I came to mathematics via physics, in part because of the reputation of physics as allowing "non-rigorous" reasoning. The subject felt more free and less anal-retentive than mathematics. This is not to say that physics does not require care, but in many physics books, proofs are scarce and derivations plentiful. Arguments are given only so far as needed to convince the student of the truth of something under the "nicest" possible conditions. When I was young, this was appealing. The sense that I only needed to argue to a point where I was convinced led me to revisit concepts and computations far more often than if some taskmaster were standing over me with a ruler waiting for me to make an error I wasn't prepared (read: mature enough) to acknowledge.
Of course, now I am a mathematics professor and I take a far different approach with my students. I require them to understand things that (based on their performance on assignments) they really do not appreciate to a degree I find satisfactory. Their basic skills are often abysmal, and the impression I get is that, if they were not forced to, they would never revisit anything mathematical. Many students see no beauty in this subject, only fear and a need to be right and get "the answer".
This condition, and its contrast with my own rather lax beginnings in the subject, makes me wonder whether we should be doing something vastly different than we are doing.
Question: Has an approach to mathematics education been tried that explicitly deemphasizes correctness and assesses almost solely based on exploration, interest and beauty? If so, how does one implement such an approach?
Of course, in the end, we want students to appreciate the need to be careful and precise. My implicit claim is that by providing an enjoyable experience for students, they will voluntarily think about topics again and again and therefore will "converge" to correctness via willing and frequent re-exposure. Is this general view valid in any sense? What research supports or critiques this idea?