Teaching for Robust Understanding. Students’ accounts of learning mathematics in problem-solving classrooms.
The aim of this thesis is to explore connections between a problem-solving approach to teaching mathematics and student learning. The student learning has not been based on test results, but on measures of activity, knowledge and skills. Student accounts from problem-solving classrooms were used to measure this. One teacher’s approach to using problem-solving activities when teaching mathematics has been used as a base for the study. Using qualitative interviews, data was collected from students in the 7th and 10th grade classes at two different schools where he teaches. These interviews were analyzed through a framework designed to identify characteristics of powerful mathematics classrooms. The theoretical perspective of this thesis connects research on knowledge types and knowledge quality to theory on what understanding is needed for- and obtained through mathematical problem solving. It is argued that there is a clear connection between the ability to solve problems, and the capability of connecting mathematical concepts, applying strategies and reasoning abilities. A qualitative approach was taken in this study, conducting interviews to collect data. Although many lessons were observed, collecting data has not been done through observation. In order to acquire the students’ perceptions on participating in problem-solving classes, using interviews were chosen as the most suitable method. Concluding the thesis, there were several findings. First, it concluded that the students in these classes learned through a sociocultural approach to learning. Working in groups and engaging in productive struggles together with peers were connected to Vygotsky’s ideas of Zone of Proximal Development and scaffolding. Collaborative work with problem solving helped the students engage with challenging tasks. Second, the students showed good abilities in applying strategies when solving problems. Although the students did not express that they had learned explicit strategies, they interacted with the tasks in ways that are connected with effective problem solving. Third, there seemed to be a correlation between the students who regularly worked with problem-solving tasks and the quality of knowledge they possessed. The interviewed students were all in possession of a variety of knowledge and skills which they were able to connect and reason with through the problem-solving tasks, making the case that the students in these classes had achieved deep learning. Last, there are implications that problem solving should be a means towards mathematical competency, and not the end result of learning mathematics.