I’m wondering how stern I should be upon the student when their concentration is waning. Should I be the friend who ushers and assists them through questions or should I be the stern teacher and push him through questions?
Stern or Friendly is up to you. However, it isn’t advisable to take either of the approaches mentioned. Neither a stern teacher nor a friend is the role of the tutor. The important question is which way would serve your student better?
As a friend you will ultimately lose credibility and the expert status you are bestowed as a tutor. The student will not take you seriously enough to do what you say. On the other hand, as a stern teacher the student may do more of what you ask but will inevitably develop a resentment or dislike toward you. Parents would not tolerate this for long; they tend to place a high value on the student enjoying the learning process or at least having positive feedback for their tutor.
A one-on-one tutoring situation is very different to traditional teaching roles. You WILL NOT succeed in tutoring a student unless it is as a team. You must enlist their support, both of you must be playing for the same team otherwise your efforts will be counterproductive. The way you achieve this is by establishing yourself as their mentor/tutor. A mentor is a “trusted counsellor or guide”, what you can do is counsel the student about mathematics. Be friendly and nice enough that they “trust” you but be stern and professional enough that they view you as a “maths counsellor”.
This means being nice and friendly but still exercising your authority to an extent – only do it in a consultative fashion rather than “stern”. If you need to be stern because the student isn’t implementing your suggestions try to involve the student in your cause. Explain what is wrong and why. Ask if student agrees with you? Have a conversation about the problem as a mentor (not stern, not friendly – just sincere and nice). You can literally say “Sam, this isn’t working as well as it should. I think it’s because of X, Y and Z which I’m not sure you agree with. Tutoring is a process and the quality of tutorials will depend on me knowing how best to work with you, so we should probably talk about this. What are your thoughts?”
Then, TOGETHER with the student decide on a course of action. Ask what he thinks is a suitable solution? Including the student in decision making will increase his commitment to the decision!
Obviously such a direct approach will work better with high school students but a softer version of this is still useful for primary math students.
