Designing tasks to aid understanding of mathematical functions

The concept of a ‘function’ can be viewed as a threshold concept in mathematics. To properly understand functions and to work with them in diverse areas of mathematics, students should be able to conceive of a function as an action, as a process and as an object in its own right. In fact, some authors have claimed that successful mathematical thinking lies in moving flexibly from one interpretation to another. However, most students find the transition of thought involved in progressing to view a function as an object troublesome. Once such ‘reification’ has taken place, it is unlikely to be forgotten, previously inaccessible means of thinking about many different mathematical concepts are opened up, and students’ conceptual understanding is transformed. The key role played by functions in mathematics justifies paying significant attention to the teaching of functions and to the types of tasks assigned to students to support their learning. Traditionally undergraduate courses in mathematics tend to be described in terms of the mathematical content and techniques students should master and theorems they should be able to prove. Moreover, recent studies have shown that many sets of mathematical tasks produced for third-level students emphasize lower level skills, such as memorization and the routine application of algorithms or procedures, rather than endeavouring to develop students’ understanding of the underlying concepts. In this paper, we will describe a set of tasks designed by the authors to help students develop a comprehensive understanding of functions and to move flexibly between different interpretations and representations of functions. The design of tasks drew on a number of frameworks, such as those of Swan (2008) and Mason and Johnston-Wilder (2004), adapted for use with undergraduate students