Where we see one function, they see two

Abstract

This study investigates when and how university students in first-semester introductory calculus interpret multiple representations of the same function. Specifically, it focuses on three tasks. The first task has students give their definitions of ‘function sameness’, the results of which suggests that many students understand a function as being determined by more than its points. The second has them assess the derivative of a piecewise-defined representation of a typical polynomial function, the results of which suggest that students see a piecewise function as two separate functions with the conditions as instructions. The third involves an instance of the fundamental theorem of calculus, the results of which suggest that students view (what we see as) the same function as two different functions. Taken together, these results suggest that many students see two functions where we as mathematicians see only one. In particular, students see various features of a function’s analytic and graphical representations as being essential to the function’s identity.