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

The liansition to formal mathematical thinking involves the use of quantified statements as definitions from which fmther properties are consliucted by formal deduction Om quest in this article is to consider how students construct meaning for these quantified statements. Dubinsky and his colleagues (Dubinsky et al, 1988) suggest that the process occms through reflective abstraction, in which a predicate with one or more variables is conceived of as a mental process that is encapsulated into a statement (a mental object) by the process of quantification In this article, we report on a case study of a university student who consttucts the fOrmalism not from processes of quantification, but from his own visuo-spatial imagery Rather than construct new objects from cognitive processes, he reflects on mental objects already in his mind and refmes them to build his own interpretation of the formal theory This example leads us to consider the development of theory in the literature, in particular Piaget's notions of pseudoempirical abstiaction (focusing on processes encapsulated as mental entities) and empirical abstiaction (focusing on the properties of the objects themselves) It has often been noted that highly successful mathematics researchers show strong preferences for different kinds of appwach (e g Poincare, 1913; Hadamard, 1945; Kuyk, 1982; Mac lane, 1994; Sfard, 1994) Some have a broad problem-solving sliategy, developing new concepts that may be useful before making appropriate definitions to form the basis for a formal theory. Others rue more formal from the beginning, working with definitions, carefully extracting meaning from them and gaining a symbolic intuition for theorems that may be liue and can be proved In a recent research study of novice mathematicians' styles of doing mathematics (Pinto, 1998; Pinto and Tall, 1999), we found analogous differences among students' stlategies for learning mathematics Some worked by extracting meaning beginning with the formal definition and consliucting properties by logical deduction This sliategy is consonant with the theory of Dubinsky and his colleagues, 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) However, our research shows that there are students who use an entirely different sliategy. In this ruticle, we focus on one successful student who builds from his imagery, giving meaning to the definition by producing a highly refined image that suppmts his formal ruguments We suggest he is not encapsulating a cognitive process into a mental object; he progresses by refining and reconstructing his existing imagery, until it is in a form that he can use to consliuct the formal theory He uses both visual and symbolic coding of ideas in a complementary manner that is characteristic of the dual-coding theory of Paivio (1971, 1986) His thinking processes resemble those of mathematicians using broad problem-solving strategies, rather than those who focus more pruticulruly on purely formal deduction The data presented here form part of a larger qualitative research study of students' understanding of real analysis (Pinto, 1998) A gmup of UK. mathematics undergraduate students was followed during the students' first two terms studying analysis, with each pruticipating in an hom's individual interview every two weeks. Interviews were recorded on tape and fully lianscribed for detailed analysis Procedmes of data collection were compatible with those used in Strauss's method of building up a theory from the data (see Strauss, 1987; Strauss and Corbin, 1990) Pinto followed a plan where each set of questions formulated for successive interviews was built upon results of the analysis of data from previous interviews, enabling a gradual transformation and emichment of the theoretical viewpoint