Learning trajectories towards strategy proficiency in multi-digit division – A latent transition analysis of strategy and error profiles

Children's proficiency in multi-digit division is based on the application of a variety of different solution strategies with different conceptual foundations. Accordingly, we assume that children can have qualitatively distinct profiles of correctly used strategies and errors, as well as different kinds of learning trajectories towards proficiency. Moreover, children's abilities in reasoning about number relations are regarded as an enabling condition for the application of solution strategies. As such, these abilities should influence both the prevalence of profiles of strategy proficiency, and the transitions between these profiles. In the current study, 208 fourth graders first discovered, then compared and practiced division strategies. We administered a test with six division problems (before and after the intervention) and a measurement of abilities in reasoning about number relations (before the intervention). Correctly used strategies and errors were coded. A latent transition analysis with children's abilities in reasoning about number relations as covariate identified three latent classes (levels) of strategy proficiency. It confirmed that both the initial class prevalences and the latent transition probabilities were influenced by the covariate. The analysis illustrated different learning trajectories, which can be explained by the specific conceptual foundations of different solution strategies in multi-digit division.