Cognitive Tendencies and Abstraction Processes

This chapter is devoted to the subject of the cognitive processes that take place during the learning of algebraic language. We analyze its close relationship with cognitive tendencies that are observed in students during the learning of algebraic syntax and its use in solving problems. Eleven cognitive tendencies are described, well identified in empirical studies, some referring to the presence of readings by learners on different levels of language or sign systems, which may obstruct the possibility of solving an algebraic task.

We explore also the processes of abstraction in terms of the theoretical notions of meaning and sense, in relation to mathematical sign systems (MSSs), seen in strata. The material for this exploration is the basic protocol or plan of two interviews, one concerning the solution of linear equations and the other concerning proportional variation. The plan of the interviews is set out and analyzed in episodes, incorporating into each episode comments related to the transformations of texts (algebraic expressions or geometric figures or propositions) for the solution of the task presented. These texts are located in more abstract or less abstract strata of MSSs. The chapter ends with a list of cognitive tendencies identified in the various episodes.