Huseyin Bahadir Yanika and Alfinio Flores have written an article called Understanding rigid geometric transformations: Jeff’s learning path for translation. The article has recently been publishedin The Journal of Mathematical Behavior. Here is the abstract of their article:
This article describes the development of knowledge and understanding of translations of Jeff, a prospective elementary teacher, during a teaching experiment that also included other rigid transformations. His initial conceptions of translations and other rigid transformations were characterized as undefined motions of a single object. He conceived of transformations as movement and showed no indication about what defines a transformation. The results of the study indicate that the development of his thinking about translations and other rigid transformations followed an order of (1) transformations as undefined motions of a single object, (2) transformations as defined motions of a single object, and (3) transformations as defined motions of all points on the plane. The case of Jeff is part of a bigger study that included four prospective teachers and analyzed their development in understanding of rigid transformations. The other participants also showed a similar evolution.
Yanik, H. B., & Flores, A. (n.d.). Understanding rigid geometric transformations: Jeff’s learning path for translation. The Journal of Mathematical Behavior, In Press, Corrected Proof. doi: 10.1016/j.jmathb.2009.04.003.