Learning Sequences

TL;DR

This paper presents algorithms for representing, generating, and assessing student knowledge states within learning spaces using sequences, with applications in adaptive learning systems and theoretical insights into their structure.

Contribution

It introduces a sequence-based representation of learning spaces, enabling efficient implementation, state generation, and integration into knowledge assessment algorithms.

Findings

Representation of learning spaces using sequences allows efficient state generation.

Algorithms for defining and decomposing learning spaces from sequences are developed.

Theoretical results include properties of projections, decomposition, and algebraic representations of learning spaces.

Abstract

We describe the algorithms used by the ALEKS computer learning system for manipulating combinatorial descriptions of human learners' states of knowledge, generating all states that are possible according to a description of a learning space in terms of a partial order, and using Bayesian statistics to determine the most likely state of a student. As we describe, a representation of a knowledge space using learning sequences (basic words of an antimatroid) allows more general learning spaces to be implemented with similar algorithmic complexity. We show how to define a learning space from a set of learning sequences, find a set of learning sequences that concisely represents a given learning space, generate all states of a learning space represented in this way, and integrate this state generation procedure into a knowledge assessment algorithm. We also describe some related theoretical results concerning projections of learning spaces, decomposition and dimension of learning spaces, and algebraic representation of learning spaces.