The school year opened with a historic picture on the front page of The New York Sun showing the president of the United Federation of Teachers, Randi Weingarten, in class with three bright-looking first-grade pupils at the union’s new charter school in Brooklyn. It’s the first time that the union has taken on the role of management. Ms. Weingarten was quoted by the Sun’s reporter as calling the school an “oasis” but complaining that this is the third year the teachers have returned without a contract.

“The talk all over is, ‘Why won’t the chancellor negotiate a contract? Why won’t the mayor negotiate a contract?’…” Ms. Weingarten said. “That’s the talk that has eclipsed everything else.”

It’s a scandalous situation, no doubt, but there is a strategy available to the teachers that I believe would bring the city to the table and help poor parents and children, to boot. I offered it to Ms. Weingarten over lunch a few weeks ago. The lunch was off the record, so I’m not at liberty to report on her reply. But my own comments at the lunch were not off the record, so I don’t mind relating what I said – which is that the way to break this standoff is for Ms. Weingarten and the United Federation of Teachers to come out in favor of vouchers.

I recognize that, at first blush, the idea might sound improbable. The union, after all, has been the foremost opponent of the idea of vouchers. Not only Ms. Weingarten herself, but most of her predecessors and colleagues at the national level have long been on the record as opposing vouchers. Their thinking has been that giving parents a choice of where to send their children to school and letting the tax breaks or voucher money follow them, well, such a scheme could only lead to a loss of funds and jobs in the public school system.

True enough, but it would also mean that the monopoly school system run by a Republican mayor and a tough-as-nails ex-prosecutor wouldn’t be the only game in town. After all, why should the management of the only game in town think it has to have a contract with the teachers, or anyone else? Where, if the government has a virtual monopoly on schools, are the teachers going to go? A strike could only work to the advantage of the schools chancellor, Joel Klein, and the mayor. So for the third year in a row the teachers are stuck in this humiliating situation of having no contract and the Tweed Courthouse telling them how to set up the desks in the classroom.

With a system of vouchers, by contrast, the power of the Tweed Courthouse would be sharply reduced. The one-time monopoly would still be a significant employer of teachers. But it would have to be much more forthcoming if it wanted to get a contract, because the teachers would have plenty of other places to go. Hundreds, perhaps thousands of private schools would be started or expanded. Not all of these would be eager to be a union shop. But there would suddenly be a teacher shortage in the private school system.

It certainly sounds like an opportunity for Ms. Weingarten – and her members – to get some bargaining power. And also a way to enhance their professional esteem. One of the issues that has arisen since Mayor Bloomberg took over the schools is frustration, even anger, on the part of the teachers, and their union, over the way they say the city has been micromanaging teaching methods. Mr. Klein makes a strong argument that if he is restricted in firing teachers, he ought to be able to have them teach the way he wants. A strong point, no doubt.

If, however, professional satisfaction is one of the things the teachers want, a way to maximize the ability of the members of the union to use their own judgment and professional skills, what possible system could be better than one that, like vouchers, maximizes the number and variety of institutions? A voucher system, in other words, maximizes choice not only for parents but also for teachers.

In addition, vouchers would relieve the teachers union of one of the burdens it carries – the fact that it has been trapped into lobbying against a reform that would give poor and middle-income parents the kinds of choices in education that wealthy parents have long had.

The idea that the teachers union needn’t be the big loser in any move to a voucher system began to form in my mind as I was sitting at the banquet in New York of the Milton and Rose Friedman Foundation honoring the Nobel laureate and his wife on the occasion of the 50th anniversary of the idea of vouchers. It was an inspiring evening. It would have been even better, I wrote at the time, had the mayor, the chancellor, and Ms. Weingarten been there.

So over lunch the other week, I extended an invitation personally to Ms. Weingarten to have lunch with Mr. and Mrs. Friedman in San Francisco. I can’t disclose whether she accepted. But I have alerted Ed and Mary Etta Moose, proprietors of Moose’s restaurant in North Beach, to be ready to set a quiet table in the corner of my favorite San Francisco eatery. I figure the worst that could come of it is a wonderful meal. And it could illuminate a landscape in which the UFT would see that its new charter school in East New York, Brooklyn, needn’t be its only oasis.