Thomas E. Kieren

Recursion and the Mathematical Experience

Mathematics for children, as they ‘do’ it and build onto their own knowledge, is a complex phenomenon. As with personal knowledge of mathematics for any person, it is multifaceted, a view supported by mathematicians (e.g. Davis and Hersh, 1986) and psychologists interested in mathematics education (e.g. Vergnaud, 1983). There are many ways to observe and hence say something about children’s mathematics. One could consider mathematical thought mechanisms used by children. One could analyze specific aspects of the complexity. However, in this essay we try to observe and talk about children’s mathematical experience as a complex whole.

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.

Orthogonal Reflections on Computer Microworlds, Constructivism, Play and Mathematical Understanding

Abstract This article contains two reflections on the research activity by both the researchers and the children in the Rational Number Project. Both reflections take persons to be bringing forth worlds in the spheres of behavioral possibility which includes their own history of action and reflection. The first reflection highlights what it means to take the childrens actions as a critical source of their mathematics and the coemergent nature of both the “mathematics of children” and “childrens mathematics” in the work of the Project. In so doing, this reflection tries to point out how the researchers own histories color the way in which they observe childrens constructions. The other reflection considers the nature of playlike activity of the children as they work in the computer environments of the Project. In particular, it points to the aesthetics in the childrens actions and reflections and to the role of childrens structures in their ability to duplicate and replicate the actions of others in ...

Toward an embodied view of the mathematics curriculum in a world of technology

Over the past 30 years, two views of curriculum in which ideas and techniques from informatics are used in mathematics have dominated. On one view the computer or multi-media device carries the curriculum to an individual and is programmed to evaluate the individual learning. On the second view the computer is a tool for personal construction of mathematics. This essay adds to the second view and contends that a curriculum in action must be observed as embodied, not only in the thinking of the individual, but in the interactions in the computer or multi-media environment and in the opportunities for mathematical practice it offers.