Mathematical connections’ network identified in solving problems about vectors in the classroom: A detailed analysis with onto-semiotic tools

Abstrak

Mathematical understanding in advanced calculus courses depends critically on the establishment of coherent connections among concepts, representations, and procedures, particularly in the study of vectors within engineering education. Despite its importance, limited empirical research has examined how such mathematical connections are constructed and coordinated by professors and students during problem solving in vector calculus, especially through a theoretically integrated lens. Addressing this gap, the present study analyzes the mathematical connections activated by a professor and his students when solving vector-related problems in an engineering vector calculus course, using the Extended Theory of Connections articulated with the onto-semiotic approach as a novel analytical framework. The aim of the study is to characterize and compare the types of connections established by the professor and the students during problem solving. A qualitative methodology was employed, involving the selection of a professor and his students, data collection through classroom explanations and a validated questionnaire focusing on vector concepts, representations, and operations, and data analysis guided by the adopted theoretical framework. The results indicate that the professor predominantly established instruction-oriented, procedural, and multiple-representation connections, whereas the students mainly activated meaning, multiple-representation, feature, part–whole, and procedural connections. In addition, one student demonstrated advanced engagement by solving a problem at the blackboard and using GeoGebra to represent the angle between vectors, thereby integrating procedural and representational connections. These findings suggest that a deep understanding of vectors emerges from the coordinated use of connections among operations, representations, and meanings, and they provide theoretical and pedagogical insights for the design of instruction that promotes meaningful learning in university-level mathematics education.