I don't think that there is any great mystery as how best to teach.
You need an excellent understanding of the subject matter, and you must be able to engage the student.
We learn things by hearing about them, reading about them, and/or doing them. By far, the most important factor in that equation is the doing.
The goal is always to get the student to answer the problems, not me.
In a classroom setting, the teacher can really only engage one student at a time, leaving the other 25 as listeners and watchers.
I encourage students to think out loud when solving problems. If they don't know where to begin, or get stuck, I can suggest the first or next step and help them work through it to the ultimate solution. Math must be learned progressively. You must know algebra to do geometry, and geometry to do trigonometry, and all three to do calculus.
Everyone is different and learns at a a different pace. I can go at whatever pace that benefits the student. If they are having trouble in one particular area, I can work with them in that area until they understand it, and then advance to the next level.
In a classroom with 20-30 other students, the teacher wouldn't have the same advantage. They would have to go at a pace determined by the average of the class. This can be most unfortunate for some, because with math, if you get lost, you generally stay lost.