Yes, Joe, I am afraid you are right on the money. I am constantly going over the deficiencies in my own education, and using the results of my research in my teaching duties. Admittedly, that is another tendency of mine, to experiment on students. If parents don't like it, they can fire me, since I am a tutor. I do get parents who fire me for one reason or another, and I am perfectly happy about it (after the passage of an appropriate time period). I also have loyal students and parents who like what I am doing. They come back for more. Naturally, I am as curious as you are as to what "works" and what doesn't, but in the end, my technique is to do what I think is best at the time. I make those decisions for a variety of reasons, and am always learning more, and observing. I often do use my teaching as much to satisfy my own curiosity as to help the student, and there is often an atmosphere of mutual discovery. I like that. I will only ruin a few lives, as opposed to being, s! ay, a classroom teacher, where I could ruin 30 lives in one blow.
Your quip about Liebniz was interesting. Who would have known that Leibniz did not admit negative numbers. His idea of infinitesimals was definitely novel. It might be of interest to see how people such as, say, Euler, Lagrange, or other famous mathematicians introduced negative numbers.