: counting colored regions of overlapping simple closed curves, counting colored triangulations of polygons, and determining the number of colors required to paint the plane so that no two points one inch apart are the same color. (MKR)

Includes lesson plans and worksheets that deal with transformational geometry, specifically reflections. The lesson for grades three to four focuses on angles of incidence and reflection and that for grades five to six involves mirror images. (MKR)

Presents an inexpensive laboratory experiment that combines the recommended techniques for teaching fractal geometry in the classroom with the standard procedures for studying electrochemical deposition of ramified patterns in the regime of low solution concentration and low applied constant driving force. Introduces students to fractal growth…

Presents a perspective on the nature of the use of proofs in high school geometry. Compares three currently used approaches to the geometry curriculum: (1) traditional geometry with no explanation of the axiomatic system; (2) hands-on geometry with no proofs until the end of the course; and (3) experimental geometry with no proofs. (DDR)

Provides suggestions for implementing the geometry standards in an elementary curriculum by including all aspects of geometry. (KHR)

Analyzes one student's thinking using the Van Hiele levels of geometric thinking. (KHR)

Describes lessons with year 7 and year 9 classes that used the ATM Active Geometry package. Presents the main activities and how to use prepared files in the geometry package for mathematics instruction. (KHR)