Intellectual need, covariational reasoning, and function: Freeing the horse from the cart

Authors (year) contended a quantitatively sophisticated image of a dynamic situation can provide students with the horse needed to pull the cart that is the mathematical properties important for the set-theoretic definition of function. In this paper, we extend our argument in two ways. First, we adopt Harel’s (2008, 2018) constructs of intellectual need and epistemological justification to describe how a student can develop a quantitatively sophisticated image of a dynamic situation. Second, we exemplify that constructing an epistemological justification can support a student in subsequently making determinations regarding ‘function’, yet there is an apparent lack of intellectual need for differentiating between functional and non-functional relationship. This latter factor leads us to call into question the value of focusing on the metaphorical cart that is the basis for the set-theoretic definition of function.