In calculus courses it is taken for granted that students cannot read the text. At most, they can read the "solved problems" at the end of each section. Even these are treated not as material to be read and understood, but behavior patterns to be imitated: given a problem of certain type, what does one do? Students' reactions have convinced me that for them the ostensibly declarative sentences in calculus are construed, not as declarations, calling for assent, but as imperatives, calling for obedience. For example, the declarative sentence: "If f( x) = x3 for each x, then f'(x)=3x2 for each x" is construed as the imperative: "In a differentiation problem, if f(x) = x3 is given, then you must write f'(x) = 3x2." I believe that for nearly all students, elementary calculus is not a body of knowledge at all; it is a repertory of imitative behavior patterns so that the question of truth hardly arises.
My Exeter colleague David Hobbs passed me this intriguing bit of text some years ago asking me if I recognised it, but I've never been able to identify the source. Any ideas? (Paul Ernest)