ABSTRACT
This chapter explains the developed terminology and theoretical framework to discuss and analyses the representational characteristics of new or potential algebra learning environments, especially those that support the linking of different algebraic representations. It discusses deliberately oriented towards future research, indeed towards the research needed to design the future. To consider research from any other perspective would contribute to the well-documented and continuing failure of school mathematics to serve students' real needs. The chapter deals with the notion of mental representation as the means by which an individual organizes and manages the flow of experience and describes the notion of representation system as a materially realizable, shared cultural or linguistic artifact. The medium in which an action notation system is instantiated has a great bearing on the kinds of actions that may be supported and the way that consequences of those actions are displayed.
