Semantic Innovation: Arithmetical and Algebraic Metaphors Within Narratives of Learning.

This paper examines a theoretical perspective on the ways in which children progress in learning mathematics. It suggests that there is a difficulty in associating teaching discourses with the mathematics they locate. This can result in an incommensurability between alternative perspectives being offered. The paper resists attempts to privilege any particular account but rather demands an analysis of these discourses and their presuppositions. In developing these themes the paper invokes Ricoeur‘s analysis of time and narrative as an analytical approach to treating notions such as transition, development and progression in mathematical learning. His notion of semantic innovation is introduced. This embraces both the introduction of a new metaphor into a sentence or the creation of a new narrative which reorganises events into a new ’plot‘. The notion is utilised in arguing that the shift in the student‘s mathematical development from arithmetic to first order linear equations with unknowns reconfigures the contextual parameters governing the understanding of these mathematical forms. It is also utilised in showing how alternative approaches to accounting for such transitions suit different and perhaps conflicting outcomes. For example, demonstrating awareness of generality or performing well in a diagnostic test featuring the solution of linear equations. [ABSTRACT FROM AUTHOR]