Linking logo,levels and language in mathematics

In his book Mindstorms (1980) Papert discusses Turtle Geometry as ego-syntonic or fitting the ways of thinking of the child as a geometric knowledge builder. The van Hiele theory (Wirzup, 1976) looks at geometric knowledge building as occuring through a necessary sequence of levels. Thus if Turtle Geometry is ego-syntonic it would seem that one could apply these van Hiele levels to better describe and understand this form of geometry. It is the first purpose of this paper to provide such an analysis.Using Logo to do Turtle Geometry requires that a child (or learner at any level) do geometry through the deliberate use of language. Thus one ought also to be able to relate theoretically levels of language use to Turtle Geometry. It is a second purpose of this paper to relate a language use framework suggested by the work of Frye (1982) to the language activities of Turtle Geometry.