Instrumenting Mathematical Activity: Reflections on Key Studies of the Educational Use of Computer Algebra Systems

This paper examines the process through which students learn to make functional use of computer algebra systems (CAS), and the interaction between that process and the wider mathematical development of students. The result of ‘instrumentalising‘ a device to become a mathematical tool and correspondingly ‘instrumenting’ mathematical activity through use of that tool is not only to extend students' mathematical technique but to shape their sense of the mathematical entities involved. These ideas have been developed within a French programme of research – as reported by Artigue in this issue of the journal – which has explored the integration of CAS – typically in the form of symbolic calculators – into the everyday practice of mathematics classrooms. The French research –influenced by socio-psychological theorisation of the development of conceptual systems- seeks to take account of the cultural and cognitive facets of these issues, noting how mathematical norms – or their absence – shape the mental schemes which students form as they appropriate CAS as tools. Instrumenting graphic and symbolic reasoning through using CAS influences the range and form of the tasks and techniques experienced by students, and so the resources available for more explicit codification and theorisation of such reasoning. This illuminates an influential North American study– conducted by Heid – which French researchers have seen as taking a contrasting view of the part played by technical activity in developing conceptual understanding. Reconsidered from this perspective, it appears that while teaching approaches which ‘resequence skills and concepts’ indeed defer – and diminish –attention to routinised skills, the tasks introduced in their place depend on another –albeit less strongly codified – system of techniques, supporting more extensive and active theorisation. The French research high lights important challenges which arise in instrumenting classroom mathematical activity and correspondingly instrumentalising CAS. In particular, it reveals fundamental constraints on human-machine interaction which may limit the capacity of the present generation of CAS to scaffold the mathematical thinking and learning of students.