MathGAP: Out-of-Distribution Evaluation on Problems with Arbitrarily Complex Proofs

Large language models (LLMs) can solve arithmetic word problems with high accuracy, but little is known about how well they generalize to problems that are more complex than the ones on which they have been trained. Empirical investigations of such questions are impeded by two major flaws of current evaluations: (i) much of the evaluation data is contaminated, in the sense that it has already been seen during training, and (ii) benchmark datasets do not capture how problem proofs may be arbitrarily complex in various ways. As a step towards addressing these issues, we present a framework for evaluating LLMs on problems that have arbitrarily complex arithmetic proofs, called MathGAP. MathGAP generates problems that follow fixed proof specifications -- along with chain-of-thought reasoning annotations -- enabling systematic studies on generalization with respect to arithmetic proof complexity. We apply MathGAP to analyze how in-context learning interacts with generalization to problems that have more complex proofs. We find that among the models tested, most show a significant decrease in performance as proofs get deeper and wider. This effect is more pronounced in complex, nonlinear proof structures, which are challenging even for GPT-4o. Surprisingly, providing in-context examples from the same distribution as the test set is not always beneficial for performance. In particular, zero-shot prompting as well as demonstrating a diverse range of examples that are less complex than the test data sometimes yield similar or higher accuracies.

LLM-based Cognitive Models of Students with Misconceptions

Accurately modeling student cognition is crucial for developing effective AI-driven educational technologies. A key challenge is creating realistic student models that satisfy two essential properties: (1) accurately replicating specific misconceptions, and (2) correctly solving problems where these misconceptions are not applicable. This dual requirement reflects the complex nature of student understanding, where misconceptions coexist with correct knowledge. This paper investigates whether Large Language Models (LLMs) can be instruction-tuned to meet this dual requirement and effectively simulate student thinking in algebra. We introduce MalAlgoPy, a novel Python library that generates datasets reflecting authentic student solution patterns through a graph-based representation of algebraic problem-solving. Utilizing MalAlgoPy, we define and examine Cognitive Student Models (CSMs) - LLMs instruction tuned to faithfully emulate realistic student behavior. Our findings reveal that LLMs trained on misconception examples can efficiently learn to replicate errors. However, the training diminishes the model's ability to solve problems correctly, particularly for problem types where the misconceptions are not applicable, thus failing to satisfy second property of CSMs. We demonstrate that by carefully calibrating the ratio of correct to misconception examples in the training data - sometimes as low as 0.25 - it is possible to develop CSMs that satisfy both properties. Our insights enhance our understanding of AI-based student models and pave the way for effective adaptive learning systems.

Towards the Pedagogical Steering of Large Language Models for Tutoring: A Case Study with Modeling Productive Failure

One-to-one tutoring is one of the most efficient methods of teaching. Following the rise in popularity of Large Language Models (LLMs), there have been efforts to use them to create conversational tutoring systems, which can make the benefits of one-to-one tutoring accessible to everyone. However, current LLMs are primarily trained to be helpful assistants and thus lack crucial pedagogical skills. For example, they often quickly reveal the solution to the student and fail to plan for a richer multi-turn pedagogical interaction. To use LLMs in pedagogical scenarios, they need to be steered towards using effective teaching strategies: a problem we introduce as Pedagogical Steering and believe to be crucial for the efficient use of LLMs as tutors. We address this problem by formalizing a concept of tutoring strategy, and introducing StratL, an algorithm to model a strategy and use prompting to steer the LLM to follow this strategy. As a case study, we create a prototype tutor for high school math following Productive Failure (PF), an advanced and effective learning design. To validate our approach in a real-world setting, we run a field study with 17 high school students in Singapore. We quantitatively show that StratL succeeds in steering the LLM to follow a Productive Failure tutoring strategy. We also thoroughly investigate the existence of spillover effects on desirable properties of the LLM, like its ability to generate human-like answers. Based on these results, we highlight the challenges in Pedagogical Steering and suggest opportunities for further improvements. We further encourage follow-up research by releasing a dataset of Productive Failure problems and the code of our prototype and algorithm.

Automated Knowledge Concept Annotation and Question Representation Learning for Knowledge Tracing

Knowledge tracing (KT) is a popular approach for modeling students' learning progress over time, which can enable more personalized and adaptive learning. However, existing KT approaches face two major limitations: (1) they rely heavily on expert-defined knowledge concepts (KCs) in questions, which is time-consuming and prone to errors; and (2) KT methods tend to overlook the semantics of both questions and the given KCs. In this work, we address these challenges and present KCQRL, a framework for automated knowledge concept annotation and question representation learning that can improve the effectiveness of any existing KT model. First, we propose an automated KC annotation process using large language models (LLMs), which generates question solutions and then annotates KCs in each solution step of the questions. Second, we introduce a contrastive learning approach to generate semantically rich embeddings for questions and solution steps, aligning them with their associated KCs via a tailored false negative elimination approach. These embeddings can be readily integrated into existing KT models, replacing their randomly initialized embeddings. We demonstrate the effectiveness of KCQRL across 15 KT algorithms on two large real-world Math learning datasets, where we achieve consistent performance improvements.

Towards Aligning Language Models with Textual Feedback

We present ALT (ALignment with Textual feedback), an approach that aligns language models with user preferences expressed in text. We argue that text offers greater expressiveness, enabling users to provide richer feedback than simple comparative preferences and this richer feedback can lead to more efficient and effective alignment. ALT aligns the model by conditioning its generation on the textual feedback. Our method relies solely on language modeling techniques and requires minimal hyper-parameter tuning, though it still presents the main benefits of RL-based alignment algorithms and can effectively learn from textual feedback. We explore the efficacy and efficiency of textual feedback across different tasks such as toxicity reduction, summarization, and dialog response generation. We find that ALT outperforms PPO for the task of toxicity reduction while being able to match its performance on summarization with only 20% of the samples. We also explore how ALT can be used with feedback provided by an existing LLM where we explore an LLM providing constrained and unconstrained textual feedback. We also outline future directions to align models with natural language feedback.

How to Engage Your Readers? Generating Guiding Questions to Promote Active Reading

Using questions in written text is an effective strategy to enhance readability. However, what makes an active reading question good, what the linguistic role of these questions is, and what is their impact on human reading remains understudied. We introduce GuidingQ, a dataset of 10K in-text questions from textbooks and scientific articles. By analyzing the dataset, we present a comprehensive understanding of the use, distribution, and linguistic characteristics of these questions. Then, we explore various approaches to generate such questions using language models. Our results highlight the importance of capturing inter-question relationships and the challenge of question position identification in generating these questions. Finally, we conduct a human study to understand the implication of such questions on reading comprehension. We find that the generated questions are of high quality and are almost as effective as human-written questions in terms of improving readers' memorization and comprehension.

Stepwise Verification and Remediation of Student Reasoning Errors with Large Language Model Tutors

Large language models (LLMs) present an opportunity to scale high-quality personalized education to all. A promising approach towards this means is to build dialog tutoring models that scaffold students' problem-solving. However, even though existing LLMs perform well in solving reasoning questions, they struggle to precisely detect student's errors and tailor their feedback to these errors. Inspired by real-world teaching practice where teachers identify student errors and customize their response based on them, we focus on verifying student solutions and show how grounding to such verification improves the overall quality of tutor response generation. We collect a dataset of 1K stepwise math reasoning chains with the first error step annotated by teachers. We show empirically that finding the mistake in a student solution is challenging for current models. We propose and evaluate several verifiers for detecting these errors. Using both automatic and human evaluation we show that the student solution verifiers steer the generation model towards highly targeted responses to student errors which are more often correct with less hallucinations compared to existing baselines.

Multilingual Trolley Problems for Language Models

As large language models (LLMs) are deployed in more and more real-world situations, it is crucial to understand their decision-making when faced with moral dilemmas. Inspired by a large-scale cross-cultural study of human moral preferences, "The Moral Machine Experiment", we set up the same set of moral choices for LLMs. We translate 1K vignettes of moral dilemmas, parametrically varied across key axes, into 100+ languages, and reveal the preferences of LLMs in each of these languages. We then compare the responses of LLMs to that of human speakers of those languages, harnessing a dataset of 40 million human moral judgments. We discover that LLMs are more aligned with human preferences in languages such as English, Korean, Hungarian, and Chinese, but less aligned in languages such as Hindi and Somali (in Africa). Moreover, we characterize the explanations LLMs give for their moral choices and find that fairness is the most dominant supporting reason behind GPT-4's decisions and utilitarianism by GPT-3. We also discover "language inequality" (which we define as the model's different development levels in different languages) in a series of meta-properties of moral decision making.

Confidence Regulation Neurons in Language Models

Despite their widespread use, the mechanisms by which large language models (LLMs) represent and regulate uncertainty in next-token predictions remain largely unexplored. This study investigates two critical components believed to influence this uncertainty: the recently discovered entropy neurons and a new set of components that we term token frequency neurons. Entropy neurons are characterized by an unusually high weight norm and influence the final layer normalization (LayerNorm) scale to effectively scale down the logits. Our work shows that entropy neurons operate by writing onto an unembedding null space, allowing them to impact the residual stream norm with minimal direct effect on the logits themselves. We observe the presence of entropy neurons across a range of models, up to 7 billion parameters. On the other hand, token frequency neurons, which we discover and describe here for the first time, boost or suppress each token's logit proportionally to its log frequency, thereby shifting the output distribution towards or away from the unigram distribution. Finally, we present a detailed case study where entropy neurons actively manage confidence in the setting of induction, i.e. detecting and continuing repeated subsequences.

DIRAS: Efficient LLM-Assisted Annotation of Document Relevance in Retrieval Augmented Generation

Retrieval Augmented Generation (RAG) is widely employed to ground responses to queries on domain-specific documents. But do RAG implementations leave out important information or excessively include irrelevant information? To allay these concerns, it is necessary to annotate domain-specific benchmarks to evaluate information retrieval (IR) performance, as relevance definitions vary across queries and domains. Furthermore, such benchmarks should be cost-efficiently annotated to avoid annotation selection bias. In this paper, we propose DIRAS (Domain-specific Information Retrieval Annotation with Scalability), a manual-annotation-free schema that fine-tunes open-sourced LLMs to annotate relevance labels with calibrated relevance probabilities. Extensive evaluation shows that DIRAS fine-tuned models achieve GPT-4-level performance on annotating and ranking unseen (query, document) pairs, and is helpful for real-world RAG development.