Those who seek a solution, shall find one.

First sessions I usually take my own mental notes and try to understand how the student understands the problem and their approach to a solution. I also assess their skill level and habits.

This question is usually not in the best interest of a tutor, but a student will have to wean off the crutch of a tutor sometime or another. However, I usually try to teach a student how he/she can overcome some of these obstacles when stuck on a problem and demonstrate troubleshooting methods.

Usually, successful test scores are the main motivation. However, when students are really down, I try and relate what their interests are to the subject material. Sometimes explaining the more advanced material that the fundamentals pertain to helps a student understand why he/she is learning such material.

Each student is different, especially in learning methods. For these scenarios, I try and demonstrate a few different methods to obtaining a solution, which can be easier to understand than maybe the textbook examples or those taught in class.

Usually drilling questions, and once a certain number of questions have been done, normally a student will begin to become aware of the patterns that are required to achieve a solution on their own. A lot of mathematics is really manipulation of equations and transforming them, in which some students just aren't using the right "tool set".

I've normally seen a boost of excitement once the student finally goes through a problem on their own, after some instruction and drilling, and is successful. A lot of times, that's the "eureka" moment. We keep going until that moment comes.

Understanding why you're learning something is the most important thing behind it. A lot of times, students have this empty objective of "I'm taking this class because it is required," and it makes the work/reward fruitless and uninspiring. Understanding the end game helps students get back on track.

Confidence is a slow progression and combination of successful questions answered and better test scores as the result.

I usually have students perform a problem they're struggling on first by themselves and watch what they do. If they get stuck, then I go to another problem that they may be more confident with and have them execute that problem. The differentiation between the two can usually expose the mistake/conceptual error more easily to me.

Pencil and paper. If available, a whiteboard and markers. I'm old school, no calculators.

What I won't do is be a "Band-Aid" tutor. Ten out of ten times the error(s)/misunderstandings of a concept are due to a misunderstanding of a previous fundamental concept. Building a stronger foundation is crucial for every student.