Exploration, innovation, proof: For students, teachers, and others who are curious, keeping your mind open and ready to investigate unusual or unexpected properties will always lead to learning something new. Technology can further this process, allowing various behaviors to be analyzed that were previously memorized or poorly understood. This article shares the adventure of one such discovery of exploration, innovation, and proof that was uncovered when a teacher tried to find a smoother way to model conic sections using dynamic technology. When an unexpected pattern regarding the locus of an ellipse's or hyperbola's foci emerged, he pitched the problem to a ninth grader as a challenge, resulting in a marvelous adventure for both teacher and student. Beginning with the evolution of the ideas that led to the discovery of the focal locus and ending with the significant student-written proof and conclusion, we hope to inspire further classroom use of technology to enhance student learning and discovery.
Chris Harrow and Lillian Chin
Irina Lyublinskaya and Dan Funsch
Symbolic geometry software, such as Geometry Expressions, can guide students as they develop strategies for proofs.
Samuel Obara and Zhonghong Jiang
Applying Zometool, vZome software, and The Geometer's Sketchpad to tetrahedrons nested in cubes enhances students' spatial visualization skills.
Christopher J. Bucher and Michael Todd Edwards
In the introductory geometry courses that we teach, students spend significant time proving geometric results. Students who conclude that angles are congruent because “they look that way” are reminded that visual information fails to provide conclusive mathematical evidence. Likewise, numerous examples suggesting a particular result should be viewed with skepticism. After all, unfore–seen counterexamples render seemingly valid conclusions false. Inductive reasoning, although useful for generating conjectures, does not replace proof as a means of verification.