Preservice teachers' mathematical understanding exhibited in problem posing and problem solving

Fostering conceptual understanding in mathematics classrooms is an important goal in mathematics education. To support this goal, we need to be able to diagnose and assess the extent to which students have conceptual understanding. In this study we employed a problem-posing task and a problem-solving task in order to diagnose and assess preservice teachers' mathematical understanding of fraction division. The results of the study show that although over 99% of the preservice teachers could correctly perform fraction division, they were much less likely to exhibit conceptual understanding of fraction division either through a problem-solving task involving graphical representation or through a problem-posing task. However, engaging in problem posing appears to more likely than making a graphical representation to elicit preservice teachers' conceptual understanding. Moreover, a conceptual cue about the meaning of fraction division appears to greatly increase the likelihood that preservice teachers will exhibit conceptual understanding of fraction division in their posed problems and generate conceptually based diagrams of fraction division. The findings highlight the usefulness of problem-posing tasks for diagnosing and assessing preservice teachers' mathematical understanding.