Learning objectives
After studying this chapter, you should be able to:
• identify content areas of the mathematics curriculum that can be learned through a function-related approach
• explain how the different settings of functions are used to develop students’ understanding
• describe learning activities that develop students’ understanding of functions
• identify and select appropriate technologies for developing students’ understanding of functions
• describe real-life situations that can be modelled with functions.
Introduction
Scholars such as Moore (2014) have suggested that quantitative reasoning and covariational reasoning are both critical for success in secondary and post-secondary mathematics education. The former is the cognition required to represent (or model) an everyday situation by measurable attributes or quantities, whereas the latter is the cognition needed to understand the relationship between two or more varying quantities. Given that a function is essentially a relationship between two varying quantities, then an emphasis on functions is likely to develop covariational reasoning in our students. Moreover, applying their mathematics to everyday situations is likely to develop students’ quantitative reasoning. Therefore, teachers can and should use functions, applied where possible to everyday situations, as they teach the curriculum. In doing so, teachers can use functions to develop those important connections that were outlined in Chapter 3, because functions are a unifying concept.
The chapter commences with a review of the concept of function and in particular it explores the different settings in which functions occur. It is argued that these settings are important as they allow students to encounter different representations of functions, thus building ‘a robust knowledge of adaptable and transferable mathematical concepts’ (ACARA, 2015, p. 6). The chapter then explores the transition to secondary school, arguing that teachers in the middle school years need to utilise the function settings that their students have already encountered, before and as they are introduced to algebra. It then explores the development of the function-related concepts through the lower secondary school, where students are progressively introduced to the different families of functions. As discussed, a number of mathematical concepts, such as rates, can and should be introduced and developed using a function-related approach because doing so will help students build important connections. The last section of the chapter explores functions in the senior secondary school, where the focus is on the algebra of functions, trigonometric functions and modelling with functions.