Construction of proof of the Fundamental Theorem of Calculus using dynamic mathematics software in the calculus classroom

The study focused on how university students constructed proof of the Fundamental Theorem of Calculus (FTC) starting from their argumentations with dynamic mathematics software in collaborative technology-enhanced learning environment. The participants of the study were 36 university students. The data consisted of participants’ written productions, dynamic materials, and the transcriptions of the participants’ argumentations for the selected groups. The analysis was based on the integration of cK¢ model and Toulmin’s model. The analysis showed that the collaborative technology-enhanced learning environment helped the participants to interpret the Mean Value Theorem (MVT) for definite integrals geometrically and use this interpretation for the proof of the FTC. They constructed proof the FTC using geometric, empirical, and symbolic conceptions involved in their argumentations supporting their conjecture about the evolution of the derivative, the MVT, and limit idea. The construction of the connections between algebraic and geometric representations regarding the FTC in a social interaction-communication process helped them to move from a geometric perspective to a theoretical perspective while constructing proof of the FTC. The mediation role of dynamic mathematics software GeoGebra provided them to construct multiple representations and verify conjectures in producing deductive argumentations about the proof of the FTC. The emergence of the social norms in classroom microculture with this integration method contributed to the evolution of participants’ representations and their reasoning on the FTC. Additionally, they made connections between the conclusions of the FTC and the differential equation, but they did not mention the continuity condition for proof of the FTC.