I once knew an otherwise excellent teacher who compelled his students to perform all their demonstrations with incorrect figures, on the theory that it was the logical connection of the concepts, not the figure, that was essential.
Man is not logical and his intellectual history is a record of mental reserves and compromises. He hangs on to what he can in his old beliefs even when he is compelled to surrender their logical basis.
One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.
Through dialogue, the teacher-of-the-students and the students-of-the-teacher cease to exist and a new term emerges: teacher-student with students-teachers.
A great teacher is not simply one who imparts knowledge to his students, but one who awakens their interest in it and makes them eager to pursue it for themselves. He is a spark plug, not a fuel pipe. The reason colleges exist is to bring students into contact with contagious personalities, for otherwise they might as well be correspondance schools.
My mom was a special-needs teacher for many years, so I knew her students. And one of my best friends from when I was growing up is a teacher for kids with autism now.
Before I met Ayn Rand, I was a logical positivist, and accordingly, I didn't believe in absolutes, moral or otherwise. If I couldn't prove a proposition with facts and figures, it was without merit.
It must be conceded that a theory has an important advantage if its basic concepts and fundamental hypotheses are 'close to experience,' and greater confidence in such a theory is certainly justified. There is less danger of going completely astray, particularly since it takes so much less time and effort to disprove such theories by experience. Yet more and more, as the depth of our knowledge increases, we must give up this advantage in our quest for logical simplicity in the foundations of physical theory.
This is what an excellent teacher does; he does not follow his disciples' fancy everywhere, but leads them to his own mind, and pulls up the thorns, and then puts the seed in, and does not answer at once in all cases to the questions put to him.
If you can develop a connection, what we call an inner connection with the teacher, then it is not even necessary to be in the physical presence of the teacher, although it will always be stronger.
The main difference in the effectiveness of teaching comes from the thoughts the teacher has had during the entire time of his or her existence and brings into the classroom. A teacher concerned with developing humans affects the students quite differently from a teacher who never thinks about such things.
It's important to know how to lead and manage a classroom with flexibility. Students of all ages are quite capable of learning these routines and contributing to their success once the teacher is comfortable guiding students in that direction.
The most successful classes are those where the teacher has a clear idea of what is expected from the students and the students know what the teacher expects from them.
I am firm in my belief that a teacher lives on and on through his students. Good teaching is forever and the and the teacher is immortal.
I travelled round the world giving demonstrations of my talents. In every country, I performed for students, professors, teachers, bankers, accountants, and even laymen who knew very little, or nothing at all, about mathematics.
For strictly scientific or technological purposes all this is irrelevant. On a pragmatic view, as on a religious view, theory and concepts are held in faith. On the pragmatic view the only thing that matters is that the theory is efficacious, that it 'works' and that the necessary preliminaries and side issues do not cost too much in time and effort. Beyond that, theory and concepts go to constitute a language in which the scientistic matters at issue can be formulated and discussed.
Excellent instruction is less about what a teacher does and more about what students can do and know as a result of the lesson.