This Small Business Innovation Research Phase I project builds on the success of The Geometer's Sketchpad(, one of very few pieces of educational software to have a real effect on the teaching of mathematics in secondary schools. The proposed research will design and prototype a next generation of this software --The Mathematician's Sketchpad --an integrated exploratory environment for algebra and geometry utilizing declarative constraints as a new mathematical modeling tool. The Geometer's Sketchpad allows users to carry out geometric constructions using Euclid's compass and straightedge tools augmented by reflection, rotation, and dilation. At any time, users may drag objects in the sketch using the computer's mouse, and all of the other objects will dynamically adjust their position and size to maintain the constructed relationships. This simple idea has profound implications. The Mathematician's Sketchpad will incorporate these tools and methods and go beyond them by allowing users to declare constraints on objects after they have been constructed, and by allowing algebraic specification of constraints. The revolution in the teaching of geometry being brought about by The Geometer's Sketchpad, reveals the opportunity to spread its fundamental ideas of constraint-based modeling and dynamic dragging to the rest of the secondary mathematics curriculum. The objectives of this Phase I research project are to design and prototype the critical elements of The Mathematician's Sketchpad and describe and prototype the support materials that will ease its introduction into classrooms.
