Children's understanding of the commutativity and complement principles: A latent profile analysis

This study examined patterns of individual differences in the acquisition of the knowledge of the commutativity and complement principles in 115 five-to six-year-old children and explored the role of concrete materials in helping children understand the prinicples. On the basis of latent profile analysis, four groups of children were identified: The first group succeeded in commutativity tasks with concrete materials but in no other tasks; the second succeeded in commutativity tasks in both concrete and abstract conditions, but not in complement tasks; the third group succeeded in all commutativity tasks and in complement tasks with concrete materials, and the final group succeeded in all the tasks. The four groups of children suggest a developmental trend – (1) Knowledge of the commutativity and of the complement principles seems to develop from thinking in the context of specific quantities to thinking about more abstract symbols; (2) There may be an order of understanding of the principles – from the commutativity to the complement principle; (3) Children may acquire the knowledge of the commutativity principle in the more abstract tasks before they start to acquire the knowledge of the complement principle. This study contributes to the literature by showing that assessing additive reasoning in different ways and identifying profiles with classification analyses may be useful for educators to understand more about the developmental stage where each child is placed. It appears that a more fine-grained assessment of additive reasoning can be achieved by incorporating both concrete materials and relatively abstract symbols in the assessment.