Will New York win the second round of the Race to the Top? We don’t know yet, but add one more item to the list of ways the state’s application has gotten stronger: More teachers unions signed on to the plan this time around, and they added fewer caveats to their endorsements.

The percentage of unions signing on to the plan is now 70%, up from 61% in the last round. That includes New York City’s United Federation of Teachers, which, though it signed on last time, added caveats along with its “yes,” as Steven Brill reported in the New York Times Magazine. One major exception was a clause saying that unions could ignore any part of the plan that violated a union contract — even though, in the same memo, the unions promised to negotiate new contracts following the plan’s main ideas.

In the first round, some judges noted the caveats and the 61% figure as a reason they docked points from the state’s application. I couldn’t find any caveats in this round’s Memorandum of Understanding documents that unions and school districts had to turn in by Tuesday.

Still, among the dissenters are some pretty major unions, including the ones in Buffalo, Rochester, Yonkers, and Albany. That’s three of the state’s “Big Five” school districts. A typical explanation why came from Buffalo’s union president earlier this month, in the Buffalo News:

“When the state Education Department and the Board of Regents start insisting on mandatory smaller class sizes — then I’ll be willing to discuss having test scores be a part of teacher evaluation,” said Philip Rumore, president of the Buffalo Teachers Federation.

He described the measure as a political and public relations effort that won’t solve the problem of poor performance resulting from poverty.

“It’s not that we have droves of poor teachers that happen to be in the poor rural and urban areas,” Rumore said. “It’s not that teachers aren’t doing a good job. They’re working their hearts out.”