K12 Mathematics Error Diagnosis and Heuristic Guidance Method Based on Lightweight Large Language Models

Abstract

Current K12 online math tutoring platforms share a common weakness: they tend to identify student errors only at a surface level, and when it comes to feedback, most systems simply hand out the correct answer. Over time, this combination chips away at the pedagogical promise of intelligent tutoring. The present work tackles both problems through a lightweight diagnostic-feedback pipeline that pairs MathBERT for semantic analysis with SymPy for symbolic verification. Why two engines? Because MathBERT reads the meaning behind a student’s response — catching conceptual slips that pattern-matching would miss — while SymPy checks whether two mathematical expressions are genuinely equivalent, which eliminates the well-documented hallucination tendencies of language models in algebraic contexts. For the feedback component, parameter-efficient fine-tuning (PEFT) first injects math-domain knowledge into a compact language model, after which direct preference optimization (DPO) steers the model toward Socratic-style questioning instead of answer-revealing behavior. Tested against the MathDial corpus, the pipeline reaches 99.17% on answer verification accuracy, posts an F1 of 0.9139 for error localization, and keeps the direct-answer rate down to 4.2% — beating every baseline on all three counts. These numbers suggest that symbolic checking paired with preference alignment offers a practical path toward tutoring systems that are both reliable and genuinely instructive, even on modest hardware.