Mathematics curriculum and policy documents in Australia, United Kingdom and the United States advocate the use of mathematical investigations as learning and assessment tools (Australian Education Council, 1991, Cockcroft, 1982; National Council for Teachers of Mathematics, 1989). Much of the research into mathematical investigations has focused on the teacher's actions in classroom implementations. Mathematical investigations also afford collaborative efforts amongst students. Therefore, there is a need for theorising the learning that occurs during these mathematical investigations. The theoretical framework for the study drew on sociocultural theories of learning, and reviewed four interpretations of the zone of proximal development. These involved scaffolding, peer collaboration, interweaving of everyday and scientific knowledge and participation in a community of practice. The second and fourth interpretations bear the most relevance to this
study, since peer collaboration and classroom cultures are especially important to understanding the nature of helping interactions and social networks.
The emergent design of the research was crucial in identifying and analysing key issues and questions as they arose. An initial exploratory study sought to identify the benefits of mathematical investigations as a teaching, learning and assessment tool, as well as the sources of assistance that students drew upon to progress with the task. This study highlighted the significance of peer helping during investigation tasks. The notion of help was further investigated in a Classroom Networks Study, which considered social arrangements in students' helping interactions. Revoicing was found to be one particularly helpful kind of interaction. The first two stages of the research took place at one secondary school, where mathematical investigations were used to broaden the scope of teaching, learning and assessment.
Most of the classes that participated in the study were from the junior school, although there was one senior class that participated in the Classroom Networks Study. Research was conducted during times when students worked on mathematical investigations as non-traditional assessment tasks. This provided the context for investigating the helping interactions between students. Multiple methods were used to gather data on features of classroom cultures and the helping interaction between students. Observations were the key data collection method used to investigate the classroom cultures and the helping interactions. Questionnaires were used to identify whom students acknowledged to be in their network of help, while interviews were conducted to obtain more detailed information on whom students sought for help and why.
The nature of mathematical investigations was explored using Ahmed's (1982) framework and Australian policy and syllabus documents as analytical
tools. The mathematics investigations resembled "rich mathematical activities", evident in the ways that students responded to them. Benefits that students experienced from participating in mathematics investigations were related to mathematics learning processes from the Years 1-10 Mathematics Syllabus (Queensland Department of Education, 1987) such as "Confidence", "A Positive Response to Use of Mathematics", "Co-operative Effort" and "Enjoyment".
Classroom cultures were also distinguished as being either "discussion-based" or "didactic" (Boaler, 1998, 2000). Characteristics of "discussion-based" classrooms closely resembled those of "community of practice" classrooms. In these classrooms, students were invited to share and contribute ideas, and were involved in looking for
relationships, and discussions of possible solutions. The findings revealed that these discussion-based classrooms seemed to provide the appropriate context for facilitating helping interactions.
Helping network categories (loose groups, cliques) were identified and analysed in terms of how students helped each other in these social arrangements. Helping interactions were investigated through transcripts of lesson segments, and qualitatively analysed, using frameworks developed by Beaumont (1999) and Webb (1991).
As revoicing appeared to be a key characteristic of helping interactions, a simulated study was carried out so as to generate rich data. This involved teaching a mathematical concept to a small group of students who were then to teach this concept to their peers. These peers were also to teach another peer the same concept. This was done to track and document episodes where revoicing occurs. The sessions
were videotaped and audio taped for accurate transcription and reporting. The present study investigated how students revoiced mathematical terms, and means of establishing identity and control.
The findings in this study have both theoretical and practical significance. In theory, the study attempts to draw together the literature on peer interaction, helping networks and classroom as a community of practice. While there have been extensive literature on each of these separate bodies, there have been limited research merging these bodies together. From a practical perspective, the study could inform teachers on how they can create a helping culture within their classroom.