Something unusual happened near the end of bargaining Thursday between the Denver teachers union and school administrators: The district offered a change to its proposal, and some teachers gathered in the audience snapped their fingers in approval.

The two sides are still far apart in terms of reaching an agreement on Denver Public Schools’ ProComp system, which offers teachers bonuses on top of their base salary for things like teaching in a high-poverty school or earning a strong evaluation. The deadline for a deal is Friday, with teachers set to vote on either strike or ratification Saturday and Tuesday. The district’s and the union’s proposals still reflect different ideas about how teachers should earn more compensation — and have very different price tags.

But Thursday’s session was a far cry from the intense frustration that marked Tuesday’s bargaining session. On that day, district officials departed the bargaining table to “process” after the union refused to make a counteroffer to the most recent district proposal. Teachers filled the room after school and waited in hot, cramped conditions for a response that never came.

Thursday’s session opened with tense verbal sparring, but by the afternoon, the union had made changes to its proposal that reduced the cost by an estimated $2.5 million. District officials said they would spend the night and early Friday morning analyzing the proposal and seeing where else they might be able to move.

The district did offer a small concession Thursday that some teachers seemed to appreciate: increasing tuition reimbursement 50 percent, to $6,000. Denver Superintendent Susana Cordova said research shows this type of incentive is a strong tool for recruiting and keeping teachers, especially teachers of color. The money for this will come from reducing the bonus that teachers get for teaching in a so-called “distinguished” school.

More often, Denver teachers have reacted with boos to offers that the district saw as significant steps toward the union position.

Beyond small steps like the tuition reimbursement, Cordova said she was “very open to considering” aspects of the union proposal, a sentiment that seemed to clear the way for more back-and-forth.

“The district has moved and DCTA was willing to change their proposal,” said Tiffany Choi, a French teacher at East High School who attended the bargaining session. “We both showed willingness to compromise, and that’s positive.”

Dana Berge, a member of the union bargaining team, said the district is “beginning to listen to us, but they are not listening to us in terms of the values in our proposal.”

Both proposals keep some bonuses, at much more modest levels, and put more money into base pay. And both proposals lay out a schedule for how Denver teachers can earn more money, both by “steps,” or number of years of service, and by “lanes,” or additional educational achievement.

The union proposal has more lanes and allows Denver teachers to start moving up by earning additional college credit or by taking the kind of professional development that teachers need to do anyway. Teachers take such training to maintain their teaching licenses — or because they see a need, for example, to learn more about helping students with trauma. The union’s counteroffer Thursday removed one of the lanes, bringing down the total cost.

The district’s first lane change comes with a master’s degree, completing 10 years of service, earning an advanced license, or earning national board certification.

Berge said the union proposal more closely mimics those in other districts and will keep “highly dedicated, highly trained, highly experienced teachers” in Denver and reduce the problem of losing more experienced teachers to better-off suburban districts while less experienced teachers concentrate in the highest-needs schools.

She argued that a more stable salary structure would do more to keep teachers in high-poverty schools than the bonuses given under the current system.

The union proposal will cost a lot more than the district proposal. Cordova said the district is in the process of identifying “deep, deep cuts” to administrative positions to redirect money to classrooms, but even those won’t provide all the money needed to close the gap between the two sides.

During the long period in which each side was working separately on its proposals, a group of religious and community leaders from the Industrial Areas Foundation arrived to offer support to teachers and then a direct message to Cordova. They criticized her for framing the disagreement as one about values.

“Please don’t dare insinuate that you care more about the children in hard-to-serve, high-poverty schools than we do,” said Susan Cooper, a retired teacher and member of the organization. “We value solidarity, not a divide and conquer approach. We value stability, which we don’t have because we can’t keep teachers in Denver.”

Billy Williams of the Denver chapter of the NAACP said parents and community members would support the teachers “if they are forced to strike.”

Cordova said it was never her intent to suggest one side held the moral high ground.

“We can have similar goals, similar values, and different ideas about how to get those done,” she said. “Good people can disagree and still care about our kids.”

The two sides expect to resume bargaining at 10 a.m. Friday.