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.
Highlights
It is potentially beneficial for teaching to have an evidence-based understanding of the different component of algebraic competency
There are many theoretical constructs that aim at structuring school algebra: Usiskin (1988) defined four conceptions of algebra, Kieran (2004) has distinguished generational activities, transformational activities, and global/meta-level activities
The same holds true for other algebraic concepts, e.g. Bardini et al (2013) distinguish different uses, and different understandings, of the equal sign, Mason and Sutherland (2002) introduce key aspects of school algebra
Summary
It is potentially beneficial for teaching to have an evidence-based understanding of the different component of algebraic competency. The same holds true for other algebraic concepts, e.g. Bardini et al (2013) distinguish different uses, and different understandings, of the equal sign, Mason and Sutherland (2002) introduce key aspects of school algebra. This list could be prolonged much more but this should suffice to give support for the following observations: There are many conceptual approaches to structure algebraic competencies, but statistical models are usually restricted to special cases, i.e. tests are constructed to explore the constructs of a special conception and it is usually not possible to apply one model to data from other tests. The full program will be outlined at the end of this paper
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