Learning and Teaching Mathematics in the First Two Years at School : Group Work, Independence, and Understanding

Abstract

In New Zealand junior classrooms the teacher's position is the key to the development of children's independent work in mathematics. This study found that the teachers used independent group work in mathematics as a means of developing children's independence, and that their practices reflected their child-centred philosophy. The findings are based on the observations of 8 classrooms in different schools in the Wellington City area over Terms 2 and 3 of 1993. The children were in their first two years at school. Field notes were taken of a total of 113 mathematics sessions, with the notes based on informal conversations with teachers and children about events which occurred during these sessions. Audiotape, videotape and photographic records of each session provided additional details. The teachers were interviewed using a semi-structured format. Interviews were audio-taped and later transcribed. An analysis, informed by emergent qualitative enquiry, focused on the issues surrounding independence and group work. The study identified a number of interrelated practices, influenced by a "child-centred" discourse, which were used by teachers in the structuring of the learning environment. Teachers used a number of micro-strategies to guide children's independent use of a number of "hands-on" activities available to each group. These strategies provided guidelines to children on appropriate classroom behaviour and were characterised by terms such as "share and take turns". Children were encouraged to take responsibility for task completion through, for example, the teacher's inquiry, "are you on task?". Teachers set up a system for social problem solving by encouraging strategies of peer support. Provided these guidelines were followed in the "child-led" independent time, children were free to interpret and adapt activities in their own way. Such practices typically involved a non-interventionist teacher position. Several of the teachers maintained this position through various micro-strategies which enabled them to teach other groups of children uninterrupted. The core, teacher-led sessions prepared children for their independent work through modelling and demonstrating activities to be used independently. The teachers thought of the core sessions as being the time for new learning of mathematical concepts, and the independent sessions as an opportunity to practise and consolidate that learning. This study argues that teachers taking this approach subsume their goal of teaching and learning mathematics with understanding to their primary goal of developing independence. A different view of independence was identified through emerging practice, and typified by a few of the teachers taking a more active position in guiding children's independent work through the use of interventionist strategies. Such teachers regarded groups more as communities of learners rather than as a management device. There was more emphasis on "learning how to learn" mathematics as a means to developing children's independence, rather than seeing independence as a goal its own right. The focus on independence was subsumed within the goal of developing children's independence through mathematical inquiry. The teachers established norms for the children's mathematical inquiry. Norms which prompted the learning of mathematics were typified by questions which might be asked by the teacher or by the children of each other as, "how will you decide?". This research raises questions about how teachers think about the exhibition by children of independence in the classroom and the use of the mathematics sessions to develop such independence. It raises questions about how we think about such independence and the ways in which this can be developed through promoting mathematical processes. In particular the study highlights what appear to be the key factors of effective small group instruction in mathematics, the place of the teacher in this, and their function as they seek to develop both independence and the children's understanding of mathematics. The study concludes that teachers need to focus on the development of mathematical understanding in the management of the classroom programme. Their desire to help children learn how to learn needs to be implemented through an explicitly mathematical framework during their mathematics sessions.