A second day of negotiations between Chicago's school board and its striking public school teachers ended Tuesday with neither side expressing optimism that an agreement was near.

"This was silly season," board President David Vitale told reporters after emerging from more than 10 hours of talks. "It is time for us to get serious."

Vitale said the board had presented the union with a "comprehensive proposal" and would resume negotiating only after "we receive a written response or a comprehensive proposal of their own."

But Barbara Byrd Bennett, interim chief education officer, said the negotiating would continue Wednesday, whatever happens. "Our team will be back here tomorrow," she said.

Chicago Teachers Union Vice President Jesse Sharkey said the day's discussions had centered on teacher evaluation and that "some substantial movement" had been made, but not enough.

"I don't want to get in the weeds, but I'd say we moved more than they did today," he said.

The board proposal would leave some 28% of teachers in danger of dismissal within a two years, he said, calling that "an insult to our profession."

"They basically dug in their heels and said if we didn't give them a comprehensive proposal, we didn't have anything to talk about," Sharkey said.

The negotiations ended after thousands of striking teachers had spent much of the day massed outside the Chicago public school system's headquarters.

Carrying signs, they chanted and marched through the streets in an expression of solidarity in their fight against the school board.

"We have a considerable way to go," union spokeswoman Stephanie Gadlin said in a news release. "This is a fact they cannot deny."

Of 49 points in the contract offer, the union has agreed to just six, she said.

"We are fighting for our students; we are fighting for education justice," she said.

Mayor Rahm Emanuel cast the strike in different terms.

"This was a strike of choice. And it's the wrong choice for the children," he told reporters.

After five months of negotiations, "we're down to two issues," he said. The sticking points are teacher evaluations and provisions dealing with jobs for laid-off teachers, said Emanuel.

The talks could have continued without a strike, which was "totally avoidable, totally unnecessary," Emanuel said.

After no deal was reached Monday, talks resumed Tuesday morning.

Late in the afternoon, U.S. Education Secretary Arne Duncan issued a statement expressing confidence "that both sides have the best interests of the students at heart, and that they can collaborate at the bargaining table -- as teachers and school districts have done all over the country -- to reach a solution that puts kids first."

Chicago's first teachers' strike in 25 years has left many parents scrambling to find alternatives for their children.

"If the kids are not in school, they're out getting into some kind of trouble," said Shatara Scaggs, a mother of a kindergartner and first grader. "They should be in school getting an education."

To prevent a possible uptick in trouble from kids on the streets, police pulled officers from desk duty to increase patrols. Dozens of churches and civic organizations have stepped in to provide activities for idle students, and the school district has opened 144 of its 578 schools for part of the day to provide a safe environment and meals to children.

Many children going to these in-school programs saw their teachers chanting, holding signs and otherwise voicing their opinions. Ola Esho, whose child attends Ray Elementary School, told CNN affiliate WBBM he "was not happy" about the commotion, which he said unnerved his children.

"I would not want to keep my children here unnecessarily, so I'm taking them back home," said Yahu Vinayaraj, whose children attend the same school in the city's Hyde Park neighborhood.

The union, which represents nearly 30,000 teachers and support staff in the nation's third-largest school district, called the strike Sunday night. The union said the two sides had been close to a deal on pay, but far apart on teacher evaluations, benefits and other issues.