From a Different Perspective: Building on Students' Informal Knowledge
Koeno P. Gravemeijer
This chapter describes the Dutch approach to geometry education, an approach that builds on students' informal knowledge about geometric aspects of everyday life situations. Most students have a great deal of informal geometrical knowledge at their disposal, and even young children can model situations with gestures and imagine events and objects from different perspectives. In this chapter, we argue that this informal knowledge can be explicated and built on in geometry education within the framework of the domain-specific instruction theory of Realistic Mathematics Education (RME), an approach developed in the Netherlands. The key characteristics--reinvention through progressive mathematization, didactical phenomenological analysis, and use of emergent models-- function as the scaffolding for an RME curriculum. Such an approach enables students to construct their own mathematical knowledge, fosters a reflective attitude toward the world, and fits in with the overall philosophy that views mathematics as a human activity. As such, this approach also accommodates the critique by Ehrenfest-( 1931), van Hiele- Geldof ( 1957), Freudenthal ( 1971), and van Hiele ( 1973) of traditional instruction of Euclidean geometry in the Netherlands.
Although other countries endorse a similar approach (see, e.g., National Council of Teachers of Mathematics [NCTM], 1989), the presentation in this chapter is limited to the Dutch curriculum. The first third of this chapter looks at ways in which the Dutch curriculum builds on informal knowledge. The next section looks at the theoretical background for Dutch mathematical education. The final third of the chapter elaborates on the theory of RME as applied to geometry education.
Every day, whether modeling a physical activity for someone else (e.g., teaching a child to tie a shoe), or mirroring the movements of another (e.g.,