Educational Cone Model in Embedding Vector Spaces

Abstract

Human-annotated datasets with explicit difficulty ratings are essential for intelligent educational systems. While embedding vector spaces are widely used to represent semantic closeness and are promising for analyzing text difficulty, the abundance of embedding methods creates a challenge in choosing the most suitable one. This study proposes the **Educational Cone Model**, a geometric framework based on the assumption that easier texts are less diverse (focusing on fundamental concepts), whereas harder texts are more diverse. This assumption leads to a cone-shaped distribution in embedding space, regardless of the embedding method used. The model frames the evaluation of embeddings as an optimization problem, aiming to detect structured difficulty-based patterns. By designing specific loss functions, the authors derive efficient, closed-form solutions that avoid costly computations. Empirical tests on real-world datasets validate the model’s effectiveness and speed in identifying embedding spaces best aligned with difficulty-annotated educational texts.