Books and teachers can give information to students, but giving students information is not always the same as teaching. While teachers are a valuable resource, it is impossible for them to effectively teach to the strengths of every single student in the class all at once. Severe time restrictions are also limiting, forcing teachers to emphasize broad instruction over hands-on activities. You learn math by doing. My job as a tutor is to help students actually do math.
My approach to tutoring is very "question oriented". Instead of talking at students and hoping that they absorb what I'm saying, I develop real life examples that will help them understand concepts in a more intuitive manner. From this intuitive notion the math follows more organically as the student begins to move through examples that build slowly on the concepts that the student does understand. Lessons move at the pace dictated by the student and their capacity to absorb and assimilate information.