This paper addresses some methodological issues concerning traditional linear approaches and shows the need for a paradigm shift in education research towards the Complexity and Nonlinear Dynamical Systems (NDS) framework. It presents a quantitative piece of research aiming to test the nonlinear dynamical hypothesis in education. It applies catastrophe theory and demonstrates that students' achievements in science education could be described by a cusp model, where two cognitive variables are implemented as controls - the logical thinking as the asymmetry and the field dependence/independence as the bifurcation respectively. The results support the nonlinear hypothesis by providing the empirical evidence for bifurcation and hysteresis effects in students' performance. Interpretation of the model is provided and implications and fundamental epistemological issues are discussed.