Concrete Models and Abstraction Processes

This chapter deals with concrete modeling in teaching the elements of algebraic syntax and the processes (abstraction) that arise both in modeling itself and in the use of the syntactic skills learned in order to solve word problems. The subject is approached by means of a clinical study case “Operación de la incógnita” (“Operating on the Unknown”), the design and experimental performance of which are within the theory of local theoretical models, following the recursive diagrams that appear in Chapter 3. The study deals with the transition from arithmetic to algebraic thought, and in the study the idea of didactic cut is introduced for this context. We also begin the discussion of the dialectic relationship between semantics and algebraic syntax. Throughout the chapter we utilize the phenomenological analysis of school algebra presented in the previous chapter and the notions of mathematical sign system and language strata, dealt with in Chapter 1.