This study introduced various nonlinear growth models, including the quadratic conventional polynomial model, the fractional polynomial model, the Sigmoid model, the growth model with negative exponential functions, the multidimensional scaling technique, and the unstructured growth curve model. It investigated which growth models effectively describe student growth in math and reading using four-wave longitudinal achievement data. The objective of the study is to provide valuable information to researchers especially when they consider applying one of the nonlinear models to longitudinal studies. The results showed that the quadratic conventional polynomial model fit the data best. However, this model seemed to overfit the data and made statistical inference problematic concerning parameter estimates. Alternative nonlinear models with fewer parameters adequately fit the data and yielded consistent significance testing results under extreme multicollinearity. It indicates that the alternative models denoting somewhat simpler models would be selected over the conventional polynomial model with more fixed parameters. Other practical issues pertaining to these growth models are also discussed.