ABSTRACT
Cognitive theorists (Case, 1985; Fischer, 1980; Halford, 1982? in preparation) have proposed, from different theoretical perspectives, that there is an upper limit to children’s capacity to process information that increases with maturation and learning. It is argued that such an upper limit determines the level at which children are able to cognize mathematical and other concepts. Halford (1982; in preparation) proposed three levels of thinking that depend on children’s increasing capacity to match systems of symbols to elements in the environment. Each of the levels of thinking proposed by Halford defines a class of tasks with a common degree of structural complexity. At Level 1R (relational mappings as described by Halford, in preparation) tasks are defined as ordered pairs that include binary relations and unary operators. At Level 2 (system mappings) tasks are defined as sets of ordered triples, including combinations of binary relations (as in transitive reasoning), ternary relations, and binary operations (as in e.g., addition of single-digit numbers). In this chapter the concern is with children at Levels 1R and 2 of thinking.
