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)

We study the interpolation capabilities of implicit neural representations (INRs) of images. In principle, INRs promise a number of advantages, such as continuous derivatives and arbitrary sampling, being freed from the restrictions of a raster grid. However, empirically, INRs have been observed to poorly interpolate between the pixels of the fit image; in other words, they do not inherently possess a suitable prior for natural images. In this paper, we propose to address and improve INRs’ interpolation capabilities by explicitly integrating image prior information into the INR architecture via deep decoder, a specific implementation of the deep image prior (DIP). Our method, which we call TITAN, leverages a residual connection from the input which enables integrating the principles of the grid-based DIP into the grid-free INR. Through super-resolution and computed tomography experiments, we demonstrate that our method significantly improves upon classic INRs, thanks to the induced natural image bias. We also find that by constraining the weights to be sparse, image quality and sharpness are enhanced, increasing the Lipschitz constant.

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.

In education applications, knowledge tracing refers to the problem of estimating students' time-varying concept/skill mastery level from their past responses to questions and predicting their future performance.One key limitation of most existing knowledge tracing methods is that they treat student responses to questions as binary-valued, i.e., whether they are correct or incorrect.Response correctness analysis/prediction ignores important information on student knowledge contained in the exact content of the responses, especially for open-ended questions.In this paper, we conduct the first exploration into open-ended knowledge tracing (OKT) by studying the new task of predicting students' exact open-ended responses to questions.Our work is grounded in the domain of computer science education with programming questions.We develop an initial solution to the OKT problem, a student knowledge-guided code generation approach, that combines program synthesis methods using language models with student knowledge tracing methods.We also conduct a series of quantitative and qualitative experiments on a real-world student code dataset to validate OKT and demonstrate its promise in educational applications.

Deep neural networks have become essential for numerous applications due to their strong empirical performance such as vision, RL, and classification. Unfortunately, these networks are quite difficult to interpret, and this limits their applicability in settings where interpretability is important for safety, such as medical imaging. One type of deep neural network is neural tangent kernel that is similar to a kernel machine that provides some aspect of interpretability. To further contribute interpretability with respect to classification and the layers, we develop a new network as a combination of multiple neural tangent kernels, one to model each layer of the deep neural network individually as opposed to past work which attempts to represent the entire network via a single neural tangent kernel. We demonstrate the interpretability of this model on two datasets, showing that the multiple kernels model elucidates the interplay between the layers and predictions.

Visual analytics techniques are widely utilized to facilitate the exploration of online educational data. To help researchers better understand the necessity and the efficiency of these techniques in online education, we systematically review related works of the past decade to provide a comprehensive view of the use of visualization in online education problems. We establish a taxonomy based on the analysis goal and classify the existing visual analytics techniques into four categories: learning behavior analysis, learning content analysis, analysis of interactions among students, and prediction and recommendation. The use of visual analytics techniques is summarized in each category to show their benefits in different analysis tasks. At last, we discuss the future research opportunities and challenges in the utilization of visual analytics techniques for online education.

High dimensionality poses many challenges to the use of data, from visualization and interpretation, to prediction and storage for historical preservation. Techniques abound to reduce the dimensionality of fixed-length sequences, yet these methods rarely generalize to variable-length sequences. To address this gap, we extend existing methods that rely on the use of kernels to variable-length sequences via use of the Recurrent Neural Tangent Kernel (RNTK). Since a deep neural network with ReLu activation is a Max-Affine Spline Operator (MASO), we dub our approach Max-Affine Spline Kernel (MASK). We demonstrate how MASK can be used to extend principal components analysis (PCA) and t-distributed stochastic neighbor embedding (t-SNE) and apply these new algorithms to separate synthetic time series data sampled from second-order differential equations.