This volume attempts to bring together a collection of reports on the Geometric Supposer, a series of computer software environments which can be a tool for exploring particulars and generalizations in geometry. The book contains the following chapters: (1) "A Personal View of the Supposer: Reflections on Particularities and Generalities in Educational Reform" (Judah L. Schwartz); (2) "The Geometric Supposer in the Classroom" (Beth Wilson); (3) "Overcoming Visual Obstacles with the Aid of the Supposer" (Michal Yerushalmy and Daniel Chazan); (4) "Generalization in Geometry" (Michal Yerushalmy); (5) "Discovery Courses Are Great in Theory, But..." (Christopher C. Heady); (6) "Instructional Implications of Students' Understandings of the Differences between Empirical Verification and Mathematical Proof" (Daniel Chazan); (7) "Posing Problems: One Aspect of Bringing Inquiry into Classrooms" (Michal Yerushalmy, Daniel Chazan, and Myles Gordon); (8) "Teachers' Thinking about Students' Thinking about Geometry: The Effects of New Teaching Tools" (Magdalene Lampert); (9) "How the Supposer Changed My Life: An Autobiography" (Richard Houde); (10) "From Recitation to Construction: Teachers Change with New Technologies" (Martha Stone Wiske and Richard Houde); (11) "Geometric Supposer Urban Network" (Grace Kelemanik and Mark Driscoll); and (12) "What Is the Supposer a Case Of?" (Myles Gordon). An appendix contains a guide to the Geometric Supposer. (MKR)