Transforming equations into equivalent equations—an empirical study of the equation transformation capability as a multidimensional construct

Abstract This study focuses on equivalent transformations of equations in the context of secondary school education. Solving (determining the solution set of an equation), normalizing (transforming an equation to reach a certain form), and reorganizing equations (isolating variables in formulae) are discussed as different possible applications of these transformations. An equation transformation capability is defined, incorporating these activities, and is empirically studied. A confirmatory factor analysis is performed to investigate the capability as a construct concerning each application (not) representing a single latent variable. Furthermore, success rates for individual applications are analyzed. Additionally, statistical implicative analysis is performed to study possible implicative relations between the different applications. The analyses are based on a test that 271 secondary school students took at the end of the 9th and 10th grades in Germany. Results suggest a three-dimensional model, with each of the mentioned applications representing individual latent variables. Regarding the reorganization of equations, it does not seem to matter whether the equation is embedded in a context or not. This application also appears to be the most challenging one, whereas the other two do not differ regarding the solution rate. The implicative analysis suggests that learners who are able to reorganize equations are more likely to solve or normalize equations compared to the reverse implications.