The Emergence of Meaningful Geometry

Abstract

This chapter is about a change in geometry education that took place in the last century. We discuss the emergence of meaningful geometry in the Netherlands. Of course, this was not an isolated reform. Worldwide, mathematicians and mathematics educators came up with new ideas as an alternative for the traditional axiomatic approach to teaching geometry. Already at the end of the 19th century, Klein had made a start with this by advocating a transformation geometry, but in this approach the axiomatic structure still played a main role for ordering activities. This was not the case in the work of Fröbel and Montessori who by building on students’ intuitions and their attention for students’ development of spatial insight were important driving forces towards a meaningful approach to geometry education. In the Netherlands, the pioneers of such a geometry were Tatiana Ehrenfest and Dieke van Hiele–Geldof. Freudenthal was a great promoter of their ideas. For him, geometry is ‘grasping space’, meaning that geometrical experiences should start with the observation of phenomena in reality. Supported by Freudenthal, from the 1970s on, experiments were carried out in the Netherlands to develop a new intuitive and meaningful approach to geometry education, in which the focus was on spatial orientation. How big the change in geometry education that resulted from these experiments was, is illustrated in this chapter by comparing geometry problems from two Dutch mathematics textbooks: one from 1976 and one from 2002.