What do students learn about functions? : a cross cultural study in England and Malaysia
This research study investigates the concept of function developed by a sample of secondary and university students in England and Malaysia studying mathematics as one of their subjects. It shows that whilst students may be able to do the 'mechanical' parts of this concept, their grasp of the 'theoretical' nature of the function concept may be tenuous and inconsistent. The hypothesis is that students develop 'prototypes' for the function concept in much the same way as they develop prototypes for concepts in everyday life. The definition of the function concept, though given in the curriculum, proves to be inoperative, with their understanding of the concept reliant on properties of familiar prototype examples: those having regular shaped graphs, such as x2 or sinx, those often encountered (possibly erroneously), such as a circle, those in which y is defined as an explicit formula in x, and so on. The results of the study in England revealed that even when the function concept was taught through the formal definition, the experiences which followed led to various prototypical conceptions. Investigations also show significant misconceptions. For example, three-quarters of a sample of students starting a university mathematics course considered that a constant function was not a function in either its graphical or algebraic forms, and three quarters thought that a circle is a function. The extension of the study in Malaysia was made with the hypothesis that there is a significant difference between the concept as perceived to be taught and as actually learned by the students. Although the intended curriculum emphasises conceptual understanding, in the perceived curriculum (curriculum as understood by the teachers), only 45% of the teachers follow this approach. The tested curriculum as reflected in the public examination questions, only emphasises the procedural skills and the results of the learned curriculum show that learning of functions is more consistent with the theory of prototypical learning. Students in Malaysia develop their own idiosyncratic mental prototypes for the function concept in much the same way as those students in the UK.