Cusp catastrophe models for students’ conceptual understanding of the rational numbers

This study investigates students’ understanding of rational numbers within Complex Dynamical Systems (CDS) framework examining the effect of two cognitive variable: logical thinking or formal reasoning and field dependence-independence. Participants (N=202) were low secondary education students in seventh to nineth grade, who were taking a mandatory course in mathematics. The hypothesis concerns potential nonlinear effects in learning outcomes and conceptual understanding of rational numbers, which were explored via catastrophe models. The analysis of empirical data showed that the cusp models were superior to the linear alternatives. The linear combination of the two control variables, formal reasoning and field dependence-independence, and specifically their sum and their difference acted as bifurcation and asymmetry factors respectively. The discontinuities in students’ conceptual understanding support the hypothesis regarding a stepwise conceptual change and add to nonlinear literature and to mathematics education having important implications for learning theories and teaching