I made a cameo appearance in my colleague Scott Nelson's presentation on how using computer software intelligently has made his Analytic Geometry course vastly more accessible. I loved his presentation. (If you teach in a member school of the California Association of Independent Schools, you will get a chance to catch it this spring at the Northern California meeting.) In his fifteen years of teaching this class, he went from "explaining" and hand-waving, which met little success, to hands-on physical models, which helped a bit, to computer models, which helped a lot. Now he does all three.

Interestingly, unlike me, Scott is not naturally drawn to computers. What led to his change of heart was the powerful impact technology had on his students (we teach in a one-to-one laptop school.) Cabri and Cabri 3D were the applications that transformed the class. (Presumably, other interactive geometry software would have worked in 2D, but there really is no alternative in 3D.) Two of his main examples were the geometry of sunrise and sunset, as seen from space, and the conic sections, which he demonstrated in the case of the ellipse, with the help of our colleague Meghan Lee.

He started from the concept of locus (the geometric location of points that satisfy a certain condition.) He used this to have students construct ellipses two ways in Cabri, and then figure out parametric and standard equations based on those constructions. (A nice touch is showing the equivalence of the equations obtained that way with the equation automatically given by Cabri.) My part was showing how basic theorems from 2D geometry provide the backbone of a straightforward proof that the ellipse is indeed a conic section. (You can see the full argument for my segment on this page of my Web site. I updated it by adding "prerequisite" 2D figures.) By the end of the unit, the students have approached the ellipse three times, and made many algebra-geometry connections. This is vastly more effective than the usual approach, which privileges symbol manipulation, and avoids any explanation of the 3D geometry.

--Henri

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