I have a motto, "My class is not an Olympic swimming pool! It is a museum full of the masterwork paintings! Relax and enjoy all of this mathematical beauty which I am drawing on the blackboard!"
By this sentence I mean that in my point of view the best mathematicians are those who enjoy beauty of math more not those who enjoy to be called as a great mathematician.
I don't like the competition atmosphere in my class because students effort to prove themselves as the best mathematicians of the class is contradictory with my teaching philosophy. Also it harms the team work morale of this small society. In this direction I try to avoid any kind of unnecessary comparison between my students because it commences a competition amongst them.
On the other hand I try to encourage them to team works for exploring mathematical wonderland and discovering more and more beauties of the teaching subject. The personal improvement of each student follows from this team works naturally and what is much more important is their ethical improvement during these group activities which learn them how to work together for the benefits of all.
Question 1. How can I encourage my students to work together as a team? What are the practical ways for achieving this goal?
Question 2. Tests in any sense (including exams, quizzes, etc.) increase the comparison (and so the competition) atmosphere of the community of my students. How can I design a test free course with fair enough marks which are acceptable for both students and me at the end of semester? In the other words, is there a more "soft" testing method different from the classic exams with an acceptable accuracy which measures the team work morale of each student?
Remark. I don't believe in numerical marks for measuring students at all. I think it is too simple-minded (and sometimes too harmful) to refer the abilities of a human just by a number. But unfortunately it seems this inappropriate method will be the common method of almost all math departments at least in the short future because there is no other known effective substitute for it.
Please explain your ideas and suggestions even if they are strange, unnatural and completely new.