In recent years, researchers have paid increasing attention to the role of language and social interaction in the learning and pursuit of mathematics (e.g. Forman & van Oers, 1998; Hoyles & Forman, 1995; Monaghan, 1999; Sfard, 2000; Sfard & Kieran, 2001; Barwell, Leung, Morgan & Street, 2005). This interest relates to the function of language in both teacher-student encounters and in peer group activities. For example, Yackel, Cobb and Wood (1991) carried out a study in which all maths instruction in a primary classroom was replaced by problem-solving activities in small groups. While emphasizing the value of the teacher’s guiding role, they found that the group activities offered valuable opportunities for allowing children to construct solutions for themselves through talk, which would not be found in whole-class instruction. Focusing more on the teacher’s role in leading classroom conversation, Strom, Kemeny, Lehrer and Forman (2001) analysed the ways that a teacher used talk to guide the development of children’s mathematical argumentation. Taking a rather different perspective, sociological researchers such as Dowling (1994) have considered how the discourse of mathematics education is constructed through pedagogic practices, and how this affects the accessibility of the subject for some children. Others, such as Barwell (2005), have argued that the tendency amongst policy makers and maths educators to stress the distinction between the precise, subject language of mathematics and more informal talk can hinder the process of inducting children into mathematical practices.
There is a well-established field of research on teacher-student interactions in classrooms, some of which has been directly concerned with the effectiveness of teachers’ discourse strategies for assisting students’ learning and development (as reviewed, for example, in Edwards & Westgate, 1994; Mercer, 1995). The study of group activities in the classroom, from the point of view of their value for assisting learning, has also become well-established. (See for example, Barnes and Todd, 1977, 1995; Bennett and Cass, 1989; Blatchford and Kutnick, 2003; and with special relevance to mathematics education, Hoyles and Forman, 1995.)
For researchers who take a neo-Vygotskian, sociocultural perspective, interest in language is related to its functions in the learning and cognitive development of individuals. Vygotsky (1978) argued for the importance of language as both a psychological and cultural tool. He also claimed that social involvement in problem solving activities was a crucial factor for individual development. As he put it, intermental (social) activity – typically mediated through language – can promote intramental (individual) intellectual development. This claim, having an obvious plausibility, has been widely accepted. However, other than our own earlier findings, any empirical evidence offered for its validity has been, at best, indirect. Our earlier research showed that the induction of children into an explicit, collaborative style of reasoning which (following Barnes and Todd, op.cit.) we call Exploratory Talk led to gains in children’s individual scores on the Raven’s Progressive Matrices test of nonverbal reasoning (Mercer, Wegerif and Dawes, 1999). These findings, first
demonstrated for children in Year 5 in British primary schools, were subsequently replicated in other year groups and in primary schools in Mexico (Rojas-Drummond, Mercer & Dabrowski, 2001). We have also demonstrated the positive influence of Exploratory Talk on children’s understanding of science and their attainment in formal science assessments (Mercer, Wegerif, Dawes & Sams, 2004). An additional important aspect of that research has been to highlight the potential significance of the role of the teacher as a ‘discourse guide’, someone who scaffolds the development of children’s effective use of language for reasoning through instruction, modelling and the strategic design and provision of group-based activities for children (Mercer, 1995; Rojas-Drummond & Mercer, 2004). As we will go on to show, we can now offer evidence of similar effects of teacher guidance and involvement in structured discussion for the study of mathematics.
As mentioned in the introduction, there are two main kinds of interaction in which spoken language can be related to the learning of maths in schools. The first is teacher-led interaction with pupils. A sociocultural account of cognitive development emphasizes the guiding role of more knowledgeable members of communities in the development of children’s knowledge and understanding and this kind of interaction can be important for their induction into the discourses associated with particular knowledge domains. Elsewhere, we have described this as ‘the guided construction of knowledge’ (Mercer, 1995). Also very relevant is the concept of ‘dialogic teaching’, as recently elaborated by Alexander (2004). His cross-cultural research (Alexander, 2000) has revealed the very different ways that teachers can make use of classroom dialogue when interacting with children. The variation he describes is not revealed by superficial comparisons of the extent to which teachers use questions or other kinds of verbal acts: rather, it concerns more subtle aspects of interaction such as the extent to which teachers elicit children’s own ideas about the work they are engaged in, make clear to them the nature and purposes of tasks, encourage them to discuss errors and misunderstandings and engage them in extended sequences of dialogue about such matters. In his 2004 publication, written mainly for a practitioner audience, Alexander suggests that dialogic teaching is indicated by certain features of classroom interaction such as:
• questions are structured so as to provoke thoughtful answers […]
• answers provoke further questions and are seen as the building blocks of dialogue rather than its terminal point;
• individual teacher-pupil and pupil-pupil exchanges are chained into coherent lines of enquiry rather than left stranded and disconnected;
(Alexander, 2004, p. 32)
Alexander’s findings resonate with our own evaluations of teachers’ interactional strategies (Rojas-Drummond & Mercer, 2004) and that of researchers in science education (Mortimer & Scott, 2004), with our own research showing that the use of more ‘dialogic’ strategies achieved better learning outcomes for children. Moreover, our research has shown that teachers can act as important models for children’s own use of language for constructing knowledge. For example, a teacher asking children why they have gone about an activity in a particular way can be very useful for revealing their perspective on the task to the teacher and for stimulating their own reflective inquiries.
The second context is that of peer group interaction. Working in pairs or groups, children are involved in interactions which are more ‘symmetrical’ than those of teacher-pupil discourse and so have different kinds of opportunities for developing reasoned arguments, describing observed events, etc. In maths education, such collaboration can be focused on solving problems or practical investigations, which also have potential value for helping children to relate their developing understanding of mathematical ideas to the everyday world. Computer-based activities can offer special opportunities for maths investigations, such as games which have mathematical principles and problems embedded in them. However, observational research in British primary schools has shown that the talk which takes place when children are asked to work together is often uncooperative, off-task, inequitable and ultimately unproductive (Bennett & Cass, 1989; Galton & Willamson, 1992; Wegerif & Scrimshaw, 1997).
A possible explanation for the doubtful quality of much collaborative talk is that children do not bring to this task a clear conception of what they are expected to do, or what would constitute a good, effective discussion. This is not surprising, as many children may rarely encounter examples of such discussion in their lives out of school – and research has shown that teachers rarely make their own expectations or criteria for effective discussion explicit to children (Mercer, 1995). Children are rarely offered guidance or training in how to communicate effectively in groups. Even when the aim of talk is made explicit – ‘Talk together to decide’; ‘Discuss this in your groups’ – there may be no real understanding of how to talk together or for what purpose. Children cannot be expected to bring to a task a well-developed capacity for reasoned dialogue. This is especially true for the kinds of skills which are important for learning and practising mathematics, such as constructing reasoned arguments and critically examining competing explanations (Strom et al., op.cit.).
On this basis, there are good reasons to expect that children studying maths would benefit from teacher guidance, in two main ways. First and most obviously, they need to be helped to gain relevant knowledge of mathematical operations, procedures, terms and concepts. Teachers commonly expect to provide this kind of guidance. Secondly, they need to be helped to learn how to use language to work effectively together: to jointly inquire, reason, and consider information, to share and negotiate their ideas, and to make joint decisions. This kind of guidance is not usually offered. One of the practical aims of the Thinking Together research has been to enable teachers to integrate these two kinds of guidance.
The results reported provide support for our first main hypothesis: that providing children with guidance and practice in how to use language for reasoning would enable them to use language more effectively as a tool for working on maths problems together. Its demonstrated that the Thinking Together programme enabled children in primary schools to work together more effectively and improve their language and reasoning skills. Now the second hypothesis: that improving the quality of children’s use of language for reasoning together would improve their individual learning and understanding of mathematics. The results support claims for the value of collaborative approaches to the learning of mathematics (Sfard & Kieran, op.cit; Yackel et al., op.cit.). We also provide evidence to support the view that the teacher is an important model and guide for pupils’ use of language for reasoning.
More generally, our results enhance the validity of a sociocultural theory of education by providing empirical support for the Vygotskian claim that language-based, social interaction (intermental activity) has a developmental influence on individual thinking (intramental activity). More precisely, we have shown how the quality of dialogue between teachers and learners, and amongst learners, is of crucial importance if it is to have a significant influence on learning and educational attainment. By showing that teachers’ encouragement of children’s use of certain ways of using language leads to better learning and conceptual understanding in maths, we have also provided empirical support for the sociocultural conception of mathematics education as successful induction into a community of practice, as discussed for example by Forman (1996) and Barwell et al. (2005). Our findings also are illustrative of the value of Alexander’s (2004) concept of ‘dialogic teaching’, as they show how judgements about the quality of the engagement between teachers and learners can be drawn from an analysis of both the structure and the pragmatic functions of teacher-student discourse. We discussed the extent to which teacher-pupil talk had a monologic or dialogic structure, the kinds of opportunities which children were offered to contribute to discussion and the ways that children’s contributions are used by teachers to develop joint consideration of a topic and the role of the teachers as a model for children’s own use of language as a tool for thinking.
The Thinking Together intervention programme was carefully designed to include both group-based peer group activities and teacher guidance. The success of its implementation supports the view that the development of mathematical understanding is best assisted by a careful combination of peer group interaction and expert guidance. Our findings indicate that if teachers provide children with an explicit, practical introduction to the use of language for collective reasoning, then children learn better ways of thinking collectively and better ways of thinking alone. However, we have also illustrated some variation in the ways that teachers enacted the dialogic principles underlying the programme as they interacted with their classes – variation which seems to have adversely affected its implementation in some classes. While it would be unreasonable to expect all teachers to give the same commitment to a research study intervention, these findings nevertheless have made us review critically the in-service training about Thinking Together we provide for teachers.
The wider programme of research we have described has already generated materials for the professional development of teachers and the implementation of the Thinking Together approach (Dawes, Mercer & Wegerif, 2003 .; Dawes & Sams, 2004) These set out the structure of the Thinking Together programme, the teaching strategies involved in its implementation and a series of activities in the form of lesson plans. Our findings have also been incorporated into an Open University on-line in-service course for teachers (Open University, 2004) and are recognized in national educational guidance for teachers (e.g. QCA, 2003a, 2003b). A related project on the use of ICT in mathematics teaching, building on the methods and results described here, has generated new software and teacher guidance for the use of ICT in maths education (Sams, Wegerif, Dawes & Mercer, 2005).
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