# MalruleLib: Large-Scale Executable Misconception Reasoning with Step Traces for Modeling Student Thinking in Mathematics

arXiv: https://arxiv.org/abs/2601.03217 · HTML: https://mlsanity.com/paper/2601.03217/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 6 of its 90 references that are in the map.
- First posted on arXiv 6 Jan 2026.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 6 of 6)

- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [gpt-oss-120b & gpt-oss-20b Model Card](https://mlsanity.com/paper/2508.10925/) (2025)
- [Phi-4 Technical Report](https://mlsanity.com/paper/2412.08905/) (2024)
- [qDKT: Question-centric Deep Knowledge Tracing](https://mlsanity.com/paper/2005.12442/) (2020)
- [Mathfish: Evaluating Language Model Math Reasoning via Grounding in Educational Curricula](https://mlsanity.com/paper/2408.04226/) (2024)

## Built on by (most cited first, 1 of 1)

- [Simulating Students or Sycophantic Problem Solving? On Misconception Faithfulness of LLM Simulators](https://mlsanity.com/paper/2605.12748/) (2026)

## Abstract (arXiv)

Student mistakes in mathematics are often systematic: a learner applies a coherent but wrong procedure and repeats it across contexts. We introduce MalruleLib, a learning-science-grounded framework that translates documented misconceptions into executable procedures, drawing on 67 learning-science and mathematics education sources, and generates step-by-step traces of malrule-consistent student work. We formalize a core student-modeling problem as Malrule Reasoning Accuracy (MRA): infer a misconception from one worked mistake and predict the student's next answer under cross-template rephrasing. Across nine language models (4B-120B), accuracy drops from 66% on direct problem solving to 40% on cross-template misconception prediction. MalruleLib encodes 101 malrules over 498 parameterized problem templates and produces paired dual-path traces for both correct reasoning and malrule-consistent student reasoning. Because malrules are executable and templates are parameterizable, MalruleLib can generate over one million instances, enabling scalable supervision and controlled evaluation. Using MalruleLib, we observe cross-template degradations of 10-21%, while providing student step traces improves prediction by 3-15%. We release MalruleLib as infrastructure for educational AI that models student procedures across contexts, enabling diagnosis and feedback that targets the underlying misconception.