| Summary: | In three consecutive cycles I adopted the role of researcher teacher in senior classes in different primary schools in Ireland. Through a continuing cycle of data collection and analysis, I developed an analytic frame that comprises four dimensions: (i) Mathematical Principles – these are the constructs that pupils could be expected to develop by engagement with the task; (ii) RBC Epistemic Actions – these actions – ‘recognising’ ‘building with’ and ‘constructing’ are the observable actions underpinning the Abstraction-in-Context (AiC) theoretical framework, which has been used to describe the construction of new mathematical ideas by individuals and groups of individuals; (iii) Hedges and Pronouns – these are aspects of ‘vague language’. Hedges refer to words and phrases such as ‘I think’, ‘maybe’, ‘possibly’. Learners often use pronouns to refer to mathematical entities which they cannot or do not name, and in particular, use the word ‘you’ as a generaliser. (iv) Teacher follow-up Moves in response to pupils’ contributions. They include ‘insert’ in which a teacher adds something to a learner’s contribution; ‘elicit’, which is similar to the funnelling pattern of interaction; ‘press’ in which the pupil is requested to justify and elaborate on his/her input; ‘maintain’ in which the learner’s contribution is maintained in the public realm; and ‘confirm’ in which teacher confirms that s/he has heard the learner correctly. Of the 32 lessons taught, I transcribed the whole-class discussions related to 20. Four lessons were then chosen for microanalysis. These lessons represented a tapestry of themes which emerged over the three cycles of research. My research supports the value of this framework for the analysis of the construction of mathematical insight by pupils in whole-class conversation. In particular, it shows that insight developed thus is usually distributed among a few participants. While a nexus of task, conjectural atmosphere and pupil engagement contributes to the fostering of mathematical insight, student agency is the overarching and critical variable.
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