Mathematics teachers' mathematical reasoning and problem solving conceptions

Problem-solving and mathematical reasoning are two important and interrelated concepts in mathematics education. The beliefs or understandings of mathematics teachers about these concepts may influence how they teach mathematics. Previous research points to the necessity for teachers to have a good understanding of the concepts that shape their teaching practices. However, few studies examined mathematics teachers' conceptions of problem-solving and mathematical reasoning. Therefore, this qualitative study focuses on mathematics teachers' problem-solving and mathematical reasoning conceptions. In the study, using semi-structured interviews, 13 volunteered mathematics teachers were asked to describe problem-solving and mathematical reasoning, as well as generate metaphors for problem-solving and mathematical reasoning. Content analysis of these interviews reveals that participant teachers have three conceptions of problem-solving: a) a means to finding a solution, b) a skill requiring steps or procedures and c) a means to achieving other ends. Similarly, findings reveal that participants have three conceptions of mathematical reasoning: a) thinking, b) problem solving and c) making inference. It was also found that participants tended to create end-product or end-product+process metaphors for problem-solving and mathematical reasoning. The results demonstrated that the majority of mathematics teachers defined mathematical reasoning as problem-solving. The results also show that the most of the participant teachers do not have high levels of awareness of problem-solving and mathematical reasoning. In addition, this study desribes teacher profiles by comparing the conceptions exhibited in problem-solving against those in mathematical reasoning. This study contributes to the existing literature by examining problem-solving and mathematical reasoning concepts together. The results of this study could be used to develop pre-service and in-service mathematics teachers' conceptions. The findings are interpreted in relation to previous research on problem-solving and mathematical reasoning conceptions.