Cultivating Online Instructional Competencies: Prospective Secondary Teachers' Learning to Teach Reasoning and Proof with Digital Technology

The rapid development of digital technologies and online learning platforms has profoundly impacted the landscape of mathematics teaching and learning. These changes raise critical questions regarding mathematics teacher preparation, as future teachers must cultivate competencies in integrating innovative technological tools with ambitious pedagogical practices to enhance students' mathematical reasoning and communication skills. An additional layer of complexity arises when considering these competencies for teaching in an online environment. In this paper, we report on two in-depth case studies conducted in the United States with prospective secondary mathematics teachers (PSTs) who taught four online lessons during the global pandemic. The teaching occurred as part of the PSTs' participation in a course, Mathematical Reasoning and Proving for Secondary Teachers. Data sources included lesson plans, Zoom video recordings of the lessons, and PSTs' written reflections. Utilizing the Mathematical Knowledge for Teaching Proof and the Instructional Technology Tetrahedron frameworks, we explored how the PSTs' competence in reasoning and proving evolved. The analysis employed two complementary perspectives: the researcher-observer and the PSTs' reflective notice of their teaching. The results reveal complex trajectories of pedagogical growth and technology integration into mathematics teaching. Throughout the semester, both PSTs' practices transitioned from teacher-centered to more student-driven engagement and sophisticated technology integration. Structured reflection, peer discussion, and iterative lesson design were vital supports in their professional learning. This study demonstrates how PSTs can develop essential competencies for teaching reasoning and proof in online environments and emphasizes the necessity for intentional preparation in both digital pedagogy and mathematical reasoning within teacher education programs.