Abstract
A clinical methodology was used to investigate the perceptions which pupils of secondary school age have concerning modes of mathematical argument which have an agreed status within the world of mathematics. The analysis of data obtained from three extended contexts led to the identification of clusters of characteristic response types. Differences were found to exist between the agreed meaning of some mathematical terms and procedures and the meaning ascribed to them by students. By considering levels of performance it was possible to identify particular components, the presence or absence of which consistently determined the capacity to structure or follow proofs and explanations.
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