What is the name of the style where research papers are written in a story-telling fashion? The following research paper is written in a story-telling manner:


One Mentor-teacher's Personal Narrative of Collaborative Research on the Teaching Practicum


What is the name of this style of research paper writing?
One word for this is an experience report. This search on Google scholar reveals that many experience reports seem to have a similar purpose to the paper you ask about.

An experience report relays some important, often qualitative information about the authors' experience in deploying a method, hypothesis, or idea in the real world, whereas a regular research paper focuses more on proposing a new method, hypothesis, or idea, and arguing why it advances the state of scientific knowledge.

Also, I think we should separate two things here: the style of the article, and the style of the writing. The style of the article is accurately described as an experience report. On the other hand, many experience reports (see the above Google scholar list) do not have such a flavor of a story -- unlike your article, which begins


  I remember sitting on the green couch...


I would say this is more a matter of writing style than actual content. I have read articles in my field (computer science) that are like this, but it is rare and unorthodox. It likely varies by field, and in some fields (I would guess that philosophy and comparative literature are examples), storytelling is a standard and accepted style of writing.