THE SOCIAL CONSTRUCTION OF MATHEMATICAL MEANINGS THROUGH COMPUTER-MEDIATED COLLABORATIVE PROBLEM SOLVING ENVIRONMENTS

Abstract

The central focus of this thesis concerns how students' personal beliefs about mathematics can be transformed from a computational or mechanical viewpoint to a disposition of considering mathematical meanings, concepts, and ideas. The- thesis describes how this transformation can take place through the social construction of mathematical meanings in the context of computer-mediated collaborative problem solving. The findings of this thesis were derived from an analysis of the construction efforts involving two students of radically different personal epistemologies. The major contributions include: (I) the peer apprenticeship learning processes; (2) a contextual framework for mathematical problem solving; (3) the characterizations of an epistemology for interpreting higher levels of mathematical meanings: (4) an episode structure for establishing closure in mathematical meanings; and (5) the implications for mathematical learning based on our findings. The meaning-context model describes three different levels of contextual influences in mathematical problem solving: first, the meaning-symbol context; second, the meaning-interpretation context: and third, the meaning-intersubjectivity context. The meaning-symbol context describes students' cognitive operations at the symbol, problem, and situational levels, commonly known as cognition. When students mechanically apply heuristic; and formulae, they perform cognitive operations mostly at the symbol level. However, when students consider the reasonableness of their solutions in terms or the mathematical problem as a whole and also consider the conceptual relationships of the problem with other mathematical situations, such a cognitive act is known as meaning--symbol dialecticism. We emphasize that students need to acquire the problem and situational level perspectives and not just possess a symbol-level epistemology. How students approach problems whether all the symbol level or otherwise, depends very much on their interpretations, disposition, and beliefs concerning mathematics. Three factors-beliefs, disposition, and interpretation-constitute the meaning-interpretation context. Students' monitoring of their own beliefs is commonly known as metacognition. It is also necessary to consider how students' beliefs concerning mathematics are cultivated by their educational, social, and cultural settings. The three social factors - world review, practice, and intersubjectivity-constitute the meaning-intersubjectivity context. lntersubjectivity can only be achieved between students when they adopt the same social practices deriving from a common world view. Sociocognition commonly occurs when students have to think about how they can establish intersubjectivity due to differences in epistemology, after having in fact recognized their mutual differences in world view and practice. The present study focuses on the social co-construction efforts between two students, Dom (age: 13) and Ming (age: 14). Both boys attend the same school and volunteered to participate in this project. Ming normally scores better than Dom in school tests and examinations. Despite being part or the same school system, both boys had very different viewpoints about problem solving. Dom's problem solving behavior was characterized by engaging in making conjectures and proving them at the problem and situational levels. In contrast, mathematics was, for Ming, an activity in which one finds rules, algorithms, and incremental patterns for solving problems. He was more concerned with symbol level manipulations and obtaining the correct answer to mathematical problems. As Ming experienced continuous difficulties with his symbol level interpretations, we found that he temporarily suspended his own efforts and began to follow the problem solving directions of Dom. In the process, Ming learned how to monitor and regulate his own mathematical interpretations according to Dom's conceptualizations, and he was able to co-construct mathematical meanings with Dom. We refer to such an effort on Ming's part to "submit" to Dom's guidance as peer apprenticeship learning. Through the use of schema representations, Dom's conceptual perspectives were made visible for Ming to follow, and the constant refinements to these representations showed that problem solving was neither straightforward nor merely procedural, but rather zig-zag, and involved making conscious guesses. Ming gradually began to "play with ideas" instead of mechanically solving problems just to obtain a solution. Through personal experimentation, Ming appropriated a problem level epistemology that complemented his own systematic approach to mathematical problem solving. We also saw evidence of Ming beginning to formulate personal theories of relationships between numbers and evaluating his solutions at the problem and situational levels, thereby anchoring closure in his understanding. As a result of this study, we see evidence that students are able to "play with ideas," formulate personal "theories," and make conjectures if they are suitably led into such practices. Students can gradually develop the disposition for conceptual meanings in mathematics if they are exposed to collaborative problem solving with other partners capable or problem-level perspectives in problem solving. School students, especially the lower achievers, need to acquire a problem-level epistemology because many of them leave school without being able to apply what they learn in mathematics to the real world.