Larry Hamilton, by day a Central High teacher of government, stood before a different kind of class the other evening wearing two buttons that said, "Teachers Do It In Politics" and "I'll Be There on Jan. 21."

The evening's lesson was "How To Win Delegates" in the Iowa caucuses Monday night. Each member of the class -- several dozen teachers from local elementary and high schools -- was armed with the phone numbers of other teachers to call, broken down by precinct.

It was an impressive scene of no-nonsense political organizing that will be repeated all over the country this year.

Over the last decade, the nation's teachers have abandoned their tradition of bland bipartisan patriotism for the rough-and-tumble of party politics.

Part of the rising tide of public employe activism, the teachers have discovered the uses of power and they want more.

The National Education Association (NEA), the largest and most powerful teachers' union, is backing President Carter. In the Iowa Democratic caucuses, this pits the 35,000 school teachers there against the 50,000 Iowa members of the United Auto Workers, who are working for Sen. Edward M. Kennedy.

Iowa illustrates at once the growing sophistication of teacher power and the internal tensions the NEA encounters as it tries to harness the political might of a huge and disparate constituency.

Iowa is one of two states so far where the NEA's state affiliates have decided to field delegates for a Republican nominee (Howard Baker) as well as for Carter, a step taken independently of the national organization.

And some teachers here still find any partisan activity at all distasteful, selfish or just downright unseemly for their profession.

Not Jean Seeland. A young first-grade teacher and political activist in Waterloo, she said she believes this new role for teachers is not "selfish politics. Teachers need good salaries and retirement benefits, but we're also interested in education funding and class size and those issues."

The NEA made its first foray into presidential politics in 1976, endorsing the ticket of Jimmy Carter and Walter Mondale. The NEA later claimed credit for making a "critical difference" in a close election, and put the squeeze on the new administration for a new department of education.

Last fall, despite strong resistance from NEA's chief rival, the American Federation of Teachers (AFT), Congress approved the department and Carter quickly signed the Bill.

There are other signs of power. Carter has increased education funding by 60 percent, the NEA reports. And when NEA representatives call the White House these days, as one of those in the Iowa gathering the other night put it, "They don't get routed to some flunky. They get right through to the man."

Indeed, last week when the NEA brought a couple of dozen political trainees to Washington they were taken to the White House for a "briefing on domestic and foreign policy" -- the first of six such visitations keyed to primaries, NEA officials said.

One of the White House staff executives they met there was a former high school math teacher named John Ryor, who from July 1975 until last year was president of the NEA. He is now associate assistant in Anne Wexler's office, working with special-interest groups on issues the president deems high priority.

"I like to think I have other attributes" than having led NEA's pro-Carter efforts, Ryor said with a smile, sitting in his big, high-ceilinged office in the Old Executive Office Building. "But it's fair to say that if I had still been a math teacher at Battle Creek High School, the president wouldn't have called me" to work for him.

Last September, when Carter was sinking to historic lows in the public opinion polls, the NEA declared its support for him again in the primaries.

"We got a lot of feedback from members saying we should have waited," one NEA official said. "But we felt that if we wanted to have an impact with Carter, with his standings in the polls where they were, we had to go then.That was the only way we could demonstrate to other politicians, as well as to Carter, that our endorsement meant something."

In its transformation from prim professional organization into a hard-nosed labor union, the 124-year-old NEA has been able to marshal remarkable natural resources that set it apart from other trade unions.

With 1.8 million members in 12,000 affiliates nationwide, it reaches into virtually every American community.

It has an average of 6,000 members in every congressional district -- members who are educated and who know and are known by a lot of people.

"They can get you into any community in a campaign, and no other union can do that," said William Romjue, the Carter-Mondale coordinator in Iowa. "In small towns especially, the teachers know everyone."

And the NEA has money to spend. In 1976, it poured more than $2.5 million into national, state and local elections.

The NEA is mobilizing its army with well-targeted objectives in mind.

It hopes to send 486 Carter delegates and alternates to the Democratic National Convention. "We'll be happy if we get 172 [the NEA had that many at the 1976 convention, plus 93 alternates] but the potential is there for more," one NEA official said.

The Democrats' affirmative action rules give the NEA an edge. With a membership that is 60 percent women and a long tradition of its own affirmative action provisions, "we can offer slates that will satisfy" the requirement for equal numbers of male and female delegates.

"Iowa is important because it's the first step," Larry Hamilton told the teachers the other night in Waterloo as he urged them to run for delegate to the county convention.

In addition to crucial phone numbers to call, the Waterloo troops got packages of written instructions on the wherefores and how-tos of the complicated caucus procedures complete to the most subtle details.

Roger White, a teacher at Longfellow elementary, encouraged the group to introduce at the caucuses resolutions in support of their goals, but he cautioned, "write them out in longhand . . . That way it looks a lot more spontaneous."

The Iowa teachers association has a record of political success, an official said, "as advanced as any of our states."

According to George Brown, head of the Iowa state organization and a Republican, 79 percent of the candidates endorsed by the teachers' political arm have won.

Their performance slipped in 1978, he conceded. They had supported Republican Robert D. Ray for governor until then, but they endorsed a Democrat that year and saw him lose to Ray. "That caused a rift that is healing now."

Political activity by teachers first began at the local and state level, officials said, as control of money for schools passed from true local control to the state legislatures.

The advent of men into the profession after World War II and the rise of the women's movement were among the social changes that contributed to the pressures for higher income for teachers, they said.

"After eight years as a high school teacher in Connecticut in the 60s, I made a wage that didn't allow me to qualify for a mortgage in the town I taught in," said Ken Melley, a teacher turned political organizer for the NEA in Washington. "I had to work weekends in a factory. I had four children. And I had a master's degree."

Many teachers today still work under similar conditions, he said. "Our most active advocate, and perhaps the least expected, is often the woman, single, who has been teaching for 25 years. She is saying, 'I've had it. I'm not taking another day of this.'"