The flow of mathematical ideas in group problem -solving sessions
This study examines the mathematical ideas of students and the representations used to express these ideas, identifies factors contributing to the development, retention, abandonment, acceptance or rejection of the ideas, and traces the movement of these ideas among students and groups in an open-ended, problem-solving activity. The 17 students in the study were high-school precalculus students involved in a summer mathematics institute that is part of an NSF funded longitudinal study of the development of mathematical thinking and proof making. The data used for the study were collected over a four-day period while the students were engaged in modeling the growth of a spiral shell. The method of analysis of the data involved the use of grounded theory and the partitioning of the process into smaller steps utilizing seven non-linear phases.
The students in the study developed representations of their mathematical ideas, and constructed mappings among the representations through the use of the overhead projector and graphing calculator.
Students retained and continued to develop ideas when the students expressed success in the development of the idea, received positive feedback concerning the idea, or received encouragement to pursue the idea from others. Factors contributing to the acceptance of another's idea included a lack of an idea to fill a need, the lack of success of one's own idea, and a researcher intervention in which the idea was suggested.
Students rejected or did not respond to others' ideas when they expressed the success of their own ideas, when the ideas were presented in a way that did not relate to the discussion at hand, or when there was a lack of a clear pronouncement or explicit expression of the others' ideas.
This study categorizes and explains behaviors of students observed in the classroom during small group work, and details the student criteria for the selection of mathematical ideas during problem solving. The conceptual framework and the method of analysis provide a foundation upon which to examine and display graphically the movement of ideas in a collaborative classroom setting.
0533: Secondary education