Overcoming Physicality and the Eternal Present: Cybernetic Manipulatives

Abstract

This chapter explores the visual and imagistic aspects of cybernetic manipulatives, how they may be designed to improve upon physical manipulatives, what their potentials and pitfalls may be, and how they fit into the larger evolution of technology use in support of mathematics learning. We will examine questions of ,physicality - of representations, dynamic connections between ,natural - and formal representations, and, especially, ways that new forms of records of actions may alter these connections and elevate levels of thinking involved in the doing of mathematics, from low-level computation to higher level planning, strategic and structural thinking. This paper extends earlier work [11, 12, 13], which also includes references to the wider literature relating to these topics, references not repeated here. It also relates closely to other papers on another aspect of the representational use of new technologies, dealing with dynamic linkages between formal mathematical notations and authentic human experience, particularly as instantiated in realistic simulations [14, 15, 16].