Locally logical mathematics: An emerging teacher honoring both students and mathematics

• Process-and-action lens is a means to relate classroom and personal mathematics.
• Process-and-action and teaching triad embed mathematics in teaching practice.
• Process-and-action lens allows distinctions between written and implemented tasks.
• Locally logical mathematics is classroom-specific teacher-influenced content.

This case study explores the mathematics engagement and teaching practice of a beginning secondary school teacher. The focus is on the mathematical opportunities available to her students (the classroom mathematics) and how they relate to the teacher's personal capacity and tendencies for mathematical engagement (her personal mathematics). We use a mathematical process-and-action approach to analyze mathematical engagement and then employ the teaching triad—mathematical challenge, sensitivity to students, and management of learning—to situate mathematical engagement within the larger context of teaching practice. The article develops the construct of locally logical mathematics to underscore the cogency of mathematical engagement in the classroom as part of a coherent mathematical system that is embedded within a teaching practice. Contributions of the study include the process-and-action approach, especially in tandem with the teaching triad, as a tool to understand nuances of mathematical engagement and differences in demand between written and implemented tasks.