My students begin my geometry for elementary education course by thinking about the geometry of the Earth's surface. Each group of students gets their own beach ball or tennis ball for hands-on experimentation.
Students create definitions of shortest distance, straight lines, parallel lines and triangles on the sphere, and then explore the consequences of those definitions. Further experiments explore the properties of intersections of lines, sums of angles on triangles, congruence theorems for triangles, and parallel properties in spherical geometry.\par Pedagogically, this project encourages students to work with each other and with the instructor at the beginning of the semester. Furthermore, it demonstrates to them the power of their own intuition and that their ideas can be more important than those of an authority.