WHAT TEACHERS LEARN FROM CHILDREN’S MATHEMATICAL ARGUMENTS IN DISCUSSION: MOVING TO A NEW PEDAGOGICAL MODEL

Julie Ryan, Derek Kassem and Charles Sarland 

Liverpool John Moores University

Paper presented at the British Educational Research Association Annual Conference, Heriot-Watt University, Edinburgh, 11-13 September 2003

Abstract

This paper is drawn from a study of teachers’ development of collaborative discussion in mathematics classrooms where children’s misconceptions, errors and methods were used as the basis for group and whole class discussion in year 5 classrooms in two contrasting schools in Liverpool. Research-based materials used to begin, sustain and extend children’s discussion were presented to two teachers during a university-based training session for the study of teachers’ understanding of children’s mathematical arguments in discussion. How the teachers adapted or rejected these materials and strategies to their own practice was tracked over a three-month period. This paper reports on one research question from the study: how do teachers begin to use children’s mathematical discussion in their classrooms. The teachers’ lessons were videotaped and these recordings were used across the three-month period to prompt teacher reflection on what they were developing. The already highly successful practice of the two teachers was securely embedded in the National Numeracy Strategy Framework and its model of ‘interactive whole class’ teaching. The introduction of children’s collaborative discussion was found by the teachers to significantly challenge this model in terms of their pedagogy. Constraints of the NNS pedagogical model were related in particular to time and the unitising of learning episodes. However, the teachers endorsed the opportunity to explore new practice within a supported research environment where they were able to take control of the agenda and develop their own practice rather than deliver the practice of an outside agency.

Keywords: mathematical discussion; accounts of practice; errors and misconceptions; teacher reflection.

Introduction

Collaborative discussion in England’s key stage 2 and 3 mathematics classrooms is a rare occurrence. The ‘numeracy hour’ or Daily Maths Lesson prescribed by the government’s National Numeracy Strategy (NNS) lays down a framework for ‘whole class interactive teaching’ designed around a timed oral/mental starter, a main activity with differentiated tasks for children of different ability and a closing plenary. Teachers are judged by their proficiency in using this model by the government’s inspection body Ofsted.

This delivery model, together with mandatory testing which establishes a school’s public success, appears to have had a profound impact on the culture of the mathematics classroom. We live in an era where education is cast narrowly as the acquisition of competences and where teachers face the same conflicts, pressures and demands behind many cycles of reform. Schon’s comment is still timely twenty years on.

Practitioners are frequently embroiled in conflicts of values, goals, purposes, and interests. Teachers are faced with pressures for increased efficiency in the context of contracting budgets, demand that they rigorously ‘teach the basics’, exhortations to encourage creativity, build citizenship, help students to examine their values. (Schon, 1983, p.17)

The NNS aims to be a complete package: curriculum, assessment, lessons, tools, artefacts and videos of exemplary practice. It is a rare teacher indeed who challenges its content and style during a school inspection regime or as an expression of professional discourse. While ‘interaction’ is promoted, it is found to be essentially teacher-centred; teacher talk dominates. The whole class interaction exemplified in NNS video materials supplied to schools does not show examples of interaction that go beyond the teacher-centred Initiate-Respond-Evaluate/Feedback model (Dillon, 1988; Mercer, 1996; Sinclair & Coulthard, 1975): significantly children are never seen to explore a misconception, develop their argument or engage in conflict and reflection which, from a constructivist perspective, are seen to be so productive for learning.

The NNS advises teachers to be aware of children’s misconceptions and errors (DfEE, 1999) and to adopt a corrective approach. How to approach ‘errors’ more productively however presents dilemmas: should errors be given centre stage in classroom dialogue, should they be corrected with immediate ‘treatment’ before they spread or can they be a source for enhancing mathematical understanding? (We use the term ‘error’ to include children’s misconceptions, alternative methods and mistakes that give some useful insight into their mathematical thinking.)

These were dilemmas we were keen to explore with classroom teachers who were already proficient in the NNS model but who were willing to explore children’s mathematical errors through discussion. How would they adapt their current teaching in the light of research-based alternative suggestions for classroom practice? We were aware that the adaptation might require a re-conceptualisation of the teaching and learning frames (Schon, 1983) in the light of the NNS system.

Our plan was to explore a shift from current practice by twinning errors (content) and discussion (method). The research we drew on stemmed from small group discussions that children had had of their errors on mathematical test items in a research environment (Ryan & Williams, 2000). The errors had been established from large scale national testing (Williams & Ryan, 2000) and the likely arguments that children used to back them had been organised into teaching planning tools (Ryan & Williams, 2003). We were keen to study what teachers could make of these research ideas.

Theoretical Perspectives

Teacher reflection on their practice was central to the study. Reflection on practice (Grimmett & Erickson, 1988; Schon, 1983, 1987a, 1987b; Smyth, 1992) is widely seen as an appropriate tool in the study of the professional development of teachers. Accounts of practice produced through narrative inquiry have been regarded as a means for teachers “to become better acquainted with their own story” (Conle, 2000, p.51).

… researcher/teacher collaboration also came to be seen as an opportunity for professional development for the teachers being studied in the research projects. To become better acquainted with their own story was indeed interesting for busy teachers who had little time for reflective writing themselves. (Conle, 2000, p.51)

Johnson (2002) however cautions researchers about the stability of accounts that teachers give of their practice and proposes “that it is entirely possible that the same teacher would produce a different version of her work, if … methods of accounting were altered” (ibid, p.22).

The teacher accounts of practice revealed in conversation with the researchers/interviewers (and in whole research group conversation) are produced within a frame of communicative action – the framing talk of the interviewer’s comments and questions – which, as Johnson (2002) warns, does not necessarily produce a definitive account. The conversational strategy as a method of enquiry is limited and can encourage teachers to produce a particular version of practice determined by the relationship between the interlocutors (Young, 1986 as cited in Johnson, 2002). However, in this study there appeared to be some stability across interviews and teacher-to-teacher conversation where accounts were confirmed or consolidated.

‘Knowing-in-action’ (Schon, 1983), or ‘tacit knowing’ (Polanyi, 1967), is found not only in everyday behaviour but especially in professional behaviour – that intriguing knowing that is sometimes more than you can put into words, that knowing that constitutes the artistry of a profession. Teachers, as with all skilled practitioners, have much of their knowledge and expertise embedded in their practice; their knowing is in their doing.

Knowing-in-action however can have some limitations. Practice can become repetitive and routine and consequently lead to boredom, burn-out or ‘over-learning’ where a practitioner has learned to be “selectively inattentive to phenomena that do not fit the categories of (their) knowing-in-action” (Schon, 1983, p.61) or where they are drawn into patterns of behaviour that they can no longer correct. Reflection-in-practice is seen as an antidote to professional over-learning (ibid.) – a provocation for new learning. Reflection on reflection-in-action is “an intellectual business, and it does require verbalization and symbolization” (Schon, 1987a). It involves making tacit knowledge explicit.

… (B)oth ordinary people and professional practitioners often think about what they are doing, sometimes even while doing it. Stimulated by surprise, they turn thought back on action and on the knowing which is implicit in action. (Schon, 1983,p.50)

Roth (2003) attempts to theorise reflection on teaching in general and the potential role of video as a medium in particular. He uses a “phenomenological perspective to articulate a number of issues that frame the opportunities and constraints of video as a tool that mediates reflection” (Roth, 2003).

First, each moment of praxis (lived experience of situated action) is marked by its intentional-horizontal character. Second, each moment of praxis is marked by a particular rhythm and tempo with which it unfolds, that is, by its characteristic temporality. Together, the two dimensions also contribute to different Selves that teachers experience while teaching and reflecting on teaching. (Roth, 2003, p.5)

The intention ‘in the moment’ of teaching and the intention in reflection ‘of the moment’ are constrained differently, most dramatically in terms of time or horizon. The first intention is to successfully complete the lesson, while the second intention may be to consider ‘how could I do this differently?’, or simply, ‘what was I doing here?’ where intention is constructed, or articulated, after the event. The video episode becomes a tool which can be used again and again for different ends: what did I do, why did I do it that way, should I be doing it differently? The temporal advantage affords the switching between “different Selves” in reflection.

We address teacher knowledge in this study from a socio-cultural perspective; knowledge conditioned by available tools, the environment of action and the language of articulation. We draw attention to these conditions mediating teacher reflection on their action. The video episode in particular can be seen as a boundary object mediating reflection on practice. The reflection on action, mediated also by the researchers’ interventions and lines of enquiry (eg. questions, comments and suggestions), is revealed as a developing social practice of collaboration.

The Study

The research study was focussed on the questions (a) how do primary teachers use research-based materials to organise classroom discussion, (b) what persuades children to change their minds in group and whole class discussion, (c) what do teachers attend to in reflection of children’s arguments in discussion and (d) what impact does it have on their practice. The materials and strategies already developed to begin, sustain and extend children’s discussion were to be presented to two teachers during a university-based training session. How teachers adapted these materials and strategies to their own practice was to be tracked over a three-month period.

The ‘ready-made’ materials however proved not to be central to the teachers’ initial interests – they were keen to first demonstrate their current practice and then to develop their ideas of what mathematical discussion could be in terms of their own securely-based NNS practice. The original aims were therefore re-conceived as the collaborative nature of the research unfolded.

The research questions were fine-tuned to become (a) how do teachers develop mathematical discussion in their classrooms, (b) what do they learn from such discussions, (c) what materials and strategies are seen to be productive, and (d) what impact does collaborative discussion have on teaching practice. This paper addresses the first question.

The two year 5 primary teachers in Liverpool worked in different schools which catered for different socio-economic communities. The two teachers, Kate and Debra, in this study were nominated because they were considered to be successful year 5 mathematics teachers who have recently gained Advanced Skills Teacher status after only four and five years teaching experience respectively. They were both in ‘Beacon’ schools which are charged with transferring good practice to other schools. Debra won her AST in mathematics and is a Leading Maths Teacher for on the exemplary teaching of mathematics; Kate won her AST for the teaching of Spanish and also advises on exemplary teaching practice. These two teachers were experienced in demonstrating their practice to visitors and were comfortable about the videotaping of their lessons. They were also experienced in evaluating other teachers’ practice and offering professional development support. They were not experienced however in terms of reflection on their own practice with outsiders.

The three researchers in the study were university educators with former school teaching backgrounds. One researcher had been involved in the development of the discussion tools and therefore had a vision of what could work, another researcher had a strongly developed democratic approach to development and therefore adopted a more open-ended start to the study, the third researcher came with a non-mathematical background and a preference for exploration of language and action research. These different perspectives affected the direction of the study.

Methodology

A case study methodology was adopted for the project given that contextual conditions were central to the study (Yin, 2003). We hoped to study the teachers’ descriptions of their practice, their beliefs about children’s learning, their reflections on changing practice, the nature of researcher interventions and the use of materials already developed for classroom discussion. A two-case design gave us the opportunity to contrast styles of development and to consider teacher-to-teacher interaction at key stages of reflection. Importantly, the two-case design afforded opportunities to consider the impact of different contexts and the possibilities of generalisability of findings in order to inform the next step: how should the research materials be adapted or further supported to make them more likely to be useful to practitioners for professional development?

The selection of teachers recognised by their schools as highly competent practitioners was deliberate. The two teachers had significant responsibility for dissemination of practice within their school regions and were highly knowledgeable of the mechanisms that were likely to support successful change in classroom practice. They were also more likely to delineate the boundaries of development.

The three-month period available for the study restricted the focus to early development only. The structure was three half-day meetings of the five participants at the university, videotaping of three lessons in the teachers’ schools and paired or whole group researcher-teacher reflection on each lesson.

The unit of analysis was accounts of practice as revealed in interview and prompted by selected videotaped segments of the teachers’ lessons. The choice of videotaped episodes for reflection was in the hands of the teachers. Attention was paid to the role of the ‘interviewer’ and the framing of questions and comments so that a researcher’s own ideology or preconceptions were either subordinated or made explicit. Videotaping protocols were also outlined to secure common terms of reference.

The teachers were given their videotapes after each lesson so that they could watch them privately and decide what parts could be used for reflection with the researchers. Two cameras were used in each taping session: one taking a fixed wide view in whole class interaction and then following the teacher and the other capturing independent student group interactions.

The researchers drew up the agendas for the meetings (with focussed questions) and made transcripts of the meetings from audiotapes. The transcripts form the basis of the analysis below. One of the teachers was also available after the conclusion of the project to analyse one of the transcripts and her comments are built into the analyses below.

Analysis of First Meeting: Baseline

In the first meeting, the researchers outlined the aims of the project and sought the teachers’ expectations. The researchers were to present ideas but the teachers were to decide what they wanted to do – in this sense there was an open agenda. The teachers described their current practice and reactions to errors and misconceptions in the classroom. Video snippets of NNS materials, where teachers responded to children’s errors in the classroom, were also discussed in order to uncover aspects of currently promoted practice or innovation.

Expectations of the project

Debra and Kate had considerable responsibility for the professional evaluation and development of their school colleagues and also demonstrated expert practice to other teachers. However, they had little opportunity to ‘view’ their own practice and they regarded this positively.

I do think it will be quite interesting to see what I do in the classroom. You never see what you do, you see what you’ve done, but you don’t know how you’ve done it. (Debra, I, 20:20)

I agree. (It will be) interesting for professional development. We watch other teachers a lot and we’re very observant and see how to develop them and their teaching, but watching yourself – we don’t have the opportunity very often. (Kate, I, 22:10)