Changes in prospective mathematics teachers' concept images of linear transformation regarding modes of thinking and description

The purpose of this study was to explore the changes in prospective mathematics teachers' concept images in a technology-supported teaching experiment on linear transformation with regard to modes of thinking and description. Nine prospective mathematics teachers studying at a state university participated in the study. The data were collected through several sources: tests, the interview form, tasks, reflective reports, and the opinion form. Before teaching experiment, prospective mathematics teachers' concept images of linear transformation were examined. Then, technology-supported teaching experiment on linear transformation was carried out with the participants. Following this, their concept images of linear transformation were examined again. The data were analyzed to reveal the changes in their concept images of linear transformation in terms of formal structure and matrix representation. Before teaching experiment, it was found that prospective teachers considered linear transformation as operational calculations that test certain conditions using the context of analytical-arithmetic thinking and algebraic mode. After teaching experiment, they considered linear transformation as a function that preserves linearity between vector spaces using the analytical-structural thinking and abstract mode. The prospective teachers could not relate linear transformation to matrices before teaching experiment; however, then they were able to generate their arguments for the transition to the matrix representation of linear transformation using analytical-structural thinking and algebraic-abstract modes. These results have suggested that after teaching experiment prospective mathematics teachers' concept images of linear transformation were compatible with the concept definition, and they moved flexibly between different modes of thinking and description as well.