The importance of language and social interaction for learning mathematics has been emphasized both theoretically (Vygotsky, 1987) and in research (e.g. Mercer & Sams, 2006; Yackel & Cobb, 1996). A shift from traditional teaching to teaching informed by the social constructivist paradigm changes the role of the teacher to become a responsive guide in developing the pupils’ own thinking. In Wood, Bruner, and Ross’s (1976) terminology such teachers are scaffolders of pupils’ learning. Anghileri (2006) describes three different levels of such scaffolding; environmental provisions (Level 1), explaining, reviewing and restructuring (Level 2) and developing conceptual thinking (Level 3). The language of the teacher is an important part of scaffolding on the second and third level. However, deciding what to attend to and what not during a dialogue with pupils is a major challenge for a teacher. Identifying important mathematical moments and using them productively is difficult even for experienced teachers (Chamberlin, 2005). Four common hypotheses of what constitutes a productive dialogue in the mathematical classroom are; 1) open questions (permitting several possible answers) are preferable to closed questions; 2) pupils should give reasons for their responses; 3) pupils, not only teachers, should comment on the expressed ideas; 4) previous ideas should be built on or elaborated. So far, existing research has been too small-scale, or in other ways restricted, to confirm these hypotheses (Howe & Abedin, 2013). However, an ESRC-funded large-scale project run by a team in Cambridge (Howe, Hennessy & Mercer) in 72 year-6 classrooms (pupils aged 10-11) shows promising results offering the necessary evidence for the importance of classroom dialogue (Howe, 2018). Even though they found no evidence that pupils should give reason for their response in teacher-pupil dialogues, reasoned dialogue amongst pupils shows to be important. Based on the findings they (Howe, 2018) recommend teachers to:

• ask some ‘open’ questions
• ask students to comment on what has been said
• note any misunderstandings that discussions reveal, so you can draw on them later
• hold back on providing correct accounts until students have reflected what they think

Research on mathematics in Norwegian classrooms indicates that there is little time for dialogues between teachers and pupils, or between pupils (Skorpen, 2006). Pupils’ interventions are often not taken up by the teacher (Bjørkås & Bulien, 2010). In a larger study with both quantitative and qualitative data, Drageset (2015) found that most of the pupil-utterances was part of sequences where the teacher controlled the process and the pupils responded to basic tasks that look like mere control questions. As the pupils normally answer the questions correctly in such IRE dialogues (Initiation-Response-Evaluation), the dialogue often ends without any further exploration of the theme.

The theory of Didactical Situations (TDS), is a scientific approach to problems related to teaching and learning of mathematics which is compatible with Vygotsky’s sociocultural theory of learning and development (Måsøval, 2011). The idea is to create a situation which sustains the pupils’ learning of a chosen target knowledge when operated on (Strømskag, 2017). Rigor in the description of different phases in the didactical model ensures both space for pupils to work on problem-solving and to formulate their own thoughts, and space for focused use of dialogues between pupils and between the pupils and the teachers. The motivation for this study is to develop knowledge on the teacher’s communication as part of the “milieu” in the “adidactical” phase where the pupils are supposed to solve problems and formulate and justify their strategies. I ask: What characterizes the communication of a skilled teacher when scaffolding young pupils working with mathematics?

The study is a case study (Stake, 1995) and the data material is audiotapes from a teacher’s communication with pairs of second-, and later third-graders during the adidactical phase in three TDS-informed lessons. In these lessons the pupils are working with: the concept ‘a half’ through halving and odd numbered set of objects; subtraction understood as difference; introduce multiplication using different contexts. To get an even deeper understanding of the teacher’s communication skills I conducted a semi-structured interview with her. Examples of topics in focus during the interview were: her role as teacher in the adidactical phase of the teaching, examples of necessary communication, the need for adapting her communication to different pupils and different tasks, and her motivation to let TDS inform her teaching in the future. To analyse the teacher’s communication, I used a framework for detailed studies of mathematical discourses on a turn-to-turn basis developed by Drageset (2015). Such sociocultural discourse analysis focuses on the functions of language for the pursuit of joint intellectual activity. The framework comprises 13 categories of teacher interventions falling into three superordinate categories: redirecting actions, progressing actions and focusing actions. Redirecting actions are actions where the teacher redirects the pupils’ attention by either asking a correcting question, advising a new strategy or putting aside a pupil’s comment. Progressing actions includes the teacher’s different ways of moving the lesson forward, either by offering a simplification of the problem, by asking an open or a closed question, or by providing a demonstration. Focusing actions are actions the teacher’s uses to put emphasis on certain aspects, either by requesting pupil input, or by pointing out. The categorization of the teacher’s utterances was followed up by a hermeneutic interpretive process where I systematically checked out the contexts where the different teacher interventions typically dominated the dialogue. This interpretative process gave me a deeper understanding of the meaning of the dialogues between the teacher and the single pair of pupils (Alvesson & Sköldberg, 2008). In addition, in the presentation the findings of the analysis are mirrored by the teacher’s utterances in the interview.

The initial findings of the deductive analysis are that the teacher often scaffolds the pupils’ work through open progress initiatives and a variation of focusing actions, by asking for an elaboration or a justification, or by focusing their attention to something in the task. Redirecting actions and progressing actions like demonstrations or simplifications are rarely used. These are findings that go well together with the idea of TDS, the teacher attends to the pupils’ own ideas and gives the pupils space for problem-solving and formulation of strategies building on their own ideas. However, the number of questions with typical one correct answer is also high. As such questions are typical for IRE dialogues in teaching with an instructional approach, this was an intriguing finding. Further analysis showed that such questions serves a different purpose in the teacher’s scaffolding in this study. TDS designs normally hold tasks where the pupils are supposed to argue for their own strategies, and communication skills regarding processes of comparison are needed to succeed. The questions that immediately appears to be closed plays a crucial part in the teacher’s modelling of how the pupils are supposed to comparis. Thus, I claim that Pam’s questions, which appear as closed progress details, model questions that the pupils need to ask to compare their thoughts. At this stage in the study I find that the teacher plays a quite active role in what is supposed to be the adidactical phase in the TDS-informed classroom. Through further interpretation using Langer’s (1995) theoretical construct ‘envisionment-building’ as a lens I will explore how Pam’s communication support the young learners’ agency.

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