An AI Prompt Framework to Support Polya’s Problem-Solving Process in Student Self-Directed Mathematics Learning

Abstract

This study addresses the lack of real-time heuristic
guidance that students encounter during after-school
mathematics self-study. To meet this need, we designed an
AI-based interactive mathematical learning prompt grounded
in Polya’s four-stage problem-solving model. The prompt
uses a five-module structure—Persona, Rules, Instruction,
Reflection, and Practice—to regulate the AI’s interaction
flow and reasoning, providing digital scaffolding that
simulates teacher-like guidance across the stages of
understanding, planning, execution, and reflection.
Using Gemini 2.5 as the testing environment, the study
conducted interactive trials on number theory and geometry
problems and incorporated qualitative feedback from teacher
interviews. Results indicate that the AI consistently
adhered to the prompt structure, delivering coherent
step-by-step reasoning and avoiding logical
inconsistencies. The Reflection module supported strategy
comparison and strengthened students’ metacognitive
awareness. Teachers noted that the interaction flow
resembled authentic classroom instruction, making the
system suitable for pre-class preparation and after-school
review, though AI may still fall short of human instructors
when students lack sufficient prior knowledge. Overall, the
study demonstrates the feasibility of prompt engineering as
instructional scaffolding in mathematics education and
suggests integrating textbook-based training and
student-level diagnosis to enhance applicability in
real-world learning contexts.