Building Formal Mathematics on Visual Imagery: A Case Study and a Theory.

The transition to formal mathematical thinking involves the use of quantified statements as definitions from which further properties are constructed by formal deduction. Our quest in this paper is to consider how students construct meaning for these quantified statements. Dubinsky and his colleagues (1988) suggest that the process occurs through reflective abstraction, in which a predicate with one or more variables is conceived as a mental process that is encapsulated into a statement (a mental object) by the process of quantification. In this paper we report a case study of a student who constructs the formalism not from processes of quantification, but from his own visuospatial imagery. Rather than construct new objects from cognitive processes, he reflects on mental objects already in his mind and refines them to build his own interpretation of the formal theory. This example leads use to consider the development of theory in the literature, in particular Piaget’s notions of pseudo-empirical abstraction (focussing on processes encapsulated as mental entities) and empirical abstraction (focussing on the properties of the objects themselves). It has often been noted that highly successful mathematics researchers show strong preferences for different kinds of approach (e.g. Poincare, 1913; Hadamard, 1945; Kuyk, 1982; Maclane, 1994; Sfard, 1994). Some have a broad problem-solving strategy, developing new concepts that may be useful before making appropriate definitions to form a basis for a formal theory. Others are more formal from the beginning, working with definitions, carefully extracting meaning from them and gaining a symbolic intuition for theorems that may be true and can be proved. In a recent research study of novice mathematicians’ styles of doing mathematics we found analogous differences between students’ strategies for learning mathematics (Pinto 1998, Pinto & Tall, 1999). Some worked by extracting meaning—beginning with the formal definition and constructing properties by logical deduction. This strategy is consonant with the theory of Dubinsky, in which multi-quantified statements are grasped by working from the inner quantifier outwards, converting a predicate (as a process) into a statement (as a mental object).