CHICAGO (CBS) — Among those who have already received the COVID-19 vaccine are thousands of Chicago Public Schools teachers — but not nearly as many as one might think. After a bitter battle between the district and the union, why haven’t more teachers raised their hands to get the shot?

The two sides cannot even agree on figuring out how many have been vaccinated. One reason is because the Chicago Teachers Union does not yet want members to tell CPS. Another is CPS’s own tracking system is incomplete.

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Albany Park’s Roosevelt High School is one of four CPS vaccination sites. It is one way the district is trying to make it easier for teachers and staff to get vaccinated, which for many is coinciding with a return to the classroom.

“We were sent emails to our CPS emails and were given opportunities in multiple locations around the city,” said CPS high school teacher Molly Lane.

Lane has gotten her first shot and believes CPS’s program so far is effective.

“As an example yesterday I was talking to a co-teacher, and she went to get her first vaccination. She said truly behind her were three other of her direct colleagues from the same school in line,” Lane said.

But tracking vaccinations is another story.

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“We don’t have the best information,” said CTU President Jesse Sharkey

Neither does CPS. Its vaccination page shows a total of 16,500 CPS staff members have been offered or scheduled appointments and just over 4,200 employees have gotten the vaccine. But it does not specifically break down to whom those appointments were offered or who got them. One reason is there is no way of knowing how many staffers got vaccines at non-CPS sites. Another data roadblock is a letter from the CTU asking that “members wait to respond to (CPS’) vaccination survey.”

“I don’t have a problem with people answering this kind of survey,” said Sharkey. “I do have a problem with CPS not bargaining it with us.”

Sharkey said another disclosure deterrent may be teachers fearful of losing their third quarter accommodations allowing them to teach from home, even though they are supposed to remain in tact.

“Now CPS appears to be saying, ‘No, we are going to review them for this quarter.’ And the problem is that people have their life set up around these accommodations,” Sharkey said.

He believes about 11,000 of CTU’s roughly 25,000 members, about 22,000 of whom are teachers, have received at least one vaccination shot.

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Sharkey said the union is set to meet with CPS leadership to discuss when union members might be given the green light to answer the survey on Monday, but there is no guarantee the matter will be settled then.