Organizations: College of Computing and Information Science, Cornell University, Ithaca, USA. · College of Connected Computing, Vanderbilt University, Nashville, USA. · Third Space Learning, Swindon, UK.
In K-12 mathematics tutoring, student-tutor dialogue provides rich evidence of learners' problem-solving processes and sources of difficulty. Learning analytics research increasingly relies on large language models (LLMs) to extract such information from dialogue for a variety of downstream tasks, including knowledge tracing, behavioral modeling, and diagnosis of student reasoning errors. However, the validity of these model-generated interpretations remains insufficiently understood. In this exploratory study, we examine the validity of LLM classifications of five student failure modes in mathematics tutoring dialogue using an operational diagnostic codebook: uncertainty, misattribution, operator selection, conceptual gap, and procedural slip. Across models, human-LLM agreement was moderate (kappa = .524-.597), while cross-model agreement was substantially higher (kappa = .755-.781; alpha = .769). These findings show that cross-model agreement can create a misleading appearance of correctness, challenging the assumption that consensus among LLMs constitutes evidence of valid learner interpretation. For learning analytics, the implication is clear: scalable labeling is useful only if the inferred constructs are valid, and model consensus cannot substitute for independent evidence of that validity.
Figures & tables
Code
Failure Mode
Definition
Example
UNC
Uncertainty
Student expresses uncertainty or reports a technical issue
“I don’t know”; “Um…”
MIS
Misattribution
Student uses numbers or context from a different problem
Answers another question than asked
OP
Operator Selection
Student chooses the wrong type of operation for this problem
Multiplies when division is needed
CON
Conceptual Gap
Reasoning reveals a wrong underlying mathematical idea
Believes 0×n=n
PROC
Procedural Slip
Correct strategy, wrong execution
Correct method with arithmetic error
NA
Not Applicable
Exclusion flag (not a failure mode)—item cannot be diagnosed
Inaudible response; lack of information
Table 1: Failure-mode taxonomy, checked in the order shown; the first matching class is taken.
Figure 1 : Agreement estimates across configurations, split into three blocks: human raters’ pre-consensus agreement (top), each model’s agreement with the human consensus (middle), and LLM–LLM agreement only (bottom). α replaces pairwise κ for the three-LLM comparison since it generalizes to more than two raters and is on the same 0–1 scale. The top two blocks cluster closely together, while the bottom block stands well apart—illustrating the central divergence this study examines.
Figure 2 : True (human consensus) vs. predicted class, pooled across the three tier-2 LLMs, row-normalized.
Figure 3 : Pooled tier-2 LLM predictions by reference class, ordered by macro-F1. Green indicates recall; P denotes precision. For every non-CON class, the most common error is misclassification as Conceptual Gap (red), indicating a shared directional bias.
Effective tutoring requires distinguishing optimal, valid but suboptimal, and incorrect student solutions, a distinction central to intelligent tutoring systems (ITS) but untested for LLM-based tutors. As LLMs are increasingly explored as conversational complements to ITS, evaluating their diagnostic precision is essential. We present a benchmark of seven LLM feedback agents in propositional logic using knowledge-graph-derived ground truth across 10,836 solution--feedback pairs and three feedback conditions. Models achieved near-ceiling performance on optimal steps but systematically over-rejected valid but suboptimal reasoning and over-validated incorrect solutions, precisely where adaptive tutoring matters most. These failures persisted across models regardless of solution context, suggesting architectural rather than informational limits. Moreover, accurate diagnosis did not reliably produce pedagogically actionable feedback, revealing a gap between diagnostic judgment and instructional effectiveness. Our findings suggest that LLMs are better suited for hybrid architectures where KG-grounded models handle diagnosis while LLMs support open-ended scaffolding and dialogue.
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.
Yanick Zengaffinen, Andreas Opedal, Donya Rooein +3
1ETH Zürich · 2Bocconi University · University of Central Florida
Selecting the correct answer from a pool of candidate reasoning chains is the engine of test-time scaling, yet the standard selectors each carry a cost: self-consistency inherits the errors of the single model it resamples, and trained reward models need labeled data and transfer poorly off-distribution. We study a third signal, free at inference time: cross-model consensus, the degree to which independently trained models, each solving the problem once, agree on a final answer. We treat the panel as an LLM-jury, in which the structure of agreement, not any model's score of another, is the verification signal. Across seven benchmarks it selects correct answers better than self-consistency and far better than a model scoring its own candidates: on competition math it closes the entire gap to an oracle selector, while self-scoring closes almost none. The mechanism is error decorrelation: independently trained models err differently, so their wrong answers scatter while the correct one accumulates agreement. We make this precise with a parameter-free law, derived in closed form, that predicts consensus accuracy from three measured panel statistics to a mean absolute error of 0.03 and exposes the method's ceiling: a shared-error floor where models share a misconception, near zero on math but non-trivial on science. Against four trained verifiers spanning discriminative, outcome, and generative reward models, the free LLM-jury matches the strongest inside their math training domain and is the top selector outside it. Cross-model consensus is thus a verifier we can characterize in advance: a law that says when to trust it, and a floor that marks where it cannot.