Modeling Teacher-AI Interactions Through Covariational Instructions in the Mathematical Working Space

The integration of Artificial Intelligence in mathematics teacher education demands new theoretical frameworks that move beyond viewing AI as a mere support tool. This study introduces the MWS-AI model, extending the Mathematical Working Space framework to analyse how covariational instructions activate distinct AI participation modes across epistemological and cognitive dimensions. Through two complementary experiences, a synchronous workshop at the University of Salerno involving prospective teachers collaboratively working with Taxicab geometry, and an asynchronous intervention at the National University of Rosario with three teachers refining instructional materials, we identify three covariational dimensions: Social (AI as collaborative participant), Weak Instrumental (AI as critical friend), and Instrumental (AI as design partner). We explore the process that emerged through covariational instructions, from the epistemological and cognitive levels, paying attention to the semiotic, discursive, and instrumental genesis dimensions. It has been possible to notice how the instructions can activate different AI participation modes and contribute to critical thinking, evaluation competencies and instructional design capabilities in prospective teachers across all the components of the model. Our findings reveal predictable task-pattern relationships that systematically foster critical thinking, evaluation competencies, and instructional design capabilities, offering practical guidelines for intentional AI integration in mathematics teacher preparation.