Developing a Talk-Moves-Based Learning Design for Systems of Linear Equations in Two Variables: A Design-Based Research Framework

Mathematical communication in systems of linear equations in two variables requires students to connect representations, explain reasoning, and respond to mathematical ideas. However, existing instructional approaches do not always integrate classroom discourse, talk moves, and communication assessment into a coherent learning design. This methodological design study develops a talk-moves-based learning design for systems of linear equations in two variables and specifies an analytical protocol for its subsequent empirical validation. The study applies a design-based research framework comprising preliminary analysis, prototype development, iterative design, and retrospective analysis. The proposed design integrates multi-representational tasks, six talk moves, anticipated student responses, presentation orchestration, and oral and written mathematical communication assessment. A simulated dataset is used solely to illustrate the planned procedures for content validity, practicality, implementation, communication outcomes, and discourse analysis. The study produces a four-lesson instructional design and an accompanying analytical protocol that specify how design quality, implementation processes, mathematical communication, and classroom discourse can be examined in future empirical research. The contribution is a theoretically coherent and testable instructional design rather than evidence of instructional effectiveness. The design provides a structured basis for future classroom validation and may be adapted to other mathematical topics, including functions, ratios and proportions, and statistics.