As a mathematics PhD student, I have seen a lot of mathematics teaching. I have been lucky enough to witness some (very) good teaching, but I have also been inflicted with so much bad teaching. Sometimes, I just wondered whether the people do even realize that their teaching is horrible, and I most often believe they are just unconscious about it, since most of the teaching they were exposed to was bad for them as well. I guess they just believe there is no way of making a good teaching of math, since most of it seems to be so bad.
I genuinely believe all graduate students in any subject should care about teaching. Even if you do not particularly care about teaching others, it makes your knowledge and understanding stronger. Just thinking at what matters (which is what you would teach), and how it would be best presented, forces you to thoroughly understand the topics to a deeper level. And at some point, you will just wonder: what are the essential elements of bad teaching? How can I avoid them?
#1 – Too many answers, too few questions
I am genuinely convinced that the most essential element of bad teaching is providing answers instead/without questions. Too often we go to class and we get lectured about some method, some theorem, some theory – and too seldom we get lectured about the path that actually led to that method, theorem or theory.
Examples: Where does the need for equations come from? When we define derivatives, what problem are we actually trying to solve? What had in his head the guy who came up with measure theory? Answering: “to solve problems”, “to measure rates of change”, to re-found the theory of integrals” is not enough. Not even “because it is important for applications”, which is just equivalent to saying “I don’t know”. Instead, we need to go through the history that motivated the inspection of the problem, how the question came up, and why it is (still) relevant.
Literally everything in mathematics comes from a question, from a problem, and failing to illustrate it is guarantee of teaching failure. It is just like being told to do something, but without the reason why we should. It leaves us purposeless, with the feeling of doing something we do not quite grasp the reason for. And what is more, often, if the problem is well-explained, bright students may just be able to come up with the solution themselves, or at least have a hint of idea.