This book shows how geometry can be learned by starting with real world problems which are solved by intuition, common sense reasoning and experiments. Gradually the more formal demands of mathematical proofs get their proper place and make it possible to explore new applications. This process helps students to feel the need for precise definitions and procedures, to contribute to the construction of an axiomatic system, and to experience the power of systematic reasoning.
The course is designed for students in a Nature& Technology strand which prepares for studying the sciences or technology at university level. Its goal was basically to reintroduce proof in a meaningful way in the late 1990s Dutch secondary education curriculum. Following the educational view of the Freudenthal Institute this is not done by stating Euclids axioms on page one, but rather a starting point is chosen in students intuitions and tentative solutions of problems that are experienced as real and relevant.
The photograph on the cover shows students exploring one of the problems from the midpart of the course in the computerlab.
Introduction; Geometry between application and proof, a general introduction; Geometry, classical topics and new applications; Given: circle with butterfly or: how do you learn proving?; Distances, edges and domains; Voronoi diagrams; Reasoning with distances; Computer practical Voronoi diagrams; A special quadrilateral; Exploring isodistance lines; Shortest paths; Example solutions; Worksheets part I; Part II: Thinking in circles and lines; Using what you know; The circle scrutinized; Finding proofs; Conjectures on screen; Proving conjectures Clues for chapter 3 and 5; Part III: Conflict lines and reflections; Preface; Edge and conflict; Parabola, ellipse and hyperbola; Analytic geometry; Conic sections; Sources of some of the illustrations.
