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Introduction In the Fall of 1997 Ohio University replaced its traditional “Foundations of Geometry” sequence with one in which the students develop their own sets of axioms and use them to establish some well-known results of plane geometry. By constructing their own axioms, the students gain a sense of both the source and the role of formal axiomatic systems. Since axioms are introduced and developed as needed, the students gain an appreciation of the significance of each axiom as it is added to the set of axioms. As the students start the course by developing their own axioms, the course is not amenable to the traditional lecture approach of developing the material. The students develop their axiom systems while working in structured cooperative groups and making use of a variety of manipulatives and software programs during their discussions. By the end of the sequence, the students have addressed all of the concepts included in the traditional course and more. They also gain, in our belief, a deeper understanding of the material than would be developed in the traditional lecture style course. The General Approach The projects described in this paper were designed for a geometry course taken primarily by prospective middle or high school teachers. The major theme of the projects is to connect experience and abstract mathematics. The early projects are designed to give students experience in working with non-Euclidean geometries while exploring the validity of certain common sense propositions in these geometries.

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