Problem solving as an approach to the teaching and learning of mathematics

This thesis presents the findings of a research project, which examined ways in which the participating teachers learned to teach mathematics effectively using a problem-solving approach. The research also examined the interactions between students during group work that were productive to their mathematical learning so that the educational aims of the problem solving lessons could be realized. The research was collaborative, and was guided by a naturalistic inquiry and an action research philosophy. The pilot study for this research was conducted in the month of August 1996, during which one teacher and a sixth form class from a school in the city of Dunedin, New Zealand were involved. The data collection period of this research spanned six months, from early May 1997 to early November 1997. Two teachers, two sixth form classes and a third form class from the same school, participated in the research. A total of 62 mathematics lessons were video-taped and the researcher also took running records of interactions in the classroom. On top of these, numerous informal discussions were held between the researcher and the teachers to resolve the dilemmas encountered by the teachers in the classroom. A number of these discussions were audio-taped. A framework was developed to analyze the data. The framework involved an initial identification of patterns of interest. Segments of the tapes where these patterns were observed were transcribed throughout the research and grouped under these patterns. This was followed by episode-by-episode analyses, which were guided by a number of themes. Finally, the comparative analysis involved a meta-analysis of the episodes to develop an overview of the progress under the various themes. Even though the participating teachers had the mathematical knowledge and the pedagogical skills required for a problem-solving approach to the teaching and learning of mathematics, a number of weaknesses were identified in their problem solving lessons at the beginning of the research. These, together with the numerous situations they encountered in the classroom that conflicted with their previous practice, and the informal discussions held to resolve them, created a context for the teachers to learn. Not only did the teachers progress dramatically in ways such as devising rich problems for a lesson and scaffolding students' learning, they were also motivated to carefully consider the appropriate classroom context required of a problem-solving approach. The professional development demonstrated by the participating teachers provides considerable support for an action research approach. Even though the use of new instructional material by the teachers themselves might accomplish certain educational objectives, the involvement of the teachers in developing new teaching skills and approaches and understanding conceptually what happens in the classroom brought about a radical change in the classroom programmes. The reciprocal obligations and expectations for the problem solving lessons were constantly negotiated between the teachers and the students. As such, the students improved in group work. The research identified the nature of exchanges between students during group work that were productive for their mathematical learning. As the teachers progressed in teaching through problem solving, they were able to capitalize on the outcomes of group work to realize the educational aims of their problem-solving lessons. The findings offer considerable support for group work as a strategy to implement problem solving in the classroom