Technology in the Upper-Level Curriculum - Geometry

Always ready to try technology, I decided to teach DePauw's junior-level geometry course with a variety of tools: Geometer's Sketchpad (Jackiw, 1992-95 ), the Lénárt Sphere (Lénárt, undated), and PoincaréDraw (The Gap Group, 2000  -- written by Robert Foote and Nathan of Wabash College to demonstrate the Poincaré disk model of hyperbolic geometry). I continually felt challenged and stimulated by these students -- I am pleased that atmosphere has been repeated during several subsequent semesters. The question I have asked myself about this course was, "Why did we all feel so energized by this course? Why did the software do so much for the students?"

My students had all studied Euclidean geometry thoroughly in high school. They had been taught how to write proofs for geometry, and it was easy for me to show them how to transform their 2-column proofs into paragraph form. I then sent them to the computer lab to see what Geometer's Sketchpad would do. What was truly exciting for them was the enhanced capacity for visualization that came from the dynamic software. This became clear when I insisted that they work out illustrations for one problem set by hand, with a straightedge and compass. They struggled with the hand sketches and invariably missed the "special cases" that made refinements of the conjectures necessary. Later, I had them work with the Lénárt Sphere, and they fairly quickly recreated some of the crisis of 19th century geometry. Finally, they learned about hyperbolic geometry and appreciated how much PoincaréDraw helped in visualizing a "different" geometry. In the lab "Inscribed and Circumscribed Circles" students work through the same constructions in all three geometries and compare the results. [For the original Scientific Notebook file, click on the icon at the right.]

Our modern students do not have much experience with visualization in earlier courses -- in either two or three dimensions -- and this limits their abilities to think about geometry. The dynamic software packages (Geometer’s Sketchpad and PoincaréDraw) enhanced their visualization and allowed them to "think gometrically." They had already understood the basic concepts and had been taught to formalize these concepts. But the new capacity to visualize, provided by the software, opened up creative avenues for them. As one student said at the end of the semester during a taped discussion, the course showed them the "wild side of math." More than in abstract algebra or real analysis, the students understood and were stimulated by the potential for mathematical exploration.