ABSTRACT
In arithmetic thinking the equals sign is conceptualized operationally. This conceptualization of the equals sign is absent of a relational sense - a sense that is pivotal for conventional treatment of algebraic equations. The design based research project investigated alternative cognitive foundations for algebraic reasoning. The researcher engaged in micro-ethnographic qualitative analysis. The analyses focused on the some experimental conditions and the extent to which the tools obfuscated or illuminated the phenomena. The analyses revealed the importance of spatially aligning the two journeys such that the relationship between the variable and integer could be identified. The critical coordination was facilitated by the participants' use of their models as a method for visualizing the two separate journeys as overlaid, almost as if these occurred simultaneously. The chapter argues that the design restructured the solution of unknown-value problems as centered on manual coordination of situated quantities.
