Reasoning Abilities in Covariational Functional Thinking: An empirically validated framework for Grade 8–10 students

Abstract Covariational functional thinking (FT), which involves conceptualizing functions as process of covariation and describing them in terms of rate of change of their variables, fosters an intuitive and dynamic understanding of functions. While covariational reasoning frameworks provide cognitive tools for analyzing and envisioning changes between quantities, little is known about the reasoning abilities that support students’ engagement with functional relationships in secondary school. This study provides and validates a model of covariational FT abilities among 13-to-16-year-old students (n = 350), identifying four key reasoning abilities: Qualitatively envisioning covariation by grasping the Direction of Change of one quantity in relation to another, numerically coordinating changes through the Calculation of Constant Rate of Change in linear functions, Discernment Between Varying and Constant Rates of Change , and developing Awareness of Rate of Change Variability for varying rates of change, graphically and numerically. Findings indicate that Direction is a strong direct predictor of Variability Awareness , with smaller additional indirect links via Calculation and Discernment (weak partial mediation). Discernment emerged as a moderate predictor, while Calculation and Direction also showed meaninful direct contributions. Furthermore, the correspondence supported by the input–output view of functions moderated the relationship between Direction and Variability Awareness . These results highlight the importance of engaging students in functional contexts that integrate qualitative and quantitative reasoning, conceptualize quantities, analyze changes, and connect explanations to graphical representations.