Towards Generalizable Representations of Mathematical Strategies

2026-04-09

AI Summary

Existing approaches struggle to generalize representations of students’ overall mathematical problem-solving strategies, often constrained by manual annotations or platform-specific behaviors and lacking cross-problem applicability. This work proposes a novel sequence representation method based on algebraic state transitions: leveraging a pretrained model to encode algebraic states, constructing transition embeddings via vector differencing, and refining strategy sequence representations through SimCSE-based contrastive learning. The approach achieves, for the first time, a generalizable modeling framework for problem-solving strategies across platforms and problems, and introduces new metrics to quantify strategy uniqueness, diversity, and consistency. Experimental results demonstrate superior performance in multi-label action classification, problem-solving efficiency prediction, and sequence reconstruction tasks. Moreover, the derived embedding metrics significantly correlate with both short- and long-term student learning outcomes, enabling scalable educational data mining and automated assessment.

Technology Category

• Machine Learning: Representation Learning
• Multiagent Systems: Distributed Problem Solving
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Pretrained encoders for mathematical texts have achieved significant improvements on various tasks such as formula classification and information retrieval. Yet they remain limited in representing and capturing student strategies for entire solution pathways. Previously, this has been accomplished either through labor-intensive manual labeling, which does not scale, or by learning representations tied to platform-specific actions, which limits generalizability. In this work, we present a novel approach for learning problem-invariant representations of entire algebraic solution pathways. We first construct transition embeddings by computing vector differences between consecutive algebraic states encoded by high-capacity pretrained models, emphasizing transformations rather than problem-specific features. Sequence-level embeddings are then learned via SimCSE, using contrastive objectives to position semantically similar solution pathways close in embedding space while separating dissimilar strategies. We evaluate these embeddings through multiple tasks, including multi-label action classification, solution efficiency prediction, and sequence reconstruction, and demonstrate their capacity to encode meaningful strategy information. Furthermore, we derive embedding-based measures of strategy uniqueness, diversity, and conformity that correlate with both short-term and distal learning outcomes, providing scalable proxies for mathematical creativity and divergent thinking. This approach facilitates platform-agnostic and cross-problem analyses of student problem-solving behaviors, demonstrating the effectiveness of transition-based sequence embeddings for educational data mining and automated assessment.

Problem

• mathematical strategies
• solution pathways
• generalizable representations
• educational data mining
• student problem-solving

Innovation

• transition embeddings
• problem-invariant representations
• SimCSE
• mathematical strategies
• educational data mining