What makes a maths tutor different? Part 1

The second great thing I think one to one maths coaching can offer, is that it alone can actually watch where you go wrong/ i spend my time in tuitions teaching certainly, but also listening y really carefully to what I am being told by my pupil as the work though a problem, but also watching their face and their pens. one therefore spends ones time actually watching thought as it happens - in all its hesitant beauty, and layered complexity.
 The job is then to work out exactly how and when one needs guide the other persons thought process. I personally am a great fan of doing a little as possible, But doing soemthing. I will tap the paper therefore on a sing, or raise an eyebrow, or sometimes re-ask the question. thee point being that i am trying to teach how one notice and then reacts to that observation in maths. the game in the subject is never not to be wrong, but merely to write down in such a way that one naturally notices or looks for ones own error.
If I get the 'correction' right, therefore it should feel like for other individual it is them that is noticing the problem, it is their eyes and their brain - I am merely showing them where and when to do it.
 And yet of course this isa  risky strategy. when it works it is great,but you have to be so careful. It is so easy to not be helpful, and merely be irritating. Each pupil is different not only in the way they  think, and the exact place they make errors, but also, and just as importantly in the way the way they like to locate those errors. one needs then to have also ready the 'stock tutor' methods which allow for one to find errors through very formal writing down of methods - a formality that is certain aspects and at certain times is astounding useful and powerful.
 What maths makes tuition so difficult, and so powerful at times (and given the right relationship), is that one can teach the real brain on skills that gets you through maths exams, skills that one cannot teach anywhere else i think, and ones always worth having - they the skills that allow one to be self critical, and reflexive.