A training session about the city’s favored teacher evaluation model went off as planned on Tuesday — but without the involvement of the city, which had worked with the teachers union on event.

Since the start of the school year, the union and city have been grappling over the Danielson Framework, the observation model the city hopes will be adopted when a new evaluation system is finalized. Over time, a tension has emerged about whether the model is meant first to help teachers improve — the union’s position — or whether it is a tool to help principals usher weak teachers out of the system, as the city’s rhetoric has sometimes suggested.

Since at least December, the city and teachers union had been planning joint training sessions for principals and union chapter leaders to clarify the model’s purpose and value.

But after Mayor Bloomberg lashed out at the United Federation of Teachers during his State of the City speech last week, declaring that he would remove half of the teachers at 33 low-performing schools, the union decided it would no longer work with the city on the trainings.

“The content of the State of the City has not been received very well by members,” Michael Mendel, a union secretary, told me Wednesday. “To do a joint training didn’t sit right.”

On Friday afternoon, union officials surprised the city by announcing that the collaboration was off.

“The UFT will continue to hold these sessions and only invite school-based personnel,” Catalina Fortino, a union vice president, told department bureaucrats in an email.

Shortly afterward, the city sent a message to principals telling them that the sessions were off and directing them to consult two department websites instead to find resources about Danielson.

“Regretfully, the UFT today has informed us that they were retracting their offer to participate with us and as such, the sessions have been cancelled,” principals who had registered for the trainings were told in an email.

It took the principals union, the Council for School Supervisors and Administrators, to let principals know that the training sessions were still happening.

“It has been brought to our attention that the DOE will not be participating in the joint UFT-DOE professional development session regarding use of the Danielson Framework for Teaching scheduled for Tuesday, January 17th in Queens,” CSA emailed its members. “However, the UFT will continue to offer this very important PD session, and we strongly encourage those who signed up, to attend with your UFT chapter chairs.”

The first session, held at the union’s Queens office in Rego Park, attracted about 200 administrators and union chapter leaders. They sat together at a time when conflict about the move to overhaul teacher evaluations has reached a fever pitch.

“I’ve never heard of a meeting like that,” said Musa Ali Shama, principal of Francis Lewis High School, who attended with his chapter leader, Arthur Goldstein. “When we talk about improving instruction and professional practice, that was a very positive step in that direction.”

Goldstein said he thought the city had twisted the intent of the framework’s architect, Charlotte Danielson. (Danielson herself has said some of the city’s practices in piloting the model have been troubling.) So he attended the session to learn about the model from the union’s perspective.

“I am glad to have learned what I did,” Goldstein said. “While the principal and I are often in adversarial situations, we both share a passion for this school, its survival and success. I would be a fool if I did not work together with him when we have a convergence of interests. In fact, we both want kids to receive the best education possible.”

The sessions are set to continue next week in Manhattan and Brooklyn.