I do, on the basis of personal experience in mathematics.

I  retired after 50 years of teaching in universities in 4 different countries with different education systems and pedagogical traditions. This experience shaped my views on our profession. I firmly believe that

1. Teaching is not a science, it is an art, and should be treated as such.
2. Students are not customers (“persons who buy”) – they are clients (“persons who seek the advice of a professional man or woman”).
3. “Good learning experience” means mastering something new and advanced. To help his/her students, a university teacher has to be able to transform and restructure highly complex material from his/her subject area into a form suitable and accessible to the learners.
4. This cannot be achieved without teachers being experts in their disciplines.
5. Successful and inspirational teaching is a highly individual skill. The choice of teaching methods should reflect not only specifics of the target audience, but also the experience, teaching philosophy and individual psychophysiological characteristics of the teacher.
6. Structuring of the learning environment, choice of teaching and assessment methods have to be subject specific.
7. Values, standards, criteria of assessment in learning and teaching have to originate in, and be set by, the professional academic communities of their particular subject areas.
8. The role of managers is to create an environment which helps professional standards to be maintained; however, managers should not interfere in setting the standards.

This Manifesto was published in a an old blog. I wish to transfer from there a comment from one of the leaders of the math circle movement. It describes a creative learning environment flourishing in math circles but absent in universities.

At some point, I have compiled a short list of reasons why I get a lot of satisfaction from teaching a math circle. I love:

-the equality and feeling of mutual respect and attention that develops between me and math circle participants
-the democracy/lack of authority that shows us the “right answer”
-seeing the value alignment and deep intellectual friendship that develops among the participants
-sharing children’s excitement when they realize their own powers
-the feeling of freedom they develop when they get rid of their own mental blocks
-the intellectual stimulation of choosing the problems and personalizing and teaching them to a particular audience
-when children realize that they feel happy from doing a challenging job
-observing their self-discovery
-observing as children come up with amazing solutions and counter-intuitive discoveries
-getting a fresh view of the beauty and awesomeness of the world we observe and create – thus multiplying my own happiness

My answer to a question on Quora:

I do not remember seeing a mathematicians who was embarrassed by their conjectures disproved.

Why? Because making conjectures and refuting them is a normal cycle of mathematics. I think 90% of conjectures die on the same writing desk where they were born, being killed by the same mathematicians who formulated them. In mathematics, it is a daily routine. Refutations are as important as proofs. There is a famous book about the role of refutations in mathematics, Imre Lakatos’ Proofs and Refutations.

And the famous Lewis Carroll’s lines in Through the Looking-Glass capture the spirit:

“I can’t believe that!” said Alice.
“Can’t you?” the Queen said in a pitying tone. “Try again: draw a long breath, and shut your eyes.”
Alice laughed. “There’s no use trying,” she said: “one can’t believe impossible things.”
“I daresay you haven’t had much practice,” said the Queen. “When I was your age, I always did it for half-an-hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.”

Proofs and refutations co-exist in the most natural way. Mathematical problems are conjectures. To solve a problem means to prove this conjecture or refute it.

Proofs are frequently done by constructing, in parallel, a counter-example: when a mathematician identifies obstacles for a proof, he/she may wish to try to use them to construct a counterexample; when this attempt at refutation encounters its own difficulties, a mathematician may try to isolate these difficulties and understand their nature – for use in the proof. In this zig-zag movement the aims — to prove a conjecture and refute it — alternate. In a happy outcome , the process converges on a definite answer: either proof or refutation.

But, if you look back at that zig-zag prowl in search of a kill, you may say that half of the time the mathematician believed impossible. Even worse, it is like lions in hunt: ten chases result in one kill; a mathematician normally solves about one problem out of ten that he or she tries.

There is one extreme case of the proof/refutation balance: the original proof of the Classification of finite simple groups. I quote Wikipedia:

The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.

As a rule, almost each of these “several hundred journal articles” contains a proof of a particular theorem, a special case or an intermediate step of the “global” statement. Since all that is about finite objects, proofs frequently use mathematical induction in a specific form: proof of non-existence of a minimal counterexample to the theorem. As a result, it makes thousands of pages of arguments about non-existent objects. At a first glance, it gives an answer to another question on Quora: What are some aspects of mathematics that are nonsense? But these arguments about eventually non-existent minimal counterexamples are not nonsense — for example, they can be re-used in proving theorems in other branches of group theory.