Variation , Covariation , and Functions : Foundational Ways of Thinking Mathematically *

In beginning this chapter we immediately faced a dilemma. There is nothing that can be called “the concept of function.” The phrase “concept of function,” regardless of its meaning, immediately calls into question whom we envision having it. Is it a mathematician, a teacher, a student, or a researcher in mathematics education? A student’s conception of function will not be as developed as that held by a mathematician, and a mathematician’s conception of function may not include detailed information that a math education researcher has about how students’ function understanding develops. Another dilemma in writing this chapter is that different researchers in mathematics education have had different conceptions of function and therefore held different norms for “students’ understanding of function.” Because there are many meanings and ways of thinking that various individuals and groups hold that could fit under the heading concept of function, we avoid speaking as if there is a standard, generally accepted meaning of function against which others should be compared. Instead, we specify the meanings and ways of thinking that we envision a person having a concept of function holds. We organized this chapter into six parts to capture the broad swath of issues surrounding the idea of covariation as a foundation for function in mathematics; the ways that covariation can be conceived among students, teachers, and researchers; and implications of various forms of reasoning covariationally. Specifically, we (1) provide a brief overview of how conceptions of function evolved historically and the central role that covariation played; (2) clarify what we mean by variational and covariational reasoning and where these meanings came from; (3) examine research on students’ and teachers’ variational and covariational reasoning in selected areas; (4) comment briefly, with a covariational lens, on past research on students’ and teachers’ conceptions of function; (5) discuss curricular treatments of function, again with a covariational lens; and (6) outline possible directions for future research.