The development and application of an interview structure on determining preservice mathematics teachers' competence in proportional reasoning

Determining preservice teachers' competence in proportional reasoning is a very complex task, and existing research methods do not provide effective tools in doing that. This study describes an interview structure and application of it in determining preservice teachers' competence in proportional reasoning. The interview structure consists of in-depth questioning and cognitive conflicts and is situated in the knowledge-in-pieces epistemological perspective. The preservice teachers' competence in proportional reasoning was determined by examining the knowledge resources that they drew upon when reasoning about proportional and nonproportional situations. Semi-structured interviews were conducted with six preservice teachers to demonstrate the effectiveness of the interview structure. The case analysis indicated the preservice teachers' attention to the various fine-grained knowledge resources. However, they mostly drew upon the qualitative relationships, cross-multiplication, and across-multiplication. The preservice teachers' over-attention to these three knowledge resources hindered their ability to distinguish proportional relationships from nonproportional relationships. Furthermore, the productivity of knowledge resources was easily influenced by the relationships presented in problems and participants' past learning experiences. Implications of the application of the interview structure in teacher education programs and further research suggestions are discussed.