ABSTRACT
The chapter focuses on three kinds of mathematical meaning which seem to be particularly critical in terms of refining what it means to do, as well as to learn, mathematics: these are formalisation, generalisation, and abstraction. Each of these forms of meaning is examined from the starting point of a vignette involving computationally-expressed mathematics. Mathematical meaning can be derived from being recognised by a computer. The chapter draws on a corpus of data derived from studying children's work in a variety of computer-based mathematical environments which attempt to facilitate mathematical expression and learning. The general problem has recently been addressed by, whose reanalysis of Piaget's water-level experiments attempts similarly to emphasise the coordination of local knowledge in building abstractions. The chapter attempts to draw together some of the findings and develops a possible theoretical framework within which to interpret them. According to Kegan Meaning is the primary human motion, irreducible.
