Finding Common Mistakes In Modelling With Mathematical Formalisms Using LLMs

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.

The reasoning ability of large language models (LLMs) is a critical factor for practical LLM-based applications. To investigate the current reasoning capability of LLMs, we clarify the types of errors that arise in LLMs' reasoning processes on mathematical datasets. We focus on problems where LLMs produce an incorrect answer. We define errors in the reasoning process as reasoning errors and manually analyze the features of reasoning errors. We defined and classified 21 error classes and identified the frequently occurring classes among them. Beyond qualitative evaluation, we leverage the evaluation results to improve the reasoning capability. We designed a prompt that explicitly focuses on eight error classes. The experiments demonstrate that this prompt effectively improves reasoning performance. Furthermore, the results suggest that the frequent reasoning errors identified in this paper are common across LLMs of comparable scale.

Effective LLM tutoring depends on correctly identifying the specific error in a student's reasoning before generating feedback. We study this problem in propositional-logic proof tutoring, where student actions can be checked against formal inference rules. We introduce a verifier-grounded architecture that separates diagnosis from language generation. Using 600 balanced student actions, we compare a zero-shot LLM detector, a fine-tuned detector, and a symbolic verifier. Each diagnosis is processed by shared rationale and feedback agents, isolating the effect of the initial diagnosis. The zero-shot detector achieves a macro-F1 of 0.191; fine-tuning raises this to 0.709 but retains systematic errors between structurally related classes. Rationales generally preserve the diagnosis supplied to them, showing that an incorrect diagnosis can be faithfully propagated through the pipeline. Feedback can likewise remain faithful to its rationale, non-revealing, and pedagogically appropriate while addressing the wrong error. Verifier-grounded feedback achieves the highest diagnostic correctness, and expert ratings largely uneven with the automatic feedback evaluations. These findings show that apparent feedback quality can conceal upstream diagnostic errors and that faithfulness must be evaluated separately from correctness. Our code is publicly available