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Abstract Studying change over time requires rigorous and sometimes novel statistical methods that can support increasingly complex applied research questions. In this article, we provide an overview of the potential of piecewise growth mixture models. This type of longitudinal model can be used to advance our understanding of group and individual growth that may follow a segmented, or disjointed, pattern of change, and where the data come from a mixture of two or more latent classes. We then demonstrate the practical utility of piecewise growth mixture models by applying it to a subsample of students from the Early Childhood Longitudinal Study – Kindergarten Cohort of 1998 (ECLS-K) to ascertain whether mathematics achievement is characterized by one or two latent classes akin to students with and without mathematics difficulties. We discuss the applicability for school psychological research and provide supplementary online files that include an instructional sample dataset and corresponding R routine with explanatory annotations to assist in understanding the R routine before applying this approach in novel applications (https://doi.org/10.1016/j.jsp.2019.03.004). [ABSTRACT FROM AUTHOR]

Effective instructional planning and intervening rely heavily on accurate understanding of students' growth, but relatively few researchers have examined mathematics achievement trajectories, particularly for students with special needs. We applied linear, quadratic, and piecewise linear mixed-effects models to identify the best-fitting model for mathematics development over elementary and middle school and to ascertain differences in growth trajectories of children with learning disabilities relative to their typically developing peers. The analytic sample of 2150 students was drawn from the Early Childhood Longitudinal Study — Kindergarten Cohort, a nationally representative sample of United States children who entered kindergarten in 1998. We first modeled students' mathematics growth via multiple mixed-effects models to determine the best fitting model of 9-year growth and then compared the trajectories of students with and without learning disabilities. Results indicate that the piecewise linear mixed-effects model captured best the functional form of students' mathematics trajectories. In addition, there were substantial achievement gaps between students with learning disabilities and students with no disabilities, and their trajectories differed such that students without disabilities progressed at a higher rate than their peers who had learning disabilities. The results underscore the need for further research to understand how to appropriately model students' mathematics trajectories and the need for attention to mathematics achievement gaps in policy. [ABSTRACT FROM AUTHOR]