A Conceptual Analysis of Early Function Through Quantitative Reasoning

Abstract

Understanding functional relationships is a critical aspect of students' algebraic reasoning, and the manner in which students develop early concepts of function provides an important foundation for later ideas in secondary and upper-level mathematics. Typical school treatment of function relies on a correspondence approach, in which function is introduced as the fixed relationship between the members of two sets. Under this approach, however, students have experienced challenges in meaningfully identifying, representing, and manipulating functional relationships. We propose an alternate approach to the development of functional relationships that leverages contexts with continuously covarying quantities. Situating students' exploration in continuous joint variation can support their abilities to identify relevant quantities, explore how those quantities change together, and ultimately create ratios, rates, and algebraic representations of functional relationships. In this chapter we provide a conceptual analysis of linear and quadratic functions, identifying for each function family a sequence of recommended conceptual activities and examples of associated student reasoning. We close with a discussion of task design principles to guide curricular decisions for introducing these function families. 3