AI-integrated didactic systems: positioning dynamics and identity transformations in mathematics education

This chapter examines how Generative AI integration fundamentally reshapes mathematical identities by introducing a fourth actor into the traditional didactic triangle of teacher-student-knowledge. Drawing on positioning theory and empirical studies involving 50 undergraduate topology students and 30 prospective mathematics teachers in Italy, we analyse how AI participation reconfigures positioning dynamics at both learning and teaching levels. At the learning level, students engaged in structured three-phase activities where AI intervened as a "peer participant" after autonomous problem-solving, triggering an alternating discussion dynamic that fostered metacognitive awareness and critical evaluation competencies. Interview data reveals students repositioning themselves from passive knowledge recipients to critical evaluators who view AI neither as authority nor threat but as a "provocative companion of thought" requiring constant epistemic vigilance. At the teaching level, covariational instructions structured through Taxicab geometry exploration activated three distinct AI participation modes within the Mathematical Working Space framework: Social (AI as collaborative problem-solver), Weak Instrumental (AI as critical friend), and Instrumental (AI as design partner). Prospective teachers developed professional identities balancing pedagogical authority with strategic AI collaboration. These identity transformations occur across cognitive, metacognitive, and affective dimensions, fundamentally altering how actors position themselves within mathematical discourse communities and demanding theoretical reconceptualization of the didactic system itself.