An investigation of the modeling processes of pre-service mathematics teachers' in the framework of mathematical working spaces

The aim of this study is to examine the mathematical modeling process of preservice elementary mathematics teachers within the Theory of the Mathematical Working Spaces, and to reveal the interaction between modeling competencies and the components of the epistemological plane and genesis processes. In this regard, the mathematical work performed by the preservice teachers during different mathematics tasks was analyzed by taking into account the components and genesis processes defined by the Theory of Mathematical Working Spaces. The study employed a qualitative research design, and the data were obtained through the responses of preservice elementary mathematics teachers to four different modeling tasks. These tasks were structured as open-ended and based on measurement of area context; data collection lasted four weeks and involved both individual and group work using worksheets, group reports, and video recordings. The data were analyzed descriptively based on the Theory of Mathematical Working Spaces. The findings of the study indicate that preservice teachers predominantly used the components of representation and artifact, while their use of theoretical references remained limited. Discursive genesis was found to be prominent across all stages of the modeling process, whereas instrumental genesis occurred with discontinuities in some groups. Furthermore, it was observed that mathematical work involving transitions between planes resulted in more holistic and qualified outcomes in terms of modeling competencies. In particular, modeling competencies were meaningfully developed in situations where artifacts were used effectively and transitions between vertical planes were flexibly established. This indicates that the preservice teachers not only constructed mathematical models but also developed their ability to restructure the process, justify their reasoning, and connect it with the context. Therefore, it was concluded that modeling activities supported by the components of the Theory of MWS play a critical role in developing preservice teachers' deep mathematical thinking and productive problem-solving skills.