* Negotiators make progress on most vexing issues
* Union opposed to evaluations based on student test scores
* Civil rights leader Jesse Jackson tries to intervene
By Renita Young
CHICAGO, Sept 12 (Reuters) - The Chicago Teachers Union and the nation's third-largest school district expressed optimism for the first time late on Wednesday that a strike could end soon, after they made good progress toward an agreement on education reforms sought by Mayor Rahm Emanuel.
But negotiators said there was still much work to do on a deal and teachers would stay out of school for a fourth day on Thursday.
"Plan for something for your children tomorrow and let's hope for something (an agreement) Friday," said a smiling Karen Lewis, the teachers union president, after talks went late into the night on Wednesday.
Lewis, who has harshly criticized the mayor and his negotiators, struck a much more conciliatory tone as she emerged from the talks just before midnight on Wednesday.
"We made a lot of progress today. We are definitely coming much closer together," Lewis said.
For the first time in days, Emanuel's chief negotiator, School Board President David Vitale, agreed with Lewis' summary of the talks. Only 24 hours earlier, Vitale had threatened not to come back to the negotiating table until the union put forward a better offer.
"We had a very productive evening," Vitale said. "We all go away hopeful that we can go come together on this."
With more than 350,000 children out of school, the patience of parents had begun to fray as hopes of a quick resolution to the biggest U.S. labor strike in a year faded.
Earlier in the day, civil rights leader Jesse Jackson, who is based in Chicago, appeared at the site where negotiations were supposed to take place on Wednesday and said that he had met with both sides separately to urge them to settle.
Lewis said the progress on Wednesday was on the two most vexing issues - using student test scores to evaluate teachers and giving more authority to principals to hire teachers.
The union is concerned that more than a quarter of its membership could be fired because the teachers work in poor neighborhoods where students perform poorly on standardized tests, which Emanuel wants to use to evaluate teachers.
Lewis also said the union fears Emanuel plans to close scores of schools, putting unionized teachers out of work.
Lewis led the walkout on Monday of more than 29,000 teachers and support staff, saying the union would not agree to school reforms it considers misguided and disrespectful.
The dispute jolted the United States, where a weakened labor movement seldom stages strikes and even less frequently wins them. Organized labor has lost several fights in the last year including Wisconsin stripping public sector unions of most of their bargaining power, Indiana making union dues voluntary and two California cities voting to pare pensions for union workers.
The strike in Barack Obama's home city has also put the U.S. president in a tough spot between his ally and former top White House aide Emanuel and labor unions Obama is counting on to win re-election on Nov. 6.
Obama has said nothing in public about the dispute, allowing administration surrogates to urge the two sides to settle.
Obama's own Education Department has championed some of the reforms Emanuel is seeking, and a win for the ambitious Chicago mayor would add momentum to the national school reform movement.
The city is operating 147 schools with non-union staff to offer meals and "keep children safe and engaged," but only a fraction of parents have been using that option, officials said.
At Disney elementary school, several dozen strikers with homemade signs targeting Emanuel and school policies picketed in cool, sunny weather on Wednesday.
Kent Barnhart, a music teacher for the past 25 years, said neighborhood parents had been supportive, offering water and opening their homes and even joining picket lines to march. But he said teachers were frustrated with the slow talks.
"It's difficult for us to understand why they have not truly discussed over the last 11 months things that have been very important," he said of school officials. "It didn't seem like they took it seriously - really important things like evaluations, health benefits and pay."
Both sides agree Chicago schools need fixing. Chicago students consistently perform poorly on standardized math and reading tests. About 60 percent of high school students graduate, compared with 75 percent nationwide and more than 90 percent in some affluent Chicago suburban schools.
The fight does not appear to center on wages, with the school district offering an average 16 percent rise over four years and some benefit improvements.
More than 80 percent of Chicago public school students qualify for free lunches at school because they come from low-income households.