Supporting Secondary Teachers\u27 Proof and Justification of Calculus Concepts Through the Intentional Use of Dynamic Technology

Learning calculus concepts plays a huge role in understanding phenomena in STEM-related disciplines. Those concepts tend to be dynamic in nature, and the visual exploration and representation of calculus concepts using paper and pencil is limited compared to pedagogically and intentionally using dynamic geometry software. As such, a primary component of this dissertation study involves the integration of dynamic technology. Additionally, previous studies have shown that students have difficulties constructing proofs related to calculus concepts. Despite the existing body of research on students\u27 comprehension of proof and justification, there has not been much focus on teachers\u27 knowledge and perception of proof and justification in connection to the ways that prospective secondary teachers can teach and learn calculus concepts. This study uses a qualitative methodology to investigate the ways in which integrating technology could help both in-service and pre-service secondary teachers gain a deeper understanding of the process of proof. Through a multiple case study approach, research participants were engaged with different mathematical tasks to explore geometric series and subsequently construct and prove conjectures through the integration of dynamic technology. This study showed that dynamic geometry software could help teachers to appreciate the value of visual representation in teaching and learning mathematics. Those technological pieces helped them with exploring different ideas, which is crucial in the process of proving. However, a lack of experience both with visual representations and constructing conjectures held participants back from using their full potential. When it comes to mathematical proofs in school mathematics, it should be considered as a process of exploring ideas, making conjectures, and checking the validity of those conjectures and not a single notion and visual representations - specifically dynamic ones that are created by technology – play a huge role in deepening teachers understanding of the process through their connection with key ideas