The didactical tetrahedron as a heuristic for analysing the incorporation of digital technologies into classroom practice in support of investigative approaches to teaching mathematics

There have been various proposals to expand the heuristic device of the didactical triangle to form a didactical tetrahedron by adding a fourth vertex to acknowledge the significant role of technology in mediating relations between content, student and teacher. Under such a heuristic, the technology vertex can be interpreted at several levels from that of the material resources present in the classroom to that of the fundamental machinery of schooling itself. At the first level, recent research into teacher thinking and teaching practice involving use of digital technologies indicates that, while many teachers see particular tools and resources as supporting the classroom viability of investigative approaches to mathematics, the practical expressions of this idea in lessons vary in the degree of emphasis they give to a didactic of reconstruction of knowledge, as against reproduction. At the final level, examining key structuring features of teaching practice makes clear the scope and scale of the situational adaptation and professional learning required for teachers to successfully incorporate use of digital tools and resources in support of investigative approaches. These issues are illustrated through examining contrasting cases of classroom use of dynamic geometry in professionally well-regarded mathematics departments in English secondary schools.