When I was thirteen, if I remember rightly, a maths teacher remarked in passing that a Pythagorean triangle with integer sides always has one of those sides divisible by 5 - as you do! What intrigued me was that I realised I simply did not know how you would go about proving something like that, so around twenty minutes later I asked him. He stopped the class and went through proof, which relies on techniques that are way off syllabus, but opened my eyes (if maybe no-one elses!) to different ways of thinking about problems. It is, perhaps, the moment I moved from 'school maths' to 'real maths'.

I agree with Steve that that is a 'good teacher'. I am also painfully aware that the ofsted-driven, preplanned approach of today would declare the opposite.