When I first started tutoring, I mainly helped students with the problems they had from class and for homework. I eventually started making my own practice problems, and it made me a much better tutor. This is best explained via example, so let’s use the algebra topic of dividing polynomials.
If a student is having trouble with synthetic division, the tutor could simply make up a synthetic division problem. However, if the student is having trouble with that, there is a good chance that they also do not understand the basics very well. Too often students memorize methods for complex problems without understanding the basic math that allows them to solve these problems without relying on memory.
Thus, instead of jumping right to the more advanced topic, I would start with a problem involving division by a monomial. If this problem is difficult for the student, we will practice and discuss it until we achieve a clear understanding. If the problem is easy for the student, great! I then step up the difficulty slightly. Next we would do division by a binomial in which the numerator is factorable and the denominator is one of the factors.
We would continue in this way, carefully checking understanding at each step up in difficulty, until the student is able to do problems at each level independently. This is the role of the tutor – to guide the student to self-sufficiency. Indeed, students can even use this technique when studying in their own using textbook problems!