A chatbot is artificial intelligence software that converses with a user in natural language. It can be instrumental in mitigating teaching workloads by coaching or answering student inquiries. To understand student-chatbot interactions, this study is engineered to optimize student learning experience and instructional design. In this study, we developed a chatbot that supplemented disciplinary writing instructions to enhance peer reviewer's feedback on draft essays. With 23 participants from a lower-division post-secondary education course, we delved into characteristics of student-chatbot interactions. Our analysis revealed students were often overconfident about their learning and comprehension. Drawing on these findings, we propose a new methodology to identify where improvements can be made in conversation patterns in educational chatbots. These guidelines include analyzing interaction pattern logs to progressively redesign chatbot scripts that improve discussions and optimize learning. We describe new methodology providing valuable insights for designing more effective instructional chatbots by enhancing and engaging student learning experiences through improved peer feedback.