The role of abstraction as a learning process in mathematical problem-solving

The purpose of this study was to deepen and elaborate the understanding of the processes of constructing conceptual knowledge during mathematical problem solving. The study focused on the internal activity of the learner with particular emphasis on the ways that learners elaborate, reorganize, and reconceptualize the meanings they give to problem statements while engaged in mathematical problem solving. In particular, a clarification of the role played by reflective abstraction was sought. This included a focus on both the activity from which the learner abstracted and the conceptual structures that resulted from the abstraction. The explanation of the role played by abstraction in mathematical problem solving was guided by a theoretical perspective which emphasized the learner\u27s internal activity. This framework was developed by examining current models of learning and problem solving. Four subjects were interviewed as they solved a set of similar algebra word problems while thinking aloud. The nonstandard format of the tasks afforded an opportunity to observe different ways that solvers construct and utilize conceptual knowledge during problem solving. First, the tasks provided opportunities to observe the solvers as they faced problematic situations. In resolving such situations the solvers were seen as having developed a structure from their solution activity. Second, the similarity among the tasks allowed an opportunity to observe how the solvers\u27 newly constructed structures influenced subsequent solution activity in similar but different situations. All interviews were videotaped. A detailed case study was prepared for each subject by analyzing their written and verbal responses together with the video protocols. Five increasingly abstract levels of solution activity were inferred from the subjects\u27 solution activity. These levels encompassed two broad categories. The first category involved levels of solution activity below the level of representation. These levels indicated solution activity of a relatively unsophisticated nature where solvers had not internalized their activity. The second category involved levels of solution activity where solvers demonstrated that they could re-present their solution activity and perform mental operations on them in thought. These levels indicated that the solvers attained a high level of awareness of the structure of their solution activity to the extent that they could reflect on potential activity in new situations and anticipate its results