What might you do in a typical first session with a student?

Assess student's needs and skills, and find out what they want us to achieve working together. Get background on what is happening in their classroom and determine how I can best help. Are they looking for enrichment, advanced learning or getting past something they are having trouble with (or any combination)? I would definitely review problems they have been working in class and provide some additional examples. I would hopefully be able to impart some additional insights/understandings that would help the student "see" how to attack the problem and succeed in doing it.

How would you help a student stay motivated?

Encouragement, and providing problems that start out easier and doable, progressing to increased complexity. Start out with lots of support (work together) until I am no longer needed and the student can work independently on them.