The Student Zipf Theory: Inferring Latent Structures in Open-Ended Student Work To Help Educators

Abstract

Are there structures underlying student work that are universal across every open-ended task? We demonstrate that, across many subjects and assignment types, the probability distribution underlying student-generated open-ended work is close to Zipf’s Law. Inferring this latent structure for classroom assignments can help learning analytics researchers, instruction designers, and educators understand the landscape of various student approaches, assess the complexity of assignments, and prioritise pedagogical attention. However, typical classrooms are way too small to witness even the contour of the Zipfian pattern, and it is generally impossible to perform inference for Zipf’s law from such small number of samples. We formalise this difficult task as the Zipf Inference Challenge: (1) Infer the ordering of student-generated works by their underlying probabilities, and (2) Estimate the shape parameter of the underlying distribution in a typical-sized classroom. Our key insight in addressing this challenge is to leverage the densities of the student response landscapes represented by semantic similarity. We show that our “Semantic Density Estimation” method is able to do a much better job at inferring the latent Zipf shape and the probability-ordering of student responses for real world education datasets.