Misconception Acquisition Dynamics in Large Language Models

Abstract

Effective educational AI depends on modeling student misconceptions. Such models enable realistic learner simulation and diagnostic, adaptive tutoring. However, instruction-tuning large language models on student responses containing misconception errors can degrade reasoning abilities, creating a tension between faithful misconception modeling and preserving correct reasoning in other contexts. To support both learner simulation and tutoring, we study two misconception-aware models: the Novice Student Misconception Model, trained to acquire a single misconception for simulating an individual student, and the Expert Tutor Misconception Model, trained on multiple misconceptions to capture the error patterns a tutor encounters across students. To study the misconception acquisition dynamics of both models, we develop MalAlgoLib, a library that generates algebra problems with correct solution traces and misconception-specific erroneous traces. Our experiments across three LLMs reveal that the student and the tutor model exhibit fundamentally different misconception acquisition dynamics. For the student model, a single misconception is not learned as a context-specific behavior. Models overapply it across problems, degrading correct-solving accuracy unless training includes correct examples to enforce boundaries. In contrast, the tutor model can learn multiple misconceptions jointly without sacrificing correct-solving accuracy. Critically, intermediate reasoning steps are the bottleneck. With final-answer supervision alone, models cannot learn where error enters the solution, so neither the student model nor the tutor model acquires misconceptions regardless of data size. Together, these results, enabled by MalAlgoLib, provide an interpretable account of misconception acquisition under instruction tuning and guidance for training misconception-aware LLMs while preserving correct reasoning.

Figures (5)

• Figure 1: Left: Problem types in MalAlgoLib. Right: Graph-based model where nodes are problem types. Each gray edge represent a reduction, similar to students' steps, from complex type to a more simple type eventually reaching the base type T1. Each red edge (only M8 shown) represents a step reduction conducted under a misconception. Such reduction structure reflected our design principle of code reuseability, easy expansion, and closely mirrors student's cognitive process of equation solving.
• Figure 2: Novice Student Misconception Model (student model) instruction-tuning dynamics for Llama3.1-8B-Instruct llama3 across five misconceptions reveal a clear trade-off. As models acquire misconceptions (MA, solid navy), their correct-solving ability degrades (OCA, dashed burgundy). Thick lines show means; thin lines show individual misconceptions. This inverse relationship motivates investigating whether degradation occurs uniformly or differs between applicable and non-applicable problem types. Similar pattern observed for Phi4-4B-mini phi4 and Qwen3-4B-Instructqwen3.
• Figure 3: Left: Decomposing Novice Student Misconception Model (student model) accuracy for Llama-3.1-8B-Instruct llama3, averaging over misconceptions, reveals that instruction-tuning degrades performance on both applicable (A, solid blue) and non-applicable (NA, dashed pink) problem types. Although $\text{Correct Accuracy (CA)}_{NA}$ declines relatively slower than $CA_A$, degrade in performance is still significant. Right: Mixing correct examples during student model instruction-tuning for Llama3.1-8B-Instruct llama3 resolves the accuracy trade-off. MA (blue) remains high ( 90-95%) demonstrating successful misconception acquisition, while CA$_{NA}$ (red) rapidly recovers with just 200-400 correct examples. Similar pattern observed for both Phi-4-4B-Miniphi4 and Qwen-3-4B-Instructqwen3.
• Figure 4: Left: Expert Tutor Misconception Model (tutor model) training on 10 misconceptions simultaneously with insufficient samples (5--80 per misconception). While OCA remains stable, average MA fails to reach the 90% threshold for most misconceptions. Right: Tutor model acquisition dynamics when trained on 10 misconceptions simultaneously with sufficient samples (100--800 per misconception). Misconception Accuracy (MA, solid blue lines) increases steadily while Overall Correct Accuracy (OCA, dashed red lines) remains stable or even improves (93% to 98%). Results shown for Llama-3.1-8B, Phi-4, and Qwen-3 models; thick lines indicate mean trends.
• Figure 5: Without step-level supervision, both the tutor model (left) and the student model (right) fail to acquire misconceptions. Even with substantial training data (up to 1600 samples for the student model, 800 for the tutor model), MA remains below 30% across all three models (Llama, Phi, Qwen). For the tutor model without steps, OCA also degrades. This demonstrates that step-level supervision is essential for misconception acquisition.

Theorems & Definitions (2)

• definition 1: Novice Student Misconception Model
• definition 2: Expert Tutor Misconception Model