Teaching Children How To Use Language To Solve Maths Problems

While carrying out my dissertation at University on the problems students face with University level Mathematics, this article published by Neil Mercer (Cambridge University) and Claire Sams (The Open University) interested me. It discusses one of the ways in which teaching Mathematics through conversational dialogue can help students overcome their fears and problems when trying to cope with new, sometimes scary concepts. It is a teaching style I have employed with some students, and the students flourished. I have attached the Introduction and Conclusions only of this very detailed article, but the full text can be found at: http://www.tandfonline.com/doi/pdf/10.2167/le678.0

TEACHING CHILDREN HOW TO USE LANGUAGE TO SOLVE MATHS PROBLEMS

It is often claimed that working and talking with partners while carrying out maths activities is beneficial to students' learning and the development of their mathematical understanding. However, observational research has shown that primary school children often do not work productively in group-based classroom activities, with the implication that they lack the necessary skills to manage their joint activity.

The research we describe investigated these issues and also explored the role of the teacher in guiding the development of children's skills in using language as a tool for reasoning. It involved an interventional teaching programme called Thinking Together, designed to enable children to talk and reason together effectively. The results obtained indicate that children can be enabled to use talk more effectively as a tool for reasoning; and that talk-based group activities can help the development of individuals' mathematical reasoning, understanding and problem-solving. The results also encourage the view that the teacher of mathematics can play an important role in the development of children's awareness and use of language as a tool for reasoning. We suggest some ways that this role can be carried out most effectively.

INTRODUCTION

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. Barwell et al., 2005; Forman & van Oers, 1998; Hoyles & Forman, 1995; Monaghan, 1999; Sfard, 2000; Sfard & Kieran, 2001). This interest relates to the function of language in both teacher-student encounters and in peer group activities. For example, Yackel et al. (1991) carried out a study in which all maths instruction in a primary classroom was replaced by problem-solving activities in small groups. While emphasising the value of the teacher's guiding role, they found that the group activities offered valuable opportunities for 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 et al. (2001) analysed the ways a teacher used talk to guide the development of children's mathematical argumentation. Taking a rather different perspective, sociological researchers such as Dowling (1998) 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 policymakers 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 e.g. Barnes & Todd, 1977, 1995; Bennett & Cass, 1989; Blatchford & Kutnick, 2003; and with special relevance to mathematics education, Hoyles & 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, 1977, 1995) we call Exploratory Talk led to gains in children's individual scores on the Raven's Progressive Matrices test of nonverbal reasoning (Mercer et al., 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 et al., 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 et al., 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.

DISCUSSION AND CONCLUSIONS

The results reported here 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. We have demonstrated that the Thinking Together programme enabled children in primary schools to work together more effectively and improve their language and reasoning skills. We also have support for 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. This finding is consistent with our earlier reported findings related to the study of science (e.g. Mercer et al., 2004). Our results support claims for the value of collaborative approaches to the learning of mathematics (Sfard & Kieran, 2001; Yackel et al., 1991). 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 are also 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, 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 & Sams, 2004; Dawes et al., 2003). 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 online in-service course for teachers (Open University, 2004) and are recognised 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 et al., 2004)