Generalization processes of mathematics teachers and pre-service mathematics teachers in a dynamic geometry software supported environment

This research examines the mathematical generalization processes of mathematics teachers and pre-service mathematics teachers. It was carried out in an environment supported by dynamic geometry software (Geometer's Sketchpad). The research, in which four mathematics teachers and four pre-service mathematics teachers, determined by the purposive sampling method, took part as participants, was designed with a case study, one of the qualitative research methods. A geometric problem was used to reveal the mathematical generalization processes of the participants. In the first stage, the participants were given a right triangle, and the relationship between the edge lengths of this triangle was asked. In what follows, the participants were expected to generalize the Pythagorean Theorem by forming a right rectangle, right pentagon, and other right polygons. The data from individual interviews recorded with video and audio were analyzed descriptively based on the generalization taxonomy. As the results of the research, all participants performed 'generalizing actions' by 'relating', 'searching', and 'extending'. While seven participants were able to reach 'reflection generalizations' by 'identification or statement' and 'definition', it was observed that one pre-service teacher could not reach the reflection generalization. Participants 'related situations or objects' and 'searched the same relationship or solution' during generalizing actions. They also extended by 'expanding the range of applicability', 'removing particulars' or 'operating', and completed the generalization actions. They performed reflection generalizations by 'identification or statement' with the 'general principle' and 'definition' with the 'class of objects'. It was observed that the participants generally completed the generalization processes and reached the upper levels of the generalization taxonomy. In addition, using dynamic geometry software in this process positively affected generalizing geometric problems.