‘Well Maybe Not Exactly, but It’s Around Fifty Basically?’: Vague Language in Mathematics Classrooms

It may come as something of a surprise to find a mathematician (albeit in the guise of a mathematics educator) writing about vagueness, since it is commonly supposed that precision is the hallmark of mathematics. Such a point of view is reflected in the landmark 1982 Report of the Committee of Inquiry into the Teaching of Mathematics in Schools (the Cockcroft Report), which asserted that, ‘mathematics provides a means of communication which is powerful, concise and unambiguous’ (Department of Education and Science 1982, p. 1), and proposed the communicative power of mathematics as a ‘principal reason’ for teaching it. There was refreshing novelty in such a claim, which seemed to be justifying the place of mathematics in the curriculum in much the same way that one might justify the learning of a foreign language, and it did much to promote and sustain interest in the place of language in the teaching and learning of mathematics. Such a view of mathematics is in contrast, however, with that expressed in a contemporary pamphlet issued by the Association of Teachers of Mathematics (ATM 1980, pp. 17–18), whose authors argued that:

Because it is a tolerant medium, everyday language is necessarily ambiguous. /…/ Now, mathematising is also a form of action in the world. And its expressions, however carefully defined, have to retain a fundamental tolerance /…/ Because it is a tolerant medium, mathematics is also necessarily an ambiguous one.