Relational Understanding and Instrumental Understanding.

One gets faux amis between English as spoken in different parts of the world. An Englishman asking in America for a biscuit would be given what we call a scone. To get what we call a biscuit, he would have to ask for a cookie. And between English as used in mathematics and in everyday life there are such words as field, group, ring, ideal. A person who is unaware that the word he is using is a faux ami can make inconvenient mistakes. We expect history to be true, but not a story. We take books without paying from a library, but not from a bookshop; and so on. But in the foregoing examples there are cues which might put one on guard: difference of language, or of country, or of context. If, however, the same word is used in the same language, country and context, with two meanings whose difference is non-trivial but as basic as the difference between the meaning of (say) ‘histoire’ and ‘story’, which is a difference between fact and fiction, one may expect serious confusion. Two such words can be identified in the context of mathematics; and it is the alternative meanings attached to these words,