The teacher's role has often been described as one of "orchestration", and this musical analogy is a powerful one in characterising the manipulation of features in the classroom setting in order to generate activity or "performance" which leads to learning. However, a classical view of orchestration would fail to recognise the extent to which effective teaching and learning make use of serendipity and improvisation--characteristics more often associated with jazz. This paper uses the characteristics of various musical genres to characterise teaching approaches observed in the authors' work in two research projects investigating the use of ICT in mathematics classrooms. In particular the authors demonstrate how jazz and other musical analogies can be useful when describing some of the more effective classrooms in which serendipitous events were exploited and performances were improvised by pupils as well as teachers. They discuss the ways in which teachers were able to use ICT to establish conditions under which more jazz-like performances were likely to occur, offering opportunities for more creative, improvised teaching and learning. They also examine lessons that can be learned by examining differences between musical and pedagogical settings. (Contains 1 table.)