Teaching the Conceptual Revolutions in Geometry

Mathematics begins in human experience thousands of years ago as empirical and intuitive experiences. It takes the deliberate naming of concepts to help crystallize and secure those observations and intuitions as abstract concepts, and to begin separating the concept of number from specific instances of objects. It takes the creation of compact symbols to enable efficient calculation and to begin raising a consciousness of this activity we call mathematics. And it takes the sustained development and discussion of mathematical conventions and practices to create entire domains of mathematical thought, such as we find in geometry. The major innovations and conceptual reformulations are few in number, but these represent perhaps the greatest challenges to learners. Historically significant transformative events have their counterpart in the cognitive growth of the individual. This article examines the interplay between these big ideas in cultural history and the deliberate processes of cognitive change that are their counterpart in the educational process.