Students’ Proof Construction through Computer-Supported Collaborative Learning: The Perspectives of Habermas’ Theory of Rationality and Duval’s Theory of Registers of Semiotic Representation

This study aims to examine how students’ performance in constructing and transforming representations during the proving of calculus rules unfolds in a computer-supported collaborative learning (CSCL) environment, within the context of rationality. The CSCL environment was designed by integrating the ACODESA method and dynamic mathematics software GeoGebra. The participants included 18 university students. The proofs and GeoGebra files of the participants and transcripts of arguments during the GeoGebra-integrated collaborative phases of ACODESA were analyzed based on Habermas’ theory of rationality and Duval’s theory of registers of semiotic representation. The students improved their proof in building a geometric representation (e.g., drawing a secant line on a constant function graph), and in the treatment within the algebraic register (e.g., during the expansion of \({(x+h)}^{n})\) ${(x+h)}^n)$ in the context of epistemic rationality. They developed their proof in the conversion (e.g., drawing a secant line in the geometric register and hence obtaining algebraic representation of the derivative function) in the context of teleological rationality. The students provided more understandable proofs due to their improvement in the context of communicative rationality. The difficulties experienced by the students in building representations, treatment, and conversion could be analyzed with the combined application of Duval’s theory of registers of semiotic representation and Habermas’ theory of rationality. Based on the results of the study, it is suggested to design GeoGebra-integrated CSCL environments to support students’ performance in constructing and transforming representations within the context of being rational during the teaching of proofs related to calculus concepts.