Challenging children to think : an investigation of the behaviours of highly effective teachers that stimulate children to examine their mathematical understandings.

The purpose of this study was to investigate teacher-student interactions that create challenging opportunities for students to learn mathematics. The behaviours used by highly effective teachers during such interactions were described in detail focusing on the ways teachers challenge young children to think mathematically. Children's accounts of these exchanges were also examined. Four teachers who taught children aged 5 to 7 years participated in the study. They were identified as highly effective teachers of mathematics through their earlier involvement in the Early Numeracy Research Project. Complementary accounts methodology was used for this study in order to capture some of the complexities of classrooms settings and to collect rich data (Clarke, 1998, 2001). The methodology involved observing, videotaping and interviewing teachers and children to collect varying accounts of classroom events. Data were collected from "ordinary" mathematics lessons in naturalistic settings. Whole mathematics lessons were observed and videotaped to enable the incidents of particular interest to be seen in context. A picture of the phenomenon under investigation was built by examining the differing accounts of events provided by the teacher, the child, and the researcher/observer. Data are presented as four case studies of the teachers, each detailing examples of the phenomena as vignettes from the classroom. Similarities and differences between teachers are discussed using a cross-case analysis. The main findings of this study show: • Mathematically challenging conversations happened on a daily basis III the classrooms of the teachers in this study. • These mathematical conversations often formed "strings" of interactions that occur over lessons, days, and even several days to develop lines of mathematical thinking. • Teachers and children remembered these intense exchanges. • Children learned mathematics when challenged to think during these interactions. • The skills displayed by teachers were complex and subtle. Highly effective teachers had a clear learning path in mind for the children in their classes, and for individuals within the class, and it was at the "cutting edge" of each child's understanding that the teachers were attempting to work. They questioned and evaluated each child's conceptual development, put that knowledge into the context of a broad knowledge of a mathematical framework, then made "on the run" judgments about how best to press for further understanding. Sometimes they offered linking ideas to connect the child to past experiences in mathematics, sometimes they offered a slight variation on the problem, and sometimes they asked a question that required the child to think generally about what had happened. They used a range of various techniques to suit the event. Perhaps the most striking quality was the flexibility and the speed with which they considered and responded to situations as they unfolded.