Many years ago I went up to Cambridge with the words of the Head of Maths ringing in my ears: ‘At Cambridge you will be able to study maths for the sake of studying maths!’ The implied contrast was with A-level preparation which consisted solely of working through old exam papers. Unfortunately he was wrong because our university course consisted of the same: lectures plus working through old papers. It was content content and more content full stop.
In my experience of teaching mathematics in primary and secondary schools pupils all too easily learn the same lesson that their highest achievement is to answer exercise questions correctly and then do the same in official examinations. I disagree. To educate pupils mathematically rather than merely train them you cannot safely or successfully ignore background setting context or perspective. Here are seven perspectives from which I believe teaching (and learning mathematics can be profitably viewed whatever the level of the student.
