Student mistakes in mathematics are often systematic: a learner applies a coherent but wrong procedure and repeats it across contexts.We introduce MALRULELIB, a learningscience-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 studentmodeling 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 MAL-RULELIB as infrastructure for educational AI that models student procedures across contexts, enabling diagnosis and feedback that targets the underlying misconception.MALRULELIB code is available here.* Equal contribution.Model CRA MRA Forward MRA Llama-3.3-70B70.4% 34.6% 39.6% Qwen3-80B-Think 70.1% 56.4% 53.9% gpt-oss-120b 65.0% 56.9% 48.8%Average 68.5% 49.3% 47.5% Forward MRA MRA (Cross-Template) System: You are simulating a student who has a specific mathematical misconception.Apply the described misconception consistently to solve the problem.System: You are an expert in identifying and understanding student mathematical misconceptions.Given an example of a student's incorrect answer, identify the systematic error and apply it to predict answers for new problems.User: A student has the following misconception: Students distribute square root over addition: √ a 2 + b 2 = a + b Apply this misconception to solve: Evaluate f (x) = √ x 2 + 4 when x = 3.What is f (3)?User: A student solved this problem incorrectly: Problem: Evaluate f (x) = √ x 2 + 25 when x = 8.Student's Answer: 13 Now predict what this same student would answer for: You walk 8 blocks east and 3 blocks north.What is the straight-line distance from your starting point?Expected: 5 (correct: √ 13 ≈ 3.61) Expected: 11 (correct: √ 73 ≈ 8.54)

High-quality conversational datasets are crucial for the successful development of Intelligent Tutoring Systems (ITS) that utilize a Large Language Model (LLM) backend. Synthetic student-teacher dialogues, generated using advanced GPT-4 models, are a common strategy for creating these datasets. However, subjects like physics that entail complex calculations pose a challenge. While GPT-4 presents impressive language processing capabilities, its limitations in fundamental mathematical reasoning curtail its efficacy for such subjects. To tackle this limitation, we introduce in this paper an innovative stateful prompt design. Our design orchestrates a mock conversation where both student and tutorbot roles are simulated by GPT-4. Each student response triggers an internal monologue, or ‘code soliloquy’ in the GPT-tutorbot, which assesses whether its subsequent response would necessitate calculations. If a calculation is deemed necessary, it scripts the relevant Python code and uses the Python output to construct a response to the student. Our approach notably enhances the quality of synthetic conversation datasets, especially for subjects that are calculation-intensive. The preliminary Subject Matter Expert evaluations reveal that our Higgs model, a fine-tuned LLaMA model, effectively uses Python for computations, which significantly enhances the accuracy and computational reliability of Higgs’ responses.

Word embedding models learn semantically rich vector representations of words and are widely used to initialize natural processing language (NLP) models.The popular continuous bag-ofwords (CBOW) model of word2vec learns a vector embedding by masking a given word in a sentence and then using the other words as a context to predict it.A limitation of CBOW is that it equally weights the context words when making a prediction, which is inefficient, since some words have higher predictive value than others.We tackle this inefficiency by introducing the Attention Word Embedding (AWE) model, which integrates the attention mechanism into the CBOW model.We also propose AWE-S, which incorporates subword information.We demonstrate that AWE and AWE-S outperform the state-of-the-art word embedding models both on a variety of word similarity datasets and when used for initialization of NLP models.