Godfrey the GPT: A Teachable Agent for Learning Precalculus

What happens when your students are the ones doing the teaching — and the students are teaching an AI?

During the Fall 2024 semester, I worked with my colleague Dr. Hui Soo Chae to develop and pilot Godfrey, a custom GPT for my Precalculus course. The idea was to use OpenAI ChatGPT to create a teachable agent (Blair et al., 2007). The teaching agent simulated a student struggling to understand inverse functions. The goal wasn’t to showcase Generative AI (GenAI). Instead it was to create an active learning environment where students are positioned as experts. I wanted students to think more carefully by having to explain ideas aloud, respond to confusion, and adjust their language for a virtual protégé (Chase et al., 2009).

Making Godfrey function like a struggling student took significant trial and error. ChatGPT is designed to provide correct answers quickly, not linger in uncertainty. My colleague Penka Marinova and I spent quite a bit of time refining the GPT Instructions so that it would act like an unsure, curious student. Getting the pacing right — when to hesitate, when to misunderstand, when to rephrase — required many iterations.

Students didn’t just “chat with the bot.” They followed a structured script that guided Godfrey through three layers of understanding: 

1. Symbolic (e.g., reversing f(x)=3x+2), 

2. Conceptual (e.g., explaining what f−1(10000)=255 means)

3. Contextual (e.g., interpreting the inverse in a real-world scenario like cost and units sold) 

   
   

This kept the conversation focused while allowing room for missteps — both Godfrey’s and students'.

Each student submitted the chat transcript and completed two brief evaluations. For the first evaluation I asked students to reflect on the learning process, specifically, whether the GPT interaction helped them think more carefully or recognize gaps. In the second evaluation I asked students to assess whether the AI stayed in character. Many students responded thoughtfully.

“Even when he answered questions correctly, he still asked questions about the process to make sure that he wasn't correct on accident but that the process was correct and could be used in the future.”

“Godfrey’s doubts when answering a question… made the process feel more real and allowed me to help throughout each question.”

Several students also highlighted how the GPT’s tone and language enhanced the realism.

“Godfrey's sense of humor, follow-up questions, and 'what if’s' were incredibly authentic.”

“He spoke like a young college student and was very relatable to my friends speaking.”

Other students focused on the emotional realism.

“His panic, overthinking, and math anxiety felt super real,” one student observed, especially in moments involving parentheses or switching x and y.

“The most authentic aspect of Godfrey's character was his openness about his confusion and his curiosity. While his self-doubt at times made him second-guess himself, it also drove him to ask questions and work through concepts, ultimately deepening his learning process.”

Other survey results echoed the same themes:

• 90% of students said they felt more engaged when they had to explain ideas to the GPT.

• 75% agreed that explaining their reasoning helped them identify gaps in their understanding.

• 75% said they learned something by seeing where the GPT got confused.

This activity wasn't about watching GenAI fail or succeed as a teaching agent. It was about giving students a virtual peer that was close enough to be plausible, but uncertain enough to make their own reasoning coherent. Teaching inverse functions demands clarity and flexibility, not only in calculation, but in how we think and explain. Structuring an interaction around a confused GenAI made that process more visible. For many students, it helped surface habits of reasoning that are rarely discussed but crucial to learning.

NOTE: In May 2025, I received a Teaching Advancement Grant from the NYU Center for Teaching and Learning. This award will enable me to enhance Godfrey and create a GPT tutor on inverse functions.

References

Blair, K., Schwartz, D. L., Biswas, G., & Leelawong, K. (2007). Pedagogical agents for learning by teaching: Teachable agents. Educational Technology, 56-61.

Chase, C. C., Chin, D. B., Oppezzo, M. A., & Schwartz, D. L. (2009). Teachable agents and the protégé effect: Increasing the effort towards learning. Journal of science education and technology, 18, 334-352.