Mathematics, viewed as a field of human endeavour, as opposed to an inert body of knowledge, offers considerable potential for intellectual risk-taking. To participate in this endeavour we are obliged to lower defences, and in speaking, to expose ourselves. Nothing ventured, nothing gained; encouraging pupils to take risks, in the form of making predictions and conjectures, might be expected to be a feature of effective teaching. Of course, there are also risks associated with public display of knowledge of any kind, even at ‘low levels’ of mathematical activity-recall of facts, rehearsal of algorithms-especially for those who have suffered ridicule at the hands of insensitive teachers or inconsiderate peers. Mathematics holds an invidious reputation within the school curriculum, being associated with fear of error and consequent public humiliation. The scars which persist into adulthood were brought sharply into relief by Sewell’s study (1985) of adults’ use of mathematics in daily life. One common perception is that the questions teachers ask their pupils are not searchlights focused to reveal truth, but traps set to expose ignorance. Janet Ainley, studying children’s perceptions of the purposes of teachers’ questions, calls such uses ‘testing questions’:
Because testing questions are so common, particularly in mathematics where answers are seen as being clearly ‘right’ or ‘wrong’, there is a danger that pupils may perceive all teacher questions in this way. Such a perception would inevitably be detrimental to attempts to encourage discussion, investigative work or problem solving in mathematics: pupils will feel that the teacher always knows the ‘right’ answers to any questions she asks, and furthermore that the teacher is always judging pupils by the answers they give. It is not surprising that pupils are reluctant to risk giving ‘wrong’ answers in these circumstances. (1988, pp. 93-94)
From a speech act perspective, Labov (1970) comments that a question is normally deemed appropriate only when the enquirer meets certain speech act ‘sincerity’ conditions-including, in this case, that s/he doesn’t know the answer, would like to know it, and has reason to believe the hearer is able to supply it. Each of these conditions would be satisfied when, for example, you make a telephone enquiry to the railway enquiries office. Labov shows that questions in classroom situations are exempt from these rules, and that the conditions governing appropriateness in the answers to such questions differ accordingly. The expectation of pupils is that ‘classroom questions’ (R.Lakoff, 1973) are not genuine requests for information, but public ‘requests for display’ (Labov and Fanshel, 1977). These differ from other questions in that the enquirer (A) already has the information sought in the question, and ‘the request is for B to display whether or not s/he has the information’ (ibid., p. 79). Given the privileged position of the enquirer in the dialogue, the ‘display’ of the respondent is a revelation of knowledge (public success) or ignorance (public failure).