Young man, in mathematics you don’t understand things. You just get used to them.
I like this quote of von Neumann’s and often mention it to tutees. Of course in the world of school maths and private tuition the focus is very much on getting pupils/tutees to understand things, as well as getting used to them.
Why get a maths tutor? What does your money get you?
The role of the teacher is to create the conditions for invention rather than provide ready-made knowledge.
An hour of conversation and problem solving (not drilling, although this could equate to getting used to things). For younger mathematicians both approaches are necessary, but private tuition provides a space where the tutee can feel free to ask questions and make errors in a non-judgemental environment. This is important and valuable and can give the boost to a child’s confidence that leads to an improvement in results. “Thinking aloud is allowed”.
For younger children I use Cuisenaire rods to help develop their number work.
For older children sitting national exams I stress the importance of understanding the basics of their previous mathematical work, as building blocks to the maths they are currently studying, and I ensure that time is given over to revision.
How to get better in math at about.com has some good advice especially the first two points regarding understanding versus memorisation, and the call to get active. Becoming an active learner is one of the many benefits of maths study, and I encourage all tutees to ask questions and make connections in maths as a means to understanding the topics we are studying.
Some feedback from my tutees.