In many algebra lessons, pupils are confined to largely passive roles. The pupils listen to, and watch, teacher explanations about algebraic manipulations, some of which may make little or no sense to the pupils. Bell (1996) explained that the symbolic language in algebra plays a crucial role in developing algebraic thinking. Hence there is need to encourage pupils’ proficiency in using symbolic language and to increase their awareness of the underlying structures in formal algebraic notation. Mercer and Sams (2006) demonstrated that the
encouragement of pupils’ language use by teachers in certain ways can lead to better learning outcomes or conceptual understanding. Therefore, it is imperative that teachers make proper,
careful and consistent use of terms in the mathematics register if their pupils are to learn the universal use of mathematical language. Through teacher-led classroom interactions, pupils can gain relevant knowledge of mathematical terms, concepts or procedures. Teachers model the correct use of mathematical language and vocabulary whilst providing pupils with
opportunities to develop communicative competence. However, the teacher cannot make the pupils learn mathematical vocabulary: pupils have to be willing to engage with the learning process if they are to grow in confidence and join the wider mathematics community (Mercer and Sams, 2006; Lee, 2006).
2.1.5 ‘Dialogic teaching’ in mathematics classrooms
From a neo-Vygotskian perspective, talk and teachers are two central and significant features of interest for research on learning in educational contexts. To Mercer (1994), through talking and listening in the classroom, alternative perspectives may be considered as information and understandings are shared, explanations or justifications are offered, and ideas are exchanged.
Dialogue with pupils in focused discussions is paramount for a teacher with a ‘connectionist’ belief orientation (Askew et al, 1997). This allows for consideration of varied understandings or interpreted meanings, and exploration of mathematical problem-solving strategies
(Ruthven et al, 2011). Following ideas grounded in counselling, Foster (2014) called for the development of listening skills in mathematics teachers to encourage pupils’ autonomy in decision-making whilst problem-solving. Lee and Johnston-Wilder (2013) stressed that although a lot of ‘talk’ ensues in mathematics classrooms, experience indicates that most of it is by teachers and there is very little pupil talk (Kanja et al, 2001). To Alexander (2008), ‘teacher talk’ may take the form of:
rote through drilling facts and routines via constant repetition
recitation in cuing pupils to work out answers through stimulated recall exposition, imparting information, explaining principles and procedures discussion, in which ideas are exchanged with a view to solving problems dialogue with common understandings achieved through exchange of structured
reasoning and discussion
The key characteristics of dialogic talk are that it is: collective in involving all participants in tasks; reciprocal through listening to each other whilst considering alternative viewpoints;
supportive of free contribution to discussions with mistakes valued; cumulative in negotiating a consensus in participants’ thinking; and purposeful in meeting the intended learning
outcomes. Alexander (2008) argued that while reorganisation of classroom settings and interactions can lead to teaching that is collective, reciprocal and supportive, being
cumulative and purposeful can present pedagogic challenges for teachers. He observed that:
“Discussion and scaffolded dialogue have by far the greatest cognitive potential. But they also, without doubt, demand most of teachers’ skill and subject knowledge. Rote, recitation and exposition give us security. They enable us to remain in firm control not just of classroom events but also of the ideas with which a lesson deals”. (p.31) Through ‘dialogic teaching’, the teacher invites pupils to share their thinking in sequences of reflective talk. Wrongly-phrased talk is used as ‘learning points’, to ask pupils more
questions, thus keeping the lines of enquiry open, extending classroom discussions, and testing both the teacher’s and the pupils’ understandings. Alexander (2008) listed characteristics of ‘dialogic teaching’ as:
1. questions structured to provide thoughtful answers
2. answers provoke further questions and are seen as building blocks of dialogue rather than terminal points
3. individual teacher-pupil and pupil-pupil exchanges are chained into coherent lines of enquiry rather than left stranded and disconnected
(p.42) According to Alexander (2008), the types of ‘talk’ within a teacher’s repertoire will
determine the quality of the opportunities to learn that will be made available to pupils who have to voice their own thinking. The classroom ‘subculture’, which depends on the learning environment created by the teacher, can determine the level of pupil participation and the kinds of interactions in lessons. Alexander stressed the need for teachers appropriately to balance the social and cognitive uses of talk in order to facilitate pupil participation and extended dialogues. He considered ‘cumulation’, in particular of the five ‘dialogic teaching’ principles, as posing the greatest challenge because it places greater demands on the
teachers’:
Mastery of content knowledge; Professional skills;
Ability to listen to and to review their pupils’ contributions, and then to judge the best way to scaffold the pupils’ thinking to achieve the intended understanding.
Alexander (2008) cautioned that ‘dialogic teaching’ and scaffolded discussions were likely to curtail the information flow from teachers to pupils through awkward questions.
Pimm (1987) stressed the need for classroom dialogues to be task-focused with an emphasis on the style and level of explicitness of talk. He argued that the amount to ‘pupil talk’ should not be seen as an end in itself: the quality rather than the quantity of talk is what is important. Extending the Vygotskian idea of ‘thinking aloud’, Mercer (1994) said that to encourage talk and communicating with others through speech is to engage in a social mode of thinking. By laying appropriate ‘ground rules’ (Edwards and Mercer, 1987) in lessons, and asking
questions that provoke reflection and thought, teachers can create the opportunity to assess whether learning may be taking place as it occurs (Hattie and Timperley, 2007; Black et al, 2003). Externalised reasoning in ‘dialogic teaching’ simultaneously evokes the
internalisation, appropriation and scaffolding of pupils’ learning within their ZPDs. In this thesis, I contend that both pupils and teachers become aware of the content intended, content taught and content learned by observing the ‘ground rules’. Collectiveness can provide a supportive atmosphere within which pupils may feel safe when invited to express their reasoned mathematical arguments: mistakes are valued as ‘learning points’. Mercer and Littleton (2007) stated that pupils trained to work in this way progress more successfully in joint problem-solving and improve their scores on written tests.
Whilst offering criticism of the establishment of ‘ground rules’ governing classroom talk, Lambirth (2006) asserted that the practice can reinforce social inequalities and disempower pupils from less privileged backgrounds rather than enhancing all pupils’ communicative and cognitive activity. Nevertheless, the effect of ‘ground rules’ in the classroom may be seen as enabling the development of resilience in activity systems. Bottrell (2009) viewed ‘resilience’ as the coping and competence strategies adapted by individuals despite adversity occasioned by the interaction of social and cultural factors. Reporting on a study of marginalized youths in Sydney, Australia, Bottrell emphasised the significance of individuals’ social identities and collective experiences to their resilience by underscoring the voices of ‘others’ in narratives of pupils’ experiences. Herein lies the recognition that the management of pupils, teachers, and technologies as classroom resources (Luckin, 2008) is important for enabling resilience. Bottrell (2009) underscored the dynamic nature of resilience, whilst pointing to the dialectical
relationship between individuals and their context, in stating that adversity requires resistance. Hernandez-Martinez and Williams (2013) considered ‘resilience’ to be the dynamic interplay between the sociocultural context and the agency of learners in transition. In this thesis, I see collaborative, computer-based mathematics classrooms as activity systems requiring explicit expectations. Hence, the criticism of ‘ground rules’ falls short of
acknowledging the pupils’ resilience arising from the developing interaction between learners and the classroom context.