Functions: Looking ahead beyond calculus

The concept of a function is a unifying foundation of modern mathematics. Increasingly, the func-tion concept also guides middle school algebra curricula, and ubiquitous research studies analyze its learning. Such recent changes have helped bridging the traditional gap between high-school algebra and college-level calculus. But teaching and learning functions in a narrow context opens and widens the rift between calculus and both post-calculus courses and real-world applications. This note pro-vides several examples to illustrate that some of these efforts miss what mathematicians consider key characteristics of functions. It is based on the observation that these school curricular changes and the majority of research studies focus almost exclusively on (real valued) functions of a (scalar) continuous variable. We focus on composition as the distinguishing operation on functions, issues relating to domain and codomain, injectivity and surjectivity, and the interplay with alternations of universal and existential quantifiers. This note is to stimulate discussion about how much teachers of algebra and curriculum designers at all levels need to understand about functions themselves, and about how deeply should students at which level understand functions.