| Schoenfeld's (1992) "near decomposition" of mathematical thinking: the knowledge base, problem solving strategies, beliefs, monitoring and control, and practices provided a framework to look at the complexity of an applied college algebra student' s mathematical thinking. Particular attention was given to the inherent and student specific relationships between these categories. An intrinsic case study was conducted during the winter of 1997. Data were collected in the form of informal interviews, classroom observations, student work, and an exit interview at the end of the course. Analysis of data followed the methods of grounded theory provided by Glaser and Strauss (1967).; The knowledge base of the student was predominantly a mixture of facts, algorithmic procedures, and informal knowledge. Belief-driven memory programs attended primarily to that information necessary for success in the applied college algebra course. Informal knowledge helped her fill in the gaps when procedural knowledge did not meet demands made from external sources.; The student relied on the "finding a related problem" strategy to work routine problems. When confronting non-routine problems, she seemed to use basic strategies that reflected Polya's sense of heuristic reasoning, rather than the ones explicitly taught in the course.; As for monitoring and control, the participant's approach to routine problems was little more than "checking the answer." However, in non-routine problems, her metacognitive repertoire included planning, testing, and even abandoning non-productive strategies. She mostly disregarded self-regulation strategies in other mathematical thinking exhibited in the interviews.; The participant seemed to believe that there were two types of mathematics: classroom mathematics and mathematics for everyday life. Another significant aspect of her belief structure focused on the real-world contexts associated with the course mathematics. The influence of these beliefs was extensive in her attempt to make sense of the situations she encountered.; Finally, the student's practices substantially differed from those espoused by the mathematics community. She seemed to have few social encounters to help shape her mathematical thinking. The participant had to rely on other resources, like intuitive knowledge and personal theories, for her interpretation and sense-making of mathematics. |
