Research on the Problem Chain of Linear Algebra Under the Main Thread of Systems of Linear Equations

Abstract

Addressing the cognitive challenges of “many abstract concepts and fragmented knowledge” in linear algebra teaching, this paper proposes to use “solving systems of linear equations” as the central main thread running through the entire course, and reconstructs the mathematics classroom using the “problem chain” teaching method based on APOS theory and deep learning concepts. The paper analyzes the epistemological dominant position of this main thread and three core design principles of the problem chain. Organized around five progressive stages—“introduction, operation, structure, sublimation, and summary”—the framework constructs an interlocking spiral problem network: beginning from system modeling and data encapsulation, progressing through the excavation of invariants in elimination and the geometric characterization of solution structures, and culminating in the search for optimal observation perspectives and the integration of the knowledge system. Through carefully designed questioning to induce cognitive conflict, obscure concepts such as matrices, rank, and eigenvalues are restored as inevitable products of overcoming bottlenecks in solving systems of equations. Teaching empirical research shows that this model significantly activates students’ internal motivation and effectively drives the educational transition from mechanical problem-solving to cultivating holistic computational thinking.

Keywords: Linear Algebra; System of Linear Equations Main Thread; Problem Chain; APOS Theory; Deep Learning; Teaching Reconstruction

DOI: 10.7176/JEP/17-4-08

Publication date: April 30^th 2026