Educational Cone Model in Embedding Vector Spaces

TL;DR

This paper introduces the Educational Cone Model, a geometric framework for evaluating text difficulty in embedding spaces, validated through empirical tests on real-world educational datasets.

Contribution

The study proposes a novel geometric model that characterizes text difficulty in embedding spaces and offers an efficient evaluation method for embedding quality.

Findings

Effective identification of difficulty-aligned embeddings

Fast, closed-form solutions for evaluation

Validated on real-world educational datasets

Abstract

Human-annotated datasets with explicit difficulty ratings are essential in intelligent educational systems. Although 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 selecting the most suitable method. This study proposes the Educational Cone Model, which is 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 the embedding space regardless of the embedding method used. The model frames the evaluation of embeddings as an optimization problem with the aim of detecting structured difficulty-based patterns. By designing specific loss functions, efficient closed-form solutions are derived that avoid costly computation. Empirical tests on real-world datasets validated the model's effectiveness and speed in identifying the embedding spaces that are best aligned with difficulty-annotated educational texts.