Do fuzzy-logic non-linear models provide a benefit for the modelling of algebraic competency?

Abstract

Statistical models used in mathematics education are often linear and latent variables are often assumed to be normally distributed. The present paper argues that by relaxing these constraints one may use models that fit the data better than linear ones and provide more insight into the domain. It combines research on statistical methodology with research on the competence structure within algebra. The methodological innovation is that models with latent variables from the unit interval are considered which allows to model relations by means of fuzzy logic. Estimation techniques for such models are discussed to the extend necessary for the present study. To assess the benefit of this modelling technique data from an algebra test is re-analyzed. It is shown that non-linear models have greater explanatory power and give interesting didactical insights. Moreover, model comparison allows to differentiate between different theoretical constructs related to algebraic understanding. Finally, a research program is outlined that aims at the development of a universal algebra competence model that can be applied to test data from various algebra tests.