Error as a Lens: Probing LLM Reasoning through Synthetic Misconception Generation

Modeling student misconceptions in a realistic manner is critical for AI in education. In this work, we examine how large language models (LLMs) reason about misconceptions when generating distractor answers for multiple-choice questions (MCQs), a task that requires producing answers that are incorrect, yet plausible. We introduce a taxonomy over reasoning strategies for distractor generation that is grounded in learning-science literature and empirical observation, which we apply to LLM-generated reasoning traces across math and science MCQs. On the math dataset, we find that models follow a misconception-based process with potentially high diagnostic value: they recover the correct solution, articulate student errors, simulate them, and select plausible candidates. On the science dataset, on the other hand, they tend to follow a less robust approach based on semantic similarity to the correct answer. We find the most frequent failure modes to be that the model is unable to generate a correct solution or that it discards plausible distractor candidates when performing selection. Providing the correct solution in the prompt yields a relative improvement of 6.4% in alignment with human-authored distractors, highlighting the critical role of anchoring distractor generation to the correct solution. Together, our findings offer an interpretable view of how LLMs model incorrect student reasoning.

Understanding student errors in the programming is a cornerstone of programming education, yet obtaining a representative set of student errors for any newly designed task remains slow and costly, since authentic submissions only accumulate after extensive classroom deployment. This paper explores whether large language models (LLMs) can serve as scalable proxies for students by simulating realistic logical errors in code submissions. Using the CodeWorkout dataset of 74,000+ unique student Java submissions across 37 problems, we evaluate five LLMs under three mainstream prompting strategies: Input-Output (IO), Chain-of-Thought (CoT), and iterative Self-Refine. We assess performance along two key dimensions: diversity (the range of distinct error patterns) and alignment (alignment with authentic student mistakes), and examine how these vary by struggling level of programming tasks. Our quantitative findings reveal that while all models generate diverse errors, their alignment to human submissions diverges: Claude Sonnet 4 achieves the most balanced performance. In addition, we conducted a blinded expert annotation study (N = 401) comparing synthetic and authentic errors. This qualitative analysis confirms that the generated errors are functionally indistinguishable from authentic student errors. Moreover, higher-struggling-level problems elicit more diverse but less student-like errors. These results highlight trade-offs in using LLMs to simulate human learners and suggest design considerations for integrating synthetic errors into teachable agents, intelligent tutoring systems, and large-scale learning analytics.

Novice math teachers often encounter students' mistakes that are difficult to diagnose and remediate. Misconceptions are especially challenging because teachers must explain what went wrong and how to solve them. Although many existing large language model (LLM) platforms can assist in generating instructional feedback, these LLMs loosely connect pedagogical knowledge and student mistakes, which might make the guidance less actionable for teachers. To address this gap, we propose MisEdu-RAG, a dual-hypergraph-based retrieval-augmented generation (RAG) framework that organizes pedagogical knowledge as a concept hypergraph and real student mistake cases as an instance hypergraph. Given a query, MisEdu-RAG performs a two-stage retrieval to gather connected evidence from both layers and generates a response grounded in the retrieved cases and pedagogical principles. We evaluate on \textit{MisstepMath}, a dataset of math mistakes paired with teacher solutions, as a benchmark for misconception-aware retrieval and response generation across topics and error types. Evaluation results on \textit{MisstepMath} show that, compared with baseline models, MisEdu-RAG improves token-F1 by 10.95% and yields up to 15.3% higher five-dimension response quality, with the largest gains on \textit{Diversity} and \textit{Empowerment}. To verify its applicability in practical use, we further conduct a pilot study through a questionnaire survey of 221 teachers and interviews with 6 novices. The findings suggest that MisEdu-RAG provides diagnosis results and concrete teaching moves for high-demand misconception scenarios. Overall, MisEdu-RAG demonstrates strong potential for scalable teacher training and AI-assisted instruction for misconception handling. Our code is available on GitHub: https://github.com/GEMLab-HKU/MisEdu-RAG.